Constrained Buckling of Variable Length Elastica 
 
 
A DISSERTATION 
SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL 
OF THE UNIVERSITY OF MINNESOTA 
BY 
 
 
Anna Liakou 
 
 
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS 
FOR THE DEGREE OF 
Doctor of Philosophy 
 
 
ADVISER 
Emmanuel Detournay 
CO-ADVISER 
Vincent Denoël 
 
 
October, 2017 
© Anna Liakou 2017
ALL RIGHTS RESERVED
Acknowledgements
I would like to express my appreciation to my academic advisor Professor Emmanuel Detournay
for his encouragement, guidance and for allowing me to grow as a research scientist. I’d like to
thank him for participating to our long conversations and the constructive feedback, I received
from him during my PhD studies. The combination of my mathematical point of view and his
physical vision, was unambiguously powerful and lead to the final form of my research. I would
also like to thank my co-advisor Professor Vincent Denoel for his vision on the subject and his
suggestions.
Now I can admit that I am more confident on conveying my research ideas in a convincing
manner, in accordance with the research community under consideration.
I would also like to thank the committee members, Professor Bojan Guzina, Professor Henryk
Stolarski, and Professor Yoichiro Mori for their insightful comments, suggestions and encour-
agement.
My sincere gratitude is reserved for Professor Yoichiro Mori whose invaluable lectures in
applied mathematics were both influential and inspirational. This allowed me to carry-on my
research, and helped me broaden my knowledge and understanding, especially in the area of
asymptotic and dimensional analysis.
Special thanks to the Civil, Environmental and Geo Engineering (CEGE) Department for
the financial support I received throughout my studies at the U of MN. I would also like to
express my sincere appreciation and thanks to the CEGE Department Head, Professor Joseph
Labuz for giving me the opportunity to carry out my doctoral research in a wonderful academic
environment, and for his invaluable advice and suggestions.
Words cannot express my gratitude and the unconditional support I received from my parents
and my brother. It was due to their unconditional support and constant encouragement that
ultimately became possible for me to see this project through to the end. I thank them for that
and for a lifetime of love and care.
In addition, I would like to thank a former PhD graduate of the U of MN and this Department
and a family friend, Dino Xykis, who was instrumental in convincing me to apply to the U of MN
for graduate studies, and his support, encouragement, and follow-up since the beginning of my
i
studies. I am also indebted to my friends Eduardo and Kaixiao for being there when I needed an
ear to listen, and for all the incredible time that we spent together during the graduate studies.
ii
Abstract
The physical understanding of the response of slender elastic bodies restrained inside constraints
under various loading and boundary conditions is of a great importance in engineering and
medical applications. The research work presented in this thesis is especially concerned with
the buckling response of an elastic rod (the elastica) subjected to unilateral constraints under
axial compression. It seeks to address two main issues: (i) the conditions that lead to the onset
of instability, and (ii) the factors that define the bifurcation diagram.
Two distinct classes of problems are analyzed; (i) the classical buckling problem of a constant
length elastica and (ii) the insertion buckling problem of a variable length elastica. Their main
difference is the generation of a configurational force at the insertion point of the sliding sleeve
in the insertion problem, which is not present in the classical problem.
The thesis describes two distinct methodologies that can solve these constrained buckling
problems; (1) a geometry-based method, and (2) an optimal control method.
The geometry-based method is used to analyze the post-buckling response of a weightless
planar elastica subjected to unilateral constraints. The method rests on assuming a deformed
shape of the elastica and on uniquely segmenting the elastica consistent with a single canonical
segment (clamped-pinned). An asymptotic solution of the canonical problem is then derived
and the complete solution of the constrained elastica is constructed by assembling the solution
for each segment.
Nevertheless, the application of the optimal control method is more generic. It can be
used to solve any constrained buckling problem under general boundary and loading conditions.
Based on Hamiltonian mechanics, the optimality conditions, which constitute the Pontryagin’s
minimum principle, involve the minimization of the Hamiltonian with respect to the control
variables, the canonical equations and the transversality conditions.
The main advantage of the optimal control method is the assumption of strong rather than
weak variation of the involved variables, which leads to the additional Weierstrass necessary
condition (“optimal” equilibrium state). Based on it, several factors such as the effect of the
self-weight of the elastica and the clearance of the walls are investigated.
iii
Contents
Acknowledgements i
Abstract iii
Contents iv
List of Tables viii
List of Figures ix
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Classes of problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Stability Analysis of Elastica: The Calculus of Variations Method 6
2.1 Buckling of elastic rods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Constrained buckling of elastic rods . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.1 Description of the constrained buckling problem . . . . . . . . . . . . . . 16
3 Constrained Buckling of Variable Length Elastica:
Solution by Geometrical Segmentation 26
3.1 Constrained buckling problem of elastica . . . . . . . . . . . . . . . . . . . . . . . 28
3.1.1 Problem definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.1.2 Governing equations of elastica . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2 Canonical problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2.1 Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.2 Asymptotic Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2.3 Unscaling the Canonical Solution . . . . . . . . . . . . . . . . . . . . . . 37
3.3 Segmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
iv
3.3.1 Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3.2 System of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.4 Validation: Constrained Pinned-Pinned Elastica . . . . . . . . . . . . . . . . . . 42
3.4.1 Validation of canonical problem . . . . . . . . . . . . . . . . . . . . . . . 42
3.4.2 Evolution of Contact Patterns . . . . . . . . . . . . . . . . . . . . . . . . 46
3.4.3 Unconstrained elastica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.4.4 Discrete contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.4.5 Continuous contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.4.6 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.5 Stability analysis based on the number of inflection points . . . . . . . . . . . . . 51
3.6 Insertion problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.6.1 Clamped-clamped elastica with symmetric walls . . . . . . . . . . . . . . 53
3.6.2 Clamped-pinned elastica with symmetric walls . . . . . . . . . . . . . . . 57
3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4 Optimal Control 61
4.1 From calculus of variations to optimal control . . . . . . . . . . . . . . . . . . . 62
4.1.1 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.2 Optimal control method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2.1 Optimal control theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2.2 Preliminaries on the pure-state constraints . . . . . . . . . . . . . . . . . 70
4.3 Numerical techniques for solving an optimal control problem . . . . . . . . . . . 75
4.3.1 Direct collocation methods . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3.2 The NLP problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5 Optimization-based stability analysis of constrained weightless elastica 89
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.2 Stability of constrained planar elastica . . . . . . . . . . . . . . . . . . . . . . . 91
5.2.1 Description of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.2.2 Displacement controlled with fixed length or variable length . . . . . . . 99
5.3 Bifurcation diagrams of planar elastica . . . . . . . . . . . . . . . . . . . . . . . 101
5.3.1 Verification example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.3.2 Clamped-clamped elastica with symmetrically located walls . . . . . . . 102
5.3.3 Pinned-pinned elastica constrained by asymmetrically located walls . . . . 107
5.4 Stability of constrained spatial elastica . . . . . . . . . . . . . . . . . . . . . . . . 109
5.4.1 Kirchhoff rod model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.4.2 Optimal control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
v
5.4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6 Optimization-based stability analysis of constrained heavy elastica 119
6.1 Loop tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.2 Heavy elastica on a horizontal foundation . . . . . . . . . . . . . . . . . . . . . . 121
6.2.1 “Long” heavy elastica on a horizontal foundation . . . . . . . . . . . . . . 123
6.2.2 “Short” heavy elastica on a horizontal foundation . . . . . . . . . . . . . 130
6.2.3 Optimal control formulation . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.3 Heavy elastica constrained by rigid horizontal walls . . . . . . . . . . . . . . . . . 136
6.3.1 Optimal control formulation of constrained heavy elastica . . . . . . . . . 138
6.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
6.3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
7 Insertion problem of spatial elastica 157
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
7.2 Derivation of Eshelby-like force . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
7.2.1 Governing equations of spatial elastica . . . . . . . . . . . . . . . . . . . . 159
7.2.2 The calculus of variations . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
7.3 Optimal control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
7.3.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
7.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
8 Conclusions 167
8.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
8.2 Major contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
8.3 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
Bibliography 174
Appendix A. Constrained Buckling of Drillstrings 191
A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
A.1.1 Vertical boreholes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
A.1.2 Horizontal/Inclined boreholes . . . . . . . . . . . . . . . . . . . . . . . . 196
A.2 Drilling equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
A.2.1 Beam-column model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
A.2.2 Buckling differential equation . . . . . . . . . . . . . . . . . . . . . . . . 204
vi
Appendix B. Supplementary material to Chapter 3 211
B.1 Principle of minimum energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
B.2 Analytical solution of constrained clamped-clamped elastica . . . . . . . . . . . 214
B.2.1 Unconstrained case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
B.2.2 Point contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
B.2.3 Onset of continuous contact . . . . . . . . . . . . . . . . . . . . . . . . . . 215
B.3 Geometry-based analysis of the constrained clamped-clamped Elastica . . . . . . 215
B.3.1 First Buckling Mode (Symmetric Case) . . . . . . . . . . . . . . . . . . . 216
B.3.2 First Buckling Mode (Anti-symmetric Case) . . . . . . . . . . . . . . . . 218
B.3.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
B.4 Clamped-pinned elastica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
vii
List of Tables
3.1 Points of evolution of the contact patterns for the bifurcation diagram for a con-
strained pinned-pinned elastica with clearance c = 0.05 . . . . . . . . . . . . . . 50
3.2 Points of evolution of the contact patterns for the bifurcation diagram for a con-
strained clamped-clamped elastica with clearance c = 0.05 . . . . . . . . . . . . 55
3.3 Points of evolution of the contact patterns for the bifurcation diagram for a con-
strained clamped-clamped elastica with clearance c = 0.1 . . . . . . . . . . . . . 55
3.4 Points of evolution of the contact patterns for the bifurcation diagram of a con-
strained clamped-pinned elastica with clearance c = 0.05 . . . . . . . . . . . . . 58
5.1 Points of evolution of the contact patterns for the bifurcation diagram for a con-
strained clamped-clamped elastica with clearance cu = −cl = 0.1 . . . . . . . . . 106
5.2 Points of evolution of the contact patterns for the bifurcation diagram for a con-
strained clamped-clamped spatial elastica with clearance c = 0.1 . . . . . . . . . 116
6.1 Points of evolution of the contact patterns for the bifurcation diagram for a con-
strained clamped-clamped elastica with self-weight w = 0, 100 and clearances
cl = 0 and cu = 0.1 for the classical problem . . . . . . . . . . . . . . . . . . . . 143
6.2 Transition points of contact patterns for w = 0, 100 consistent with Figures 6.12
and 6.13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
6.3 Points of evolution of the contact patterns of the bifurcation diagram for a con-
strained clamped-clamped elastica with self-weights w = 0, 100 and clearances
cl = cu = 0.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.4 Points of evolution of the contact patterns of the bifurcation diagram for a con-
strained pinned-pinned elastica with self-weights w = 10 and clearances cl = cu =
0.05 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
7.1 Points of evolution of the contact patterns for the bifurcation diagram of a con-
strained clamped-clamped spatial elastica of variable length with clearance c = 0.1 164
viii
List of Figures
1.1 Classes of problems: a) the classical buckling problem Domokos et al. (1997);
Manning and Bulman (2005) and b) the insertion buckling problem Ro et al.
(2010). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Contact patterns (equilibrium configurations) for different applied axial loads. . 4
2.1 Pitchfork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Comparison planar-spatial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Elastic rod; (a) uniform and (b) non-uniform . . . . . . . . . . . . . . . . . . . . 15
2.4 An elastica constrained by two rigid horizontal walls, which are symmetrically
located with respect to the centerline. The left end is fixed while the right end is
clamped. Sliding is allowed along the horizontal axis. . . . . . . . . . . . . . . . 17
3.1 An elastica constrained by two rigid horizontal walls with clearances Cu, Cl lo-
cated with respect to the centerline with the left end fixed and clamped; (a)
Classical problem: the right end is clamped on a slider restricted to move along
the horizontal axis, (b) Insertion problem: while the elastica is gradually inserted
through a fixed frictionless sleeve at the right end. . . . . . . . . . . . . . . . . . 29
3.2 a) Canonical problem and b) Phase portrait . . . . . . . . . . . . . . . . . . . . 35
3.3 Free standing fold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.4 Graphs of the axial force Q/pi2, the end-shortening δ1 and the vertical displace-
ment ∆y1 with respect to ψ[1] = θ[1] for α1 = 0 . . . . . . . . . . . . . . . . . . . 45
3.5 Graphs of the axial force Q/pi2, the end-shortening δ1 and the vertical displace-
ment ∆y1 with respect to ψ[1]; (i) α1 = 0.1 and (ii) α1 = 0.4 . . . . . . . . . . . 46
3.6 First buckling mode of pinned-pinned elastica . . . . . . . . . . . . . . . . . . . 47
3.7 Second and third buckling mode of pinned-pinned elastica . . . . . . . . . . . . 48
3.8 Bifurcation diagram for a constrained pinned-pinned elastica with clearance c = 0.05 51
3.9 Bifurcation diagram for a constrained clamped-clamped elastica with clearance
c = 0.05; (1) classical stability problem (dashed line) and (2) insertion stability
problem with dashed line for the applied load Q and dotted line for the axial force
R cosα. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
ix
3.10 Bifurcation diagram for a constrained clamped-clamped elastica with clearance
c = 0.1; (1) classical stability problem (solid line) and (2) insertion stability
problem with dashed line for the applied load Q and dotted line for the axial
force R cosα. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.11 Bifurcation diagram for a constrained clamped-pinned elastica with clearance
c¯ = 0.05; solid line for the applied load Q and dashed line for the axial force
R cosα . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.1 Description of the classical stability problem for a clamped-clamped elastica . . 92
5.2 Bifurcation diagram for a constrained pinned-pinned elastica of constant length
with clearances cu = −cl = 0.05 . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.3 Axial compressive force Q/pi2 with respect to the angles; (i)α, (ii) β and (iii) θ(0)
for a discrete contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.4 Axial compressive force Q/pi2 with respect to the length of the continuous contact
`c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.5 Bifurcation diagram for a pinned-pinned elastica of constant length . . . . . . . 102
5.6 Bifurcation diagram for a constrained clamped-clamped elastica of constant length
with clearances cu = −cl = 0.05 . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.7 Adjoint variables for distinct values of end-shortening δ . . . . . . . . . . . . . . 105
5.8 Bifurcation diagram for a constrained clamped-clamped elastica with clearance
cu = −cl = 0.1; solid line for the applied load Q and dashed line for the axial
load R cosα. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.9 Bifurcation diagram for a constrained pinned-pinned elastica of constant length
with clearances cu = 0.125 and cl = 0 . . . . . . . . . . . . . . . . . . . . . . . . 109
5.10 Method 3− 2− 1 of Euler angles . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.11 Bifurcation diagram for a constrained clamped-clamped elastica of constant length
with clearance c = 0.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.12 Deformation shapes; a) Spatial one point (Q = 8.05pi2, δ = 0.04, κ1 = 0.21), b)
Point-point (Q = 9.33pi2, δ = 0.1, κ1 = 0.64), c) Point-line-point (Q = 11.04pi2,
δ = 0.16, κ1 = 1.21), d) Line contact (Q = 11.95pi2, δ = 0.22, κ1 = 1.8) , e)
Point-line-point contact (Q = 8.31pi2, δ = 0.37, κ1 = 3.37) and f) Point-point-
point contact (Q = 3.95pi2, δ = 0.7, κ1 = 4.2) . . . . . . . . . . . . . . . . . . . . 117
6.1 Folds of elastica i) Case 1, ii) Case 2 and iii) Case 3 . . . . . . . . . . . . . . . . 120
6.2 Problem description a) heavy elastica of length L, b) ruck formation of length Lr 124
6.3 Steady motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.4 Problem description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.5 Slipping process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
x
6.6 a) Bifurcation diagram for a clamped-clamped elastica for different values of self-
weight w with one-sided horizontal foundation and b) height of the formed ruck
for two distinct values of self-weight w = 0, 100 . . . . . . . . . . . . . . . . . . . 135
6.7 ”Symmetry-breaking bifurcation” for a horizontal foundation . . . . . . . . . . . 136
6.8 Deformation shapes for different values of end-shortening δ for self-weight w = 200
and foundation slope β = pi/3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.9 Buckling problem of elastica constrained by a) asymmetrically and b) symmetri-
cally located rigid walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
6.10 Bifurcation diagram for a constrained clamped-clamped elastica with self-weight
w = 0 and clearances cl = 0 and cu = 0.1 for the classical problem . . . . . . . . 144
6.11 Bifurcation diagram for a constrained clamped-clamped elastica with self-weight
w = 100 and clearances cl = 0 and cu = 0.1 for the classical problem . . . . . . . 145
6.12 Bifurcation diagram for a constrained clamped-clamped elastica with self-weight
w = 0 and clearances cl = 0 and cu = 0.1 for the insertion problem; (i) solid line
denotes the evolution of the internal horizontal force Rx/pi2 and (ii) dashed line
denotes the evolution of the applied load Q/pi2 . . . . . . . . . . . . . . . . . . . 147
6.13 Bifurcation diagram for a constrained clamped-clamped elastica with self-weight
w = 100 and clearances cl = 0 and cu = 0.1 for the insertion problem; (i) solid
line denotes the evolution of the internal horizontal force Rx/pi2 and (ii) dashed
line denotes the evolution of the applied load Q/pi2 . . . . . . . . . . . . . . . . 148
6.14 Bifurcation diagram for a constrained clamped-clamped elastica with self-weights
w = 0, 100 and clearances cl = cu = 0.1 . . . . . . . . . . . . . . . . . . . . . . . 151
6.15 Bifurcation diagram for a constrained clamped-clamped elastica with self-weight
w = 10 and clearances cl = cu = 0.1 . . . . . . . . . . . . . . . . . . . . . . . . . 152
6.16 Bifurcation diagram for a constrained clamped-clamped elastica with self-weight
w = 10 clearances cl = cu = 0.05 . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
6.17 Bifurcation diagram for a constrained pinned-pinned elastica with self-weight w =
10 and clearance cl = cu = 0.05 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
7.1 A spatial elastica constrained by a cylindrical wall with clearance C located with
respect to the centerline with the left end fixed, while the elastica is gradually
inserted through a fixed frictionless sleeve at the right end. . . . . . . . . . . . . 158
7.2 Bifurcation diagram for a constrained clamped-clamped elastica of variable length
with clearance c = 0.1; (1-2) Planar one point (P), (3-4) Spatial one point,
(4-5) Two discrete contacts (P-P), (5-6) Three discrete contacts (P-P-P), (6-7)
Point-Line-Point (P-L-P), (7-8) Line contact (L) and (8-9) Line-Line-Line (L-L-L)
(Same holds for the corresponding points denoted by ′). The black and grey lines
represent the evolution of the internal force Rx/pi2 and the applied load Q/pi2
with respect to the change of the inserted length δ¯, respectively. . . . . . . . . . 165
xi
A.1 Configurations of the drillstring constrained by a vertical well; (a) Initial straight
configuration, (b) first-order planar shape, (c) second-order planar shape, (d)
spatial shape, (e) spatial shape with a continuous segment, and (f) helical shape
Huang et al. (2016). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
A.2 Configuration of a weightless drillstring constrained in a straight borehole Huang
et al. (2015) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
B.1 Bending energy U with respect to δ¯ . . . . . . . . . . . . . . . . . . . . . . . . . . 213
B.2 Bending energy U with respect to Q . . . . . . . . . . . . . . . . . . . . . . . . . 213
B.3 Symmetrical first buckling mode of clamped-clamped elastica . . . . . . . . . . . 216
B.4 Secondary buckling of clamped-clamped elastica. . . . . . . . . . . . . . . . . . . 218
B.5 Asymmetrical first buckling mode of clamped-clamped elastica . . . . . . . . . . 219
B.6 Asymmetrical configuration of clamped-clamped elastica . . . . . . . . . . . . . 220
B.7 Bifurcation diagram for a constrained clamped-clamped elastica with clearance
c = 0.05 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
B.8 Bifurcation diagram for a constrained clamped-pinned elastica with clearance
c = 0.053 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
xii
Chapter 1
Introduction
1.1 Motivation
Rods are slender elastic bodies characterized by lengths that are much greater than their di-
ameters, L  D. They can therefore be considered as one-dimensional structures, which can
experience bending, torsion, extension and shearing. This thesis is concerned with the buckling
response of an elastic rod under the presence of unilateral constraints.
The foremost application of this research concerns the petroleum industry, which relies on
a kilometers long elastic drillstring to transmit axial force and torque to the bit, to drill deep
boreholes. The history of deformation of a rod under compression, constrained by a borehole,
is complex as the derived contact conditions (discrete and/or continuous) depend on multiple
factors, such as the geometry of the wellbore, the boundary conditions and the clearance between
the rod and the conduit. The stability of drillstrings has been extensively investigated by a
number of authors (Miller, 2014; Wicks et al., 2008; Gao and Huang, 2015), starting with the
pioneering work of Lubinski (1950).
There are also medical applications to the problem of a constrained rod. The insertion of
guidewire and catheters for interventional radiology procedures requires very precise understand-
ing of their deformation behavior as any potential mistake can lead to damage of vital tissues
and blood vessel walls (Alderliesten et al., 2007; Konings et al., 2003; Li et al., 2011a). In this
area of research, however, the interest is limited to real-time simulations of constrained rods
rather than stability analysis.
This thesis describes methodologies that can be applied to investigate the buckling response
of elastic rods in the presence of rigid discrete and longitudinal obstacles. It seeks to address
some important questions: (i) what are the conditions leading to the loss of stability of an elastic
rod inside a rigid straight conduit and (ii) which are the factors that define the complete form
of the bifurcation diagram.
1
2The thesis explains the difficulties that arise from the presence of the unilateral constraints,
first in the derivation of an equilibrium state and most importantly in the stability analysis.
The reasons behind the failure of classical techniques are clarified and alternative approaches
that overcome these numerical barriers are provided. In few words, the thesis suggests two
distinct methodologies; (1) a geometry-based method and (2) an optimal control method. Both
methodologies are novel and their application is analyzed in the following Chapters.
1.2 Classes of problems
The post-buckling analysis of constrained elastic rods under axial compression is the main con-
cern of the thesis. A buckling problem always involves two parts: (i) derivation of a particular
equilibrium state depending on the loading and the boundary conditions; and (2) stability
analysis of the equilibrium state. The former part requires a first-order analysis that can be
performed in a variety of ways, using analytical, semi-analytical, and numerical approaches. A
novel semi-analytical geometry-based approach is suggested in Chapter 3.
To extract information regarding the stability of a particular equilibrium configuration fur-
ther analysis is required. If we apply the calculus of variations, a second-order analysis should
be performed. Nevertheless, this classical method involves several numerical difficulties due to
the presence of unilateral constraints, as explained in detail in Chapters 2 and 4. Therefore an
optimization-based stability analysis is a more appropriate choice and can provide the solution
of the constrained buckling problem in a simple and elegant manner.
Through this study, we analyze two distinct classes of problems. The first class of problems
involves a weightless planar unit-length elastica resting in its unstressed configuration at mid-
distance between two frictionless rigid horizontal walls (see Figure1.1 a)). The left end of the
elastica is fixed while the right end is constrained to move horizontally. The extremities of the
elastica are either clamped at zero inclination or pinned. We seek to determine the post-buckling
behavior of the elastica under the action of a predefined compressive force R (force-controlled)
or an imposed relative displacement δ = 1 − x(1) at the right end (displacement-controlled).
This problem is referred to as the “classical buckling problem”. The graph of the compressive
force in terms of the end-shortening δ is a bifurcation diagram, where every equilibrium position
corresponds to a particular contact pattern (see Figure 1.2).
This problem can be slightly modified. We consider that the horizontal distance between the
two supports is fixed x(l) = 1 and the elastica is inserted gradually from the right end. This
implies that the length of the elastica is variable. Hence, in this case, we are looking for the
post-buckling behavior of the elastica assuming a fixed compressive force R (force-controlled),
or a predefined change of the arc-length of the elastica (displacement-controlled). This leads to
a different bifurcation diagram as we derive the compressive force in terms of the change of the
arc-length of the elastica. This class of problem is denoted as “insertion buckling problem” (see
3Figure 1.1b)).
The above buckling problems are our main concern. Several extensions are also included.
One extension includes the analysis of the post-buckling response of a spatial rather than planar
elastica. Another extension involves the investigation of the weight effect in the constrained
stability analysis. Detailed analysis of all examples can be found in Chapters 3-7.
( )s

( 1 )x
2P
1P
R
c
c
y x
( )s

( ) 1x l =
2P
1P
R
c
c
y x
l
Figure 1.1: Classes of problems: a) the classical buckling problem Domokos et al. (1997); Man-
ning and Bulman (2005) and b) the insertion buckling problem Ro et al. (2010).
41 - 2 2 - 3
3 - 4 4 - 5
5 - 6 6 - 7
Figure 1.2: Contact patterns (equilibrium configurations) for different applied axial loads.
1.3 Outline
The thesis starts with a state-of-the-art literature review on the post-buckling analysis of elas-
tic rods in the absence of unilateral constraints, see Chapter 2. More attention is placed on
inextensible-unshearable elastic rods (planar or spatial elastica). The difference between the
post-buckling response of a planar and a spatial elastica is illustrated, while several factors are
also investigated. In the second part of Chapter 2, the constrained buckling problem is then
analyzed by applying variational calculus. Through a concrete example, we then illustrate the
limitation of the calculus of variations when unilateral constraints are present.
In Chapter 3, we study the post-buckling response of a weightless planar elastica in the
presence of two symmetrically located rigid frictionless walls. Both classes of problems are
analyzed. The main focus is placed on the insertion problem, where a configurational force is
generated at the insertion point and leads to a reduced value of the applied force. The proposed
method is a geometry-based technique. More specifically, assuming a particular buckling mode,
a sequence of contact patterns of the elastica is derived by starting from an unconstrained
shape and leading gradually to shapes with discrete and/or continuous contact. For every
known configuration of the planar elastica, the phase portait of the inclination θ- curvature θ′ is
derived. Based on it, the elastica can be divided into distinct segments, which are consistent with
a single canonical problem with clamped-pinned boundary conditions. Instead of using elliptic
integrals, an asymptotic solution of the canonical problem is derived. After solving all distinct
canonical subproblems, the complete solution is obtained by satisfying continuity properties at
5the common extremities. Several bifurcation diagrams of the constrained elastica are computed
by adopting this novel approach. A simple geometry-based rule for the stability analysis of every
configuration state is also suggested based on the number of inflection points (i.e., θ′ = 0) and
the nature of the contact conditions.
A novel approach for the stability analysis of constrained elastic structures is presented in
Chapter 4. Inspired by Maurer and Mittelmann (1991) and Liberzon (2012), we show that a
constrained buckling problem can be solved by applying an optimal control method (Berkovitz,
1961). First, we illustrate the advantage of adopting an optimal control approach rather than
a calculus of variations method and their equivalence. Based on Hamiltonian mechanics, the
general framework of an optimal control problem with pure-state constraints is described. This
formulation is consistent with our constrained buckling problem. The optimality conditions are
then derived by applying the Pontryagin’s minimum principle. The available numerical methods
for the solution of an optimal control problem are also described, with the main focus on the
direct method.
Based on the optimal control theory, we investigate the buckling response of elastic rods
constrained by frictionless rigid longitudinal obstacles in Chapters 5, 6 and 7. In particular, in
Chapter 5, we compute the bifurcation diagram of a planar elastica constrained by two rigid
horizontal walls. Both the classical and the insertion buckling problems are analyzed. Force-
controlled and displacement-controlled conditions are studied, separately. The same buckling
problems are then solved for the spatial case. The derived results are compared with previous
numerical and experimental works (Manning and Bulman, 2005; Ro et al., 2010; Fang et al.,
2013; Domokos et al., 1997).
Chapter 6 involves the analysis of the buckling response of a planar elastica with non-
negligible self-weight (i.e., heavy elastica) constrained by either a rigid one-sided straight foun-
dation, or two rigid horizontal walls. The one-sided foundation case is a classical problem, which
was first studied by Wang (1986). A comprehensive literature review around this problem, also
called the ribbon problem, is first performed. Its solution by the optimal control method is also
included. Then we study the constrained stability problem with two-sided foundations. Several
factors are analyzed, such as the effects of the boundary conditions, the clearance of the walls
and the self-weight of the elastica.
The insertion problem for the spatial elastica is then analyzed in Chapter 7. Based on the
calculus of variations, the configurational or Eshelby-like force is first computed. A concrete
example is accordingly solved by applying the optimal control method, which illustrates the
effect of this configurational force.
The thesis concludes with a short summary of the most remarkable results of the study.
Other examples that can be solved by the geometry-based method and the optimal control
method are also mentioned. The advantages and limitations of both numerical techniques when
compared to other approaches are also explained.
Chapter 2
Stability Analysis of Elastica: The
Calculus of Variations Method
In this Chapter, we first perform a comprehensive literature review about the buckling response
of elastic rods under axial compression. The description starts with the stability analysis of
an unconstrained elastic rod. Classical and novel approaches are presented. The main focus
is placed on the post-buckling response of an inextensible-unshearable elastic rod, the planar
or the spatial elastica. The assumption of the inextensibility and unshearability constraints is
appropriate as we further investigate the buckling response of an elastic cylindrical rod with
a large length/diameter ratio inside a rigid conduit with a clearance much smaller than the
length of the rod. This constrained buckling problem is then studied by applying the calculus
of variations. Through a concrete example, we also illustrate the limitations arising from the
application of the calculus of variations in the analysis of a constrained buckling problem.
2.1 Buckling of elastic rods
The stability analysis of an inextensible and unshearable elastic rod —the elastica— is nowadays
a classical problem that can be found in many books (Bigoni, 2016; Antman, 2005; Love, 2013;
Timoshenko, 2009).
The equations of the elastica were first derived by J. Bernoulli (1692) for the particular
case of the so-called rectangular elastica. The next important step took place in 1742 with
the recognition by D. Bernoulli that the general problem of the elastica could be treated as an
isoperimetric problem. This was the origin of variational calculus. Based on this pioneering
idea, Euler (1744) computed a family of curves, known as Euler’s elastica figures, which were
then verified experimentally by Born (1906) (see (Levien, 2008) for an historical account). The
6
7governing equation of the planar elastica is analogous to the equation of motion of a pendulum
(Goss, 2008), while for the spatial case, Kirchhoff (1859) recognized that the deflection curve of
the elastica is governed by the same set of differential equations as the motion of a heavy spinning
top (Lazarus et al., 2013; Davies and Moon, 1993). The elastica has gained a lot of interest as
it is used for the design of multiple mechanical systems from hair simulation (Spillmann and
Teschner, 2007; Daviet et al., 2011) to the simulation of drill-strings in oil-drilling operations
(Miller, 2014; Wicks et al., 2008; Su et al., 2013) and the animation of a locomotion (Zhou et al.,
2015).
A short review of the methods used in the stability of a planar elastica is included herein.
The stability analysis of an inflectional elastica is only presented as Born (1906) has shown that
any elastic arc without inflection points is stable (Sachkov, 2008a,b).
Let us analyze the buckling response of a planar elastica under the action of an axial compres-
sive load. The most classical problem is the Dirichlet-Dirichlet case (Gelfand, 2000; Liberzon,
2012; Love, 2013). The dimensionless potential energy of the elastica is given by
Π (θ, θ′) =
∫ 1
0
F (θ, θ′) ds = 1
2
∫ 1
0
θ′2ds+Q
∫ 1
0
(cos θ − 1) ds (2.1)
where s = σ/L ∈ [0, 1] is the normalized arc-length with L the total length of the elastica, θ(s)
is the inclination angle with boundary conditions θ(0) = θ(1) = 0, θ′(s) is the curvature and
Q = QL2/EI is the normalized axial load. The first term is the bending elastic energy, while
the second term is the external work of the applied force Q.
We assume a variation of the form θ(s, ε) = θ∗(s) + εη(s), where ” ∗ ” denotes the extremal
variable, η(s) is a sufficiently smooth function that vanishes at end-points η(0) = η(1) = 0 and
0 < ε 1 is an independent parameter. Based on it, the first variation of the potential energy
δΠ (θ, θ′) = lim
ε→0
dΠ/dε can be derived as
δΠ =
∫ 1
0
{
∂F (θ∗, θ′∗)
∂θ
η +
∂F (θ∗, θ′∗)
∂θ′
η′
}
ds (2.2)
which after applying integration by parts becomes
δΠ =
∫ 1
0
{θ∗′′ +Q sin θ∗} ηds (2.3)
An extremal solution is then obtained by vanishing the first variation (2.3). This equilibrium
state θ∗′′+Q sin θ∗ = 0 is the Euler-Lagrange equation. Its analytical solution is given in terms
of elliptic integrals.
Several studies are restricted to the first order analysis from which either the critical (lin-
earized) buckling load is computed (i.e. first bifurcation point assuming sin θ∗ ≈ θ∗) (Wang,
81997; Wang et al., 2004; Timoshenko, 2009), or a post-buckling analysis is performed (Mikata,
2007; Wang, 1997). For this problem the critical buckling load is Q = pi2, which is identical
to the first buckling load of a pinned-pinned elastica. This value becomes Q = 4pi2, when the
isoperimetric constraint
∫ 1
0
sin θds = 0 is additionally imposed.
Nevertheless the stability of an extremal solution can be determined by the second variation
of the functional (2.1), which is given by
δ2Π =
1
2
∫ 1
0
{
∂2F (θ∗, θ′∗)
∂θ∂θ
η2 + 2
∂2F (θ∗, θ′∗)
∂θ∂θ′
η′η +
∂2F (θ∗, θ′∗)
∂θ′∂θ′
η′2
}
ds (2.4)
Let us transform (2.4) in a more convenient form. Applying an integration by parts, we get∫ 1
0
2
∂2F (θ∗, θ′∗)
∂θ∂θ′
η′ηds = −
∫ 1
0
d
ds
(
∂2F (θ∗, θ′∗)
∂θ∂θ′
)
η2ds
and substituting into (2.4) the following expression is obtained
δ2Π =
∫ 1
0
{
P(s)η′2 + G(s)η2
}
ds or δ2Π =
∫ 1
0
{
− d
ds
(P(s)η′) + G(s)η
}
ηds (2.5)
where
P (s) =
1
2
∂2F (θ∗, θ′∗)
∂θ′∂θ′
, G(s) =
1
2
(
∂2F (θ∗, θ′∗)
∂θ∂θ
− d
ds
(
∂2F (θ∗, θ′∗)
∂θ∂θ′
))
Hence, the second variation of the potential energy of the planar elastica becomes
δ2Π =
∫ 1
0
{
η′2 −Q cos θ∗η2} ds (2.6)
A necessary condition for deriving the minimum of the functional Π (θ, θ′) is the Legendre
condition P (s) ≥ 0. An extremal that satisfies the strengthened Legendre condition P (s) > 0
(strict inequality) is also said to be regular and it implies that θ and its variation η are of class
C2. In the above example, the strengthened Legendre condition is always satisfied as P (s) = 1.
Nevertheless the Legendre condition is not a sufficient condition for the quadratic functional
δ2Π to be ≥ 0 for all admissible variations η(s). If we satisfy δ2Π > 0, or δ2Π < 0, the
corresponding extremal solution is stable, or unstable, respectively. In the case δ2Π = 0, higher
order variation is required.
The positive definiteness of the second variation can be studied in a variety of ways; (i)
considering the solution of an auxiliary Riccati equation (Gelfand, 2000; O ’ Reilly and Peters,
2011), (ii) applying the conjugate point theory (Manning et al., 1998; Manning, 2009), or (iii)
computing the eigensolution of an auxiliary Sturm–Liouville problem (Kuznetsov and Levyakov,
2002; Levyakov and Kuznetsov, 2010).
9The oldest method in the derivation of the sign of the second variation of energy is based
on a very simple mathematical idea. If the second variation of energy can be transformed to
a perfect square, its sign is always non-negative (Gelfand, 2000). To make this possible, a
vanishing quantity is added into (2.5)∫ 1
0
d
ds
(
w(s)η2
)
ds = 0
where w(s) is an arbitrary differentiable function. A new form of (2.5) can be accordingly
obtained
δ2Π =
∫ 1
0
{
P (s)
(
η′ +
w
P (s)
η
)2}
ds
if and only if a bounded solution of the following Riccati equation can be found
P
(
G+
dw
ds
)
= w2 (2.7)
which, in the case of the planar elastica, is simply given by w2 = dw/ds−Q cos θ∗. The Riccati
equation can be further reduced to a linear differential equation if we apply a simple change of
variables. In fact, setting w(s) = − (Pdu/ds) /u into (2.7), it gives
− d
ds
(P (s)u′) +G(s)u = 0 (2.8)
which is the Euler equation corresponding to the functional (2.5) with boundary conditions
u(0) = u(1) = 0. A point α ∈ (0, 1) is conjugate to the point s = 0 if (2.8) has a solution, which
vanishes at s = 0 and s = α but it is not identically zero. If there is at least one conjugate
point, the corresponding extremal solution is unstable.
In summary, the necessary and sufficient conditions for the stability of the elastica (i.e.,
δ2Π > 0 for all admissible non-trivial variations η(s)) involve the following requirements; (i) the
strengthened Legendre condition P (s) > 0 and (ii) the absence of the conjugate points at the
interval [0, 1], which can be established by solving the Euler equation, or deriving a bounded
solution of the Riccati equation.
An alternative transformation of (2.4) has been suggested by Manning (2009), who applied
a different type of integration by parts
1
2
∫ 1
0
{
∂2F (θ∗, θ′∗)
∂θ∂θ′
η′η +
∂2F (θ∗, θ′∗)
∂θ′∂θ′
η′2
}
ds =
−
∫ 1
0
d
ds
(
∂2F (θ∗, θ′∗)
∂θ∂θ′
η +
∂2F (θ∗, θ′∗)
∂θ′∂θ′
η′
)
ηds
10
that leads to
δ2Π =
∫ 1
0
{
D(s)− d
ds
D(s)
}
ηds = 〈η, Sη〉
where 〈·, ·〉 denotes the L2 inner product 〈h, g〉 = ∫ 1
0
h(s)g(s)ds and S is a self-adjoint operator
which is a natural analogue of a Hessian matrix
Sη = D(s)− d
ds
D(s)
and
D(s) = ∂
2F (θ∗, θ′∗)
∂θ∂θ′
η +
∂2F (θ∗, θ′∗)
∂θ′∂θ′
η′
To deduce whether an extremal solution is stable, the spectrum of S is studied. In particular,
there is a conjugate point α if we can find the solution of Sη = 0 with boundary conditions
η(0) = η(α) = 0. On the other hand the absence of conjugate points implies 〈η, Sη〉 > 0 and
thus positive definiteness of the second variation of the potential energy. For unconstrained
problems, Morse (2013) showed that the Morse index corresponds exactly to the number of
conjugate points. Quite recently, Manning et al. (1998)) showed that the classical index theory
is also applicable to an isoperimetrically constrained problem except for Neumann-Neumann
boundary conditions, where some adjustment is required (Manning, 2009).
Another methodology is to start with (2.6) and apply an integration by parts in order to get
δ2Π = −
∫ 1
0
{η′′ +Q cos θ∗η} ηds (2.9)
For every admissible η(s), the sign of (2.9) is studied by considering the decomposition η =
∞∑
i=1
Aifi, where Ai are arbitrary coefficients, fi are the eigenfunctions of the problem with boundary
conditions f(0) = f(1) = 0 and λi are the eigenvalues that solves the following Sturm-Liouville
problem
f ′′ + λQ cos θ∗f = 0 (2.10)
For an imposed eigenvalue λi, the corresponding eigenfunction fi satisfies f ′′i +λiQ cos θ∗fi = 0.
If we multiply this expression by fi and integrate from s = 0 to s = 1, we get∫ 1
0
Q cos θ∗f2i ds =
1
λi
∫ 1
0
f ′i
2ds
Taking also into account the orthogonality condition
∫ 1
0
Q cos θ∗fifjds = 0 for i 6= j, we arrive
11
at the following expression
δ2Π =
∞∑
i=1
A2i
(
1− 1
λi
)∫ 1
0
f ′i
2ds
If all eigenvalues lie outside the interval (0, 1), the second variation of energy is positive and
the extremal solution becomes stable. The opposite holds if at least one eigenvalue λi ∈ (0, 1).
If there is an eigenvalue λi = 0, or λi = 1, higher variations should be evaluated to deduce
whether the equilibrium state is stable. These stability criteria formulate a theorem, whose
complete proof is contained in (Levyakov and Kuznetsov, 2010).
To solve (2.10), the interval s ∈ [0, 1] is divided into m-subintervals leading to a constant
step hi = 1/m. For every subinterval, the solution of the second order ordinary differential
equation (ODE) is a trigonometric function. For a fixed value of λ ∈ (0, 1), (2.10) is solved
piece-by-piece as an initial value problem. Starting from s = 0 and imposing the left boundary
condition f(0) = 0 and an arbitrary value of f ′(0) = f
′
o, we piecewisely compute the solution
of the problem at the right end. If the boundary condition f(1) = 0 is satisfied, the solution is
complete. Otherwise, a different assigned value λ should be chosen until convergence. If there
is no eigenvalue λ ∈ (0, 1) that can satisfy the boundary conditions, the corresponding extremal
solution is stable. More details can be found in (Levyakov and Kuznetsov, 2010).
The stability behavior of the elastica can be also studied by taking into account the fun-
damentals of catastrophe theory. If we apply a spectral decomposition method (i.e., a Fourier
expansion) to the first variation of the potential energy (2.3), we get a function equivalent to
an algebraic equation λx − x3 = 0, which exhibits a pitchfork bifurcation point at λ = 0. The
addition of a parameter α 6= 0 into the algebraic equation α + λx − x3 = 0 can be understood
as an initial imperfection (e.g., initial curvature) that leads to a modification of the bifurcation
diagram (see Figure 2.1) (Peterson and Manning, 2010). In this manner, we can indirectly
deduce about the stability of the elastica by analyzing an equivalent algebraic equation with
known stability behavior (i.e., unfolding).
Figure 2.1: Pitchfork
12
As explained in detail in (Lazarus et al., 2015, 2013), an elastic structure can be also dis-
cretized by using methods, such as finite differences and finite elements. The stability, or insta-
bility of an equilibrium state can be then deduced by the positive, or negative definitiness of the
Hessian matrix, respectively.
The classical problem of Dirichlet-Dirichlet boundary conditions can be further extended
to any type of boundary conditions. When isoperimetric constraints are included, we cannot
solve the stability problem by solving a Riccati equation and the “unfolding” is also impossible
for similar reasons. The other numerical techniques, mentioned above, are applicable, but the
most efficient approach, even for Newmann-Newmann boundary conditions, has been proposed
recently by Levyakov and Kuznetsov (2010). A comprehensive review of the general buckling
problem of a planar elastica under the action of an axial compressive load can be found in
(Bigoni, 2016).
Quite recently, the sufficient conditions of the stability of the elastica have been also studied
by taking into account the number of inflection points along the arc-length of the elastica. In
particular, Jin and Bao (2008) investigated the case of a planar elastica clamped at one end
and guided at the other end and showed that any equilibrium configuration with a number of
inflection points less or equal to one is stable. This simple geometry-based criterion for the
derivation of the stability of the elastica for a variety of boundary conditions has been also used
by Sachkov (2007, 2008a,b), who applied a method of topology optimization.
For the stability analysis of a spatial elastica, or Kirchhoff rod model, multiple studies can be
found (Maddocks, 1984; Majumdar et al., 2012; Goyal et al., 2005). For an isotropic Kirchhoff
elastic rod with circular cross section, the energy functional can be written as
Π (θ, φ, ψ, θ′, φ′, ψ′) =
1
2
∫ 1
0
(
κ21(s) + κ
2
2(s)
)
ds+
γ
2
∫ 1
0
τ2(s)ds
+Q
∫ 1
0
cos θ cosψds
where s = σ/L ∈ [0, 1] is the normalized arc-length with L the total length of the elastica,
θ(s), φ(s), ψ(s) are the three Euler angles and Q = QL2/EI is the applied axial load. The energy
functional involves the bending strain energy with curvatures {κ1(s), κ2(s)}, the torsional strain
energy with torsion τ (Antman, 2005; Majumdar et al., 2012) and γ the ratio of torsional to
bending moduli and the external work of the applied force Q. For clamped clamped boundary
conditions, we have x(0) = y(0) = z(0) = y(1) = z(1) = 0 and ψ(0) = ψ(1) = θ(0) = θ(1) =
φ(0) = φ(1) = 0. The detailed derivation can be found in Chapter 5.
Goyal et al. (2005) studied the post-buckling behavior of a clamped-clamped spatial elastica
and compared its response with the post-buckling response of a planar elastica (see Figure 2.2).
The initial response of the planar elastica coincides with the minimum root of the first buckling
13
mode. When the applied force becomes Q/4pi2 ≈ 2.18, the two ends come in contact (end-
shortening δ = 1) and its shape is the so-called “figure-of-eight” Euler (1744). Beyond this
point, a transition to the second root of the first buckling mode is obtained with a second type
of “figure-of-eight”. van der Heijden et al. (2003) explains that the first shape is held at the edge
and requires a compression, while the second shape is held at the center and requires a tension.
As shown in Figure 2.2, the post-buckling behavior of the spatial elastica with γ = 5/7
coincides with the response of the planar elastica until δ = 0.559. Beyond this value, an out-of-
plane deformation initiates with gradually decreasing horizontal force, because bending strain
energy is exchanged for torsional strain energy (Goyal et al., 2005). This branch continues
until the rod deforms into a circular loop with δ = 1 and Q/4pi2 slightly tensile. An analytical
description of this problem can be found in Goyal et al. (2005) and in van der Heijden et al.
(2003) and is also used as a verification example in Chapter 5.
The onset of the out-of plane deformation depends on the value of γ. In particular, van der
Heijden et al. (2003) computed the initiation of the out-of-plane bifurcation for clamped-clamped
boundary conditions, which is represented by the following parametrized form:
δ = 2
(
1− E(k)
K(k)
)
γ =
K(k)
[
2(1− k2)K(k) + (4k2 − 3)E(k)]
2(1− k2)K(k) + (4k2 − 5)K(k)E(k) + 3E2(k)
where the applied load is Q = 4(n + 1)2K2(k), n is the buckling mode, K(k) and E(k) are
the complete elliptic integrals of the first and the second kind and k (0 ≤ k ≤ 1) is the elliptic
modulus.
So far, we have only described the stability behavior of isotropic Kirchhoff rods. Nevertheless,
factors such as anisotropy and non-uniformity can affect the buckling reponse of an elastic rod.
The effect of anisotropy in the stability behavior of an elastic rod has been investigated by
Maddocks (1984), who showed that an anisotropic rod can only undergo planar buckling in
either its plane of minimum, or maximum stiffness. Independently of the boundary conditions,
any extremal in the plane of maximum stiffness is unstable to out of the plane perturbations.
With regard to spatial stability, Maddocks (1984) deduced that equilibria of an anisotropic rod
are at least as stable as those of an isotropic rod, whose constitutive relation coincides with that
of the anisotropic rod in its weakest plane.
14
0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 1 . 4 1 . 6 1 . 8- 2
- 1
0
1
2
3
4
5


/4pi
2
δ
 S p a t i a l P l a n a r
Figure 2.2: Comparison planar-spatial
On the other hand, the stability behavior of a non-uniform rod remarkably differs from a
uniform elastic rod. One representative example is shown in Figure 2.3. In Figure 2.3a), the solid
and dashed lines represent the first symmetrical mode of a uniform rod and its perturbed con-
figuration, respectively. This equilibrium state is unstable, while the equilibrium configuration
shown in Figure 2.3b) (dark lines denote stiffer parts than the light) is stable.
Another factor, the extensibility effect in the stability analysis of the elastica has been in-
vestigated by (Jin and Bao, 2013; Manning, 2014). Taking into account the extensibility effect,
Jin and Bao (2013) showed that the first critical load is higher than Euler buckling load derived
in the classical case, while their difference diminishes if the elastica becomes more slender.
The effect of a twist in the stability behavior of elastica has gained recently a lot of interest,
especially in the DNA loop formations (see (Goyal et al., 2005; Manning et al., 1998; van der
Heijden, 2001; Lazarus et al., 2013; Clauvelin et al., 2009)). Nevertheless the current study is
restricted to the compression buckling of the elastic rods in the absence of a twist and thus the
demonstration of this effect is omitted here.
15
Figure 2.3: Elastic rod; (a) uniform and (b) non-uniform
2.2 Constrained buckling of elastic rods
The buckling of a slender elastic body constrained by discrete or continuous walls is a problem
widely encountered in engineering and medical applications. Examples include the constrained
buckling of drill-pipes in deep drilling for oil and gas (Wicks et al., 2008; Gao and Huang, 2015;
Huang et al., 2015; Sun et al., 2015; Miller et al., 2015), the uplift buckling of a heavy elastic
sheet from a one-sided rigid foundation in handling operations (Kolinski et al., 2009; Vella et al.,
2009; Wang, 1981b, 1984b), and the insertion of a guide-wire or catheter into blood vessels
(Duriez et al., 2006; Li et al., 2011b; Tang et al., 2012; Lenoir et al., 2006). There are actually
two distinct classes of constrained buckling problems. The classical problem, which deals with
constant length structures (Domokos and Holmes, 1993; Domokos et al., 1997; Holmes et al.,
1999; Manning and Bulman, 2005), and the insertion problem, which is concerned with variable
length slender objects (Denoel and Detournay, 2011; Huynen et al., 2016; Lu and Chen, 2008;
Ro et al., 2010).
Due to the existence of unilateral constraints, most studies are restricted to a first-order
analysis using either a discrete (i.e., finite differences or finite elements (Lazarus et al., 2015)) or
a continuous (i.e., closed-form solutions (Domokos et al., 1997; Holmes et al., 1999; Chai, 1998))
formulation.
Methods that include second-order analysis have recently been proposed (Manning and Bul-
man, 2005; Chen and Wu, 2011; Ro et al., 2010). Manning and Bulman (2005) have reformulated
Domokos et al. (1997)’s problem by replacing the unilateral constraint with a smooth poten-
tial. Furthermore, they have assessed the stability of an equilibrium state on the basis of the
conjugate point theory. Ro et al. (2010) provide an Eulerian description of the problem that
takes into account the variation of the contact points between the elastica and the walls during
vibration. Accordingly a shooting method is applied from which natural values of the lowest
frequencies can be derived, when the equilibrium configuration is stable.
Nevertheless, only preliminary studies can be found (see (Miersemann and Mittelmann, 1989;
16
Miersemann et al., 1986; Miersemann and Mittelmann, 1991)) that attempt to solve the con-
strained buckling problem by applying variational calculus. The reason behind that is the
presence of the unilateral constraints, which affect the space of allowed variations. More specif-
ically, Manning and Bulman (2005) state that “ the space of allowed variations is no longer a
vector space, but only a half-space”. Hence, a second-order analysis cannot be easily performed,
especially for the solution of the general constrained buckling problem with a priori unknown
contact conditions.
In the following Section, we limit our attention to the constrained buckling response of a
constant length elastica with known contact conditions. Based on this assumption, the unilateral
constraints are transformed to bilateral constraints (i.e., closed contacts). Using the known
positions of the contact points, the elastica is divided into distinct segments. For every distinct
segment, we then derive the associated potential energy and its variations, while continuity
requirements at the common extremities are also applied in order to get the complete solution
of the problem.
The presentation starts with the simplest possible case, the presence of a single discrete
contact. The general problem with a variety of discrete and continuous contacts is also demon-
strated. The aim of this analysis is to illustrate the limitation of the classical calculus of variation
method for the analysis of a constrained buckling problem, which can be overcome by applying
an optimal control method, as explained in Chapter 4.
2.2.1 Description of the constrained buckling problem
Based on the calculus of variation, we analyze the buckling response of a weightless planar elas-
tica of length L and uniform bending stiffness EI, which is constrained by two rigid horizontal
walls. In the initial state, we assume that the elastica rests at mid-distance between two friction-
less rigid horizontal walls. The left end of the elastica is fixed while the right end is constrained
to move horizontally. The extremities of the elastica are either clamped at zero inclination or
pinned, i.e., free to rotate.
A system of coordinates (X,Y ) is defined with origin at the left extremity of the elastica
and with the X-axis pointing towards the elastica right end. Relative to it, the two horizontal
walls are located at Y = +Cu and Y = −Cl. To describe an equilibrium state of the elastica,
the arc-length coordinate S is introduced with its origin S = 0 at X = 0 and Y = 0, and the
local inclination angle Θ(S) of the elastica on the X-axis. This implies that, in its unstressed
configuration, the elastica is described by Θ(S) = 0 with S ∈ [0, L]. The cartesian components
are then derived by applying the inextensibility and unshearability constraints. In particular,
a deformed configuration of the elastica X¯(S), Y¯ (S) is obtained by the local inclination angle
Θ(S) and the conditions X¯(0) = 0 and Y¯ (0) = 0. In addition Y¯ (L) = 0, the imposed relative
displacement is ∆ = L− X¯(L) and the presence of the two walls is described by the inequality
17
condition −Cl ≤ Y¯ (S) ≤ Cu with S ∈ [0, L].
The internal force transmitted by the elastica is denoted by R(S), while its inclination is
α(S) on the X-axis, as shown in Fig. 3.1. The functions R(S) and α(S) become piecewise
uniform whenever there is a discrete or a continuous contact with the walls. The axial force
Q = R cosα is uniform, however, due to the assumed frictionless nature of the contact.
Figure 3.1 illustrates a particular post-buckling configuration for an elastica with clamped
ends, which touches the top wall at a single point P1.
Θ( S )Y X X ( L )C l C u
P 1
QLR α ∆
Figure 2.4: An elastica constrained by two rigid horizontal walls, which are symmetrically located
with respect to the centerline. The left end is fixed while the right end is clamped. Sliding is
allowed along the horizontal axis.
Scaling follows from adopting L as length scale and EI/L2 as force scale;
s =
S
L
, x =
X
L
, y =
Y
L
, δ =
∆
L
, cu,l =
Cu,l
L
, R = RL
2
EI
, Q = QL
2
EI
(2.11)
The inextensibility and unshearability constraints are then given by ∂x¯(s)/∂s = cos θ(s) and
∂y¯(s)/∂s = sin θ(s). The cartesian components are accordingly derived by imposing the bound-
ary conditions x¯(0) = y¯(0) = 0, leading to
x¯(s) =
∫ s
0
cos θdξ, y¯(s) =
∫ s
0
sin θdξ (2.12)
Stability analysis of the elastica for a single discrete contact
The potential energy of the elastica involves its bending energy
U = 1
2
∫ 1
0
θ′2(s)ds (2.13)
18
under the presence of the equality and inequality constraints∫ s
0
sin θdζ − cu ≤ 0
− ∫ s
0
sin θdζ + cl ≤ 0∫ 1
0
sin θds = 0
(2.14)
while θ(0) = θ(1) = 0, θ′(0) = θ′(1) = 0 denote the clamped-clamped and pinned-pinned
boundary conditions, respectively. There are two distinct loading conditions; (1) force-controlled
conditions, where the axial compressive force Q is predefined and (2) displacement-controlled
conditons, where an axial distance between the supports is imposed. The potential energy also
involves
Q
(∫ `
0
cos θ(s)− 1
)
ds
which is the external work of the axial compressive forceQ for force-controlled loading conditions.
On the other hand, for displacement-controlled loading conditions, the axial force Q plays the
role of a Lagrange multiplier for the associated bilateral constraint, which is the imposed axial
distance between the extremities of the elastica.
To study the stability behavior of the constrained elastica, we assume that the positions of
the discrete, or continuous contacts are known. To better explain the procedure, let us first
investigate the simple case of a single discrete contact at s = `1 with the upper wall (see Figure
3.1). This means that we get the following bilateral constraint∫ `1
0
sin θdζ − cu = 0 (2.15)
while the other inequality constraint (2.14) ii) is inactive. Based on this discrete contact, the
elastica can be divided into two distinct segments with inclination angles θ1(s1) for s1 ∈ [0, `1]
and θ2(s2) for s2 ∈ [`1, 1], respectively. Hence, the potential energy can be splitted into two
parts;
Π(θ1, θ
′
1, θ2, θ
′
2) =
∫ `1
0
F1(θ1, θ′1)ds1 +
∫ 1
`1
F2(θ2, θ′2)ds2 (2.16)
where ∫ `1
0
F1(θ1, θ′1)ds1 =
1
2
∫ `1
0
(
θ
′
1
)2
ds1
+Q
{∫ `1
0
cos θ1ds1 − `1
}
19
+µ1
{∫ `1
0
sin θ1ds1 − cu
}
(2.17)
∫ 1
`1
F2(θ2, θ′2)ds2 =
1
2
∫ 1
`1
(
θ
′
2
)2
ds2
+Q
{∫ 1
0
cos θ2ds2 − (1− `1)
}
+µ2
{∫ 1
`1
sin θ2ds+ cu
}
(2.18)
are the potential energies of the two distinct segments and µ1, µ2 are the Lagrange multipliers
associated with the bilateral constraints, derived by combining (2.14)(iii) with (2.15).
To perform a stability analysis, a weak variation of the involved variables is assumed, which
is consistent with the 1-norm;
‖χ‖1 = max0≤s≤1 |χ(s)|+ max0≤s≤1 |χ
′(s)| (2.19)
where χ : [0, 1]→ R is some arbitrary variable. A variation of the position of the contact point
`1 + εγ is also assumed. In this manner, the variations of the involved variables can be written
as
θi(si, ε) = θ
∗
i (si) + εηi(si), i = 1, 2 (2.20)
Q = Q∗ + εQ¯ (2.21)
µi = µ
∗
i + εµ¯i, i = 1, 2 (2.22)
where θ∗i (si) , Q∗, µ∗i are the extremal values of the inclination angles, the axial compressive
load and the Lagrange multipliers, respectively, while their corresponding variations are ηi(s),
Q¯ and µ¯i and ε 1 is a positive parameter.
The continuity at the discrete contact is given by
θ1(`1 + εγ, ε) = θ2(`1 + εγ, ε) (2.23)
which leads to
θ∗1(`1 + εγ) + εη1(`1 + εγ) = θ
∗
1(`1 + εγ) + εη1(`1 + εγ) = 0 (2.24)
using (2.20). Taking the first and second derivatives of (2.24) with respect to ε and setting
20
ε→ 0, we get the following compatibility equations
γθ∗
′
1 (`1) + η1(`1) = γθ
∗′
2 (`1) + η2(`1) = 0 (2.25)
γ2θ∗
′′
1 (`1) + 2γη
′
1(`1) = γ
2θ∗
′′
2 (`1) + 2γη
′
2(`1) (2.26)
where the first derivative is zero in order to satisfy the boundary condition at the discrete contact
(i.e., zero inclination angle) after its variation.
Replacing `1 by `1 +εγ and substituting (2.20-2.22) into (2.16), the potential energy becomes
a function of ε. The first variation of the potential energy is then obtained by utilizing the
following derivation
δΠ(θ1, θ
′
1, θ2, θ
′
2) = lim
ε→0
dΠ
dε
(2.27)
Based on the Leibniz rule, the first variation of (2.17) for the two distinct segments is first
obtained
δ
∫ `1+εγ
0
F1(θ1, θ′1)ds1 =
∫ `1
0
[
−θ∗′′1 −Q∗ sin θ∗1 + µ∗1 cos θ∗1
]
η1ds1
+ θ∗
′
1 (`1)η1 (`1)− θ∗
′
1 (`o)η1 (`o)
+ γF1
(
θ∗1(`1), θ
∗′
1 (`1)
)
− γQ∗
+ Q¯
{∫ `1
0
cos θ∗1ds1 − `1
}
+ µ¯1
{∫ `1
0
sin θ∗1ds1 − cu
}
(2.28)
δ
∫ 1
`1+εγ
F2(θ2, θ′2)ds2 =
∫ 1
`1
[
−θ∗′′2 −Q∗ sin θ∗2 + µ∗2 cos θ∗2
]
η2ds2
+ θ∗
′
2 (`2)η2 (`2)− θ∗
′
2 (`1)η2 (`1)
− γF2
(
θ∗2(`1), θ
∗′
2 (`1)
)
+ γQ∗
+ Q¯
{∫ 1
`1
cos θ∗2ds− (`2 − `1)
}
+ µ¯2
{∫ 1
`1
sin θ∗2ds+ cu
}
(2.29)
21
where `o = 0 and `2 = 1 and leads to the following final form
δΠ(θ1, θ
′
1, θ2, θ
′
2) =
2∑
i=1
∫ `i
`i−1
[
−θ∗′′i −Q∗ sin θ∗i + µ∗i cos θ∗i
]
ηidsi
+
2∑
i=1
(−1)i+1γFi
(
θ∗i (`1), θ
∗′
i (`1)
)
+
2∑
i=1
[
ηi (`i) θ
∗′
i (`i)− ηi (`i−1) θ∗
′
i (`i−1)
]
+
2∑
i=1
Q¯
{∫ `i
`i−1
cos θ∗i dsi − (`i − `i−1)
}
+
2∑
i=1
µ¯i
{∫ `i
`i−1
sin θ∗i dsi + (−1)i cu
}
(2.30)
The additional compatibility requirements at the discrete contact involve the Weierstrass corner
conditions, which are given by
∂F1
∂θ′
(
θ∗1(`1), θ
∗′
1 (`1)
)
=
∂F2
∂θ′
(
θ∗2(`1), θ
∗′
2 (`1)
)
⇒
θ∗
′
1 (`1) = θ
∗′
2 (`1) (2.31)
θ∗
′
1 (`1)
∂F1
∂θ′
(
θ∗1(`1), θ
∗′
1 (`1)
)
−F1
(
θ∗1(`1), θ
∗′
1 (`1)
)
= (2.32)
θ∗
′
2 (`1)
∂F2
∂θ′
(
θ∗2(`1), θ
∗′
2 (`1)
)
−F2
(
θ∗2(`1), θ
∗′
2 (`1)
)
Equations (2.31)-(2.32) are automatically satisfied as the inclination angles and the curvatures
are continuous at the contact point, θ∗1(`1) = θ∗2(`1) and θ∗
′
1 (`1) = θ
∗′
2 (`1). This implies that
the second term of (2.30) vanishes. Then by vanishing the first variation of the potential energy
δΠ = 0, the equilibrium state of the elastica is derived;
θ∗
′′
i +Q∗ sin θ∗i − µ∗i cos θ∗i = 0, i = 1, 2 (2.33)
with boundary conditions
ηi (`i−1) θ∗
′
i (`i−1) = 0, i = 1, 2 (2.34)
ηi (`i) θ
∗′
i (`i) = 0, i = 1, 2 (2.35)
22
and the following additional requirements for the bilateral constraints
Q¯
{∫ `i
`i−1
cos θ∗i dsi − (`i − `i−1)
}
= 0, i = 1, 2 (2.36)
µ¯i
{∫ `i
`i−1
sin θ∗i dsi + (−1)i cu
}
= 0 i = 1, 2 (2.37)
For force-controlled loading conditions, the axial compressive force is fixed Q¯ = 0, while for
displacement-controlled conditions, the associated isoperimetric constraint remains unchanged.
On the other hand, the bilateral constraint related to the vertical distance of the extremities of
every discrete segment should be always satisfied. This implies a non-vanishing variation of the
Lagrange multiplier µ¯i, except the special case, where the elastica first touches one of the walls
(i.e., no shear force is present).
Combining (2.25) with (2.34-2.35), there are two potential boundary conditions; (1) constant
position of the contact point γ = 0, which leads to η1(`1) = η2(`1) = 0 and (2) change of the
position of the contact point γ 6= 0 that gives η1(`1) = η2(`1) = θ∗′1 (`1) = θ∗
′
2 (`1) = 0. The
derived boundary conditions imply that variation of a contact point can be assumed only if a
continuous contact is present. In other words, configurations with variable positions of discrete
contacts cannot be studied by following this derivation.
To deduce about the stability behavior of a particular equilibrium state of the elastica, the
second variation of the potential energy is required, which is denoted as
δ2Π(θ1, θ
′
1, θ2, θ
′
2) = lim
ε→0
d2Π
dε2
(2.38)
and involves the following two parts
δ2
∫ `1+εα
0
F1(θ1, θ′1)ds =
∫ `1
0
[
−η′′1 −Q∗ cos θ∗1η1 − µ∗1 sin θ∗1η1
]
η1ds1
+
[
η1 (`1) η
′
1(`1)− η1 (`o) η
′
1(`o)
]
+ γθ∗
′
1 (`1)
{
γθ∗
′′
1 (`1) + 2η
′
1(`1)
}
+ γ [−Q∗ sin θ∗1(`1) + µ∗1 cos θ∗1(`1)]
{
γθ∗
′
1 (`1) + 2η1(`1)
}
+ 2Q¯
[
−
∫ `1
0
sin θ∗1η1ds1 − γ (1− cos θ∗1 (`1))
]
+ 2µ¯1
[∫ `1
0
cos θ∗1η1ds1 + γ sin θ
∗
1(`1)
]
23
δ2
∫ `2
`1+εα
F2(θ2, θ′2)ds =
∫ `2
`1
[
−η′′2 −Q∗ cos θ∗2η2 − µ∗2 sin θ∗2η2
]
η2ds2
+
[
η2 (`2) η
′
2(`2)− η2 (`1) η
′
2(`1)
]
− γθ∗′2 (`1)
{
γθ∗
′′
2 (`1) + 2η
′
2(`1)
}
− γ [−Q∗ sin θ∗2(`1) + µ∗2 cos θ∗2(`1)]
{
γθ∗
′
2 (`1) + 2η2(`1)
}
+ 2Q¯
[
−
∫ `2
`1
sin θ∗2η2ds2 + γ (1− cos θ∗2 (`1))
]
+ 2µ¯2
[∫ `2
`1
cos θ∗2η2ds1 − γ sin θ∗2(`1)
]
Considering θ∗1 (`1) = θ∗2 (`1) = 0, γθ∗
′
1 (`1) = γθ
∗′
2 (`1) = 0 and η1(`1) = η2(`1) = 0, the second
variation of energy (2.38) takes the following form
δ2Π(θ1, θ
′
1, θ2, θ
′
2) =
2∑
i=1
∫ `i
`i−1
[
−η′′i −Q∗ cos θ∗i ηi − µ∗i sin θ∗i ηi
]
ηidsi
+
2∑
i=1
[
ηi (`i) η
′
i(`i)− ηi (`i−1) η
′
i(`i−1)
]
−
2
2
∑
i=1
Q¯
∫ `i
`i−1
sin θ∗i ηidsi
+
2
2
∑
i=1
µ¯i
∫ `i
`i−1
cos θ∗i ηidsi (2.39)
which becomes
δ2Π(θ1, θ
′
1, θ2, θ
′
2) =
2∑
i=1
∫ `i
`i−1
[
−η′′i −Q∗ cos θ∗i ηi − µ∗i sin θ∗i ηi
]
ηidsi (2.40)
under the following variations of the boundary conditions and the bilateral constraints
ηi(`i−1)η′i(`i−1) = 0, i = 1, 2 (2.41)
ηi(`i)η
′
i(`i) = 0, i = 1, 2 (2.42)
Q¯
∫ `i
`i−1
sin θ∗i ηidsi = 0, i = 1, 2 (2.43)
µ¯i
∫ `i
`i−1
cos θ∗i ηidsi = 0, i = 1, 2 (2.44)
24
The sign of the second variation of energy (2.40) determines the stability of an equilibrium state
of the elastica (a positive sign implies a stable solution).
Stability analysis of the elastica for a multiple number of contacts
If the elastica comes in contact with the wall at N contact points (i.e., discrete contacts and/or
entry and exit points of continuous contacts), we divide the elastica into N + 1 segments. The
start and end segments (i.e., {1} and {N + 1}) involve a contact point and a pinned, or clamped
end, depending on the boundary conditions of the elastica. The intermediate segments always
involve two contact points with one of the walls, or a combination of them.
Following the above approach, we can generalize the expressions for the first and second
variations of the potential energy. By vanishing the first variation of the total potential energy
of the elastica, the equilibrium configurations can be written as
θ∗
′′
i +Q∗ sin θ∗i − µ∗i cos θ∗i = 0, i = 1,N + 1 (2.45)
under the boundary conditions and the bilateral constraints
ηi (`i−1) θ∗
′
i (`i−1) = 0, i = 1,N + 1 (2.46)
ηi (`i) θ
∗′
i (`i) = 0, i = 1,N + 1 (2.47)
Q¯
{∫ `i
`i−1
cos θ∗i ds− (`i − `i−1)
}
= 0, i = 1,N + 1 (2.48)
µ¯i
{∫ `i
`i−1
sin θ∗i ds± (βucu ± βlcl)
}
= 0, i = 1,N + 1 (2.49)
where c = cu, or c = cl depending on the contact conditions, while βu = {−1, 0, 1}, βl =
{−1, 0, 1}. For instance, if there is a segment with its ends in contact with the same wall,
βu = βl = 0.
Similarly to (2.40), the second variation of the potential energy of the elastica is given by
δ2Π =
N+1∑
i=1
∫ `i
`i−1
[
−η′′i −Q∗ cos θ∗i ηi − µ∗i sin θ∗i ηi
]
ηidsi (2.50)
with variations of the boundary conditions and the bilateral constraints
ηi(`i−1)η′i(`i−1) = 0, i = 1,N + 1 (2.51)
25
ηi(`i)η
′
i(`i) = 0, i = 1,N + 1 (2.52)
Q¯
∫ `i
`i−1
sin θ∗i ηidsi = 0, i = 1,N + 1 (2.53)
µ¯i
∫ li
li−1
cos θ∗i ηidsi = 0, i = 1,N + 1 (2.54)
Limitation of the calculus of variation
To perform the stability analysis of every distinct segment, we can derive the eigensolution of an
associated Sturm-Liouville problem, as explained by (Levyakov and Kuznetsov, 2010). O Reilly
and Tresierras (2011) explains that if at least one segment is unstable, the elastica loses stability
as a whole. This stability criterion is usually valid for force-controlled conditions. Nevertheless,
for displacement-controlled conditions, there are cases, where a “stable segment” stabilizes the
instability of another neighboring “unstable segment”, leading to a final stable solution. This
phenomenon is met in asymmetrical configurations, or symmetrical configurations, such as the
“free-standing fold”.
Nevertheless, these configurations cannot be studied by applying the above procedure as a
variation of the contact position should be included, which violates (2.25). This is the main
limitation of the calculus of variation. We cannot also avoid the analysis of these particular
configurations for two main reasons; (1) the onset of them typically coincides with critical points,
such as bifurcation points or limit points and (2) their analysis can make clear the distinction
between force- and displacement-controlled loading conditions.
The above limitation becomes even more evident, if we analyze the stability behavior of an
elastica with a non-negligible self-weight, where the greater number of the equilibrium states is
asymmetrical, or symmetrical with variable positions of the contact points (see Chapter 6).
Chapter 3
Constrained Buckling of Variable
Length Elastica:
Solution by Geometrical
Segmentation
The main focus of this Chapter is the development of a method to calculate the equilibrium
states of a variable-length elastica (the constant length elastica being a particular case), which
is loaded along its axis and constrained by rigid longitudinal obstacles.
Numerous papers have been published on the unconstrained buckling of thin elastic structures
(Wang et al., 2004; Timoshenko, 2009; Bigoni, 2016) in contrast to the relatively small number
of studies concerned with constrained buckling. The unconstrained deflection of a variable-
length elastica has also been the subject of a few recent investigations (Humer and Irschik,
2011; Pulngern et al., 2013; Thongyothee and Chucheepsakul, 2015; Chucheepsakul et al., 2003;
Athisakul et al., 2011). The a priori unknown length of an inserted elastica loaded by a com-
pressive force introduces an additional challenge compared to the classical problem of a constant
length elastica and several methods have been proposed to address this specific issue (Chucheep-
sakul et al., 1995; Chucheepsakul and Monprapussorn, 2000; Chucheepsakul and Phungpaigram,
2004; Humer and Irschik, 2009, 2011). Furthermore, a configurational or Eshelby-like force at
the insertion point, not present in the classical problem, needs to be accounted as recently
discovered (Bosi et al., 2015; Bigoni et al., 2015).
In the constrained buckling problem, the unknown position and nature of the contact con-
ditions add new numerical difficulties. Discrete methods have been proposed to compute the
26
27
deformed configuration of a constant length elastica with unilateral constraints (Lazarus et al.,
2015; Klarbring, 1988; Schulz and Pellegrino, 2001), but these methods face numerical issues if
weakly active constraints are present, or when the contact patterns evolve discontinuously.
Therefore, most methods of solution are based on a continuous formulation of the constrained
elastica problem. One approach is to solve the constrained problem analytically using elliptic
integrals (Vaillette and Adams, 1983; Domokos et al., 1997), or assuming small displacements
(Chai, 1998). An alternative to the classical continuation method is the simplex method, typi-
cally used to solve linear programming problems. In combination with analytical solutions, the
simplex method makes it possible to compute all possible equilibrium states in a given domain,
the adjacent branches and even isolated solutions (Domokos and Holmes, 1993; Domokos et al.,
1997; Holmes et al., 1999).
These methods are restricted, however, for solving the constrained buckling of a constant
length elastica. Approaches specifically developed for simulating the insertion of an elastica in-
side a conduit have been proposed by a few authors (Huynen et al., 2016; Denoel and Detournay,
2011; Ro et al., 2010). In particular, an Eulerian formulation of the constrained planar elastica
in combination with a shooting method has been suggested (Huynen et al., 2016; Denoel and
Detournay, 2011). The Eulerian formulation provides a simple definition of the distance between
the elastica and an obstacle when compared to a Lagrangian-based method, but the application
of this technique is not straightforward. In addition the shooting method requires an accurate
guess of the involved variables, which is necessary for the solution of the initial value problem,
and a precise definition of the contact pattern. The same numerical issues are also present in a
similar technique suggested by Ro et al. (2010).
This Chapter proposes a novel technique to calculate the equilibrium configurations of a
planar weightless elastica subjected to unilateral constraints that overcomes the above numerical
difficulties and in particular the issue related to the a priori unknown length of the elastica in
insertion problems under force control. The method is contingent on assuming a deformed shape
of the elastica that is consistent with an assumed buckling mode and given unilateral constraints.
The key concept, which can be traced to Euler (Love, 2013) and Domokos et al. (1997), is to
divide the elastica into segments, each one being characterized by a zero curvature at one end.
The segmentation takes into account the contact constraints, the number of inflection points,
and the boundary conditions of the elastica for a specific buckling mode. This technique enables
the construction of the elastica solution, by assembling the solution for each segment, each one
being a particular realization of the same canonical problem. The method further depends
on deriving a closed-form asymptotic solution for the canonical problem using a perturbation
technique. The solution assembly entails solving a nonlinear system of equations that embodies
continuity conditions between the segments and the contact constraints.
The proposed approach is first validated by computing the post-buckling response of a
pinned-pinned elastica constrained by two symmetrically located walls, a particular instance
28
of the class of classical problems. The solution is shown to be in excellent agreement with
previously published numerical and experimental results (Domokos et al., 1997). Also, a de-
tailed comparison of the asymptotic solution of the canonical problem — the building block of
the segmentation technique — with the closed-form solution based on elliptic integrals and an
asymptotic expansion of this solution is documented. This comparison shows that the proposed
asymptotic solution is the most efficient within the framework of the segmentation technique
and at the same time is accurate enough for solving the constrained buckling problem for a
significant range of clearances.
The proposed approach is then applied to solve related constant and variable length elastica
problems. The similarities and differences between the two classes of problems are discussed.
For small clearances and displacements, these two problems are comparable. Nonetheless, the
development of a configurational force in the insertion problem leads to a reduction of the
external axial compressive force. Other factors such as the effect of the boundary conditions and
the loading conditions (i.e., force- or displacement-controlled conditions) are also investigated
for the insertion problem.
Special attention is placed on the bifurcation points, when either the bending moment at a
discrete contact vanishes, or a continuous contact with the wall buckles as the first buckling mode
of a clamped-clamped elastica. The first case is observed in the constrained buckling problem of
a clamped-clamped or clamped-pinned elastica with symmetrical walls, while the second case is
obtained in the classical buckling problem of a pinned-pinned elastica with symmetrically located
walls. Taking into account these geometrical characteristics, the corresponding equilibrium
configurations of the elastica can easily be obtained by applying the proposed geometry-based
method. Then, based on the principle of minimum energy (see Appendix B), the most favorable
or optimal configurations of the elastica among candidates can be determined. Throughout this
study, an optimal configuration is also called stable. A simply geometry-based stability criterion,
based on the number of the inflection points of the elastica, is also included. A more rigorous
stability analysis is required, however, which is beyond the scope of this study.
3.1 Constrained buckling problem of elastica
While the proposed method to calculate the equilibrium configuration of a planar elastica subject
to unilateral constraints is very general, we introduce it within the context of a specific problem
involving either a constant or a variable length elastica.
3.1.1 Problem definition
Consider a weightless planar elastica of uniform bending rigidity EI, whose deflection is con-
strained by two frictionless rigid horizontal walls that are symmetrically located on either side
29
of the elastica in its unstressed configuration (Fig. 3.1). Two different problems are under con-
sideration: (i) the classical problem, where the elastica has a fixed length L (Fig. 3.1a); and
(ii) the insertion problem, where the inserted length L¯ changes with the loading (Fig. 3.1b).
In both problems, one end of the elastica is fixed and is either clamped or pinned. In the first
problem, the other end of the elastica —either clamped or pinned — is constrained to move
horizontally. In the second problem the elastica is inserted through a frictionless sliding sleeve
that functions as a clamped end, with the distance L between the fixed end of the elastica and
the point of insertion maintained constant. In both cases, the axial loading is either under force
or displacement control.
Θ( S )Y X X ( L )C l C u
P 1
QLR α ∆
X ( L )Θ( S ) C l C uY X
P 1
L t QL
R
α
∆
Figure 3.1: An elastica constrained by two rigid horizontal walls with clearances Cu, Cl located
with respect to the centerline with the left end fixed and clamped; (a) Classical problem: the
right end is clamped on a slider restricted to move along the horizontal axis, (b) Insertion
problem: while the elastica is gradually inserted through a fixed frictionless sleeve at the right
end.
For both problems, we seek to determine the post-buckling behavior of the elastica to an
axial loading applied under either static or kinematic control. For force-controlled conditions, an
axial compressive force Q is progressively applied. For displacement-controlled conditions, either
the displacement ∆ at the right end of the elastica or the inserted length increment ∆¯ = L¯− L
is gradually increased. Specifically, we aim to establish the relation Q(∆) or Q(∆¯) between the
axial load and the end displacement or the change in inserted length. This relation reflects the
buckling mode, the pattern of contacts, and the nature of the boundary conditions at the ends
of the elastica.
30
We define a system of coordinates (X,Y ) with origin at the left extremity of the elastica
and with the X-axis pointing towards the elastica right end. The initial clearances between
the elastica and the upper and lower walls are denoted as Cu and Cl, respectively. Thus in
a representation of the planar elastica as a 1D object, the two horizontal walls are located at
Y = +Cu and Y = −Cl.
To describe the post-buckling configuration of the elastica, we introduce the arc-length co-
ordinate S with its origin S = 0 at X = 0 and Y = 0, and the local inclination angle Θ(S) of
the elastica on the X-axis. In its rest configuration, the elastica is thus described by Θ(S) = 0
with S ∈ [0, L] for the classical problem and S ∈ [0, L¯t] for the insertion problem, where L¯t
is the total length of the elastica. In view of the inextensibity and unshearability constraints
characterizing the elastica, its deformed configuration Xˆ(S), Yˆ (S) is fully defined by the func-
tion Θ(S) and the condition that the end S = 0 is fixed, i.e., Xˆ(0) = Yˆ (0) = 0. Additionally,
Yˆ (L) = 0 or Yˆ (L¯) = 0 in accordance with the imposed boundary conditions, and ∆ = L−X(L)
or ∆¯ = L¯−X(L¯). The unilateral constraints associated with the presence of the two walls are
represented by the inequality condition −Cl ≤ Yˆ (S) ≤ Cu with S ∈ [0, L] or S ∈
[
0, L¯
]
.
Let R(S) denote the magnitude of the internal force transmitted by the elastica and α(S) its
inclination on the X-axis, see Fig. 3.1 for the sign convention. The functions R(S) and α(S) are
actually piecewise uniform, as they experience only a jump at contact points (either discrete or
at the ends of continuous contacts). The axial force Q = R cosα is uniform, however, due to the
assumed frictionless nature of the contact for the classical problem. In the insertion problem,
the applied compressive force Q does not coincide with the axial force R cosα. More specifically,
an Eshelby-like force develops at the point of insertion (Bosi et al., 2015; Bigoni et al., 2015).
The existence of a configurational force implies that the external force Q is smaller than the
axial force R cosα.
3.1.2 Governing equations of elastica
Assuming that the elastica is unstressed in its initial state, L represent either the initial distance
between the ends of the elastica in the classical problem or the initial inserted length in the other
problem. Hence, scaling follows from adopting L as length scale and EI/L2 as force scale and
defining the dimensionless quantities
s =
S
L
, x =
X
L
, y =
Y
L
, cu,l =
Cu,l
L
, R = RL
2
EI
, Q = QL
2
EI
(3.1)
that are common to both problems, as well as
δ =
∆
L
or δ¯ =
∆¯
L
where δ represents the end-shortening and δ¯ the change of inserted length of the elastica.
31
The cartesian parametrization of the elastica, xˆ(s), yˆ(s), is deduced from θ(s) through inte-
gration
xˆ(s) =
∫ s
0
cos θdξ, yˆ(s) =
∫ s
0
sin θdξ (3.2)
where the boundary conditions xˆ(0) = yˆ(0) = 0 have been taken into account.
Insertion Problem
Under force-controlled conditions, the dimensionless form of the potential energy of the elastica
for the insertion problem can be written as
Π =
1
2
∫ ρ
0
θ′2(s)ds−Q
(
ρt −
∫ ρt
0
cos θ(s)ds
)
(3.3)
where ρ = L¯/L = 1+ δ¯, ρt = L¯t/L are the inserted and total lengths of the elastica, respectively.
The first term of Π is the elastic bending energy of the elastica, while the second term is the
external work of force Q under the following equality and inequality constraints∫ s
0
sin θdξ − cu ≤ 0, s ∈ [0, ρ] (3.4)
−
∫ s
0
sin θdξ + cl ≤ 0, s ∈ [0, ρ] (3.5)∫ ρ
0
sin θds = 0 (3.6)∫ ρ
0
cos θds = 1 (3.7)
and the boundary conditions, either θ(0) = 0 or θ′(0) = 0, and θ(ρ) = 0. For displacement-
controlled conditions, the external forceQ is the Lagrange multiplier associated with the bilateral
constraint ρt −
∫ ρt
0
cos θ(s)ds = δ¯.
The equations governing the equilibrium configuration of the elastica are now derived for a
configuration involving a single discrete contact with the upper wall, see Fig. 3.1b. The deriva-
tion can be readily extended to multiple contact points. Taking into account the constraints
(3.4)-(3.7), Π can be expressed as
Π =
1
2
∫ ρ
0
θ′2(s)ds−Q
(
ρt −
∫ ρt
0
cos θ(s)ds
)
− λx
(
1−
∫ ρ
0
cos θds
)
(3.8)
+ λy
∫ ρ
0
sin θds− F
(∫ `l
0
sin θdξ − cu
)
32
where λx, λy, F are the Lagrange multipliers for the corresponding equality constraints, F is
the contact force at the discrete contact P1 and `l is the length of the segment from s = 0 to
the discrete contact P1, as shown in Fig. 3.1b.
The variations of the involved variables can be written as
θ(s, ε) = θ∗(s) + εη(s) (3.9)
ρ(ε) = ρ∗ + εγ (3.10)
`l (ε) = `
∗
l + εγl (3.11)
where θ∗(s) , ρ∗, `∗l are the extremal values of the inclination angle, the inserted length of the
elastica and the length of the segment from s = 0 to the discrete contact P1, respectively, while
their corresponding variations are η(s), γ , γl and ε 1 is a positive parameter.
Taking into account that θ(ρ) = θ∗(ρ∗) = 0 and θ (`l) = θ∗ (`∗l ) = 0 and using (3.9), we
obtain the following compatibility equations
γ
dθ∗(ρ∗)
ds
+ η(ρ∗) = 0 (3.12)
γl
dθ∗(`∗l )
ds
+ η(`∗l ) = 0 (3.13)
Then the first variation of the potential energy is obtained as
δΠ(θ, θ
′
) =ε→0
∣∣∣∣dΠdε (3.14)
Using Leibniz rule, the first variation of (3.8) can be written as
δΠ(θ, θ
′
) =
∫ ρ∗
0
dθ∗
ds
dη
ds
ds− (Q+ λx)
∫ ρ∗
0
sin θ∗ηds
+ [λy − FH (s− `l)]
∫ ρ∗
0
cos θ∗ηds
+ γ
[
1
2
(
dθ∗(ρ∗)
ds
)2
+ λx cos θ
∗ (ρ∗)
]
+ γλy sin θ
∗ (ρ∗)− γlF sin θ∗ (`∗l ) (3.15)
where H(s) is the Heaviside function. Integrating by parts the first term of (3.15) and considering
the compatibility equation (3.12) yield
∫ ρ∗
0
dθ∗
ds
dη
ds
ds =
dθ∗
ds
η
∣∣∣∣ρ∗
0
−
∫ ρ∗
0
d2θ∗
ds2
ηds
33
= −γ
(
dθ∗(ρ∗)
ds
)2
−
∫ ρ∗
0
d2θ∗
ds2
ηds (3.16)
After substituting (3.16) into (3.15) and noting that θ∗ (ρ∗) = θ∗ (`∗l ) = 0, the first variation of
the potential energy becomes
δΠ(θ, θ
′
) = −
∫ ρ∗
0
{
d2θ∗
ds2
+ [Q+ λx] sin θ∗ − [λy − FH (s− `l)] cos θ∗
}
ηds (3.17)
+ γ
[
−1
2
(
dθ∗(ρ∗)
ds
)2
+ λx
]
The governing equation of the elastica is then obtained by setting δΠ(θ, θ
′
) = 0
d2θ∗
ds2
+
[
Q+ 1
2
(
dθ∗(ρ∗)
ds
)2]
sin θ∗ − [λy − FH (s− `l)] cos θ∗ = 0
Omitting ∗, the final form of the governing equation of the elastica can be written as
d2θ
ds2
+R cosα sin θ −R sinα cos θ = 0 (3.18)
where R cosα = Q+(dθ(ρ)/ds)2 /2, R sinα = λy−F[2]H (s− `l) and s ∈ [0, ρ] (i.e., the absolute
values of the resultant force R and its inclination angle are constant along the arc-length of the
elastica as the configuration is symmetrical). The difference between the applied load and the
axial force P = R cosα−Q = (dθ(ρ)/ds)2 /2 is an Eshelby-like force that appears at the point
of insertion (Bigoni, 2016; Bigoni et al., 2015; Bosi et al., 2015; Misseroni et al., 2015).
Classical Problem
For the classical problem, the total length of the elastica is ρ = 1 and thus the variation of
the total length (3.10) is not applied. In addition (3.7) is replaced by
∫ 1
0
cos θds = δ, which
is identical to the displacement produced by the applied compressive force Q. Hence, there is
no Eshelby-like force and Q = R cosα. Following the above derivation, the same governing
equation of the elastica (3.18) is obtained with boundary conditions θ(0) = 0 or θ′(0) = 0 and
θ(1) = 0 or θ′(1) = 0.
3.2 Canonical problem
In this section we formulate and solve a canonical problem that constitutes the building block
for the segmentation technique described in Section 3.3. Indeed, this technique involves dividing
the elastica into contiguous segments, each one of them having characteristics similar to those
34
of the canonical problem. The solution of the canonical problem can thus serves as a general
solution for each segment. Construction of the solution for the whole elastica is then reduced to
assembling the segment solutions that have to satisfy various conditions, such as continuity of
the solution between two successive segments or the unilateral constraints. The whole approach
thus hinges on obtaining a solution of the canonical problem in terms of three numbers that
fully define it.
3.2.1 Problem Definition
Consider the unit length elastica depicted in Fig. 3.2. Let ξ ∈ [0, 1] denote the arc length
coordinate. The elastica is pinned at ξ = 0 and clamped at ξ = 1, i.e., ϑ′(0) = 0 and ϑ(1) = ϑ1.
A dimensionless compressive force R˜ inclined by an angle α ∈ [−pi/2, pi/2] on the x-axis is
applied at extremity ξ = 1. Note that the force has here been scaled by EI/`2 with ` denoting
the physical length of the segment. The solution ϑ(ξ) is obtained by solving
d2ϑ
dξ2
+ R˜ cosα sinϑ− R˜ sinα cosϑ = 0 (3.19)
subjected to ϑ′(0) = 0 and ϑ(1) = ϑ1, given also R˜ and α. The form of (3.19) suggests to
rewrite this equation in terms of the relative inclination ψ(ξ) = ϑ(ξ) − α. Furthermore, we
replace the input parameter R˜ by the boundary condition ϑ(0) = ϑ0. The canonical problem is
thus reformulated as
d2ψ
dξ2
+ R˜ sinψ = 0 (3.20)
with ψ′(0) = 0, ψ(0) = ψ0 = ϑ0 − α, and ψ(1) = ψ1 = ϑ1 − α, noting that R˜ is now part of the
solution.
Once ψ(ξ) has been determined, the cartesian components x˜(ξ), y˜(ξ) of a point ξ of the
elastica are given by
x˜(ξ) = x˜(0) +
∫ ξ
0
cos(ψ + α)dζ, y˜(ξ) = y˜(0) +
∫ ξ
0
sin(ψ + α)dζ (3.21)
The canonical problem depends therefore on three angles: α, ϑ0, and ϑ1. It can in principle
be solved analytically, but the presence of incomplete elliptic integrals in the solution makes it
difficult to compute, see Appendix. However, an asymptotic solution to this problem can readily
be obtained by transforming the nonlinear equation (3.20) into a system of linear equations,
which can be solved as boundary value subproblems.
35
)i
ϑo
y˜(1)
x˜(1)R˜
R˜
M˜
ϑ1
ϑ
)i i
ϑο
ϑ'
ϑ' 1
Figure 3.2: a) Canonical problem and b) Phase portrait
3.2.2 Asymptotic Solution
Inspired by (Challamel et al., 2015), we seek an asymptotic expansion of the solution in the
form
R˜ = R˜(0) + εR˜(1) + ε2R˜(2) + ε3R˜(3) +O(ε4) (3.22)
ψ(ξ) = εΨ(1)(ξ) + ε2Ψ(2)(ξ) + ε3Ψ(3)(ξ) +O(ε4) (3.23)
where ε = ψ0 is the small parameter. The second expansion starts with an order of O(ε) because
of the post-buckling nature of the analysis. It results from this choice of expansion that R(0) is
the Euler buckling for a pinned-clamped beam subjected to an inclined load.
Inserting the asymptotic expansions (3.22) and (3.23) into (3.20), after first replacing sinψ
by ψ − 16ψ3, and balancing terms of same order yields a sequence of linear ODEs that replaces
the original nonlinear ODE. Limiting the enforcement of (3.20) to the first three orders, we
obtain
O(ε) : d
2Ψ(1)
dξ2
+ R˜(0)Ψ(1) = 0 (3.24)
O(ε2) : d
2Ψ(2)
dξ2
+ R˜(0)Ψ(2) = −R˜(1)Ψ(1) (3.25)
O(ε3) : d
2Ψ(3)
dξ2
+ R˜(0)Ψ(3) = −R˜(2)Ψ(1) − R˜(1)Ψ(2) + 16R˜(0)Ψ(1)3 (3.26)
36
The first order solution Ψ(1) satisfies the boundary conditions
Ψ(1)(0) = 1,
d
dξ
Ψ(1)(0) = 0, Ψ(1)(1) = β (3.27)
where β is defined as
β =
ψ1
ψ0
(3.28)
while the higher order solutions Ψ(i), satisfy the homogeneous boundary conditions
Ψ(i)(0) = 0,
d
dξ
Ψ(i)(0) = 0, Ψ(i)(1) = 0, i > 1 (3.29)
Solving (3.24) with (3.27) yields
Ψ(1)(ξ) = cos
(√
R˜(0) ξ
)
(3.30)
with R˜(0) given by
R˜(0) = arccos2 β (3.31)
which reduces to R˜(0) = pi2/4 if β = 0. The next order solution Ψ(2)(ξ) is of the form
Ψ(2)(ξ) = − R˜
(1)
2
√
R˜(0)
ξ sin
(√
R˜(0) ξ
)
(3.32)
However, necessarily R˜(1) = 0, in order to satisfy the boundary condition at ξ = 1. Finally,
the terms Ψ(3)(ξ) and R˜(2) of the asymptotic expansion of the solution are obtained by solving
(3.26) with (3.29)
Ψ(3)(ξ) =
1
192
[
cos
(√
R˜(0) ξ
)
− cos
(
3
√
R˜(0) ξ
)]
+
R˜(0) − 8R˜(2)
16
√
R˜(0)
ξ sin
(√
R˜(0) ξ
)
(3.33)
R˜(2) = 18 arccosβ
[
β
(
1− β2)
3
√
1− β2 + arccosβ
]
(3.34)
with R˜(2) = pi2/32 if β = 0.
In summary, the asymptotic expansion of the solution reads
R˜ = R˜(0) + ψ20R˜(2) +O(ψ40) (3.35)
ψ(ξ) = ψ0Ψ
(1)(ξ) + ψ30Ψ
(3)(ξ) +O(ψ50) (3.36)
37
3.2.3 Unscaling the Canonical Solution
As shown above, three numbers completely define the solution of the canonical problem, e.g.,
ϑ0, ϑ1, and α. Indeed force R˜ and moment M˜ = d2ϑ/dξ2 at ξ = 1 depend on ϑ0 − α and
ϑ1 − α, while ∆x˜ and ∆y˜ depend on ϑ0, ϑ1, α. An additional parameter, the scaled length ` of
the segment (i.e., the physical length of a segment scaled by the length of the elastica) needs
to be known, however, to translate the solution of the canonical problem in terms of quantities
defined for the original elastica.
It is easy to show that
R = `−2R˜(ϑ0 − α, ϑ1 − α)
M = `−1M˜(ϑ0 − α, ϑ1 − α)
∆x¯ = `∆x˜(ϑ0, ϑ1, α)
∆y¯ = `∆y˜(ϑ0, ϑ1, α)
Hence four parameters define the solution of a segment.
3.3 Segmentation
3.3.1 Principle
The proposed approach to solve the elastica is based on presupposing a configuration of the
elastica corresponding to a given buckling mode, and on slicing the elastica into a series of
contiguous segments. The segmentation is carried out in such a way that each segment has
a zero curvature (θ′ = 0) at one end, a non-zero curvature (θ′ 6= 0) and an expression for θ
at the other end that is either implicit or explicit. The segment is further characterized by
a monotonically varying curvature, with the trivial case of a straight segment being the only
exception. The length of the segments is unknown a priori, except for particular cases where the
configuration exhibits symmetry or anti-symmetry. As each segment is a particular realization
of the canonical problem, the segmentation technique leads to the formulation of a system of
equations to be solved for the unknown quantities that determine inclination ϑ(ξ) and length `
of each segment.
Defining as nodes the two ends of the elastica and the interior points at which the elastica
is segmented, the elastica is thus marked by an alternate sequence of “pinned” (θ′ = 0) and
“clamped” nodes (θ′ 6= 0). The interior pinned nodes are inflection points of the elastica. The
segmentation leads to the definition of m segments which forms a set I` = {1, 2, ..m} and
n = m+ 1 nodes with a set I = {1, 2, ..n} consisting of nc clamped nodes and np pinned nodes
with nc = np+nk−1, where nk denotes the number of ends of the elastica at which the inclination
38
θ is prescribed (i.e., nk = 0, 1, or 2 depending on whether the elastica is pinned-pinned, pinned-
clamped, or clamped-clamped). All the segments have boundary conditions consistent with the
canonical problem presented in Section 3.2.
Let s[i] denote the arc length coordinate of node i, i = 1, n, with s[1] = 0 and s[n] = 1. The
length `i of segment i, bounded by nodes i and i+1 is thus given by `i = s[i+1]−s[i]. Denoting by
θi(s) the inclination of the elastica along segment i, it is evident that determining the deformed
shape of the elastica is equivalent to computing function θi(s) and length `i for each segment
i = 1,m. Solution θi(s), s[i] ≤ s ≤ s[i+1], for each segment can be expressed in terms of four
parameters, using the explicit asymptotic expansion of the solution of the canonical problem. As
discussed in Section 3.2.3, these four parameters can be chosen as length `i, inclinations θi(si)
and θi(si+1), and inclination αi of the force transmitted by segment i. Solving the constrained
elastica problem is thus transformed, with the segmentation technique, into determining the 4m
parameters that completely define the global solution.
3.3.2 System of Equations
The 4m primary unknowns are computed by imposing a series of conditions that reflect the
boundary conditions (imposed inclination or curvature at the end nodes, horizontal displacement
of force at the first node), continuity of the solution, force equilibrium, and several constraints
(imposed length of the elastica, no vertical offset between the two ends of the elastica, constraint
on the vertical displacement due to the existence of walls). In particular, if there are nw nodes in
contact with the walls, the nodal reaction forces represent nw additional secondary unknowns,
which are determined by imposing a known displacement at these nodes. Thus, there is a total
of 4m+nw unknowns. We present next the 4m+nw equations to be solved for the 4m primary
and nw secondary unknowns. These equations can be grouped in 8 categories, as follows.
1. Inclination at the clamped nodes (nc equations):
G[i](θ[i]) = 0, (3.37)
where relation G[i](θ[i]) provides either an implicit or an explicit expression for θ[i], the
inclination of the elastica at node i.
2. Continuity of inclination between two segments at an interior node (n− 2 equations):
θi(s[i+1]) = θi+1(s[i+1]), i ∈ Iin (3.38)
where Iin = {2, .., n− 1}
39
3. Continuity of curvature between two segments at a clamped interior node (nc − nk equa-
tions):
θ′i(s[i+1]) = θ
′
i+1(s[i+1]) (3.39)
4. Force equilibrium at an interior node (2n− 4 equations):
Ri cosαi = Ri+1 cosαi+1, i = 1,m− 1 (3.40)
Ri sinαi = Ri+1 sinαi+1 + F[i+1]δi+1,k k ∈ Iw (3.41)
where F[i+1] denotes the vertical reaction force at node i + 1 in contact with a wall, δi,k
denotes the Kronecker delta, and Iw is the index set for the constrained nodes.
5. Imposed vertical displacement at the nw constrained interior nodes (nw equations):
k∑
i=1
∆yi = c, k + 1 ∈ Iw+ and
k∑
i=1
∆yi = −c, k + 1 ∈ Iw− , (3.42)
where
∆yi =
∫ s[i+1]
s[i]
sin θi(s) ds. (3.43)
and Iw± denotes the set containing the index of the nodes in contact with the upper (+)
and the lower (−) walls (Iw = Iw− ∪ Iw+). Conditions (3.42) provide the additional
equations required to solve the vertical reaction force at the nw constrained nodes.
6. Constraint on the segment lengths (1 equation):
m∑
i=1
`i = 1. (3.44)
for the classical problem and
m∑
i=1
`i = 1 + δ¯. (3.45)
for the insertion problem
7. Zero vertical offset between the two ends of the elastica (1 equation):
m∑
i=1
∆yi = 0. (3.46)
40
8. Imposed horizontal force or displacement at the first node (1 equation):
R1 cos θ[1] = Q or
m∑
i=1
∆xi = δ, (3.47)
for the classical problem and
R1 cos θ[1] = Q or
m∑
i=1
∆xi = 1, (3.48)
for the insertion problem, where
∆xi =
∫ s[i+1]
s[i]
cos θi(s) ds. (3.49)
It can readily be verified that the total number of equations is indeed 4m + nw, after noting
that m = 2nc − nk.
3.3.3 Example
Consider the configuration of a pinned-pinned elastica of constant length, illustrated in Fig.
3.3. This configuration is characterized by two symmetric discrete contacts between the elastica
and the upper wall and a free standing fold (Roman and Pocheau, 1999; Pocheau and Roman,
2004). In view of the symmetry, only half of the elastica, consisting of 3 segments (12, 23, and
34), needs to be considered. Nodes 1 and 3 are pinned, while nodes 2 and 4 are clamped. The
length `i of the segments is a priori unknown but are constrained by
∑3
i=1 `i = 1/2, noting
that `2 = `3 on account of symmetry. There is a vertical reaction force at node 2. Assuming
a vertical position for node 4 (to replace constraint (3.46)), the horizontal force-displacement
relationship at node 1 can in principle be calculated from the system of equations (3.37)-(3.49).
Expressions of these equations for this particular example is given below.
Referring to Figure 3.3, the inclinations, relative inclinations and curvatures for the segments
can be expressed as follows
θ1(s) = ψ1(s) + α1 (3.50)
θ2(s) = ψ2(s) (3.51)
ψ1(s) =
(
θ[1] − α1
)
cos
√
R(0)1 s s ∈
[
s[1], s[2]
]
(3.52)
ψ2(s) = θ[3] cos
√
R(0)2
(
s[3] − s
)
s ∈ [s[2], s[3]] (3.53)
41
κ1(s) = −
(
θ[1] − α1
)√R(0)1 sin√R(0)1 s s ∈ [s[1], s[2]] (3.54)
κ2(s) = θ[3]
√
R(0)2 sin
√
R(0)2
(
s[3] − s
)
s ∈ [s[2], s[3]] (3.55)
where R(0)2 = R(0)1 cosα1 = Q(0), θ[1] > 0, θ[3] < 0 are the inclinations at nodes 1, 3 respectively
and α1 is the inclination angle of the resultant force R(0)1 at node 1.
The system of equations is then given by
F(φ) =

κ1(s[2])− κ2(s[2])∫ s[2]
s[1]
θ1(s)ds−∆y1∫ s[3]
s[2]
θ2(s)ds−∆y2√
Q(0)/ cosα1 − arccos (β1) /`1√
Q(0) − pi/2`2

(3.56)
with unknown vector φ (i.e., φi ≥ 0, i = 1, 5) defined as
φ =

θ[1]
θ[3]
α1
`1
Q(0)

In the above, β1 = −α1/
(
θ[1] − α1
)
, ∆y1 = c is the constant clearance between the upper wall
and the centerline and −c ≤ ∆y2 ≤ 0 is predefined. Length `2 of the second segment is given
by `2 = 1/4− `1/2.
The system of equations (3.56) is solved numerically using the nonlinear least-squares MatLab
solver “lsqnonlin”. With this solver, convergence of the solution is achieved even with a poor
initial guess. If all terms up to the second order (i.e., ε2) are kept, the second term of the
resultant force R(2)1 can be obtained by (3.34) by setting β = β1.
For the insertion problem, the total length of the elastica is not known a priori. Hence, the
second segment ¯`2 and the constraint (3.48) have to be included in φ and F(φ), respectively.
Alternatively, the total length of the elastica can be predefined, while the vertical displacement
becomes the additional unknown variable.
42
F [ 6 ]F [ 2 ]
32 7
654
C l a m p e dP i n n e d
1 c
Figure 3.3: Free standing fold
3.4 Validation: Constrained Pinned-Pinned Elastica
Validation of the proposed technique entails demonstrating the effectiveness and accuracy of the
geometrical segmentation technique and of the asymptotic solution of the canonical problem.
The segmentation technique is validated by computing the post-buckling response of a pinned-
pinned elastica constrained by two symmetrically located walls at y = ±c, a classical problem
solved by Domokos et al. (1997). Detailed comparison of the asymptotic solution of the canonical
problem with the closed-form solution based on elliptic integrals and an asymptotic expansion
of this solution is documented in Section 3.2. The possible configurations of the elastica are then
described as functions of the buckling mode and the applied load. Each specific configuration is
accordingly analyzed by applying the segmentation technique. Lastly, the constrained problem
is solved for clearance c = 0.05, to enable comparison with published numerical and experimental
results of (Domokos et al., 1997).
3.4.1 Validation of canonical problem
The canonical problem, solved in Section 3.2 using a pertubation technique, has also a closed-
form solution. However, as stated by Domokos et al. (1997) “The explicit solution involves
incomplete elliptic integrals and is awkward to work with, except in a second limiting case when
the internal moment at the contact point drops to zero”. This statement can be easily confirmed
if the analytical solution is adopted to calculate a sequence of equilibrium states of the elastica
for force- or displacement-controlled loading conditions for a fixed clearance c. In particular,
the problem becomes highly nonlinear when incomplete elliptic integrals are involved. For this
reason, the root-finding process, used to construct the whole solution, usually fails, especially if
the initial guess is not close to the correct solution. To overcome this numerical difficulty, there
are two possible approaches: (1) an approximate solution of the problem by expanding onto a
Taylor series the elliptic integrals; and (2) a semi-analytical solution derived from asymptotic
43
considerations.
In this Section, these approaches are applied and compared with the analytical solution for
the the first buckling mode of a pinned-pinned elastica with or without a discrete contact at
its midpoint position. The solution of this particular problem does not require a root-finding
process.. Based on the segmentation technique, there is a single canonical segment of length
`1 = 0.5 with boundary conditions ψ1(s[1]) = θ[1] − α1, ψ′1(s[1]) = 0 and ψ1(s[2]) = θ[2] − α1,
where s[1] = 0, s[2] = `1, θ[1] ≥ 0, α1 ≥ 0, θ[1] ≥ 2α1 and θ[2] = 0.
The detailed derivation of the analytical solution can be found in Domokos et al. (1997).
The results are only summarized here. More specifically, the resultant force R1 is given by
K
(
k¯
)− F (φ; k¯) = √R1`1 (3.57)
where φ = arcsin [sin (ψ(`1)/2) / sin (ψ(0)/2)], the cartesian components ∆x1, ∆y1 at the right
end of the elastica can be expressed as
∆x1 =
∫ `1
0
cos(ψ1 + α1)dξ, ∆y1 =
∫ `1
0
sin(ψ1 + α1)dξ (3.58)
where ∫ `1
0
cosψ1ds =
2
[
E(k¯)− E(φ, k¯)]√R1 − `1,
∫ `1
0
sinψ1ds =
2k¯√R1
cosφ (3.59)
and the compressive axial load becomes Q = R1 cosα1.
As shown in (3.57-3.59), the analytical solution involves incomplete elliptic integrals of first
and second kind which can be written as
F
(
φ; k¯
)
=
∫ φ
0
dϕ√
1− k¯2 sinϕ
, E
(
φ; k¯
)
=
∫ φ
0
√
1− k¯2 sinϕdϕ
where k¯ is the modulus of the elliptic integral. If φ = pi/2, the complete elliptic integrals of first
and second kinds are simply obtained as K(k¯) = F
(
pi/2; k¯
)
and E(k¯) = E
(
pi/2; k¯
)
, respectively.
After setting χ = sinϕ, a Taylor series expansion of the incomplete elliptic integrals can be
derived, leading to
F
(
φ; k¯
) ≈ φ+ 1
2
k¯2
∫ φ
0
sin2 ϕdϕ+
3
8
k¯4
∫ φ
0
sin4 ϕdϕ+ ... (3.60)
E
(
φ; k¯
) ≈ φ− 1
2
k¯2
∫ φ
0
sin2 ϕdϕ− 1
8
k¯4
∫ φ
0
sin4 ϕdϕ+ ... (3.61)
44
while the complete elliptic integrals become
K(k¯) ≈ pi
2
(
1 +
1
4
k¯2 +
9
64
k¯4 + ...
)
(3.62)
E(k¯) ≈ pi
2
(
1− 1
4
k¯2 +
3
64
k¯4 + ...
)
(3.63)
On the other hand the asymptotic solution of the canonical problem can be found in Section
3.2.2. For convenience, the Taylor series method is called T1, T2, when the first and second
order expansions are used, respectively. In a similar way, A1, A2 are the asymptotic solutions
of the canonical problem that include the first and second order expansions, respectively. These
approaches are compared with the analytical solution for three distinct inclination angles; (1)
α1 = 0, (2) α1 = 0.1 and (3) α1 = 0.4. The first case, α1 = 0, corresponds to the unconstrained
pinned-pinned elastica. The remaining cases describe configurations of a constrained pinned-
pinned elastica with a single discrete contact at the midpoint position. Obviously, for the higher
value of inclination angle α1 = 0.4, higher values of clearances c = ∆y1 are obtained.
When θ[1] ≈ 2α1 or ψ[1] ≈ α1, the moment at the discrete contact vanishes, leading to
the onset of the continuous contact. In this limited case, the derived axial force coincides
with the Euler-buckling load and the asymptotic solutions are identical to the analytical solu-
tion, independently of the calculated vertical displacement ∆y1. This coincidence can be easily
understood by the fact that the analytical solution only involves complete elliptic integrals,
verifying Domokos et al. (1997)’s remark. This conclusion also implies that the comparison of
these methods with the analytical solution should be performed by using the relative inclination
angle ψ[1] rather than the inclination angle θ[1]. In Figs. 3.4-3.5, the diagrams of the axial force
Q/pi2, the end-shortening δ1 and the vertical displacement ∆y1 with respect to ψ[1] for the three
distinct clearances are presented.
For the comparison of these approaches with the analytical solution, the relative error with
max |e| = 0.1 is used. Based on it, the upper bounds of the relative inclinations ψ[1] = θ[1]−α1 are
obtained for all cases. The results show that the T1 fails even for relatively small displacements
(see Figs. 3.4-3.5). Furthermore, the T2 is better for the derivation of the axial force with
maxψ[1] = 0.8 when compared to A2 with maxψ[1] = 0.6. Nevertheless, the latter method
is more accurate for the derivation of the end-shortening and the vertical displacement with
maxψ[1] = 1.25 when compared to the former method with maxψ[1] = 0.8.
This comparison suggests that T2 is the best method for the derivation of the axial force,
especially when the elastica is unconstrained and large displacements are present ψ[1] > 0.6.
However the derivation of a constrained elastica with a fixed clearance requires a root-finding
process. The use of T2 is not efficient due to the nonlinear dependence of φ on the modulus of the
elliptic integral k¯, the relative inclination ψ1 and the inclination angle α1, and the complex form
45
of the expansion, which is given by (3.60)-(3.61). For clearance c ≤ 0.2 (i.e, 1/5 of the total length
of the elastica), ψ[1] ≤ 0.6 and A2 is the most efficient and at the same time accurate enough
method for the solution of the constrained problem with fixed clearance. According to Figs.
3.4-3.5, for the same range of clearance, the use of A1 for the derivation of the displacements
and the application of A2 for the calculation of the axial force simplifies the solution of the
problem, without sacrificing much accuracy.
0.4 0.6 0.8 1.0 1.2
1.00
1.05
1.10
1.15
1.20
1.25
0.4 0.6 0.8 1.0 1.2
0.00
0.05
0.10
0.15
0.20
0.25
0.4 0.6 0.8 1.0 1.2
0.12
0.16
0.20
0.24
0.28
0.32
0.36
0.40
T2
T1
A2
A1
δ 1
Analytical
∆
y 1
Q/
⇡
2
 [1]
 [1]
 [1]
Figure 3.4: Graphs of the axial force Q/pi2, the end-shortening δ1 and the vertical displacement
∆y1 with respect to ψ[1] = θ[1] for α1 = 0
46
0.2 0.4 0.6 0.8 1.0 1.2
1.2
1.3
1.4
1.5
1.6
1.7
1.8
0.2 0.4 0.6 0.8 1.0 1.2
0.00
0.05
0.10
0.15
0.20
0.25
0.2 0.4 0.6 0.8 1.0 1.2
0.1
0.2
0.3
0.4
T2
T1
A2
A1
Analytical
δ 1
i)
∆
y 1
 [1]
 [1]
 [1]
Q/
⇡
2
0.6 0.7 0.8 0.9 1.0
1.6
1.8
2.0
2.2
0.6 0.7 0.8 0.9 1.0
0.05
0.10
0.15
0.20
0.25
0.6 0.7 0.8 0.9 1.0
0.25
0.30
0.35
T2
T1
A2
A1
Analytical
ii)
δ 1
∆
y 1
Q/
⇡
2
 [1]
 [1]
 [1]
Figure 3.5: Graphs of the axial force Q/pi2, the end-shortening δ1 and the vertical displacement
∆y1 with respect to ψ[1]; (i) α1 = 0.1 and (ii) α1 = 0.4
3.4.2 Evolution of Contact Patterns
For an assumed buckling mode, the critical axial force at which stability is lost is given by
the classical buckling solution for beam-columns (Timoshenko, 2009). Increasing the axial load
beyond the critical value causes continuing deformation of the elastica, which remains initially
unconstrained as illustrated in Fig. 3.6i) for the first buckling mode and in Figs 3.7i) and 3.7iv)
for higher buckling modes. Eventually the elastica comes in contact with the wall at some
discrete points (first contact event), causing the appearance of a reaction force at every contact
point j ∈ Iw. Given the buckling mode, the exact positions of the initial contact points are
known, see Figs. 3.6ii, 3.7ii and 3.7v.
At some stage, the direction of the resultant applied force Ri, i ∈ I \ Iw at every pinned
node i ∈ I \ Iw passes from the adjacent contact point j ∈ Iw (i.e., j = i− 1 and j = i+ 1 for
i ∈ Iin, j = i + 1 for the i = 1 and j = i − 1 for i = n). When this happens, the curvature
at contact point j vanishes, leading to the onset of a line contact (second contact event), as
shown in Figs. 3.6iii and 3.7iii. The contact length increases with continued loading. For the
47
first buckling mode, this contact pattern persists until secondary buckling occurs (third contact
event), as illustrated in Fig. 3.6iv.
Configurations of the elastica characterized by no contact, discrete, and continuous contacts
are discussed next. For the calculations, inclination θ[1] at the elastica fixed end is used as a
loading parameter in lieu of δ or Q.
i i i )
n d = 2n p = 4n c = 3
n d = 2n p = 4n c = 3
n d = 1n p = 2n c = 1
i )
n d = 1n p = 2n c = 1
i i )
C l a m p e d
i v )
P i n n e d
Figure 3.6: First buckling mode of pinned-pinned elastica
48
n d = 1n p = 4n c = 3
n d = 1n p = 4n c = 3
n d = 2n p = 7n c = 6
n d = 1n p = 3n c = 2
i ) n d = 1n p = 3n c = 2
i i )
i i i )
i v )
C l a m p e d
v )
I n f l e c t i o n
Figure 3.7: Second and third buckling mode of pinned-pinned elastica
3.4.3 Unconstrained elastica
The post-buckling response of an unconstrained elastica is well-known (Love, 2013). However,
an approximate solution can be deduced from the asymptotic analysis of the canonical problem
described in Section 3.2. In particular, inclination θ1(s) of the first segment and horizontal force
R1 of the first segment for the k−buckling mode depends on θ[1] according to
R1 = (kpi)2
[
1 +
θ2[1]
8
]
(3.64)
θ1(s) = θ[1]
{
cos [kpis] +
θ2[1]
192
(cos [kpis]− cos [3kpis])
}
(3.65)
Because the force transmitted by the unconstrained elastica is uniform, inclination θi(s), i ∈
I` of the other segments is easily derived from (3.65) using symmetry and/or antisymmetry
49
arguments, see Figs. 3.6i, 3.7i and 3.7iv. The above expressions are applicable until the elastica
touches one of the horizontal walls.
3.4.4 Discrete contact
Further increasing θ[1] past the first contact event causes contact force F[j] > 0, j ∈ Iw to
increase. Because of symmetry, the magnitudes of the transverse forces Ri sinαi, i ∈ I \ Iw are
identical and equal to half the contact forces F[j], where j = i − 1 and j = i + 1 for i ∈ Iin,
j = i+ 1 for i = 1 and j = i− 1 for i = n.
The positions of the contact points on the elastica remain unchanged with further loading.
Due to symmetry, the solution of a single clamped-pinned segment of length `i = 1/2k, i ∈ I` is
only required for which ∆yi = c, i ∈ I` and ∆xi = (1− δ)/m, i ∈ I` are satisfied. This contact
pattern is present until θ[i] = 2αi, which leads to the onset of the continuous contact.
3.4.5 Continuous contact
Past the second contact event, the elastica consists of two types of segments: pinned-discrete
contact elements and straight elements in continuous contact with the wall. Noting that the
moment vanishes at the ends of the continuous contact zones, the original 2k pinned-discrete
point elements are divided in half by assigning a clamped node at its midpoint and the k straight
elements are also divided in half in order to be consistent with the canonical problem. After
subdivision, there are thus m = 6k segments.
The solution for the whole elastica is constructed for a given θ[1], once the solution for one
of the 4k pinned-clamped segments has been obtained. Due to symmetry, the solution for these
segments satisfies
∆yi = c/2, ∆xi = (1− δ − s¯)/4k, i ∈ I \ Iw
where s¯ denotes the combined length of the line contact segments.
s¯ =
k∑
i=1
(
s[6i−1] − s[6i−3]
)
Furthermore, forces Ri, i ∈ I \ Iw are equal and satisfy Ri sinαi = F[j], where j = i − 1 and
j = i + 1 for i ∈ Iin, j = i + 1 for the i = 1 and j = i − 1 for i = n. Taking into account that
the resultant force R1 passes through the first contact point (node 3), a geometric relation can
be obtained
α1 = arctan
(
2kc
1− δ − s¯
)
In the special case of the first buckling mode k = 1, this contact pattern is present until secondary
buckling (i.e., post-buckling of line contact segment) is reached with `i ≈ 1/4k, i = 1,m.
50
3.4.6 Numerical Results
The post-buckling response of a pinned-pinned elastica constrained by two symmetrically located
rigid walls is calculated for a clearance c = 0.05. The response is expressed as relationships
between compressive axial force Q and end displacement δ for various admissible configurations
of the elastica, see the bifurcation diagram illustrated in Fig. 3.8. the solution — in particular,
the parametric relationshipsQ(θ[1]), δ(θ[1]) — is calculated for different admissible configurations
by varying the imposed angle θ[1].
The asymptotic method with terms up to the second order captures accurately the analytical
solution of Domokos et al. (1997); no difference could be detected if both solutions were plotted
in the bifurcation diagram of Fig. 3.8. The discrepancy between analytical and the experimental
results in Fig. 3.8, is attributed to some initial imperfections of the elastica (Domokos et al.,
1997). As the description of the sequence of deformation shapes can be found in (Domokos
et al., 1997), only the main results are presented here. In particular, the bifurcation diagram is
shown in Fig. 3.8 and the points at the contact pattern change are summarized in Table 6.1.
Points θ[1] δ Q/pi2
2 0.1575 0.0062 1.003
3 0.2 0.0075 3.985
4 0.406 0.0155 15.8
7 0.407 0.35 15.8
10 0.48 0.0616 9
Table 3.1: Points of evolution of the contact patterns for the bifurcation diagram for a con-
strained pinned-pinned elastica with clearance c = 0.05
51
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07
0
5
10
15
20
25
11
109
88
4 77
66
5
4
3
2
Experimental (Domokos, 97)
Stable (D.C.)
Unstable
δ
Stable
1
Q/
⇡
2
1 - 2 2 - 3
3 - 4 4 - 1 0
6 - 7
7 - 89 - 1 0
5 - 6
1 0 - 1 1
Figure 3.8: Bifurcation diagram for a constrained pinned-pinned elastica with clearance c = 0.05
3.5 Stability analysis based on the number of inflection
points
The stability of an unconstrained elastica can be determined from the number of inflection
points (Sachkov, 2008a; Jin and Bao, 2008, 2014). A somewhat similar geometry-based stability
criterion is conjectured to hold for constrained elastica. The criterion depends on the number
of the inflection points and on the nature of the contact conditions. Although a formal proof
of this conjecture does not exist yet, its predictions are confirmed by results from other studies
52
(Ro et al., 2010; Manning and Bulman, 2005; Chen et al., 2015).
For the purpose of the stability analysis, the elastica is now sectioned into elements based
on the contact positions (Holmes et al., 1999). There are thus two types of elements: (i) the
two end-elements — connecting one of the elastica ends to a discrete contact or to the endpoint
of a continuous contact; and (ii) the mid-elements — either elements of continuous contacts, or
elements in between two discrete contacts. The total number of elements is m(d)e = nw + 1 for
discrete contacts and m(l)e = 2nw/3 + 1 for continuous contacts.
Continuous and discrete contacts may not coexist in the case of a weightless elastica. Indeed,
the contact patterns are consistent with the configurations derived from the (Euler-)buckling
modes under consideration (Timoshenko, 2009). Due to the contact constraints, a symmetrical
evolution of the configurations is derived for clamped-clamped and pinned-pinned boundary
conditions leading to identical types of contact conditions (discrete or line contacts). Even for
asymmetrical cases the contact conditions are limited to discrete contacts (e.g., clamped-pinned
elastica) since either the deformation shape evolves gradually to a symmetrical configuration with
higher number of discrete contacts, or it loses stability at the onset of a continuous contact.
The stability analysis for force-controlled and displacement-controlled boundary conditions
have to be investigated separately. For force-controlled conditions the negative slope of a branch
(i.e., softening) is a sufficient condition for instability, while for displacement-controlled condi-
tions the softening branches can be either stable or unstable based on the current configuration
of the elastica.
The stability criteria can be expressed solely in terms of np —the number of inflection points,
and me — the number of elements. For configurations with discrete contacts, the elastica is
unstable if n(d)p > m
(d)
e for force-controlled conditions and n
(d)
p > m
(d)
e + 1 for displacement-
controlled conditions. For configurations with continuous contacts, the elastica loses stability if
n
(l)
p > 2m
(l)
e − 2 for either force- or displacement-controlled conditions.
The stability of an unconstrained elastica in the post-buckling regime is well-known (Love,
2013)(Bigoni, 2016). Provided that the two end supports are on the axis of the elastica in its
unstressed state, the elastica becomes unstable when n(u)p > me + 1 (or n
(u)
p > 2), irrespective
of the boundary condition (index (u) is used unconstrained).
3.6 Insertion problem
Here we first investigate the post-buckling responses of a clamped-clamped elastica, considering
both cases of constant and variable length and compare the bifurcation diagrams for these two
cases. Next we analyze the problem of inserting a clamped-pinned elastica in a conduit and we
compare the sequence of equilibrium configurations with results for the insertion of a clamped-
clamped elastica. The difference between force- and displacement-controlled conditions is also
contrasted for the problems considered.
53
3.6.1 Clamped-clamped elastica with symmetric walls
The similarities and differences between the post-buckling behavior of a constrained clamped-
clamped elastica for a classical and an insertion problem are discussed in details for a clearance
c = 0.05 (see also (Manning and Bulman, 2005)) and are outlined for c = 0.1. The corresponding
bifurcation diagrams illustrated in Figs 3.9 and 3.10 are expressed not only in terms of the axial
force Q, but also with respect to the internal axial force R cosα. In this manner the additional
effect of the configurational force is clearly demonstrated.
Consider first displacement-controlled conditions. The behavior for the first symmetric buck-
ling mode shares some similarities with the pinned-pinned case examined in Section 3.4.6 (see
Fig. 3.8). Due to symmetry the elastica comes into contact with the wall at the midpoint posi-
tion at configuration C2 for the classical problem, and at C2¯, C2′ for the insertion problem, see
Table 3.2 for the characteristics (θ[2], δ, Q/pi2, R cosα/pi2) of the critical configurations at which
a change of pattern takes place. (The symbol Ci stands for the configuration of the elastica at
Point i of the bifurcation diagram, where i i′, i¯ denote the corresponding points of the graphs
Q = R cosα for the classical problem and Q, R cosα for the insertion problem, respectively.)
The following observations can be made from a comparison of the two problems; (1) almost
identical values of δ and δ¯, with δ¯ < δ, are obtained at the onset of the first contact and (2) the
post-buckling response is consistent with a super-critical bifurcation for the classical problem
and sub-critical bifurcation for the insertion problem. The second observation can easily be
rationalized by considering the first order term of the asymptotic expansion of the axial force
R(0)/pi2 = 4/ (1 + δ¯)2; this term indicates that the internal axial force decreases with increasing
length of the elastica and the applied axial force is even lower Q(0)/pi2 =
(
1− θ2[1]
)
R(0)/pi2 due
to the configurational force. In contrast, the first order term in the classical problem is equal
to the Euler-buckling load Q(0)/pi2 = 4, while the second order term has an additional positive
effect, leading to a gradual increase of the axial force with end-shortening δ. Hence, the branches
1 − 2 and 1¯ − 2¯, or 1′ − 2′ have positive (i.e., hardening branch) and negative (i.e., softening
branch) slopes, respectively; they are both stable for displacement-controlled conditions.
The contact pattern 2 − 3 remains unchanged until C3, or C3 = C3′, see Table 3.2. The
axial applied and internal forces coincide at Points 3¯ and 3′, because the bending moment at
the insertion point of the sliding sleeve vanishes, leading to the vanishing of the configurational
force. This configuration also corresponds to the vanishing of the moment at the discrete contact.
Nevertheless it does not evolve into a continuous contact. In addition the simultaneous change
of the moments of both ends is not feasible. Hence, beyond Point 3 the sign of the moment at
the right end changes, leading to an asymmetrical configuration with a single discrete contact
(see pattern 3− 4). More explanation can be found in Appendix B.
Along Branch 3−4, 3¯− 4¯, or 3′−4′, the horizontal load decreases with increasing δ or δ¯ (i.e.,
softening branch). This branch is present until a second discrete contact occurs at C4, C4¯, or C4′,
54
where the configuration becomes anti-symmetric with two discrete contacts. To understand the
slight difference between Points 4 and 4¯, the second-root of the first buckling load of a clamped-
clamped elastica has to be analyzed. This corresponds to an anti-symmetric unconstrained
configuration of the elastica (see shape 8 − 4) with initial C8 = C8¯ = C8′. This branch is
unstable. Theoretically, if δ (δ¯) is gradually increased, Point 4 (4¯) is eventually reached. As
explained above, this clearly shows the super-critical and sub-critical bifurcations and explains
the reason behind the difference between the Points 4 and 4¯. An additional reduction arises
from the configurational or Eshelby-like force, which leads to a lower value of the applied force
Q of Point 4′. Beyond these points, the calculated configuration 4−5 with two discrete contacts
remains unchanged until C5, or C5¯ = C5′. Then an asymmetrical configuration is obtained
with shape 5−6, for similar reasons described above. Past this point, the difference between the
classical and the insertion problems starts increasing. See for example the difference between
C6, C6¯ and C6′ in Table 3.2, both corresponding to the onset of three discrete contacts.
To complete the presentation, the distinction between the displacement- and force-controlled
conditions should be also clarified. For force-controlled conditions, the softening branches 1¯− 2¯,
or 1′−2′ and 3−4, 3¯− 4¯, or 3′−4′ and 5−6, 5¯− 6¯, or 5′−6′ are unstable. Instead, the solution
jumps from Points 1¯, or 1′ and 3, 3¯, or 3′ and 5, 5¯, or 5′ to the branches 2¯ − 3¯, or 2′ − 3′ and
4− 5, 4¯− 5¯, or 4′ − 5′ and 6− 7, 6¯− 7¯, or 6′ − 7′. These jumps are indicated by arrows in Fig.
7.2. Upon unloading, the response follows the force-controlled type of sequence of deformations
shapes in the opposite way (i.e., from shape 6 − 7 to the shape 4 − 5 and then 2 − 3 until its
unconstrained mode).
From the bifurcation diagram, it is shown that the effect of the configurational force becomes
significant along the softening branches. This is evident due to the increasing values of the
bending moments at the insertion point of the sliding sleeve with increasing displacement, which
leads to higher values of the configurational force. On the other hand, the Points 3¯, 3′ and 5¯, 5′
coincide, respectively, because the bending moment vanishes at the insertion point, implying
that Q = R cosα.
We conclude this Section with a few remarks on the case with clearance c = 0.1 (see also
(Ro et al., 2010)). The sequence of the deformation shapes is identical to the case c = 0.05.
However, the difference between the bifurcation values of the parameters δ and δ¯ at Points 2−6
(2¯ − 6¯, or 2′ − 6′) increases, especially for large displacements. The bifurcation diagrams of
the constrained elastica for both problems are given in Fig. 5.11, while the particulars of the
solution (θ[2], δ, Q/pi2, R cosα) at the bifurcation points 2−6 (2¯− 6¯, or 2′−6′) are summarized
in Table 7.1.
55
Points Classical problem Insertion problem
θ[2] δ Q/pi2 = R cosα/pi2 θ[2] δ¯ R cosα/pi2 Q/pi2
2, 2¯, 2′ 0.16 0.0062 4.01 0.1565 0.00615 3.96 3.91
3, 3¯, 3′ 0.2 0.0075 15.90 0.2 0.0074 15.30 15.3
4, 4¯, 4′ 0.26 0.026 8.30 0.253 0.02564 7.85 7.45
5, 5¯, 5′ 0.29 0.034 33.00 0.29 0.033 32.80 32.8
6, 6¯, 6′ 0.367 0.0606 16.35 0.347 0.0571 14.67 13.24
8, 8¯, 8′ 0 0 8.18 0 0 8.18 8.18
Table 3.2: Points of evolution of the contact patterns for the bifurcation diagram for a con-
strained clamped-clamped elastica with clearance c = 0.05
Points Classical problem Insertion problem
θ[2] δ Q/pi2 = R cosα /pi2 θ[2] δ¯ R cosα /pi2 Q/pi2
2, 2¯, 2′ 0.318 0.025 4.05 0.311 0.025 3.86 3.67
3, 3¯, 3′ 0.407 0.030 15.78 0.394 0.030 14.86 14.86
4, 4¯, 4′ 0.536 0.113 8.50 0.484 0.100 6.97 5.72
5, 5¯, 5′ 0.560 0.190 27.80 0.517 0.152 24.03 24.03
6, 6¯, 6′ 0.790 0.296 17.60 0.628 0.219 11.46 8.00
Table 3.3: Points of evolution of the contact patterns for the bifurcation diagram for a con-
strained clamped-clamped elastica with clearance c = 0.1
56
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07
0
5
10
15
20
25
30
35
Stable D.C.
3
3,3'
9,9,9'
5,5,5'
7,7' 7
6'
6
6
4' 4
48,8,8'
2,2,2'
δ,δ
1,1,1'
Stable
Unstable
Q/
⇡
2
,R
co
s(
↵
)/
⇡
2
1 - 2 2 - 3
3 - 4 4 - 5
5 - 6 6 - 7
8 - 4 9 - 6
Figure 3.9: Bifurcation diagram for a constrained clamped-clamped elastica with clearance
c = 0.05; (1) classical stability problem (dashed line) and (2) insertion stability problem with
dashed line for the applied load Q and dotted line for the axial force R cosα.
57
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
5
10
15
20
25
30
Unstable
Stable D.C
5,5'
5
7'
7
3,3'
39,9,9'
7
6
6
6'4
4
  4'
8,8,8'
2
22'
δ,δ
1,1,1'
Stable
Q/
⇡
2
,R
co
s(
↵
)/
⇡
2
1 - 2 2 - 3
3 - 4 4 - 5
5 - 6 6 - 7
8 - 4 9 - 6
Figure 3.10: Bifurcation diagram for a constrained clamped-clamped elastica with clearance
c = 0.1; (1) classical stability problem (solid line) and (2) insertion stability problem with
dashed line for the applied load Q and dotted line for the axial force R cosα.
3.6.2 Clamped-pinned elastica with symmetric walls
Here we analyze the post-buckling behavior of a clamped-pinned inserted elastica and compare
it with the response of a clamped-clamped inserted elastica presented above. For both cases,
clearance c = 0.05.
58
Initially, the elastica deflects according to the first buckling mode of a clamped-pinned elas-
tica. As the elastica is gradually inserted in the conduit, a discrete contact appears at C2 or
C2¯, as shown in Table 5.3.2. This contact pattern remains unchanged until the moment at the
discrete contact vanishes at C3 or C3¯. Beyond this point, a change of the sign of the moment
occurs leading to the asymmetrical deformation shape 3 − 4. This softening branch gradually
evolves, until a second discrete contact with the opposite wall is achieved at C4 or C4¯. This new
deformation shape 4 − 5 evolves until the moment vanishes at both discrete contacts at C5 or
C5¯. In a similar manner, a new asymmetrical configuration 5 − 6 is then obtained. Along this
branch, the axial force decreases and a third discrete contact appears at the midpoint position at
C6 or C6¯ with a symmetrical shape 6−7. For force-controlled conditions, all softening branches
(i.e., 1−2, 1¯− 2¯ 3−4, 3¯− 4¯ 5−6, 5¯− 6¯) are unstable. Hence, at points 1, 1¯, 3, 3¯, 5, 5¯ the evolution
of the deformation shapes is shown by the arrows in Fig. 5.8.
Points Insertion problem
θ[1] δ¯ Q/pi2
2 0.2 0.0064 2.03
3 0.3 0.0085 8.93
4 0.36 0.0254 6.35
5 0.495 0.0329 23.84
6 0.505 0.0563 12.63
Table 3.4: Points of evolution of the contact patterns for the bifurcation diagram of a constrained
clamped-pinned elastica with clearance c = 0.05
59
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07
0
5
10
15
20
25
30
Stable D.C.
7,7
6
6
5,5
4
4
3,3
2,2
δ
1,1
Stable
Q/
⇡
2
,R
co
s(
↵
)/
⇡
2
1 - 2
5 - 6
4 - 53 - 4
2 - 3
6 - 7
Figure 3.11: Bifurcation diagram for a constrained clamped-pinned elastica with clearance c¯ =
0.05; solid line for the applied load Q and dashed line for the axial force R cosα
3.7 Conclusions
This Chapter has introduced a method that enables accurate and rapid determination of the
different equilibrium states of a constrained elastica of constant or variable length. The method
hinges on a segmentation of the elastica that defines sub-problems, each consistent with a sin-
gle canonical problem, and on a closed-form asymptotic solution of the canonical problem.
60
The equilibrium solution of the elastica is then constructed by assembling the solutions of the
subproblems while satisfying various continuity and constraints requirements. The complete
solution requires solving a nonlinear system of equations. The principle of minimum energy is
applied to determine the optimal sequence of equilibrium configurations associated with mono-
tonic loading of the elastica under either force- or displacement-control (see Appendix B). In
addition a simple geometry-based criterion to assess the stability of an equilibrium has been
conjectured. The criterion is based on the number of inflection points and on the nature of the
contact. A formal proof of this conjecture remains to be done, however.
The asymptotic solution of the canonical problem is compared with the analytical solution
and another asymptotic method, both based on the elliptic integrals. The proposed asymptotic
solution is shown to be the most efficient and is accurate enough to compute the post-buckling
response of constrained elastica for clearances c ≤ 0.25. The proposed segmentation method is
then validated by computing the constrained response of pinned-pinned elastica. The solution
is shown to be in full agreement with previously reported numerical results (Domokos et al.,
1997).
The method has then been applied to the solution of several classical and insertion problems.
The two classes of problems are first compared for a particular constrained buckling problem of
a clamped-clamped elastica. Their main difference is the development of a configurational force
at the insertion point of the sliding sleeve, which is not present in the classical problem. This
leads to an applied axial load lower than the internal axial force. Even though this distinction is
present, the analysis shows that the derived solutions are almost identical for small displacements
and clearances. Discrepancy appears for large displacements, because the difference between
the end-shortening for the classical problem and the change of the length of the elastica for the
insertion problem becomes significant. Factors such as the boundary and loading conditions
have also been investigated.
Chapter 4
Optimal Control
The application of the calculus of variations in the stability analysis of elastic structures sub-
jected to unilateral constraints is questionable. The main source of the problem is the assumption
of the 1-norm (i.e., assumption of piecewise-smooth functions) when nonsmooth conditions are
present. This issue is resolved by applying an optimal control method, as explained in Section
4.1.
The equivalence between the calculus of variations and the optimal control is first shown
by summarizing the derivation of Berkovitz (1961). We then provide a detailed description of
the optimal control theory with pure-state constraints, including some preliminary studies on
the derivation of the second-order sufficient conditions for optimality (Malanowski et al., 2004;
Hermant, 2009b). Numerical methods for solving general optimal control problems are also
presented, with the main focus on the direct methods.
The content of the chapter is not original. It is a compilation of materials from (Berkovitz,
1961, 1962; Bonnans and Hermant, 2008; Betts, 1998, 2010; Bryson, 2016; Berkovitz and Medhin,
2012; Bryson et al., 1963; Bryson and Denham, 1964; Hartl et al., 1995; Hermant, 2009b,a;
Elnagar et al., 1995; Ross, 2015; von Stryk, 1993; Wachter and Biegler, 2005; Biegler and Zavala,
2009; Kameswaran and Biegler, 2007; Barclay et al., 1998; Gill et al., 2002; Liberzon, 2012;
Mesterton-Gibbons, 2009; Nocedal, 2006; Waltz et al., 2005; Byrd et al., 2006; Goldfarb, 1970;
G. Fasano, 2012; Bulirsch and Stoer, 2010; Goh and Teo, 1988; Kraft, 1985; Malanowski et al.,
2004; Maurer and Pickenhain, 1995; Maurer and Oberle, 2002). To make the text more readable,
quotation marks on specific sentences copied from papers were not used as well as attribution
to the corresponding sources. This chapter was included to provide background material on the
methodology used to solve the constrained buckling problems described in Chapters 5-7, where
the interface ICLOCS (Falugi P. and van Wyk, 2010) connected with the NLP software IPOPT
(Biegler and Zavala, 2009; Wachter and Biegler, 2005) is used.
61
62
4.1 From calculus of variations to optimal control
A classical calculus of variations problem is an equivalent optimal control problem even if in-
equality constraints are present. It is shown below that the necessary conditions for a calculus
of variations problem can also be translated into necessary conditions for the optimal control if
some supplementary conditions are included (Berkovitz, 1961, 1962). It is also shown that the
Pontryagin’s Minimum Principle is a direct translation of the usual Weierstrass condition.
The main distinction between calculus of variations and optimal control is the choice of
the norm in the derivation of the local optimality. In particular, on the space of C1 curves
x : [a, b]→ R, there are two candidates for the norm
‖x‖0 = max
a≤s≤b
|x(s)| (4.1)
‖x‖1 = max
a≤s≤b
|x(s)|+ max
a≤s≤b
|x′(s)| (4.2)
where (4.1) is the 0-norm ‖x‖0 and (4.2) is 1- norm ‖x‖1.
When the 0-norm or the 1-norm is assumed the extremal of the corresponding functional is
called strong or weak extremal, respectively. In the field of calculus of variations, the 1-norm is
adopted. This means that piecewise-smooth functions are assumed in order to derive the first
and second variation of a functional. This restriction is relaxed in the optimal control theory
where the 0-norm is assumed, which implies that piecewise-continuous functions are also allowed.
In Section 4.1.1, we show the equivalence between the calculus of variations and the optimal
control under the presence of unilateral constraints. The theorem of Pontryagin’s Minimum
Principle is accordingly deduced.
4.1.1 Analysis
A general nonlinear optimal control problem P with mixed - pure state constraints (Ross, 2015)
is given by 
Minimize J [x(·),u(·), s] = ∫ s`
0
L (x(s),u(s)) ds
subject to
x′ = f(x(s),u(s))
h(x(s),u(s)) ≤ 0
φ (x(0),x(s`)) = 0
(4.3)
which involves the derivation of the control and state spaces {x(·),u(·)}, where x(·) ∈ RNx
are piecewise-smooth functions and u(·) ∈ RNu are piecewise-continuous functions. In the
system of equations (4.3), L : RNx × RNu → R is the Lagrangian, or the running cost, f :
RNx ×RNu → RNx is the spatial dynamics, φ : RNx ×RNx → RNφ are the boundary conditions,
63
and h :∈ RNh are mixed-pure state constraints. We assume that all functions are k-times
continuously differentiable Ck with respect to their arguments (k ≥ 2) and the dynamics f is
Lipschitz continuous.
The functional J [x(·),u(·), s] is of Lagrangian form since it only includes a Lagrangian, or
running cost. Other forms of the functional J [x(·),u(·), s] are also possible (Liberzon, 2012).
In particular, the functional J [x(·),u(·), s] is of Bolza form, when it involves both a running
cost and an endpoint cost, while it is of Mayer form, when it includes only an endpoint cost
(Ross, 2015). A problem can be easily transformed from one to another form if it is necessary.
When an isoperimetric constraint is also present∫ s
0
ϕ(σ, x, u)dσ ≤ c (4.4)
a new state variable is included with associated spatial derivative x′ = ϕ(s, x, u), initial condition
x(0) = xo and constraint x(s) ≤ c.
A classical optimal control problem can be transformed to a calculus of variations problem
if we replace the control variables u(·) ∈ RNu by y′(·) ∈ RNu and assign the associated spatial
derivative y′(s) = u(s) with initial conditions y(0) = 0. If inequality constraints are also present,
they can be transformed to equality constraints by adding slack variables. In this manner the
optimal control problem P becomes a new problem Pc in the calculus of variations;
Minimize J [x(·),y′(·), s] = ∫ s`
0
L (x(s),u(s)) ds
subject to
f(x(s),y′(s))− x′ = 0
h(x(s),y′(s)) + ξ′2 = 0
φ (x(0),x(s`),y(0),y(s`), ξ (0) , ξ (s`)) = 0
(4.5)
where ξ ∈ RNh is the slack variable and the boundary conditions φ (x(0),x(s`),y(0),y(s`), ξ (0) , ξ (s`)) =
0 are given by
x(0) = xo, y(0) = 0, ξ (0) = 0
x(s`) = x`, y(s`) = y`, ξ (s`) = ξ`
(4.6)
The augmented Lagrangian can be expressed as
L¯(t,x,y,x′,y′, ξ′,λ,µ) = L+ λ (f − x′) + µ (h + ξ′2) (4.7)
where λ(s) ∈ RNx and µ(s) ∈ RNh are the Lagrange multipliers of the equality constraints
f(x(s),y′(s))− x′ = 0 and h(x(s),y′(s)) + ξ′2 = 0, respectively.
Let us denote the extremal variables by the symbol ” ∗ ”. More specifically, the optimal
64
control is given by u∗ = y
′∗(s) , while the arc Z is defined by {x∗(s),y∗(s), ξ∗(s)}, where(
ξ∗
′)2
= −h (s,x∗, y∗′) , ξ∗(0) = 0 (4.8)
furnishes a minimum for the problem Pc.
Assuming a 1-norm (4.2), we can consider the family of trial curves
y(s) = y∗(s) + εηy(s)
x(s) = x∗(s) + εηx(s)
ξ(s) = ξ∗(s) + εηξ(s)
(4.9)
where ε  1 is a positive parameter and ηx, ηy, ηξ are arbitrary admissible functions that
satisfy the boundary conditions (4.6). The Euler-Lagrange equations along Z are then derived
separately for the variables {x(s),y(s), ξ(s)}
dL¯x′
ds
= L¯x, dL¯y
′
ds
= L¯y, dL¯ξ
′
ds
= L¯ξ (4.10)
If
{
x
′∗(s),y
′∗(s), ξ
′∗(s)
}
are not continuous everywhere along Z, additional requirements should
be satisfied. In particular, the du Bois-Reymond equation is derived by the vanishing first
variation of the functional
L¯x′ =
∫ s`
0
L¯xds+ Cx (4.11)
where Cx is some arbitrary constant (Mesterton-Gibbons, 2009).
Let us suppose that there is at least one sc ∈ (0, s`) at which x′∗ is discontinuous. For the
sake of definiteness, suppose that the left- and right-hand limits of x
′∗ at the corner sc are given
by ω1 = x
′∗(s−c ) = lim x
′∗(s)
s→s−c
and ω2 = x
′∗(s+c ) = lim x
′∗(s)
s→s+c
with ω1 6= ω2. Now, even though
the integrand on the right-hand side of the du Bois-Reymond equation (4.11) is discontinuous
at the corner sc, the integral itself is continuous. Hence, the right-hand side of (4.11) must be
continuous at sc, which implies the continuity of the “momentum”
L¯x′ (sc,x∗ (sc) , ω1) = L¯x′ (sc,x∗ (sc) , ω2) (4.12)
This is the so-called first Weierstrass-Erdmann corner condition. The consequence is that any
broken extremal must be consistent with the Euler-Lagrange equation, except at corners where
(4.12) should be satisfied.
In addition we can assume that the other variables y
′∗(s), ξ
′∗(s) are also discontinuous at
the same position sc with left- and right-hand limits ϕ1 = y
′∗(s−c ), ϕ2 = y
′∗(s+c ), χ1 = ξ
′∗(s−c )
and χ2 = ξ
′∗(s+c ), respectively. Taking into account (4.10) and considering that L¯y = L¯ξ = 0,
65
the momenta L¯y′ and L¯ξ′ are constant values. This implies that the continuity of L¯y′ and L¯ξ′
is trivially satisfied everywhere.
To derive the second Weierstrass-Erdmann corner condition, we assume two distinct classes
of trial curves with a predefined corner point sc. The first trial curve x(1)(s) can be expressed
as follows
x(1)(s) =

x∗(s) if 0 ≤ s ≤ sc
x∗(s) + ω1(s− sc) if sc ≤ s ≤ sc + ε
x∗(s) + {x
∗(sc)+ω1ε−x∗(sc+ε)}(s`−x)
(s`−sc−ε) if sc + ε < s ≤ s`
(4.13)
where 0 ≤ ε < s` − sc, while the second trial curve x(2)(s) is
x(2)(s) =

x∗(s) + {x
∗(sc)−ω2ε−x∗(sc−ε)}x
(sc−ε) if 0 ≤ s ≤ sc − ε
x∗(s)− ω2(sc − s) if sc − ε ≤ s ≤ sc
x∗(s) if sc < s ≤ s`
(4.14)
where 0 ≤ ε < sc − so. Similar functions can be expressed for y(s) and ξ(s) .
Considering the augmented functional J¯ (1) = ∫ s`
0
L¯ds, the minimization of the first variation
J¯ ′(1)(0) ≥ 0 is first deduced by imposing (4.13). The augmented functional can be accordingly
decomposed as
J¯ (1)(ε) = J¯1 + J¯2 (ε) + J¯3 (ε)
where J¯1 =
∫ sc
0
L¯ds, J¯2 (ε) =
∫ sc+ε
sc
L¯ds and J¯3 (ε) =
∫ s`
sc+ε
L¯ds. This leads to J¯ ′1 (0) = 0 and
J¯ ′2 (0) = L¯ (sc,x∗(sc),y∗ (sc) , ξ∗ (sc) ,ω1,ϕ1,χ1) (4.15)
J¯ ′3 (0) = −L¯ (sc,x∗(sc),y∗ (sc) , ξ∗ (sc) ,ω2,ϕ2,χ2)− (ω1 − ω2) L¯x′ (sc,x∗ (sc) , ω1) (4.16)
− (ϕ1 − ϕ2) L¯y′ (sc,y∗ (sc) , ϕ1)− (χ1 − χ2) L¯ξ′ (sc, ξ∗ (sc) , χ1)
The same derivation is repeated by applying the second trial curve (4.14) and taking into account
the continuity of the momenta (4.12), which gives J¯ (2)′(0) = −J¯ (1)(0) ≥ 0. This means that
J¯ (2)′(0) = −J¯ (1)(0) = 0. This provides the second Weierstrass-Erdmann corner condition at
the corner point sc, namely the continuity of L¯ − x′L¯x′ − y′L¯y′ − ξ′L¯ξ′ at any corner point.
So far, we considered a weak variation of the involved variables. If we relax this restriction
and assume the 0-norm (4.1), a new trial curve x = x(s, ε) should be predefined with the
66
following structure (Mesterton-Gibbons, 2009)
x(s, ε) =

x∗(s) if 0 ≤ s ≤ sc
x∗(s) + ω(s− sc) if sc ≤ s ≤ sc + ε
x∗(s) + {x
∗(sc)+ωε−x∗(sc+ε)}(sg−x)
(sg−sc−ε) if sc + ε < s ≤ sg
x∗(s) if sg ≤ s ≤ s`
(4.17)
where sc is not a corner, ω may be any real number and 0 ≤ ε < sg−sc < s`−sc. This is a strong
variation since the trial function fails to satisfy lim
ε→0+
∣∣∣xs (s, ε)− x′∗(s)∣∣∣ = 0 for all s ∈ [0, s`].
Based on the trial function (4.17) and assuming similar expressions for the other variables,
the augmented functional can be divided into four terms J¯ (ε) = J¯1 + J¯2 (ε) + J¯3 (ε) + J¯4
where J¯2 (ε) =
∫ sc+ε
sc
L¯ds and J¯3 (ε) =
∫ sg
sc+ε
L¯ds. The first variation of the augmented sub-
functionals yields to (4.15) and (4.16) if we replace {ω1, ϕ1, χ1} by {ω, ϕ, χ} and {ω2, ϕ2, χ2} by{
x∗
′
,y∗
′
, ξ∗
′}
. In this way, we get the Weierstrass E function
E(s,x∗,y∗, ξ∗,ω,ϕ,χ,x∗′ ,y∗′ , ξ∗′ ,λ,µ) ≥ 0 (4.18)
where
E = L¯ (s,x∗,y∗, ξ∗,ω,ϕ,χ)− L¯
(
s,x∗,y∗, ξ∗,x∗
′
,y∗
′
, ξ∗
′)
− (ω − x∗′)L¯x′ − (ϕ− y∗′)L¯y′ − (χ− ξ∗
′
)L¯ξ′ (4.19)
with the functions L¯x′ , L¯y′ and L¯ξ′ being evaluated at (s,x∗,y∗, ξ∗,x∗′ ,y∗′ , ξ∗
′
). This condition
(4.18) is the so-called Weierstrass necessary condition.
We can now examine the terminal conditions φ (x(0),x(s`),y(0),y(s`), ξ (0) , ξ (s`)) = 0.
Assuming strong variations for the trial curves we get x = x(s, ε) with x∗(s) = x(s, 0), ηx (s) =
xε (s, 0) and x′ = xs (s, ε). Then the terminal conditions are obtained s` = s` (ε), x` (ε) =
x` (s` (ε) , ε), y` (ε) = y` (s` (ε) , ε) and ξ` (ε) = ξ` (s` (ε) , ε). The vanishing first variation leads
to (L¯ − x′L¯x′ − y′L¯y′ − ξ′L¯ξ′) s`ε + L¯x′x`ε + L¯y′y`ε + L¯ξ′ξ`ε + Eε = 0 (4.20)
where the subscript ε denotes partial derivative with respect to ε. Because the terms are in
general y`ε 6= 0, ξ`ε 6= 0, the momenta are given by L¯y′ = 0 and L¯ξ′ = 0 and are constant
everywhere. Using L¯ξ′ = 0, we get µh = 0, while the other Euler-Lagrange equations (4.10)
become
∂L
∂y′
+ λ
∂f
∂y′
+ µ
∂h
∂y′
= 0 (4.21)
−λ′ = ∂L
∂x
+ λ
∂f
∂x
+ µ
∂h
∂x
(4.22)
67
A final necessary condition is the Clebsch-Legendre condition. Before we derive this condition,
three additional constraint conditions are necessary. First, if Nh > Nu, then at most Nu
components of h(x(s),u(s)) can vanish. Second, the Jacobian ∂hi/∂uj where i ranges over
those indices such that hi = 0 and j = {1, 2, ..., Nu} has maximum rank. Lastly, we assume that
the equations f(x(s),y′(s)) − x′ = 0 and h(x(s),y′(s)) + ξ′2 = 0 are independent; that is the
matrix [
fy′ −I 0
hy′ 0 2Ξ
′
]
(4.23)
has rank (Nx+Nh) along Z where I is the Nx-dimensional identity matrix and 2Ξ′ is an Nh×Nh
with entries 2
(
ξi
)′
on the diagonal. This is the so-called full rank condition (Berkovitz, 1961,
1962).
Then for every vector (pi,ρ,κ) 6= 0 with pi∈ RNx , ρ∈ RNu and κ∈ RNh , a unique solution
of the linear system of equations can be obtained by the full-rank condition (4.23){
fy′ρ− Ipi = 0
hy′ρ+ 2Ξ
′
κ = 0
(4.24)
if the following inequality holds
piL¯x′x′pi + ρL¯y′y′ρ+ 2κµκ ≥ 0 (4.25)
Since L¯x′x′ = 0, (4.25) becomes ρL¯y′y′ρ+ 2κµκ ≥ 0. If hi < 0, then by µihi = 0 we get µi = 0.
On the other hand, If hi = 0 at this point, µi ≥ 0 in order to satisfy (4.25). This gives the
complementarity condition due to the inequality constraint
µi =
{
= 0 if gi < 0
≥ 0 if gi = 0
(4.26)
Based on the assumption of maximum rank of the matrix ∂h/∂y′ for the closed constraints
hi = 0, the solution of the (4.24) implies the vanishing vector κ∈ RNh . Hence, the final form of
the Clebsch-Legendre condition becomes
ρL¯y′y′ρ ≥ 0 (4.27)
where the vector ρ satisfies
Nu∑
i=1
∂hi
∂yj′
ρj = 0, i =
{
1, 2, ..., N
′
h
}
(4.28)
with N
′
h ≤ Nh the dimension of the closed constraints hi = 0 for ∀i ∈ N
′
h.
68
To make a clear connection with a classical optimal control problem, the above derivation
should be expressed in terms of a new function, the Hamiltonian. Based on a Legendre trans-
formation (i.e., L = H − λx′), we can introduce the simple Hamiltonian
H(t,x,y′,λ) = L(t,x,y′) + λf (t,x,y′) (4.29)
The augmented Lagrangian (4.7) can be then reformulated in terms of the Hamiltonian (4.29)
L¯ = H − λx′ + µ (h + ξ′2) (4.30)
Along the arc Z defined by {x∗(s),y∗(s), ξ∗(s)} and combining (4.30) with (4.21-4.22), the
following system of equations is obtained
x′(t) = Hλ
λ′ = − (Hx + µhx)
Hy′ + µhy′ = 0
µihi, µi ≥ 0
(4.31)
Since L¯y′ and L¯ξ′ vanish along Z, the continuity of L¯ − x′L¯x′ − y′L¯y′ − ξ′L¯ξ′ becomes L¯ −
x′L¯x′ = H, which implies continuity of the simple Hamiltonian at corner points. In addition
the transversality condition (4.20) is given by
Hs`ε + λx`ε + Eε = 0 (4.32)
while the Weierstrass’s necessary condition (4.19) reads
H(t,x∗,ϕ,λ) ≥ H(t,x∗,y∗′ ,λ) (4.33)
which can be translated as minimization of the Hamiltonian with respect to the control variable
y′ = u along Z where µh = 0.
Finally, the Clebsch-Legendre condition is simply given by
ρ
(
(H + µh)y′y′
)
ρ ≥ 0 (4.34)
for any solution vector ρ of the system (4.28).
Replacing y′ by u, the above derivation is completely equivalent to the Pontryagin’s Mini-
mum Principle applied on the original problem P which is summarized below.
69
Pontryagin’s Minimum Principle
Let {x∗(·),u∗(·)} be an optimal pair of problem P such that the full-rank condition holds for
{s, x∗(s), u}. Then there exist right-continuous vector functions, λ (·) ∈ [0, s`]→ RNx and right
continuous covectors, µ (·) ∈ [0, s`] → RNg such that (λ,µ) 6= 0 for every s (i.e., nontriviality
condition), the optimality conditions (4.31) and (4.33) hold on [0, s`] and the transversality
conditions (4.32) are satisfied at s = 0 and s = s`.
4.2 Optimal control method
Optimal control is usually encountered in different research areas, such as mechanical and
aerospace engineering (Bryson, 2016; Paiva and Fontes, 2015) with the main focus on the con-
trollability, or the optimization of a dynamical system. Quite recently, Liberzon (2012); O’Reilly
and Peters (2012); Sachkov (2008a) introduced the optimal control concept in a different con-
text, such as the stability analysis of a mechanical system. In particular, Liberzon (2012) gives
a precise introduction in calculus of variation and optimal control and explains their connec-
tion through concrete examples, while O’Reilly and Peters (2012); Sachkov (2008a) analyze the
stability behavior of elastic rods by applying optimal control.
Inspired by Liberzon (2012) and Maurer and Mittelmann (1991), we apply the optimal
control method for the analysis of a constrained buckling problem (see Chapters 5, 6 and 7).
In particular, in the following Section, we describe a general optimal control problem with
unilateral constraints. Multiple studies regarding the optimal control that includes (mixed-
state and pure-state) constraints can be found in literature (Maurer and Pickenhain, 1995;
Malanowski, 1997; Maurer and Malanowski, 1998; Augustin and Maurer, 2001; Malanowski and
Maurer, 2001; Maurer and Oberle, 2002; Malanowski et al., 2004; Bryson, 2016; Berkovitz and
Medhin, 2012). Nevertheless, our attention is placed on pure-state constraints because of the
nature of the unilateral constraints present in the constrained buckling problem. The derivation
of the conditions that constitute the minimum Pontryagin’s principle are then provided while the
preliminary studies in the derivation of second-order optimality conditions are also discussed.
4.2.1 Optimal control theory
Inspired by Liberzon (2012), we demonstrate an optimal control problem with pure-state con-
straints (i.e., functions of the state variables) of order q > 1 (Ross, 2015). The functional
to be optimized is the potential energy of the mechanical system, which is the Lagrangian
of the problem. The dynamics involves spatial derivatives rather than temporal derivatives.
Based on the above assumptions, our main concern is the computation of the state and control
spaces {x(·),u(·)} composed by piecewise-smooth and piecewise-continuous functions respec-
tively, which solve the following optimal control problem P
70

Minimize J [x(·),u(·)] = ∫ `
0
L (x(s),u(s)) ds
subject to
x′ = f(x(s),u(s))
φ (x(0),x(`)) ≤ 0
g(x(s)) ≤ 0
(4.35)
where s is the arclength defined on the interval I = [0, `], x ∈ RNx is the state vector, u ∈ RNu
is the control vector, L : RNx × RNu → R is the Lagrangian, or the running cost, f : RNx ×
RNu → RNx represents the spatial dynamics, φ : RNx ×RNx → RNφ characterize the boundary
conditions, or events, and g :∈ RNg characterize the pure-state constraints. We assume that all
functions are k-times continuously differentiable Ck with respect to their arguments (k ≥ 2),
and the dynamics f is Lipschitz continuous.
The objective of the study is not only to derive a feasible pair {x(·),u(·)} that satisfies the
problem P but also to minimize globally the functional J [x(·),u(·), s]. In this case, the feasible
pair {x(·),u(·)} is called the optimal pair, denoted by {x∗(·),u∗(·)}. The optimal pair can be
derived by applying the Pontryagin’s minimum principle (for more details see (Hartl et al., 1995;
Bonnans and Hermant, 2007b, 2009; Hermant, 2009b,a)).
Due to the presence of pure-state constraints, special care in the derivation of the Pontryagin’s
minimum principle is required. Hence, we start with some necessary definitions for the pure-
state constraints and then we complete our analysis with the presentation of the Pontryagin’s
minimum principle for problem P. The contact and regularity conditions for second-order pure-
state constraints, which are involved in our problem, are also included.
4.2.2 Preliminaries on the pure-state constraints
The order of a pure-state constraint provides us with information about the nature and evolution
of the contact patterns. Based on it, we are also aware of the continuity properties of the adjoint
variables.
The order of a pure-state constraint is derived if we differentiate the pure state constraint
with respect to arclength as many times as required until it depends explicitly on the control
variable. If we denote g(x(s)) =: g0, the i− th order (i.e., i ≥ 1) can be written as
gi (x,u) =
dgi−1 (x,u)
ds
= ∂xg
i−1(x,u)>x′ (4.36)
= ∂xg
i−1(x,u)>f(x,u)
If the order of the pure state constraint is q, it means that
∂ug
i (x,u) = 0 for 0 ≤ i ≤ q − 1 and ∂ugq (x,u) 6= 0 (4.37)
71
Let us denote 0 ≤ ζ1 < ζ2 ≤ `. An interval (ζ1, ζ2) is an unconstrained segment (or interior arc)
if g(x(s)) < 0 and it is a continuous segment (or boundary arc) if g(x(s)) = 0 for s ∈ [ζ1, ζ2].
In the latter case, we denote ζen ≡ ζ1 and ζex ≡ ζ2 the entry and exit points, respectively. On
the other hand, a discrete contact ζc at the endpoint of an unconstrained segment can be met
if g(x(ζc)) = 0 holds. Entry, exit, and contact points are called junction points (Bonnans and
Hermant, 2007b).
Let us consider the maximal interval I = [0, `]. Suppose that there is a finite set of junction
points denoted by Z = Zen ∪ Zex ∪ Zc, where Zen =
{
ζ1en, ζ
2
en, .., ζ
N`
en
}
, Zex =
{
ζ1ex, ζ
2
ex, .., ζ
N`
ex
}
and Zc =
{
ζ1c , ζ
2
c , .., ζ
Nc
c
}
. The pairs of ζjen, ζjex with ζjen ∈ Zen,ζjex ∈ Zex for j = {1, 2, .., N`}
define the union of continuous segments I` ⊆ I = ∪N`j=1
[
ζjen, ζ
j
ex
]
.
Pontryagin’s minimum principle
Before we formulate the necessary conditions of Pontryagin’s minimum principle, we introduce a
new function, the Hamiltonian function (Gelfand, 2000) which is derived by applying a Legendre
transformation of the Lagrangian function L. More precisely, the Hamiltonian is given by
H(λ,x,u) = L (x,u) + λ>f(x,u) (4.38)
Taking into account the presence of the pure-state constraints g(x(s)), the simple Hamiltonian
H(λ,x,u) is replaced by the augmented Hamiltonian
H¯ (µ,λ,x,u) = H(λ,x,u) + µ>g(x) (4.39)
where µ ∈ RNg is the vector of Lagrange multipliers that satisfies the complementarity condition
µi =
{
= 0 if gi (xo,xf ) < 0
≥ 0 if gi (xo,xf ) = 0
, i ∈ Ng, (4.40)
also denoted as µ⊥g.
The necessary conditions for global optimality, which constitute the Pontryagin’s minimum
principle (Hartl et al., 1995), involve the following properties: (1) the minimization of the
Hamiltonian with respect to the control which should be consistent with the control space; (2)
the canonical, or adjoint equations of the augmented Hamiltonian; and (3) the transversality
conditions (i.e., satisfaction of the boundary conditions).
Since only pure-state constraints are present and there are no constraint functions with
respect to the control, the control space is simply u ∈ RNu . The Hamiltonian minimization
condition (HMC) is then derived in terms of the simple Hamiltonian as far as HMC does not
72
violate the complementarity condition (4.40);
∂uH¯ = ∂uH = 0 with µ⊥g (4.41)
The second necessary condition involves the adjoint, or canonical equations, which are derived
by the augmented Hamiltonian H¯
−λ′ = ∂H¯/∂x
= ∂H/∂x + (∂g/∂x)
>
µ µ⊥g (4.42)
and
x′ = ∂H¯/∂λ = f(x(s),u(s)) (4.43)
Based on the complementarity condition (4.40), we get a non-vanishing value of the correspond-
ing covector µi 6= 0 at a closed contact ζ ∈ Z , where gi (x (ζ)) = 0. Considering that there
is a finite number of contact points, a finite number of non-vanishing elements of µ(s) is also
present. This implies that µ(s) are piecewise-continuous vector functions.
If we assume that the vectors λ,µ are right-continuous vector functions, the adjoint variables
can be written as a Lebesgue-Stieltjes integral in the following way
λ
(
`+
)− λ (0+) = −∫
(0,`]
(∂g/∂x)
>
µds
=
∫
(0,`]
(∂g/∂x)
>
dµ˜ (4.44)
where µ˜ is a function of bounded variation, which can be written as the sum of two functions;
(1) an absolutely continuous function µ˜(1) and (2) a singular function µ˜(2) and µ(s) = − ˙˜µ for
every s for which ˙˜µ exists (Hartl et al., 1995). Based on this decomposition, the integral at the
right side of (4.44) can be evaluated as follows∫
(0,`]
(∂g/∂x)
>
dµ˜ =
∫ `
0
(∂g/∂x)
> ˙˜µ(1)ds+
∫
(0,`]
(∂g/∂x)
>
dµ˜(2)
=
∫ `
0
(∂g/∂x)
> ˙˜µ(1)ds+
∑
ζi∈(0,`]
(∂g/∂x)
> [
µ˜
(
ζ+i
)− µ˜ (ζ−i )]
while a single jump condition at a discrete contact ζ is given by
λ
(
ζ−
)− λ (ζ+) = (∂g
∂x
)∣∣∣∣
x(ζ)
[
µ˜
(
ζ−
)− µ˜ (ζ+)] (4.45)
=
(
∂g
∂x
)∣∣∣∣
x(ζ)
η (ζ)
73
This implies that an adjoint variable λi (associated with the state variable xi) with i ∈ Nx is a
piecewise absolutely continuous function, if at least one pure-state constraint gj with j ∈ Ng is
a function of the state variable xi and is closed at least in one junction point. Otherwise, in the
absence of the state constraints, the adjoint variable is an absolutely continuous function.
To complete the necessary conditions of the Pontryagin minimum principle, we need to
satisfy the existing boundary conditions. Let us denote by ν ∈ RNφ the Lagrange multipliers,
or covectors of the boundary conditions φ (x(0),x(`)) ≤ 0 which satisfy the complementarity
condition ν⊥φ and the associated scalar function Φ = ν>φ (x(0),x(`)). Based on it, the
transversality conditions can be written as
λ (0) = − ∂Φ
∂xo
, λ (`) =
∂Φ
∂x`
(4.46)
where ν⊥φ is always satisfied.
In the optimal control problem P, the Hamiltonian and the pure-state constraints are not
explicitly dependent on the arc-length. This implies that the augmented Hamiltonian is constant
(an integral of motion dH¯/ds = 0), which is an important property of our system (Goldstein
et al., 2002).
The canonical equations (4.43-4.43) together with the transversality conditions (4.46) con-
stitute a two-point boundary value problem, while the minimization of the Hamiltonian with
respect to the control (4.41) forms an algebraic system. This means that an optimal control
problem that satisfies the first-order optimality conditions involves a differential-algebraic system
of equations.
Minimum Pontryagin’s principle - Theorem
Let {x∗(·),u∗(·)} be an optimal pair of problem P. Then, there exist a right-continuous vector
functions λ (·) ∈ [0, `] → RNx , µ (·) ∈ [0, `] → RNg and ν ∈ RNφ associated with the boundary
conditions φ : RNx × RNx → RNφ such that (λ,µ(`)− µ(0),ν) 6= 0 for every s ∈ I (i.e.,
nontriviality condition), the optimality conditions (4.41)-(4.44) hold almost everywhere on [0, `]
and the transversality conditions (4.46) are satisfied at the extremities. In addition, for any
junction point ζ ∈ Z, the jump condition should also hold (4.45).
Contact and regularity conditions for second-order state constraints
In case of a scalar control variable and a scalar pure state constraint of order q, Bonnans and
Hermant (2007b) provide a result (Proposition 2.5) based on (Jacobson et al., 1971), which
explains the regularity conditions of the control variable at the junction points depending on
the order of the state constraint. Our focus is the second-order pure-state constraint q = 2. If
the trajectory {u(s),x(s)} is a regular Pontryagin extremal, then u and its q−2 first derivatives
74
are continuous at ζ ∈ Z. The only special case is when the contact point ζc first appears without
the presence of a jump (i.e., η = 0), where u and its q first derivatives remain continuous at
ζ ∈ Zc. In addition Hartl et al. (1995) categorizes the types of contact conditions, depending on
the order of a pure state constraint. For q = 2, a point contact occurs first, which is gradually
transformed to a continuous contact.
Second-order optimality conditions
The general solution of an optimal control problem involves the satisfaction of the minimum
Pontryagin’s principle. By minimizing the Hamiltonian with respect to the control variable
and deriving the canonical equations of the augmented Hamiltonian, we arrive at a differential-
algebraic system (DAE). This system can be solved by a shooting method. The stability of
an optimal control problem is typically investigated by the shooting structure of the problem
(Malanowski et al., 2004; Bonnans and Hermant, 2007b).
For an unconstrained, or a control-constrained optimal control problem, the shooting method
can be easily applied. In this case, the shooting structure (e.g., Jacobian) can be directly derived
for the unconstrained case, while when mixed-pure state constraints are present, the shooting
method can be applied once the structure of active constraints is known (Bulirsch and Stoer,
2010; Betts, 2010). Nevertheless for state-constrained optimal control problems, the shooting
structure cannot be easily obtained, especially if the order of a pure-state constraint is of high
order q > 2. The existence of set-valued functions of the adjoint variables and point and/or
continuous contacts makes the optimal control problem very challenging.
The first attempt to establish necessary conditions for an optimal control problem with pure
state constraints is attributed to (Bryson et al., 1963; Bryson and Denham, 1964). In particu-
lar, Jacobson et al. (1971) presented the first results on the regularity of the multipliers and on
contact conditions, which are associated with the order of the involved pure-state constraints.
More analyticaly, this result shows that the spatial derivative of the control is continuous un-
til an order relative to the order of the constraint and the nature of the contact conditions
(continuous or point contact), as also explained in Section 4.2.2 for a pure-state constraint of
order q = 2. This analysis of the junction conditions was considered the starting point in the
derivation of second-order sufficient conditions. Various versions of Pontryagin’s principle with
state constraints were summarized by (Hartl et al., 1995).
Second-order sufficient conditions for first-order state constraints were first obtained by
Malanowski (1995) using the implicit function theorem, which were later modified by Malanowski
(1997) taking into account only the strictly active constraints. Then Malanowski et al. (2004);
Malanowski and Maurer (2001) derived sensitivity results by applying the implicit function the-
orem to the shooting mapping for a parametric optimal control problem with first-order and
75
higher-order pure-state constraints, where the following assumptions hold; (1) the reference so-
lution has a regular structure with a finite number of boundary subintervals and contact points
and (2) strict complementarity conditions are satisfied. They realized that if the order of the
pure-state constraint is q ≥ 2, the derived solution can become sensitive due to the potential
variation of entry contact points. Based on (Malanowski and Maurer, 2001; Malanowski et al.,
2004) numerical examples can be found in (Augustin and Maurer, 2001). From this analysis it
became even more apparent that additional contact conditions are required for a constrained
stability analysis.
Inspired by Jacobson et al. (1971), Bonnans and Hermant (2007b) investigated an optimal
control problem with a scalar pure-state constraint and established conditions for well-posedness
of the shooting equations. In particular, Bonnans and Hermant (2007b) showed that the shooting
algorithm is well-posed (i.e., invertible Jacobian) iff the no-gap second-order sufficient optimality
condition holds. As explained in (Bonnans and Hermant, 2007a), a no-gap condition holds,
when the only change between necessary and sufficient second-order optimality conditions is
between a strict and non strict inequality, respectively. Bonnans and Hermant (2007a) provided
additional contact conditions that should be satisfied for this purpose. The extension of it to the
optimal control problem with a vector-valued state constraint of arbitrary orders can be found
in (Bonnans and Hermant, 2009).
4.3 Numerical techniques for solving an optimal control
problem
Numerical methods that solve an optimal control problem are presented herein (Bryson, 2016;
Berkovitz and Medhin, 2012; Kornienko et al., 2014; Paiva and Fontes, 2015). These involve two
different approaches; (1) the indirect and (2) the direct methods. An indirect method attempts
to solve the optimal control necessary conditions. Based on Pontryagin’s minimum principle,
the differential-algebraic system (DAE) is obtained, which is solved by applying a shooting, or a
collocation method. On the other hand, in a direct method, the optimality conditions are only
indirectly obtained. Discretization of the control and/or the state variables is initially applied
and the problem is then formed as a nonlinear optimization problem (NLP) subjected to (mixed-
state and/or pure-state) constraints. More precisely, a direct method involves three main steps;
(1) the transcription of the optimal control problem using a collocation method, (2) the solution
of the derived nonlinear programming sub-problem (NLP) by applying a NLP solver (Nocedal,
2006) and (3) repetition of (1)-(2) until convergence of the problem. The derived solution is
then used to check if the necessary conditions, the minimization conditions and the canonical
equations are all satisfied.
Essentially, the indirect methods are more accurate than the direct methods as the solution
76
is based on the optimality conditions. Nevertheless the direct methods are preferable, because
several numerical difficulties arise when an indirect method is applied. The necessary conditions
of optimality involve a differential-algebraic system, namely the algebraic system by the mini-
mization of the Hamiltonian with respect to the control and the first order ODE with respect
the state and adjoint variables. To derive the solution, the partial derivatives of the Hamiltonian
∇xH, ∇uH should be obtained which are usually computationally difficult, especially when the
problem is complex and high dimensional (Betts, 2010). Another difficutly arises in the solution
of the algebraic equation ∇uH = 0, as its derivation depends on the condition number of the
matrix ∇uuH. Only if ∇uuH is non-singular and specifically positive to get minimization of
the Hamiltonian with respect to the control, a unique solution of the control variables can be
obtained.
Even if the Hamiltonian is convex with respect to the control ∇uuH > 0, the solution
of the system of the first order ODE with respect to the state and adjoint variables is non-
trivial. This system together with the boundary and transversality conditions forms a two-point
boundary value problem. To achieve its solution an initial guess of the involved variables is
required. A classical shooting method is very sensitive to any noise, which is usually produced
by an erroneous initial guess of the adjoint variables. Even with a reasonable guess of the adjoint
variables, the numerical solution of the adjoint equations can be very ill-conditioned, as explained
in detail by Betts (2010). Another numerical issue is met in optimal control problems with pure-
state constraints. In fact, if the number, position and sequence of constrained/unconstrained
subarcs are unknown, then its solution is infeasible.
The direct method is presented in detail in the following Section. The presentation involves
(1) the application of a direct collocation method and (2) the numerical methods that are
applied for solving a nonlinear optimization problem (NLP). Computational details, such as the
exploitation of sparsity in the NLP, or the methodology in the computation of derivatives of the
objective and constraint functions are omitted, see, for instance, (Jorge Nocedal, 2006; Bryson,
2016; Betts, 2010).
4.3.1 Direct collocation methods
An optimal control problem can be solved by two distinct classes of direct methods; (1) the
simple, or multiple shooting method and (2) the collocation method. The direct shooting method
shares several similarities with the numerical difficulties that arises in an indirect shooting
method, such as the solution of the involved initial value problem. This is not the case in a
direct collocation method. The initial value problem is avoided as the differential equations
are approximated by collocation. Even though the discrete approximation of the variables can
lead to a highly dimensional problem, the sparsity pattern, both in the Hessian matrix of the
Lagrangian function and in the Jacobian matrix of the constraint functions does not make the
77
final form of the problem computationally cumbersome.
For these reasons, a direct collocation method is preferable. In particular, in a direct colloca-
tion method, the state and/or control of the original optimal control problem are approximated
in some appropriate manner. In the case where only the control is approximated, the method
is called a control parameterization method (Goh and Teo, 1988). When both the state and
control are approximated, the method is called a state and control parameterization method
(von Stryk, 1993). In either case, the optimal control problem is converted into a minimiza-
tion problem of an algebraic equation subjected to finite-dimensional equality and inequality
constraints. In other words, the collocation method is the bridge between an optimal control
problem that involves continuous functions and an NLP that consists of a finite set of variables.
This stage is called transcription process (Kraft, 1985).
Direct collocation methods involve the discretization of the differential equations using for
instance, trapezoidal, Hermite–Simpson, or pseudospectral approximations (Elnagar et al., 1995;
Betts, 1998, 2010). Consider a fixed interval [so, sf ] which is divided by N equally-sized subin-
tervals with the following intermediate nodes
so < s1 ≤ ... ≤ sN = sf (4.47)
In this manner, we derive the segments [sk, sk+1] with k = {0, 1, ...N − 1}. Based on this
subdivision, the control variables u ∈ RNu are usually chosen as piecewise linear interpolating
functions between u(sk) and u(sk+1) (von Stryk, 1993);
u¯(t) ≈ u(sk) + s− sk
sk+1 − sk (u (sk+1)− u (sk)) , s ∈ [sk, sk+1] (4.48)
and the state variables x ∈ RNx are approximated by applying aM - degree piecewise polynomial
x¯(s) ≈
M∑
m=0
a(k)m (s− sk)m , s ∈ [sk, sk+1] (4.49)
where the coefficients a(k)m are theM+1 polynomial coefficients for the interval [sk, sk+1]. These
polynomial coefficients are computed by equating the continuous function of the state variable
to its corresponding polynomial approximation at the endpoints of the interval, that is
x¯(sk) = x(sk), x¯(sk+1) = x(sk+1) (4.50)
and by matching the spatial derivatives x′(s) = f(x,u, s) with s ∈ [so, sf ] in a similar manner;
dx¯(sk)
ds
= f(x (sk) ,u(sk), s),
dx¯(sk+1)
ds
= f(x (sk+1) ,u(sk+1), s) (4.51)
78
Equations (4.50) and (4.51) constitute the (Lobatto) collocation conditions.
For instance, the trapezoidal method (see G. Fasano (2012)) is based on a quadratic inter-
polation polynomial. Hence, (4.49) can be written as
x¯(s) = a(k)o + a
(k)
1 (s− sk) + a(k)2 (s− sk)2 (4.52)
Computing (4.52) at node sk, we get x¯(sk) = x(sk) = a
(k)
o . The other polynomial coefficients
are deduced by the spatial derivatives at sk and sk+1, as
dx¯(sk)
ds
= f(x (sk) ,u(sk), s) = fk = a
(k)
1 (4.53)
dx¯(sk+1)
ds
= f(x (sk+1) ,u(sk+1), s) = fk+1 = a
(k)
1 + 2a
(k)
2 (s− sk) (4.54)
Substituting (4.53), (4.54) into (4.50) and using x¯(sk) = x(sk) = a
(k)
o , we get
x¯(s) = x¯(sk) + fk(s− sk) + 1
2
(fk+1 − fk)
(sk+1 − sk) (s− sk)
2 (4.55)
which at node sk+1 leads to
x¯(sk+1) = x(sk) +
1
2
(fk+1 + fk) (sk+1 − sk) (4.56)
while the spatial derivative is simply written as
dx¯(sk)
ds
= fk +
(fk+1 − fk)
(sk+1 − sk) (s− sk) (4.57)
Since x¯(sk+1) = x (sk+1), the defect constraint ζ (sk) at node sk is finally derived
ζ (sk) = x (sk+1)− x(sk)− 1
2
(fk+1 + fk) (sk+1 − sk) (4.58)
and considering the nodes j = {0, 1, ...N − 1}, a vector of defect constraints with dimension
NxN is obtained.
In a similar way, other (Lobatto) collocation methods, such as the Hermite-Simpson method
with a cubic interpolating polynomial can be also adopted. Nevertheless the matching of the
spatial derivatives can be also applied at different nodes rather than at the endpoints of the
interval. In fact, in a Gauss mathod, the collocation conditions are imposed somewhere in the
interior of the interval, while in a Radau method at most one of the endpoints sk, or sk+1 is a
collocation point (Betts, 2010, 1998).
79
Quite recently, the standard collocation methods were extended to orthogonal, or speu-
dospectral methods. Even though the mathematics around the orthogonal collocation methods
was established by (Reddien, 1979), only recently they gained a lot of interest. There are sev-
eral classses of methods; (1) the Legendre pseudospectral method (Elnagar et al., 1995), (2)
the Radau orthogonal collocation method (Kameswaran and Biegler, 2007), (3) the Chebyshev
pseudospectral method (Vlassenbroeck and Dooren, 1988), and (4) the Gauss pseudospectral
method (Benson et al., 2006). Comparison of the orthogonal, or pseudospectral collocation
methods can be found in (Huntington and Rao, 2008).
The main difference between an orthogonal and a standard collocation method is the way the
collocation points are chosen. Specifically, in an orthogonal collocation method, the collocation
points are not predefined as in the standard collocation methods. Instead, they are indirectly
derived by the roots of an orthogonal polynomial. In this manner, we can compute a more
accurate quadrature approximation of an integral when compared to a standard collocation
method. A comprehensive description of speudospectral, or orthogonal collocation methods can
be found in (Elnagar et al., 1995).
Applying a standard, or a speudospectral collocation method, we can transform an optimal
control problem to a finite-dimensional minimization problem with inequality constraints
Minimize J (y)
subject to
Z (y) ≤ 0
E (y) ≤ 0
G(y) ≤ 0
(4.59)
where y ∈ RNy with Ny = (N+1)(Nx+Nu) is the vector, which represents the values of the state
and control variables at the nodes of the interval [so, sf ], Z : RNy → RNζ with Nζ = NxN is the
vector of the defect constraints and E : RNy → R(N+1)Ng , G : RNy → R(N+1)Ng are the discrete
representations of the endpoints cost and the equality/inequality constraints, respectively. This
is a classical inequality-constrained nonlinear programming problem (NLP), which can be solved
by applying a NLP solver, such as IPOPT (Wachter and Biegler, 2005; Biegler and Zavala, 2009),
SNOPT (Barclay et al., 1998; Gill et al., 2002), and KNITRO (Byrd et al., 2003, 2006). A short
description of these solvers is shown below.
4.3.2 The NLP problem
A nonlinear optimization problem (NLP) is a minimization problem of a nonlinear objective
function f : Rn → R with respect to x ∈ Rn under the presence of bilateral (equality) and
80
unilateral (inequality) constraints g : Rn → Rm with m ≤ n, which can be written as
min
x∈Rn
f(x)
subject to gi(x) = 0, i ∈ IE
gi(x) ≥ 0, i ∈ II
(4.60)
where IE ,II are the index sets for the equality and inequality constraints, respectively.
There are various techniques that can be applied to solve (4.60). If the constraints are linear
and the objective function is quadratic, the optimization problem is called quadratic (QP). A
linear equality, or inequality constraint can be formulated as Ax = b, or Ax ≥ b, where A is an
m× n matrix and b ∈ Rm. A quadratic objective function is given by f(x) = x>Hx/2 + x>c,
where c ∈ Rn is a vector and H is a n × n matrix which is called the Hessian matrix. If the
Hessian matrix is positive (semi)-definite, the quadratic problem is called convex and its solution
is simpler than a non-convex program, where several stationary points and local minima can be
present. In particular, if the Hessian matrix is positive (semi)-definite and the KKT conditions
are satisfied, then a unique global solution is always derived.
Alternative methodologies for the solution of the constrained optimization problem (4.60) are
also available, especially when the objective function is not quadratic. These methodologies have
a common motivation; to replace the constrained optimization problem by a single function that
includes the objective function with an additional term for every constraint function, leading
to an unconstrained optimization problem (Jorge Nocedal, 2006). These approaches are the
penalty and augmented Lagrangian methods. For example, the exact penalty function is given
by
f(x) + µ
∑
i∈IE
|gi(x)|+ µ
∑
i∈II
[gi(x)]
− (4.61)
where [gi(x)]− = max {0,−gi(x)}. On the other hand, the augmented Lagrangian function
LA(x, λ, µ) is a combination of the quadratic penalty function and the Lagrangian function. For
instance, for an equality-constrained problem, it can be written as
LA(x, λ, µ) = f(x)−
∑
i∈IE
λigi(x) +
µ
2
∑
i∈IE
g2i (x) (4.62)
where the first two terms constitute the Lagrangian function L(x,λ) and λ ∈ Rm is the Lagrange
multiplier vector, associated with the equality constraints g : Rn → Rm.
Nevertheless when the constraints are nonlinear, the optimization problem is highly dimen-
sional, or nonconvex issues are present, alternative approaches are applied, such as sequential
quadratic programming (SQP) (or active-set) and interior-point (or barrier) methods. These
approaches are currently considered the most powerful numerical techniques for large-scale non-
linear inequality-constrained programming.
81
One main difference between the SQP approach and the interior point methods is the proce-
dure that is adopted to consider the inequality constraints. SQP algorithm initiates by making a
guess of the optimal active set IA, which contains only the strictly active constraints gi(x) = 0.
This implies that the optimal active set is a subset of the involved inequality constraints IA ⊆ II .
The corresponding optimization problem is solved by utilizing only equality constraints and ig-
noring the rest of the inactive constraints. As the problem then iterates, an update of the active
set is required in order to satisfy the KKT conditions. The appropriate choice of the initial
active set and its accurate update in later stages constitute a very challenging problem in non-
linear optimization that is called “combinatorial” difficulty of inequality constrained problems
(Nocedal, 2006).
A different methodology is followed by interior-point (or barrier) methods as all equality-
inequality constraints are included. In particular, the problem is solved in a manner that it stays
inside the feasible region defined by the constraints, which is achieved by applying a barrier
function. As the optimal solution is approached, the effect of the barrier function is gradually
diminished such that an accurate result is obtained. Hence, the interior-point method is superior
to SQP approach in this aspect, because the combinatorial issues of nonlinear programming are
avoided. In this method, however, the whole set of constraints is hold even if a subset of these
inequality constraints does not probably affect the result of the problem, leading to higher
dimension of the optimization program and accordingly affecting the computational cost.
Nevertheless numerical tests show that interior-point methods are usually faster than active-
set SQP methods on large problems, particularly when the number of free variables is also large
(Nocedal, 2006). On the other hand, the opposite holds if the number of active constraints is
nearly as large as the number of variables, implying a small number of free variables. In this
particular case, SQP method can be faster and more robust when compared to the interior-point
method, especially if the latter is badly scaled.
Before the presentation of the two methodologies some complementary material is included.
This involves the classical Newton method of an equality-constrained optimization problem,
traditionally applied for the solution of subproblems of both approaches. A short illustration of
the merit and filter functions is also included, which are applied to decide whether an iteration
step is acceptable, or further improvement is required.
Newton method
In practice, the inequality-constrained problem (4.60) is always transformed to an equality-
contrained problem of the form;
min
x∈Rn
f(x)
subject to g(x) = 0
(4.63)
82
by adopting two distinct methodologies; (i) the whole set of inequality constraints is transformed
into equality constraints using slack variables, or (ii) the active set is only included, which
involves, by definition, only closed constraints. An additional necessary condition is essential for
inequality-constrained problem, namely the non-negative value of their correspondent Lagrange
multipliers. More specifically, for an inactive inequality constraint gi > 0 a zero value of Lagrange
multiplier λi = 0 is implied, while for an active constraint gi = 0 a positive Lagrange multiplier
is expected λi > 0. This is the well-known complementarity condition giλi = 0.
The Lagrangian function of the problem is L(x,µ) = f(x)−λ>g(x), where λ ∈ Rm are the
Lagrange multipliers. Let us denote as G(x) the Jacobian matrix of the constraints
G(x)> = [∇g1(x),∇g2(x), ...,∇gm(x)] (4.64)
where gi(x) is the ith component of the constraint vector g(x). The first-order optimality (KKT)
conditions can be written as a (n+m)-system of equations
F(x,µ) =
[
∇f(x)− G(x)>λ
g(x)
]
= 0 (4.65)
with unknowns x ∈ Rn and λ ∈ Rm.
Let us now describe the most classical methodology to solve (4.63) for a single step k.
Deriving the Jacobian of the system, the Newton’s method from a known pair of (xk, µk) of the
k-step can be expressed as[
∇2xxLk −G>k
Gk 0
]{
4xk
4µk
}
= −
{
∇fk − G>k λk
gk(x)
}
(4.66)
where f(xk) = fk, xk+1 = xk+4xk and λk+1 = λk+4λk. The matrix on the left side of (4.66)
is called the Karush–Kuhn–Tucker (KKT) matrix. To derive a unique solution for a Newton
step, the condition number of the Karush-Kuhn-Tucker (KKT) matrix is crucial. In fact, the
non-singularity of this matrix quarantees the uniqueness of the solution. This can be achieved if
the Jacobian G(x) has full row rank, which implies linear independence constraint qualification
(LICQ) and the Hessian matrix H = ∇2xxL (x,µ) is positive definite v>∇2xxL (x,µ) v > 0 in the
tangent space of the constraints, defined by G(x)v = 0. In case, however, the Hessian ∇2xxLk
is not positive definite, or the active-set is not properly defined, numerical issues can be met.
Multiple numerical techniques can be applied in these cases, such as a quasi-Newton method
when for instance a large matrix is involved and its inverse is computationally cumbersome.
There are two widely used quasi-Newton techniques; (i) the BFGS method, where a rank-two
positive secand update is assumed (Fletcher, 1970; Goldfarb, 1970; Shanno, 1970) and (ii) the
SR1 method, where a symmetric rank-one is applied. Even though a recursive update is easier
83
to implement, the correspondent quasi-Newton method exhibits a superlinear convergence when
compared to the classical Newton method, which provides a quadratic convergence. This implies
that a larger number of iterations are usually required in the case of a quasi-Newton method.
After the completion of a single step, some additional criteria are required in order to ensure
the acceptance of the step, or to improve the size of it. These criteria are described in the
following Section.
Criteria for step size control and acceptance/rejection
Let us suppose that a Newton step is applied, as explained in Section 4.3.2. If the value of
the objective function is reduced, or increased at the end of the step while the violation of
the unilateral constraints is increased, or decreased, respectively, it is not easy to decide if this
step is acceptable. In other words, the main goal is to find the best methodology to achieve a
simultaneous reduction of the objective function and the violation of the constraints at every
single Newton step. To this point, methods for correcting deficiencies of Newton’s methods are
available, which are called “globalization strategies”. In particular, to solve this issue, merit
functions and/or filter methods are typically applied.
In merit functions, the cost function and the constraints are combined into a single mini-
mization problem. For instance, a merit function is simply given by
φ1 (x;µ) = f(x) + µ
∑
i∈IE
|gi(x)|+ µ
∑
i∈II
|gi(x)|− (4.67)
where µ is a positive penalty parameter. The particular merit function (4.67) is called `1
penalty function, which is not differentiable because of the presence of inequality constraints.
Nevertheless the choice of an appropriate value of the penalty parameter µ is a difficult task.
Instead, a filter technique can be applied, where the minimization of the cost function and the
set of the constraints are applied seperately. In particular, a measure of infeasibility is defined
h(x) =
∑
i∈IE
|gi(x)|+
∑
i∈II
[gi(x)]
− (4.68)
which includes only the constraints. For every step, a new pair of
{
min
x
f(x),min
x
h(x)
}
is derived.
The list of these pairs constitutes the so-called filter. If the current pair is not dominated by
any other pair in the filter, the particular step is accepted.
Some algorithms based on merit functions, or filters may fail to converge rapidly because
they reject steps that make good progress toward a solution. This undesirable phenomenon
is typically called the Maratos effect (Nocedal, 2006). To eliminate this effect second-order
correction steps (SOC) can be applied, which lead to a fast local convergence (Wachter and
Biegler, 2005). We can also use a merit function, such as Fletcher’s augmented Lagrangian
84
function, which does not suffer from the Maratos effect (Ulbrich, 2003).
These merit and/or filter methods are applied in combination with a line-search method, or
a trust-region method. More specifically, in line search methods, the magnitude of the step is
adjusted by applying criteria defined by a merit, or a filter method. In trust-region methods,
a merit, or filter method aims to determine whether the step is accepted, or rejected and if
the trust-region radius should be adjusted. This also implies that in trust-region methods, a
simultaneous correction of the magnitude and the direction of a step can be applied. The choice
of the approach is based on the problem under consideration.
In a line-search method, the size of a step 4xk is derived by the following expression
αmaxk = max {α ∈ (0, 1] : xk + α4xk ≥ (1− τ)xk} (4.69)
with τ ∈ (0, 1) (a typical value of τ is 0.995). (4.69) is the maximum length that satisfies a
sufficient decrease of a merit function, or ensuring acceptability of a filter at the end of a step
4xk. The new iterate is then defined as
xk+1 = xk + α
max
k 4xk (4.70)
If the Hessian∇2xxLk is not necessarily positive definite and Jacobian singularities Gk are possibly
present, a trust-region method is a better choice. In this case, a trust-region constraint is
included which can be written as
‖4xk‖ ≤ ∆k (4.71)
and indicates that every step 4xk should lie inside the trust region, defined by the circle of
radius ∆k. The trust-region constraint (4.71) is an additional constraint of the problem with
its associated Lagrange multiplier τ , also called Levenberg parameter (Levenberg, 1944). The
classical methodology is to assign a constant radius ∆k and update the Levenberg parameter
τ such that ‖4xk‖ = ∆k. Nevertheless the choice of the value of ∆k is under question. If the
assigned value of ∆k is not appropriate, the solution can be remarkably affected. This numerical
issue is eliminated by using a merit, or a filter method, as mentioned above and explained in
detail in (Bonnans, 2006; Nocedal, 2006).
Active-set or SQP approach
One approach to solve (4.60) for a single step 4xk is the sequential quadratic programming
method (SQP), or active-set method, which can be written in the following form
85
min
4xk∈Rn
fk +∇f>k 4xk + (4xk)>∇2xxLk4xk/2
subject to gik + Gik4xk = 0, i ∈ IE
gik + Gik4xk ≥ 0, i ∈ II
(4.72)
This formulation is obtained by applying a linear approximation of the equality/inequality con-
straints g(x) (4.72b),c)) and a quadratic approximation of the cost function, or a pure quadratic
approximation of the Lagrangian function Lk if equality constraints are only included.
There are two main procedures for solving (4.72), the inequality-constrained QP (IQP) ap-
proach and the equality-constrained (EQP) QP method. In the former case, there is an initial
guess of the active set of the inequality constraints, which remains unchanged during iteration.
For this active set, a Newton method is then applied. The second approach selects a subset
of constraints at each iteration to be the so-called working set IW , and solves only equality-
constrained subproblems of (4.72), where the constraints in the working sets are imposed as
equalities and all other constraints are ignored.
A possible difficulty with SQP methods is that the linearizations (4.72) of the nonlinear
constraints may lead to an infeasible subproblem. To overcome this difficulty, the nonlinear
optimization program can be formulated as a `1 penalty program
min
x∈Rn
f(x) + µ
∑
i∈IE
(ri + si) + µ
∑
i∈IE
ti
subject to gi = ri − si, i ∈ IE
gi ≥ −ti, i ∈ II
r, s, t ≥ 0
(4.73)
where r, s, t are slack variables that allow relaxation of the inequality constraints and µ is a
penalty parameter with necessarily positive value. The quadratic subproblem (4.72) associated
with (4.73) is always feasible. This formulation is used by Barclay et al. (1998); Gill et al. (2002)
in the SNOPT software package.
Another alternative methodology, the so-called sequential linear-quadratic programming
(SLQP) method has been proposed by Byrd et al. (2003, 2006) in KNITRO package, where
a two-step procedure is suggested. In particular, a linear program (LP) is first solved to calcu-
late the working set IW , which consists of the active inequality-equality constraints. This linear
program (LP) includes the linear approximation of the main problem (i.e., the Hessian ∇2xxLk
is not included) and a trust-region constraint (4.71) is applied for improving the magnitude and
the direction of the first iteration step. For the derived working set IW , the EQP associated
problem of (4.73) is then solved with a new trust-region constraint (4.71).
SQP methods often use a merit function in combination with a line search, or a trust region
method to decide whether a trial step should be accepted. A variety of merit functions have
86
been used in SQP methods, such as nonsmooth penalty functions and augmented Lagrangians.
Interior point methods
The interior-point method, or barrier method is formulated in a different manner. The nonlinear
inequality-constrained optimization problem (4.60) is transformed to the following equality-
constrained optimization problem
min
x∈Rn
f(x)
subject to gi(x) = 0 i ∈ IE
gi(x)− si = 0 i ∈ II
s ≥ 0
(4.74)
where s ∈ RI is a vector of slack variables. If an inequality of the form g(x) < 0 is included,
(4.74c) becomes g(x) + s = 0 with s ≥ 0.
Interior-point methods can be seen as continuation methods, or as barrier methods. In
a continuation (or homotopy) method, a perturbed KKT condition is derived, which can be
written as
∇f(x)− G>E (x) y − G>I (x) z = 0
Sz− µe = 0
gE(x) = 0
gI(x)− s = 0
s ≥ 0, z ≥ 0
(4.75)
with ν ≥ 0 and gE(x),gI(x) are the equality and inequality constraints with associated Jacobians
GE (x) ,GI (x) and Lagrange multipliers y, z, respectively. We define S and Z to be the diagonal
matrices whose diagonal entries are given by the vectors s and z, respectively, and let e =
(1, 1, ..., 1)
> (Jorge Nocedal, 2006; Nocedal, 2006). For a fixed value of µ a Newton method
is applied, which in this case is called a primal-dual system. The homotopy approach consists
of solving (4.75) for a sequence of positive parameters {µj} until it converges to zero, while
maintaining s, z > 0. If µ = 0 is reached, the KKT conditions are derived. Furthermore, by
requiring the iterates to decrease a merit function (or to be acceptable to a filter), the iteration
can possibly converge to a minimizer, not simply a KKT point (Bonnans, 2006; Nocedal, 2006).
The second derivation of interior-point methods associates (4.74) with a barrier function
min
x∈Rn
f(x)− µ
m∑
i=1
log si
subject to gi(x) = 0 i ∈ IE
gi(x)− si = 0 i ∈ II
(4.76)
87
where µ is a positive parameter and log(·) denotes the natural logarithm function. The barrier
function log si is very large when si → 0 and consequently it can be viewed as a “barrier” in the
minimization process. Problem (4.76) does not include the additional constraint s ≥ 0 that is
required in (4.74). The barrier approach consists of finding successive solutions of the problem
(4.76) for a sequence of positive barrier parameters µj until the KKT conditions are satisfied
(i.e., µj → 0), as above.
Specifically, the derivation of (4.76) can be translated as follows; the logarithmic barrier
function is given by
ϕµ (x) = f(x)− µ
m∑
i=1
log (gi(x))
subjected to the perturbed complementarity condition (perturbed KKT system)
gi(x)λi = µ (4.77)
where µ is the barrier, or homotopy parameter. The gradient of barrier function is then derived,
as follows
∇ϕµ (x) = ∇f(x)− µ
m∑
i=1
∇gi(x)
gi(x)
(4.78)
If we substitute (4.77) into (4.78), we get
∇ϕµ (x) = ∇f(x)− λ∇g(x) = 0 (4.79)
which means that the gradient of f(x) should lie in the subspace, spanned by the gradient of the
unilateral constraint. This system of primal-dual equations (4.79) subject to (4.77) is repeatedly
solved until the convergence of the problem (i.e., µ = 0).
Wachter and Biegler (2005); Biegler and Zavala (2009) have suggested an interior-point
software package, IPOPT that includes a line-seach method in combination with a filter method
and second-order corrections for a faster convergence of the above barrier problem. This is the
NLP solver, applied in the current study. On the other hand, alternative software packages are
also available such as KNITRO, which is a barrier method with a line search, or a trust-region
algorithm, see (Chen et al., 2014; Waltz et al., 2005).
88
Chapter 5
Optimization-based stability
analysis of constrained
weightless elastica
5.1 Introduction
The stability problem of a constrained elastic structure is widely encountered in engineering and
medical applications. In particular, the constrained buckling of the drill-pipes is a crucial issue
in oil drilling operations (Gao and Miska, 2009b,a; Wicks et al., 2008; Miller, 2014; Gao and
Huang, 2015; Huang et al., 2015; Sun et al., 2015). To solve the constrained buckling problem
simplified analyses based on a spectral decomposition method are typically used (Wicks et al.,
2008; Gao and Huang, 2015; Gao and Miska, 2009a). Similar applications can be found in the
insertion of a guidewire, or a catheter into blood vessels (Duriez et al., 2006; Li et al., 2011b;
Tang et al., 2012; Lenoir et al., 2006). In the latter case, the focus is placed on an accurate
real-time simulation rather than a stability analysis.
While the buckling of a slender object is a classical topic (Love, 2013; Bigoni, 2016; Wang
et al., 2004; Wang, 1997), the constrained stability problem has not been investigated extensively.
Two distinct classes of problems can be found: (i) the constrained buckling problem of an elastic
structure of constant length with a movable end (classical stability problem) (Domokos et al.,
1997; Holmes et al., 1999) and (ii) the stability problem of an elastica of variable length inside
a conduit (insertion stability problem) (Denoel and Detournay, 2011; Huynen et al., 2016; Lu
and Chen, 2008).
Maurer (1979) was the first to derive the buckling response of a nonlinear beam under the
presence of unilateral constraints by applying an optimal control method. Nevertheless Maurer
89
90
and Mittelmann (1991)’s innovative idea remained almost unknown for several years and only
recently O’Reilly and Peters (2012) and Sachkov (2008a) revisited the optimal control concept
for the buckling analysis of elastic rods, limited only to unconstrained conditions.
Inspired by Maurer (1979) and Liberzon (2012), we extend this idea to the stability problem
of thin elastic structures constrained by rigid obstacles. Instead of using the calculus of varia-
tions, the optimal control method is applied. The direct method is adopted for the solution of
the constrained stability problem. In particular, the interface ICLOCS (Falugi P. and van Wyk,
2010), which is connected with a interior-point filter line-search algorithm IPOPT (Biegler and
Zavala, 2009; Wachter and Biegler, 2005), is used. Details can be found in Chapter 4.
The optimal control problem consists of formulating the potential energy of the elastica
subject to the spatial derivatives of the state variables (i.e., spatial dynamics of the system) and
the inequality constraints. The state variables involve the cartesian coordinates of the elastica
and the Euler angles. The evolution of the cartesian coordinates in space is derived by applying
the inextensibility-unshearability constraints, while the spatial derivatives of the Euler angles
are the control variables of the problem. Lastly, the unilateral constraints represent the presence
of rigid longitudinal obstacles. These unilateral constraints are imposed by predefining lower
and upper bounds on the corresponding state variables and are particularly second-order pure-
state constraints. As explained in detail in (Maurer and Mittelmann, 1991; Malanowski and
Maurer, 2001; Bonnans and Hermant, 2009; Hartl et al., 1995; Bryson et al., 1963; Bryson and
Denham, 1964; Jacobson et al., 1971), the existence of a second order pure state constraint leads
to set-valued functions of the adjoint equations. This discontinuity is physically understood due
to the presence of pointwise contact forces, whenever the elastica comes in contact with the wall.
A more detailed analysis of an optimal control problem with other types of constraints can be
found in (Bryson, 2016; Berkovitz and Medhin, 2012; Ross, 2015).
At the beginning of the Chapter, the classical stability problem is analyzed. From the gov-
erning equations of the problem we show step-by-step the equivalence of a classical calculus
of variations problem to an optimal control problem, including a concrete example. The corre-
sponding optimal control formulation for the insertion stability problem is also briefly presented.
These two classes of stability problems are investigated by varying several factors, such as the
clearances of the walls, the boundary and loading conditions. The method is shown to reproduce
numerical results published in the literature for a planar elastica of either constant or variable
length, for any combination of boundary conditions and different types of longitudinal obstacles
(Domokos et al., 1997; Manning and Bulman, 2005; Ro et al., 2010; Fang et al., 2013).
In the final Section, we complete our study with the investigation of the constrained stability
of a spatial elastica, or Kirchhoff rod. Initially, the optimal control problem of a spatial elastica
is formulated and its optimality conditions are demonstrated. Validation of the optimal control
formulation is performed for contact-free conditions. A particular example of a clamped-clamped
spatial elastica constrained by a cylindrical wall is then solved and its derivation is compared
91
with published results (Fang et al., 2013).
5.2 Stability of constrained planar elastica
The optimal control problem is formulated to study the stability of a planar elastica of constant
length constrained by two rigid horizontal walls (classical stability problem). This particular
problem for force-controlled conditions is described below. The methodology of optimal control
is clarified and its equivalence to a calculus of variation problem is described. A representative
example is also included.
5.2.1 Description of the problem
A weightless planar elastica of constant length L and uniform bending rigidity EI is considered.
The left end of the elastica is fixed while the right end is free to move horizontally. Its initial
configuration is stress-free (i.e., straight elastica). The supports of the elastica are either clamped
at zero inclination, or pinned. The planar elastica is constrained by symmetrically, or non-
symmetrically located horizontal longitudinal obstacles of clearances Cu, Cl, where u, l hold for
the upper and lower walls, respectively. The concern of this study is the post-buckling behavior
of the elastica under the action of an axial compressive force Q at the right end. This action leads
to shortening of the distance between the two supports. The derived displacement is denoted
by ∆. This is the so-called classical stability problem for force-controlled conditions.
The problem is described in a system of coordinates (X,Y ) with associated inclination angle
Θ(S) about the X-axis. More specifically, we define a system of coordinates (X,Y ) with origin
at the left end of the elastica, see Figure 5.1. We also introduce the arc-length coordinate S
with its origin S = 0 at X = 0 and Y = 0. In its initial configuration, the elastica is straight
implying that Θ(S) = 0, 0 ≤ S ≤ L. Due to the inextensibity and unshearability constraints, its
deformed configuration Xˆ(S), Yˆ (S) is fully defined by the function Θ(S) and the origin of the
system. In view of the imposed boundary conditions, the additional condition of the cartesian
coordinate Yˆ (L) = 0 should be also satisfied. The end-shortening is expressed as ∆ = L− Xˆ(L)
and the rigid horizontal longitudinal obstacles are formulated mathematically by imposing the
unilateral constraints Cl ≤ Yˆ (S) ≤ Cu, 0 < S < L.
Figure 5.1 illustrates a particular configuration of a clamped-clamped elastica with two dis-
crete contacts at P1 and P2. The internal force R(S) with its inclination α(S) about the X-axis
is also shown. The relation between the applied force Q and the internal force R(S) is simply
given by Q = R cosα = Rx. The horizontal force is uniform since the effect of the friction
is neglected, while the shear force Ry = R sinα is piecewise uniform as it is discontinuous at
contact points.
92
∆

R
( )SΘ
( )X L
2P
1P
QuC
lC
Y X
Figure 5.1: Description of the classical stability problem for a clamped-clamped elastica
Scaling of the problem follows from adopting L as length scale and EI/L2 as force scale
s =
S
L
, x =
X
L
, y =
Y
L
, cu,l =
Cu,l
L
, δ =
∆
L
, R = RL
2
EI
, Q = QL
2
EI
(5.1)
The cartesian coordinates of the elastica xˆ(s), yˆ(s) are derived by θ(s) through integration
xˆ(s) =
∫ s
0
cos θdξ, yˆ(s) =
∫ s
0
sin θdξ (5.2)
taking into account the boundary conditions xˆ(0) = yˆ(0) = 0. At the right end xˆ(1) = 1− δ and
yˆ(1) = 0, while the unilateral constraints are given by cl ≤ yˆ(s) ≤ cu, 0 < s < 1.
The dimensionless form of the potential energy of the elastica can be written as follows
Π =
1
2
∫ 1
0
θ′2(s)ds−Q(1−
∫ 1
0
cos θ(s)ds) (5.3)
where 12
∫ 1
0
θ′2(s)ds is the elastic bending energy of the elastica and Q(1− ∫ 1
0
cos θ(s)ds) is the
external work of the axial compressive force Q under the presence of the equality and inequality
constraints ∫ s
0
sin θdζ − cu ≤ 0
− ∫ s
0
sin θdζ + cl ≤ 0∫ 1
0
sin θds = 0
(5.4)
and the boundary conditions which are θ(0) = θ(1) = 0, θ′(0) = θ′(1) = 0 for clamped-clamped
and pinned-pinned boundary conditions, respectively.
The equation governing of the extremal solution is accordingly derived by vanishing the
first variation of the potential energy subject to the constraints and the boundary conditions.
The derivation of an extremal solution is based on the choice of the norm. There are different
93
candidates of this norm. In calculus of variations, the 1-norm is traditionally assumed
‖χ‖1 = max0≤s≤1 |χ(s)|+ max0≤s≤1 |χ
′(s)| (5.5)
where χ : [0, 1] → R is some curve. If we denote by ∗ the extremal values, closeness (or
boundness) in the sense of the 1-norm implies that not only χ is close enough to its extremal
solution χ∗, but also its derivative. Using this norm, by definition, a weak (local) extremal
solution is obtained (Mesterton-Gibbons, 2009). In our case, considering that Q = R cosα and
the fact that the Lagrange multiplier associated with the equality constraint is the shear force
R sinα, the governing equation of the elastica (i.e., Euler-Lagrange equation) is derived
d2θ
ds2
+R cosα sin θ −R sinα cos θ = 0 (5.6)
for the unconstrained case. When the inequality constraints (5.4i)-ii)) are also present, pointwise
shear forces (i.e., associated Lagrange multipliers) are developed, modifying accordingly the
governing equation of the elastica (i.e., non-uniform resultant force R and inclination angle α
but the product R cosα is always equal to the applied force Q as friction is neglected).
The solution of a particular configuration of the elastica with a known contact pattern can be
derived analytically, or numerically (Domokos et al., 1997; Holmes et al., 1999). The elastica is
divided into segments with respect to the contact points and every segment is solved separately.
The complete solution is then constructed by assembling the solution for each segment. This
involves the enforcement of the continuity conditions, namely identical moment, or curvature and
horizontal force at the common extremities of the segments (at the contact points). To derive
the stability of these equilibrium configurations, a second (weak) variation of the functional is
required. Nevertheless this derivation involves some mathematical difficulties when unilateral
constraints are present due to the assumption of a 1-norm (see Chapter 2). This restriction of
the 1-norm is relaxed in an optimal control formulation, where a different norm is considered to
be more suitable. In particular, the 0-norm is assumed
‖χ‖0 = max0≤s≤1 |χ(s)| (5.7)
for a curve χ : [0, 1] → R. Closeness in the sense of the 0-norm means that only χ should
be close enough to its extremal solution χ∗. This implies that the variable χ can be piecewise
continuous.
With the 0-norm (5.7), by definition, a strong (local) extremal of a functional is obtained
(Mesterton-Gibbons, 2009). The difference between a weak and a strong (local) extremal solu-
tion is that in the latter case, the Weierstrass necessary condition should be additionally satisfied,
94
as explained by Liberzon (2012). This means that we do not only derive an equilibrium configu-
ration, but an equilibrium state that is also consistent with the Weierstrass necessary condition.
A detailed analysis of the equivalence between an optimal control formulation and a calculus of
variation problem with unilateral constraints can be found in Chapter 4.
Let us now construct the optimal control formulation for the particular stability problem. To
do so, we need to choose the state and control variables of the problem in an appropriate manner.
The functional J coincides with the potential energy of the elastica (7.4). One state variable is
clearly the inclination angle θ(s). The integral (isoperimetric) constraints are also accounted for
by adding the cartesian coordinate y(s) =
∫ s
0
sin θdζ with associated spatial derivative y′(s) =
sin θ, as a state variable. Hence, the problem involves two state variables, the inclination angle
θ(s) and the cartesian coordinate y(s) and one control variable, the curvature of the elastica
u(s) = θ′(s). The minimum continuity requirements for these funcrions are: C0 for θ′(s) and
hence C1, C2 for θ(s) and y(s), respectively. After making these substitutions, we arrive at the
following optimal control problem for force-controlled conditions
Minimize J [x(·), u(·), s] = 12
∫ 1
0
u2(s)ds−Q(1− ∫ 1
0
cos θ(s)ds)
subject to
y′(s) = sin θ(s)
θ′(s) = u(s)
y(0) = y(1) = θ(0) = θ(1) = 0 clamped B.C. or
y(0) = y(1) = θ′(0) = θ′(1) = 0 pinned B.C
y(s)− cu ≤ 0
−y(s) + cl ≤ 0
(5.8)
where the state and control variables are given by
x =
{
y
θ
}
∈ R2, u = θ′ ∈ R (5.9)
The pure-state constraints read
g(y) =
{
y − cu
−y + cl
}
≤ 0 (5.10)
The first and second spatial derivatives of (5.10) are given by
g1(y) =
{
sin θ
− sin θ
}
g2(y, u) =
{
u cos θ
−u cos θ
}
(5.11)
where g1u(y) = 0 and g2u(y, u) = {cos θ,− cos θ}> at contact points which implies that the pure
95
state constraints (5.10) are of second order.
The normal Hamiltonian is given by
H(λ,x, u) =
1
2
u2(s) +Q (cos θ(s)− 1) + λy sin θ(s) + λθu(s) (5.12)
where λ = {λy, λθ} ∈ R2 are the covectors, or costate variables of the dynamics f = {sin θ, u} ∈
R2. The set of the boundary conditions are described by the scalar function
Φ = ν1y(0) + ν2y(1) + ν3θ(0) + ν4θ(1) (5.13)
for clamped-clamped boundary conditions and
Φ = ν1y(0) + ν2y(1) + ν3θ
′(0) + ν4θ′(1) (5.14)
for pinned-pinned boundary conditions, where ν is the associated covector, or the vector of
Lagrange multipliers which satisfy the complementarity condition for the corresponding argu-
ments.
The Lagrangian of the Hamiltonian, or augmented Hamiltonian is given by
H¯ (µ,λ,x, u) = H(λ,x, u) + µ1 (y(s)− cu) + µ2 (−y(s)− cl) (5.15)
where
µi =
{
= 0 if gi (xo,xf ) < 0
≥ 0 if gi (xo,xf ) = 0
(5.16)
The minimization of the Hamiltonian in terms of the control variable gives
∂H¯
∂u
=
∂H
∂u
= 0⇒ u+ λθ = 0 (5.17)
which leads to the optimal control variable uˆ = −λθ. Using this result, we can eliminate the
control variable from (5.12) in order to get the lower, or minimized Hamiltonian H, which
depends only on the state and costate variables
H = Q (cos θ − 1) + λy sin θ − 1
2
λ2θ (5.18)
This remains constant for every s ∈ [0, 1], since it does not depend explicitly on s. The same
also holds for the augmented Hamiltonian (6.41).
96
The adjoint equations are given by
−λ′y =
∂H¯
∂y
=

0 if g < 0
µ1 if y(ζc) = cu
−µ2 if y(ζc) = −cl
(5.19)
−λ′θ =
∂H¯
∂θ
= −Q sin θ + λy cos θ (5.20)
where ζc denotes the position of a discrete contact (i.e., closed contact). The initial and terminal
transversality conditions are λy(0) = −ν1, λθ(0) = −ν3 and λy(1) = ν2, λθ(1) = ν4 for clamped-
clamped boundary conditions, respectively. In a similar manner the same expressions can be
derived for pinned-pinned boundary conditions.
For the unconstrained case λy(s) = −ν1 = ν2. This means that a uniform lateral, or shear
force is present. If we denote α the inclination angle of the resultant force R with respect to
the x-axis, then Q = R cosα, λy = Ry = R sinα and (5.20) becomes λ′θ = R sin (θ − α). If we
substitute λθ = −uˆ = −θ′, we get
θ′′ +R sin (θ − α) = 0 (5.21)
with boundary conditions θ(0) = θ(1) = 0 for a clamped elastica. Equation (6.45) is identical
to (5.6), derived by applying the calculus of variations. Nevertheless, in optimal control, the
assumption of the 0-norm introduces the additional requirement of the Weierstrass necessary
condition. This condition is equivalent to the minimization of Hamiltonian with respect to the
control variable, the curvature of the elastica. For instance, for a predefined axial compressive
force Q (i.e., force-controlled conditions), the internal force and the inclination angle are both
unknown except of their product, Q = R cosα. Hence, we may derive more than one equilibrium
state that satisfy (6.45). The equilibrium state with the mimimum Hamiltonian energy with
respect to the control variable is the optimal solution (i.e., stable solution).
Let us now assume that we have a single contact point ζc where y(ζc) = cu is satisfied.
In this case, the shear force is λy(s) = −ν1, for s ∈ [0, ζc), λy (ζ−c ) − λy (ζ+c ) = η1 (i.e.,
η1 =
∫ ζc+ε
ζc−ε µ1ds) at the contact point ζc and λy(s) = −ν1 − η1 = ν2, for s ∈ (ζc, 1]. On the
other hand, λθ (ζ−c )− λθ (ζ+c ) = 0 since the pure state constraint is independent of θ. This also
means that λθ = −uˆ = −θ′ is continuous. However −λ′θ = −uˆ′ is discontinuous at the contact
points (5.20), because it depends on the shear force λy(s). When a continuous contact is present
the only difference is the vanishing control variable u(ζ) = 0 for ζ ∈ [ζen, ζex], where ζen and ζex
denote the entry and exit points of a continuous contact.
For force-controlled conditions, a predefined value of the axial compressive force is assigned.
Then the displacements are obtained by the solution of the optimization problem, while the
adjoint variables are calculated indirectly by the adjoint equations (5.19)-(5.20).
97
Illustrative example with an elastica of constant length
To clarify the application of the optimal control method to solve a constrained buckling problem,
a simple example is presented. Let us consider a planar elastica of constant length, which is
constrained by two symmetrically located walls with clearances cu = −cl = 0.05. Pinned-
pinned boundary conditions are assumed and the elastica is initially unstressed. Force-controlled
conditions are applied.
Following the mathematical derivation of Section 5.2.1, the initial and terminal transversality
conditions are λy(0) = −ν1, λθ(0) = 0 and λy(1) = ν2, λθ(1) = 0 for pinned-pinned boundary
conditions, respectively. Taking into account the canonical equations (5.19)-(5.20), λy(s) =
−ν1 = ν2 for the unconstrained case. Using additionally the minimization of the Hamiltonian
with respect to the control variable λθ = −uˆ = −θ′, we arrive at equation θ′′ +R sin θ = 0 with
boundary conditions θ′(0) = θ′(1) = 0.
Instead, if there is a contact point ζc, y(ζc) = cu is satisfied and the shear force becomes
discontinuous. In particular, λy(s) = −ν1, for s ∈ [0, ζc), λy (ζ−c ) − λy (ζ+c ) = η1 and λy(s) =
ν2 = −ν1 − η1, for s ∈ (ζc, 1]. Due to symmetry η1 = 2ν1. When a continuous contact
is present the control variable u(ζ) vanishes for ζ ∈ [ζen, ζex] and the shear force becomes
λy(s) = −ν1, for s ∈ [0, ζc), λy (ζ−en)− λy (ζ+en) = η1, λy (ζ−ex)− λy (ζ+ex) = η2 and λy(s) = ν2 =
−ν1 − η1 − η2, for s ∈ (ζc, 1] with η1 = η2 = ν1.
For symmetrical reasons only half of the elastica has to be solved. If the elastica is in contact
with the wall, we solve θ′′ +R sin (θ − α) = 0 with boundary conditions θ′(0) = θ′(1/2) = 0, or
θ′(0) = θ′(ζen) = 0 for a discrete, or a continuous contact at ζen, respectively.
Let us now analyze the post-buckling behavior of the pinned-pinned elastica, as derived by
the optimal control problem. The elastica remains straight until an axial compressive force of
Q/pi2 = 1 is applied which corresponds to the first buckling load of a pinned-pinned elastica with
deformed shape (1− 2), as shown in Figure 7.2. In this case the elastica is unconstrained until
C2 (Q/pi2 = 1.03, δ = 0.0062) where a discrete contact at the midpoint position s = 0.5 occurs.
Beyond C2 the position of the contact point remains unchanged with increasing axial compressive
force while a contact force also appears with increasing magnitude. When C3 (Q/pi2 = 3.98,
δ = 0.0075) is reached, the moment, or the curvature at the discrete contact vanishes leading to
the onset of a continuous contact. Then the length of the continuous contact starts increasing
with maximum value of `c = 0.498 ≈ 0.5 at C4 (Q/pi2 = 15.85 ≈ 16, δ = 0.0154). At C4, the
values `c = 0.5 and Q/pi2 = 16 imply that the continuous segment buckles according to the first
buckling mode of a clamped-clamped elastica (i.e., the Euler buckling load 4pi2/`2c is overcome
under further loading).
98
0 . 0 0 0 0 . 0 0 4 0 . 0 0 8 0 . 0 1 2 0 . 0 1 6 0 . 0 2 00
2
4
6
8
1 0
1 2
1 4
1 6 4
3
2


/pi2
δ
1
Figure 5.2: Bifurcation diagram for a constrained pinned-pinned elastica of constant length with
clearances cu = −cl = 0.05
The inclination α of the internal force R(0), the inclination angle θ(0) and the angle β =
arctan [cu/x (ζen)] are also shown in Figure 5.3 after the onset of the discrete contact. From
these angles some notable conclusions are deduced, which have clear geometrical interpretation,
as also explained by (Domokos et al., 1997). Beyond C2 the inclination α increases from zero
with α < β to α = β = 0.1 at C3. This means that at C3 the internal force R(0) passes through
the position of the discrete contact, implying therefore a vanishing bending moment at that
point. Past C3, α = β or in other words the curvature at the entry point ζen remains equal to
zero until C4, where α = β = 0.2. On the other hand, the inclination angle at s = 0 is θ(0) > 2α
along the branch 2− 3 and θ(0) = 2α when the deformation shape 3− 4 initiates. In summary,
there are two geometrical restrictions; α ≤ β and α ≤ θ(0)/2.
The new deformation shape after C4 is not derived herein since the scope of this Section is
to briefly describe the optimal control method. Further analysis of this particular example can
be found in (Manning and Bulman, 2005; Domokos et al., 1997).
The benefit of the assumption of strong variation of a functional, which is incorporated in
the optimal control formulation, is not obvious through this particular example. Nevertheless
its advantage becomes evident in Section 5.3.
99
0.0 0.1 0.2 0.3 0.4 0.5
0
2
4
6
8
10
12
14
16
β
Angles
θ(0)
α
α θ(0) β x(ζen)
cuy
x
Q/
⇡
2
R
Figure 5.3: Axial compressive force Q/pi2 with respect to the angles; (i)α, (ii) β and (iii) θ(0)
for a discrete contact
0.0 0.1 0.2 0.3 0.4 0.5
0
2
4
6
8
10
12
14
16
Q/
⇡
2
`c
Figure 5.4: Axial compressive force Q/pi2 with respect to the length of the continuous contact
`c
5.2.2 Displacement controlled with fixed length or variable length
The constrained stability problem of the planar elastica for force-controlled conditions is ex-
plained in detail in Section 5.2. If we assume displacement-controlled conditions, the external
work of the axial compressive force Q is removed from the potential energy of the planar elastica
since the value of the applied force is unknown. Instead, the distance between the two supports
should be predefined. To impose a specific value of the coordinate x(1), or end-shortening δ,
a new state variable x(s) with its associated spatial derivative x′(s) = cos θ(s) is added with
boundary conditions x(0) = 0 and x(1) = 1− δ. Hence, the nonlinear optimal control problem
100
Pδ of the constrained planar elastica for displacement-controlled conditions is formulated as
follows

Minimize J [x(·),u(·), s] = 12
∫ 1
0
u2(s)ds
subject to
y′(s) = sin θ(s)
x′(s) = cos θ(s)
θ′(s) = u(s)
y(0) = y(1) = x(0) = θ(0) = θ(1) = 0 clamped B.C. or
y(0) = y(1) = x(0) = θ′(0) = θ′(1) = 0 pinned B.C
x(1) = 1− δ
y − cu ≤ 0
−y − cl ≤ 0
(5.22)
where y is the vertical displacement, x is the horizontal displacement, θ is the inclination angle,
δ is the imposed end-shortening and cu, cl are the clearances of the upper and lower hori-
zontal walls, respectively. The adjoint equations can be derived in a similar manner to the
force-controlled problem. Nevertheless the axial compressive force Q is computed here by the
corresponding adjoint variable λx of the additional state variable x(s) with λ′x = 0. Similarly
to the force-controlled problem, the graph of the axial compressive force Q/pi2 is also derived in
terms of the end-shortening δ.
Another constrained buckling problem can be also solved using a similar set-up. In particular,
we consider that the elastica is clamped or pinned at the left end, while it is inserted through a
fixed frictionless device that functions as a clamped end at the right end. Hence, the arc-length
of the elastica varies while the distance between the fixed end of the elastica and the point of
insertion is constant. This is the insertion stability problem, which can be described by the new
optimal control problem P¯δ
Minimize J [x(·),u(·), s] = 12
∫ 1+δ¯
0
u2(s)ds
subject to
y′(s) = sin θ(s)
x′(s) = cos θ(s)
θ′(s) = u(s)
y(0) = y(1 + δ¯) = x(0) = θ(1 + δ¯) = 0
θ(0) = 0 or θ′(0) = 0
x(1 + δ¯) = 1
y − cu ≤ 0
−y − cl ≤ 0
(5.23)
101
where δ¯ is the change of the inserted length. For the solution of the above problem, a prede-
fined value of the arc-length is assigned, while the horizontal position is always x(1 + δ¯) = 1.
The solution is similar to the classical stability problem for displacement-controlled conditions.
However the solution of this problem provides the internal axial force R cosα rather than the
applied force Q. As explained in Chapter 3, the applied axial force Q is accordingly obtained
by Q = R cosα−P , where P = θ′(1 + δ¯)2/2 is a configurational or Eshelby-like force generated
at the insertion point of the sliding sleeve. In addition the graph of the axial compressive force
Q/pi2 and the internal axial force R cosα is computed in terms of the change of the arc-length
δ¯ rather than the end-shortening δ that is used above.
5.3 Bifurcation diagrams of planar elastica
Here we apply the optimal control method for several cases. Initially we verify the use of
the optimal control in the derivation of the post-buckling behavior of an unconstrained elas-
tica with pinned-pinned boundary conditions. The complete solution of this problem has been
first obtained by (Kuznetsov and Levyakov, 2002) and it is compared with the optimal control
formulation.
Next the constrained stability problem is investigated. Three distinct examples are presented.
The first example involves the classical stability problem of an elastica with clamped-clamped
boundary conditions, which is constrained by two symmetrically located walls with clearances
cu = −cl = 0.05. Force-controlled and displacement-controlled conditions are imposed sepa-
rately and their effect in the post-buckling response of the elastica is investigated.
Then we investigate the insertion stability problem, where the length of the elastica varies
while the horizontal distance between the two supports is fixed. The particular constrained
problem has been first performed by (Ro et al., 2010) for clearances cu = −cl = 0.1. In this
case the bifurcation diagram is expressed in terms of the change of the arc-length δ¯ instead of
the end-shortening that is used above.
Finally we examine the effect of the boundary conditions and the asymmetrical location of
the walls. In particular, we investigate the post-buckling behavior of a pinned-pinned elastica of
constant length constrained by two asymmetrically located rigid walls with clearances cu = 0.125
and cl = 0. The same problems have been first solved by (Ro et al., 2010; Manning and Bulman,
2005; Domokos et al., 1997) and they are used for comparison.
5.3.1 Verification example
The post-buckling behavior of an unconstrained pinned-pinned elastica of constant length is
investigated assuming that the left support is fixed at x(0) = 0 while the right end is free to
move along the x-axis. The bifurcation diagram is given in Figure 5.5. The sequence of the
102
configurations of the elastica is also illustrated. The minimized Hamiltonian (i..e, mimimized
total energy) initially increases with increasing applied horizontal force, or end-shortening until
it reaches a plateau with Q/pi2 = 2.18. At this point the two supports come in contact meaning
that the end-shortening is δ = 0. Beyond this point the solution is unstable. For force-controlled
conditions, the solution jumps to C4 with δ = 2 (see Figure 5.5), while for displacement-
controlled conditions, it moves to C2′ with tensile load Q/pi2 = −2.18 which is obtained by a
180 rotation of the previous configuration. For the same absolute value of forces, a configuration
under the action of the tensile force can be derived by the corresponding configuration under
the action of the compressive load if we add to the latter inclinations angles an angle of pi.
In two-dimensional space, we cannot recognize the difference between the configurations. This
solution is in agreement with the solution by (Kuznetsov and Levyakov, 2002), who derived the
same problem using a semi-analytical solution of a Sturm-Liouville boundary value problem.
0.0 0.5 1.0 1.5 2.0
-5
-4
-3
-2
-1
0
1
2
3
4
5
3'
4'
1'
2'
4
3
2
δ
1
Q/
⇡
2
Figure 5.5: Bifurcation diagram for a pinned-pinned elastica of constant length
5.3.2 Clamped-clamped elastica with symmetrically located walls
The post-buckling behavior of a clamped-clamped elastica constrained by two symmetrically
located walls is now studied. The classical and insertion stability problems are examined sepa-
rately. The results are compared with previous works (Manning and Bulman, 2005; Ro et al.,
2010).
103
Classical problem
A clamped-clamped elastica is constrained by two symmetrically located walls with clearances
cu = −cl = 0.05. The elastica is of constant length (classical stability problem) and both force-
and displacement-controlled conditions are applied which are considered.
For displacement-controlled conditions, the elastica is initially unconstrained and its defor-
mation shape (1− 2) corresponds to the minimum root of the first buckling mode of a clamped-
clamped elastica, as shown in Figure 5.6. With further increase of δ, the elastica comes in
contact with the wall at its midpoint position s = 0.5 at C2 for symmetrical reasons. When the
horizontal force reaches 3, the moment vanishes at the discrete contact. Past this point there are
three distinct possibilities; (i) onset of a continuous contact, (ii) simultaneous change of the sign
of moments at both end supports and (iii) change of the sign of moment at the one end. The
third case is optimal, which leads to the asymmetrical deformation shape (3 − 4). Along this
branch the axial force starts decreasing until a second contact point on the opposite wall appears
at 4. Accordingly the axial force increases gradually until 5. Beyond this point a new change of
sign of the moment at the one end appears, leading to a new asymmetrical deformation shape
(5−6) with decreasing applied load. This branch evolves until a third contact point develops at
6. Due to symmetry the second contact point is located at s = 0.5 and the other contact points
are symmetrically located at the opposite walls.
The adjoint functions for representative values of δ are also presented in Figure 5.7 for
displacement-controlled conditions. Due to the absence of any external pressure, the nature
of the boundary conditions and the symmetrical location of the walls, the functions of the
adjoint variables and accordingly the cartesian coordinates of the elastica can be also reversed,
or changed in sign without affecting the bifurcation diagram.
In the case of a force-controlled problem, the elastica deforms in an identical manner at the
initial stage, see deformation shapes (1− 2) and (2− 3). Nevertheless past C3 the asymmetrical
deformation shape (3− 4) is unstable as the axial compressive force decreases (i.e., the branch
is softening). Instead the configuration jumps to the branch (4 − 5) with two symmetrically
located contact points to the opposite walls. For the same reason, it jumps to the deformation
shape (6 − 7) beyond C5, as shown in Figure 5.6. The bifurcation diagram is shown in Figure
5.6 and the points of evolutions of the contact patterns are summarized in Table 5.3.2.
Points of evolution of the contact patterns for the bifurcation diagram for a constrained
clamped-clamped planar elastica with clearances cu = −cl = 0.05
104
0 . 0 0 0 . 0 1 0 . 0 2 0 . 0 3 0 . 0 4 0 . 0 5 0 . 0 6 0 . 0 70
5
1 0
1 5
2 0
2 5
3 0
3 5 7
6
5
4
3
2


/pi2
δ
1
1 - 2 2 - 3
3 - 4 4 - 5
5 - 6 6 - 7
Figure 5.6: Bifurcation diagram for a constrained clamped-clamped elastica of constant length
with clearances cu = −cl = 0.05
105
0.0 0.2 0.4 0.6 0.8 1.0
-15
-10
-5
0
5
10
15
0.0 0.2 0.4 0.6 0.8 1.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
λ
y
s
λ
θ
s
(2 3) : Q/⇡2 = 13.1,  = 0.007
0.0 0.2 0.4 0.6 0.8 1.0
-15
-10
-5
0
5
10
15
0.0 0.2 0.4 0.6 0.8 1.0
-3
-2
-1
0
1
2
3
λ
y
s
λ
θ
s
(3 4) : Q/⇡2 = 9.3,  = 0.015
0.0 0.2 0.4 0.6 0.8 1.0
-60
-50
-40
-30
-20
-10
0
10
20
30
0.0 0.2 0.4 0.6 0.8 1.0
-5
-4
-3
-2
-1
0
1
2
3
4
5
λ
y
s
λ
θ
s
(4 5) : Q/⇡2 = 22.8,  = 0.03
0.0 0.2 0.4 0.6 0.8 1.0
-60
-50
-40
-30
-20
-10
0
10
20
30
0.0 0.2 0.4 0.6 0.8 1.0
-5
-4
-3
-2
-1
0
1
2
3
4
5
λ y
s
λ θ
s
(5 6) : Q/⇡2 = 20.8,  = 0.04
0.0 0.2 0.4 0.6 0.8 1.0
-100
-80
-60
-40
-20
0
20
40
60
80
100
0.0 0.2 0.4 0.6 0.8 1.0
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
λ
y
s
λ
θ
s
(6 7) : Q/⇡2 = 30.6,  = 0.065
Figure 5.7: Adjoint variables for distinct values of end-shortening δ
106
Insertion problem
The post-buckling behavior of a clamped-clamped elastica of variable length constrained by
symmetrically located walls with cu = −cl = 0.1 is investigated. In this case the horizontal
distance between the two supports is assumed to be fixed x(1 + δ¯) − x(0) = 1 while the arc-
length gradually increases. The same problem has been solved by applying the geometry-based
technique in Chapter 3. In Figure 5.8, the applied and internal axial forces are illustrated with
respect to the change of the length of the elastica. The applied axial force is lower than the
internal axial force as a configurational or Eshelby-like force is generated at the insertion point
of the sliding sleeve. More details about the development of the Eshelby-like force can be found
in Chapter 3.
One main difference between the constrained elastica of constant length and variable length
is the initial post-buckling behavior of the elastica in its unconstrained condition. In the former
case the value of the horizontal applied force increases (i.e., super-critical bifurcation), while
in the latter case the opposite behavior is observed (i.e., sub-critical bifurcation). However the
sequence of the contact patterns is exactly the same compared with the case of a displacement-
controlled problem of a constrained elastica of constant length. The value of the horizontal
force at C3 is approximately the same as in the previous example for a change of arc-length
δ¯ = 0.029. The decrease of the horizontal force occurs until C4. Beyond C4 two contact points
are present until C5. Then the deformation shape becomes asymmetrical until C6, where three
contact points appear. The subsequent deformation shapes are shown in Figure 5.8 and the
corresponding points of evolution of the contact patterns are summarized in Table 6.1.
Points Insertion problem
θ[2] δ¯ R cosα /pi2 Q/pi2
2, 2¯ 0.311 0.025 3.86 3.67
3, 3¯ 0.394 0.030 14.86 14.86
4, 4¯ 0.484 0.100 6.97 5.72
5, 5¯ 0.517 0.152 24.03 24.03
6, 6¯ 0.628 0.219 11.46 8.00
Table 5.1: Points of evolution of the contact patterns for the bifurcation diagram for a con-
strained clamped-clamped elastica with clearance cu = −cl = 0.1
107
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
5
10
15
20
25
30
5,5
3,3
7
7
6
6
4
  4
2
2
δ
1,1
Q/
⇡
2
,R
co
s(
↵
)/
⇡
2
1 - 2 2 - 3
3 - 4 4 - 5
5 - 6 6 - 7
Figure 5.8: Bifurcation diagram for a constrained clamped-clamped elastica with clearance
cu = −cl = 0.1; solid line for the applied load Q and dashed line for the axial load R cosα.
5.3.3 Pinned-pinned elastica constrained by asymmetrically located
walls
A pinned-pinned elastica of constant length is constrained by two asymmetrically located rigid
walls with clearances cu = 0.125 and cl = 0. The problem is solved for both force-controlled
and displacement-controlled conditions.
Initially, the elastica is unconstrained and its configuration (1 − 2) is consistent with the
first buckling mode of a pinned-pinned elastica with axial force Q/pi2 = 1, see Figure 5.9. Due
to symmetry the elastica comes in contact with the upper wall at s = 0.5 corresponding to
C2. This discrete contact remains unchanged until C3, where the moment at the contact point
vanishes. This leads to the onset of the continuous contact with deformation shape (3 − 4),
which is a stable configuration in the case of a pinned-pinned elastica. The length of the
continuous contact increases until C4. Past this point the continuous segment buckles as the
108
first buckling mode of a clamped-clamped elastica forming a free-standing fold, see deformation
shape (4−5). This branch is stable for displacement-controlled conditions, while it is unstable for
force-controlled conditions, because a decrease of the load is observed. Hence for displacement-
controlled conditions the axial force decreases until a new discrete contact is met at s = 0.5
with the lower wall at C5. This yields to the new symmetrical configuration (5 − 6). On the
other hand, for force-controlled conditions, beyond C3 the elastica jumps directly to the branch
(5− 6).
The whole sequence of the contact patterns is illustrated in Figure 5.9 and the points of
evolution of the contact patterns are summarized in Table 7.1. The same problem has been also
solved by (Domokos et al., 1997) and it is in full agreement with the current solution.
Points of evolution of the contact patterns for the bifurcation diagram for a constrained
pinned-pinned planar elastica with clearances cu = 0.125 and cl = 0
109
0.00 0.05 0.10 0.15 0.20 0.25 0.30
0
2
4
6
8
10
12
14
16
18
20
6
5
4
3
2
δ
1
Q/
⇡
2
1 - 2 2 - 3
3 - 4 4 - 5
5 - 6
Figure 5.9: Bifurcation diagram for a constrained pinned-pinned elastica of constant length with
clearances cu = 0.125 and cl = 0
5.4 Stability of constrained spatial elastica
Now we examine the post-buckling behavior of a spatial elastica of constant length (classical
stability problem) constrained by a cylindrical wall. A short description of the geometrical
characteristics and the potential (strain) energy of an isotropic inextensible-unshearable elastic
rod, the so-called Kirchhoff rod or spatial elastica, is first provided. Based on the Kirchhoff rod
model we construct the new form of the optimal control problem for displacement-controlled
conditions. Validation of the formulation is then achieved by solving a well-known stability
problem of an unconstrained spatial elastica with clamped-clamped boundary conditions and for
a material parameter γ = 5/7 which is the ratio of the shear to the bending moduli (Maddocks,
1984; Majumdar et al., 2012; Goyal et al., 2005; Majumdar and Goriely, 2013). Keeping the same
110
material parameter γ and assuming the same boundary conditions, the post-buckling response
of a spatial elastica constrained by a wall with clearance c = 0.1 is solved and the results are
compared with similar results obtained by (Fang et al., 2013; Chen and Fang, 2013).
5.4.1 Kirchhoff rod model
A Kirchhoff rod is a special case of a Cosserat rod, which is inextensible and unshearable. As
Bernoulli law (i.e., the cross section remains plane and rigid) is applicable in this case, this
problem can be formulated as a one dimensional continuum (Weiss, 2002).
The Kirchhoff rod is parametrized by s the arc-length paramater with s ∈ [0, L]. Its initial
configuration is unstressed. Then the configuration of the rod is described by its centerline r(s)
and the orthonormal director basis {d1,d2,d3}. The director basis is related to the reference
configuration {e1, e2,e3} by a rotation matrix D(s)
di(s) =D(s)ei (5.24)
and satisfies the following property
d1(s) = d2(s)× d3(s) (5.25)
(5.25) indicates that d1(s) is orthogonal to the cross-section plane defined by {d2(s),d3(s)}.
The director basis {d1,d2,d3} changes smoothly with respect to the reference configuration
{e1, e2,e3}. This change is described by the following kinematic relation
∂sdk = κ× dk (5.26)
where κ = {κ1, κ2, κ3} is called Darboux vector (or strain vector) which involves the twist (κ1)
and the bending (κ2, κ3) of the rod about the three directors.
Considering that the Kirchhoff rod is inextensible and unshearable, the kinematics of the
centerline is given by
∂sr = v = d1 (5.27)
where v is a vector-valued function whose components along the three directors v = vidi =
{v1, v2, v3} describe the extension v1 = 1 and the shear v2 = v3 = 0 of the rod.
The resulting strain energy of an isotropic Kirchhoff rod becomes
U = 1
2
∫ L
0
GJκ21(s) + EI
(
κ22(s) + κ
2
3(s)
)
ds (5.28)
where I = piα4/4 is the second moment of area and J = piα4/2 is the polar moment of area for
a cylindrical rod of uniform radius α.
111
k 3
φ
 j 3   
c )  R o l l
i 2  ≡  i 3  
k 2
 j 2   
b )  P i t c hk 2
i 2
θ
k 1
j 1  ≡  j 2   
i 1
j 1
i 1
ψ
k ≡ k 1
j
a )  Y a w
i
ψ
θ
φ
Figure 5.10: Method 3− 2− 1 of Euler angles
To derive the components of the Darboux vector, we need to define the rotation matrix
D(s) which is described by a set of Euler angles {θ(s), φ(s), ψ(s)} with respect to the reference
configuration {e1, e2, e3}. Let us denote that i = e1, j = e2 and k = e3. In this study, the three
Euler angles are obtained applying the method (3 − 2 − 1) (Atanackovic, 1997), which yields
to the simplest possible form of the director axis d1, as shown in Figure 5.10. In fact, we first
apply a rotation ψ around k (i.e., yaw) which leads to
{
i(1), j(1),k(1)
}
coordinate system. A new
rotation θ around j(1) (i.e., pitch) is then imposed which gives the
{
i(2), j(2),k(2)
}
coordinate
system. A third rotation φ around i(2) (i.e., roll) finally gives the
{
i(3), j(3),k(3)
}
coordinate
system which coincides with the director axes {d1,d2,d3}. This sequence of rotation matrices
can be expressed as follows
Dx =
 1 0 00 cosφ sinφ
0 − sinφ cosφ
 , Dy =
 cos θ 0 − sin θ0 1 0
sin θ 0 cos θ
 , Dz =
 cosψ sinψ 0− sinψ cosψ 0
0 0 1

(5.29)
leading to the total rotation matrix D = DxDyDz. Based on (5.24) the director axes d1,d2,d3
become
d1 = cos θ cosψi + cos θ sinψj− sin θk (5.30)
d2 = (sin θ sinφ cosψ − cosφ sinψ) i
+ (sin θ sinφ sinψ + cosφ cosψ) j
+ sinφ cos θk (5.31)
d3 = (sin θ cosφ cosψ + sinφ sinψ) i
112
+ (sin θ cosφ sinψ − sinφ cosψ) j
+ cosφ cos θk (5.32)
and applying the method (3− 2− 1) the Darboux vector κ can be described as follows
κ = ψ′k + θ′j(1) + φ′i(3) (5.33)
where the j(1) = j(2),k axes can be expressed about the director axes as j(1) = cosφd2− sinφd3
and k = − sin θd1 + sinφ cos θd2 + cosφ cos θd3, respectively while i(2) ≡ d1. This leads to
κ1 = φ
′ − ψ′ sin θ
κ2 = ψ
′ sinφ cos θ + θ′ cosφ
κ3 = ψ
′ cosφ cos θ − θ′ sinφ
(5.34)
where κ1 = τ is the torsion, κ2 and κ3 are the curvatures about the director axes.
5.4.2 Optimal control
The optimal control formulation of a spatial elastica constrained by a cylindrical wall is derived
herein. The potential energy of the spatial elastica is given by (5.28) and involves the torsion
and bending curvatures (7.5). Adopting L as length scale and EI/L2 as force scale (see also
(7.2)), the strain energy becomes
U = 1
2
∫ 1
0
(
κ22(s) + κ
2
3(s)
)
ds+
γ
2
∫ 1
0
κ21(s)ds (5.35)
where γ = GJ/EI is the ratio of shear to bending moduli. Taking into account the director axes
(5.30-5.32) and considering that ∂sr = d1(5.27), the inextensibility-unshearability constraints
can be written as follows
x′ = cos θ cosψ
y′ = cos θ sinψ
z′ = − sin θ
(5.36)
where {x, y, z} are the cartesian coordinates of the Kirchhoff rod. If we consider a cylindrical
wall of clearance c, the unilateral constraint can be expressed with respect to the cartesian
coordinates {y, z};
y2(s) + z2(s) ≤ c2 (5.37)
For displacement-controlled conditions we predefine the boundary condition x(1) = 1−δ where δ
is the end-shortening and the optimal control problem for clamped-clamped boundary conditions
113
takes the following form
Minimize J [x(·),u(·)] = 12
∫ 1
0
(
κ22(s) + κ
2
3(s)
)
ds+ γ2
∫ 1
0
κ21(s)ds
subject to
x′ = cos θ(s) cosψ(s)
y′ = cos θ(s) sinψ(s)
z′ = − sin θ(s)
ψ′ = u1
θ′ = u2
φ′ = u3
x(0) = y(0) = z(0) = y(1) = z(1) = 0
ψ(0) = ψ(1) = θ(0) = θ(1) = φ(0) = φ(1) = 0
y2(s) + z2(s) ≤ c2
(5.38)
where the state variables are the cartesian coordinates {x, y, z} of the elastica and the Euler
angles {ψ, θ, φ} and the control variables are the spatial derivatives of the Euler angles {ψ′, θ′, φ′}
while the rotational strains become
κ1 = u3 − u1 sin θ (5.39)
κ2 = u1 sinφ cos θ + u2 cosφ (5.40)
κ3 = u1 cosφ cos θ − u2 sinφ (5.41)
Substituting (5.39-5.41) into the potential energy of the spatial elastica (5.35), the final form of
the functional is obtained
J [x(·),u(·)] = 1
2
∫ 1
0
(
u21 cos
2 θ + u22
)
ds+
γ
2
∫ 1
0
(u3 − u1 sin θ)2 ds (5.42)
The normal, or simple Hamiltonian can be then expressed as follows
H(λ,x,u) =
1
2
(
u21 cos
2 θ + u22
)
+
γ
2
(u3 − u1 sin θ)2
+ λx cos θ cosψ + λy cos θ sinψ+
− λz sin θ + λψu1 + λθu2 + λφu3
and the augmented Hamiltonian is given by
H¯ (µ,λ,x,u) = H(λ,x,u) + µ
(
y2(s) + z2(s)− c2) (5.43)
114
where µ † g is always satisfied. Based on the assigned boundary conditions the scalar function
Φ is described as follows
Φ = ν1x(0) + ν2x(1) + ν3y(0) + ν4y(1) + ν5z(0) + ν6z(1)
+ ν7ψ(0) + ν8ψ(1) + ν9θ(0) + ν10θ(1) + ν11φ(0) + ν12φ(1) (5.44)
The minimization of the Hamiltonian with respect to the control variables gives
H¯u =

u1
(
cos2 θ + γ sin2 θ
)− γu3 sin θ + λψ
u2 + λθ
γ (u3 − u1 sin θ) + λφ
 = 0
which leads to the optimal control variables uˆ1 = − (λψ + λφ sin θ) / cos2 θ, uˆ2 = −λθ, uˆ3 =
− (λψ + λφ sin θ) sin θ/ cos2 θ−λφ/γ. The adjoint equations derived by H¯ are written as follows
−λ′x =
∂H¯
∂x
= 0 (5.45)
−λ′y =
∂H¯
∂y
=
{
0 if unconstrained
µc sinβ if y
2
c + z
2
c = c
2
(5.46)
−λ′z =
∂H¯
∂z
=
{
0 if unconstrained
µc cosβ if y
2
c + z
2
c = c
2
(5.47)
−λ′ψ =
∂H¯
∂ψ
= cos θ (−λx sinψ + λy cosψ) (5.48)
−λ′θ =
∂H¯
∂θ
=
(γ − 1)
2
u21 sin 2θ − γu1u3 cos θ
− sin θ (λx cosψ + λy sinψ)− λz cos θ (5.49)
−λ′φ =
∂H¯
∂φ
= 0 (5.50)
where yc = c sinβ, zc = c cosβ and µc = 2cµ. The adjoint variables have a clear physical
meaning. In particular, λx, {λy, λz} and {λψ, λθ, λφ} provide the axial compressive force, the
shear forces and the moments (twisting and bending), respectively.
The initial and terminal transversality conditions are given by λx(0) = −ν1, λy(0) =
−ν3,λz(0) = −ν5, λψ(0) = −ν7, λθ(0) = −ν9, λφ(0) = −ν11, λx (1) = ν2, λy (1) = ν4,
λz(1) = ν6, λψ(1) = ν8, λθ(1) = ν10 and λφ(1) = ν12. Using the equations of the adjoint
variables, λx(s) = ν2 = −ν1 and λφ(s) = ν12 = −ν11.
115
Taking into account uˆ3 − uˆ1 sin θ = −λφ/γ and λ′φ = 0
d
ds
(uˆ3 − uˆ1 sin θ) = 0⇒ κ′1 = 0
which implies that the torsion κ1 is uniform for an isotropic cylindrical rod independently of the
loading conditions.
For displacement-controlled conditions the axial compressive force Q is indirectly derived
by the adjoint variable λx. Instead, if we apply force-controlled conditions, the external work
Q ∫ 1
0
cos θ(s) cosψ(s)ds with predefined axial compressive force Q is included in the functional
(5.42) while the end-shortening δ is indirectly calculated by the solution of the problem. In the
current study displacement-controlled conditions are only assumed.
5.4.3 Results
The stability behavior of an unconstrained spatial elastica is initially studied in order to validate
the optimal control formulation for the spatial case . More specifically, we investigate the post-
buckling behavior of the clamped-clamped spatial elastica for a ratio of shear to bending moduli
γ = 5/7. Its analytical description is not included here, as it can be found in Chapter 2.
The classical constrained stability problem is then analyzed. A particular example is solved
and compared with Fang et al. (2013).
Constrained clamped-clamped spatial elastica
Next we investigate a clamped-clamped spatial elastica constrained by a cylindrical wall of radius
c = 0.1 and γ = 5/7. The same problem has been solved by (Chen and Fang, 2013). The elastica
is initial unstressed (i.e., straight elastica) and δ is gradually increased.
The elastica remains straight until it reaches a load of 4pi2, which is consistent with the
first symmetrical buckling mode of a planar clamped-clamped elastica. The elastica does not
touch the wall until C2, where a point contact with the midpoint position of the elastica is
met. This is a planar deformation shape and it remains unchanged until C3, where an out-of
plane deformation initiates with the same contact condition. Very rapidly the discrete contact
becomes a continuous contact at C4 whose length slightly increases until C5. Past this point, the
continuous contact splits into two point contacts symmetrically located left and right from the
midpoint position s = 0.5. At C6 the point contact at midpoint position appears again, which
implies a three-contact-point deformation shape. Then the contact point at s = 0.5 becomes a
continuous, or line contact at C7. The line contact gradually increases until the point-line-point
deformation becomes a single continuous contact at C8. This reaches a maximum horizontal
force Q/pi2 = 12.03 with end-shortening δ = 0.239. Accordingly a reduction of the horizontal
load is observed. At C9 the single continuous contact is splitted into three continuous contacts.
116
As the horizontal force continues to decrease, the two continuous contacts next to the midpoint
position s = 0.5 become discrete contacts at C10). Hence, a point-line-point deformation shape
is obtained. The midpoint continuous contact also becomes a point contact leading to a point-
point-point deformation shape at C11, which remains until C12.
As shown in Figure 5.11, the bifurcation diagram is identical to previous work performed by
Fang et al. (2013). Nevertheless there are two minor differences. The first difference is that we
derive the deformation shape (4− 5) with a continuous contact while Fang et al. (2013) jumps
from the deformation shape (3 − 4) directly to (5 − 6) without the presence of a continuous
contact. The second difference is around the plateau. Using optimal control the deformation
shape (8−9) starts before the plateau, while Fang et al. (2013) derives the same contact pattern
after the peak. The rest of the deformation shapes and their sequence are identical. The
deformation shapes for distinct values of end-shortening δ are summarized in Figure 5.12.
Classical Problem
Points 2 3 4 5 6 7 8 9 10 11 12
δ 0.025 0.0261 0.048 0.057 0.15 0.155 0.22 0.34 0.36 0.4 0.7
Q/pi2 4.05 8.02 8.068 8.123 10.8 10.93 11.95 9.647 8.73 7.038 4.01
Table 5.2: Points of evolution of the contact patterns for the bifurcation diagram for a con-
strained clamped-clamped spatial elastica with clearance c = 0.1
0 . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 70
2
4
6
8
1 0
1 2
1 4
7
P - P - P
P
P L
P - L - P
8
P - P
L
L - L - L
P - L - P
1 0
1 2
1 1
9
6
543
2


/pi2
δ
1 P - P - P
Figure 5.11: Bifurcation diagram for a constrained clamped-clamped elastica of constant length
with clearance c = 0.1
117
Figure 5.12: Deformation shapes; a) Spatial one point (Q = 8.05pi2, δ = 0.04, κ1 = 0.21), b)
Point-point (Q = 9.33pi2, δ = 0.1, κ1 = 0.64), c) Point-line-point (Q = 11.04pi2, δ = 0.16,
κ1 = 1.21), d) Line contact (Q = 11.95pi2, δ = 0.22, κ1 = 1.8) , e) Point-line-point contact
(Q = 8.31pi2, δ = 0.37, κ1 = 3.37) and f) Point-point-point contact (Q = 3.95pi2, δ = 0.7,
κ1 = 4.2)
118
5.5 Conclusions
From this study it is evident that the optimal control is an efficient tool for the stability analysis
of elastic rods in the presence of unilateral constraints. Further analysis could also include
second-order optimality conditions. This would require three distinct steps; (i) derivation of
the solution adopting a direct method, (ii) use of this solution to define properly an initial
guess of the involved variables such that an indirect shooting method can be easily derived for
further accuracy and (iii) application of a sensitivity analysis of the shooting mapping which can
provide the second-order optimality conditions through the solution of the Riccati equations, as
explained in detail in (Malanowski and Maurer, 2001; Maurer and Oberle, 2002). Even though
we do not examine the second-order optimality conditions, the assumption of strong variations in
the derivation of the first-order optimality conditions, which includes the Weierstrass necessary
condition, is sufficient to derive properly the bifurcation diagrams for all conditions without
difficulty.
Chapter 6
Optimization-based stability
analysis of constrained heavy
elastica
The constrained buckling problem of a planar elastica with a non-negligible self-weight, also
known as a heavy elastica (Wang, 1984b), is analyzed here. Its buckling response differs re-
markably from the buckling problem of a weightless elastica, especially if unilateral constraints
are also present, thus special attention is placed to this effect.
The Chapter consists of three parts; (1) the loop tests, (2) the buckling response of a heavy
elastica lying on a rigid horizontal or inclined foundation and (3) the buckling behavior of a
heavy elastica constrained by two rigid horizontal walls. The first part includes a short review
of the available loop tests, which are widely used in the manufacturing of thin elastic sheets
for measuring their material properties (see (Plaut, 2015)). In the second part, a step-by-
step derivation of the buckling problem is provided, including the effect of the friction. The
same problem is then solved by applying an optimal control method. A similar formulation is
then suggested for the analysis of the constrained buckling problem with two-sided foundations.
Several factors are investigated, such as the effect of the boundary and loading conditions and
the clearance of the walls.
6.1 Loop tests
The folding of slender elastic objects with a negligible self-weight is important in manufacturing
of papers and textiles (Lloyd et al., 1978; Mahadevan and Keller, 1999). A long thin sheet can be
considered inextensible and unshearable. Hence, a simple model of a planar weightless elastica
119
120
is a good approximation for the response of slender objects, which undergo only planar bending.
Based on the governing equation of the elastica and considering as length scale lb = (EI/V )
1/2,
where EI is the bending stiffness of the elastica and V is the vertical load, (Wang, 1981a)
derived similarity solutions for three distinct cases; (1) the elastica is compressed by two parallel
approaching plates, (2) it is folded by two symmetric rollers moving to the right, or (3) it is
folded between a moving roller and a flat plane, as shown in Figure 6.1. The dimensionless
quantities α, β, `f denote the width, height and fold length of the elastica, respectively and are
given by 
α =
√
2, β = 1.69443, `f = 1.85407, Case 1
α = 2.67181, β = 1.28591, `f = 3.14844, Case 2
α = 2.70745, β = 1.34972, `f = 5.26292, Case 3

i )
i i i )
i i )






Figure 6.1: Folds of elastica i) Case 1, ii) Case 2 and iii) Case 3
If the weight effect is essential, the length scale lb = (EI/V )
1/2 is replaced by lg = (EI/w)
1/3,
where w is the weight per unit length. The length scale lg, also called bending length, is a measure
of the ratio of its flexural rigidity to its weight. Similarly to the case 3 (see 6.1 iii)), Stuart (1966)
proposed a loop test that consists in laying down on a horizontal surface a sufficiently long strip
with length ` (` ≥ 5lg) in the form of a loop. In that instance, the dimensionless quantities
α, β, `f are given by
α = 3.24324, β = 0.9066, `f = 4.6833 (6.1)
121
This experimental test, called “free-standing fold” (Plaut, 2015), is widely used to measure the
Young modulus E of “heavy” strips. In particular, for a given weight per unit length w and
considering the known geometry of the invariant loop (6.1), Young modulus E can be indirectly
computed.
The invariance of the loop can be also verified by solving the following optimal control
problem 
minimize
u(s)
J = 12
∫ `
0
u2(s)ds+
∫ `
0
y(s)ds
subject to
y′(s) = sin θ(s)
x′(s) = cos θ(s)
θ′(s) = u(s)
y(0) = 0, θ(0) = −pi/2
y(`) = −c, θ(`) = −pi, θ′(`) = 0
−y(s)− c ≤ 0
(6.2)
where the functional J consists of the bending energy of the elastica and the gravitational
potential energy, x, y are the cartesian coordinates, θ is the inclination angle with respect to
the x-axis, u is the control variable, ` > `f = 4.6833 is the total length and c is the clearance,
which should be small enough such that a contact with the wall is feasible. Assigning any
appropriate combination of `, c, the invariant loop with geometrical characteristics, given by
(6.1), is obtained. More details for the optimal control formulation can be found in Chapters 4
and 5.
Alternative experimental tests can be also applied for the derivation of the bending stiffness
of thin elastic sheets, such as the heart loop, the standing wrinkle and the clamped fold. More
details can be found in (Lloyd et al., 1978; Plaut, 2015; Mahadevan and Keller, 1999).
6.2 Heavy elastica on a horizontal foundation
The buckling behavior of a slender elastic object with a non-negligible self-weight, which is
constrained by one-sided foundation, is a problem widely encountered in a variety of applications.
For instance, the upheaval buckling problem of a heavy elastic sheet from a rigid foundation
and/or rigid walls gained a lot of interest in oil drilling operations and rail tracks (Maltby
and Calladine, 1995a,b; Chen and Lin, 2013; Wicks et al., 2008; Miller et al., 2015). Other
applications involve the textile fabric and wrinkling of papers during handling operations (Hunt
and Blackmore, 1997; Kopp et al., 2000; Lloyd et al., 1978; Jae Ho Jung et al., 2004).
The first major contribution in the buckling problem of a heavy elastic sheet from a flat
rigid horizontal substrate is attributed to (Wang, 1981b, 1984a). Wang (1986) applied a simple
model of a planar elastica to analyze its buckling response under the action of a compressive
122
force. Based on the length of the elastica, Wang (1984b) proposed two definitions in accordance
with its buckling behavior; (1) the elastica is called “short” if the buckled sheet does not touch
the rigid foundation, except at its extremities and (2) the elastica is called “long” if a localized
“ruck” is formed, while the rest (also, called flat-lying section) remains in contact with the rigid
foundation. These definitions are quite universal nowadays (Domokos et al., 2003; Santillan
et al., 2006; Chen and Lin, 2013; Vella et al., 2009; Kolinski et al., 2009).
The main focus, however, was placed on the buckling response of the “long” heavy elastica
lying on a horizontal foundation. More specifically, Domokos et al. (2003) solved this buckling
problem numerically by using a global search algorithm and compared the theoretical results with
experimental observations. The main conclusion is that as the uplift of the formed ruck increases,
its symmetrical shape gradually transforms into an asymmetric configuration. This transition
point is the “symmetry-breaking bifurcation” and occurs before a self-contact is established. The
same result was also deduced by Chen and Lin (2013). In particular, Chen and Lin (2013) used
an Eulerian description of the problem, taking into account the variation of the contact points
during vibration, and accordingly applied a shooting method from which natural values of the
lowest frequencies imply a stable equilibrium state. A similar methodology was followed by
Santillan et al. (2006), who studied the buckling behavior of heavy elastic strips resting on an
inclined rigid foundation.
Quite recently, the motion of the localized ruck along a horizontal substrate was also inves-
tigated. The major results that arise from experimental observations (see Vella et al. (2009))
are the steady speed and the steady shape of the ruck, except at initial and final stages of
the experiments. Vella et al. (2009) also noticed that the velocity of the travelling ruck is not
uniquely dependent on its shape. This observation is in full aggreement with the numerical
analysis by Santillan et al. (2006); Chen and Lin (2013), who calculated a zero fundamental
frequency that implies that the ruck can translate at any velocity. For an inclined substrate,
Kolinski et al. (2009) also observed the same phenomenon (i.e., steady motion) and found that
velocity increases as we increase the slope of the foundation.
Another important factor is the friction between the “heavy” elastica and the rigid substrate.
Comninou (1978) was the first who discovered that a rug can be moved from one place on the
floor to another by pulling along one of its edges until friction is overcome and simply making it
slip. These ruck modes in polymer surfaces were first discovered by Schallamach (1971) and are
called Schallamach waves (Barquins, 1985). Viswanathan et al. (2015) showed that Schallamach
waves are qualitatively similar to dislocations and carpet rucks, even though three different
length scales (nm,microns and m) are present and the corresponding materials have unrelated
structural properties.
To understand the effect of friction, Vella et al. (2009) then performed experiments by simply
shifting the one end of the strip vertically, then moving it rapidly downwards and keeping this
end fixed afterwards, while the rest is resting on the frictional substrate. In this manner, a ruck is
123
generated from the “fixed” end that moves towards the free end. The experimental results showed
that the slipping of the flat-lying section causes the ruck height to reduce and consequently a
ruck velocity to increase as the free end is gradually approached. A further characteristic of the
slipping ruck behaviour was identified by O’Keefe and Gooch (2015). When the elastic strip is
sufficiently long, the ruck was observed to reduce to a certain “residual” height with a constant
“residual” speed. This condition remains unchanged until reaching the free end.
Even though this buckling problem has been extensively studied, its physics is not completely
understood yet. In particular, there are two main issues; (1) why theory predicts an infinite
compressive force for the formation of the ruck and (2) what is the dissipation mechanism
in the motion of the ruck, the rolling or the sliding friction. The first paradox was recently
investigated by Lee et al. (2015), who suggested that the reason behind this unorthodox result
is the assumption of inextensibility. Relaxing this property, a finite value of the compressive
horizontal force can be derived. The second issue is still unclear. For instance, as we mention
above, Vella et al. (2009) and O’Keefe (2015) take only into account the effect of a sliding
friction, while Kolinski et al. (2009) believes that the rolling friction is dominant in the case of
an inclined substrate.
6.2.1 “Long” heavy elastica on a horizontal foundation
The main focus of the current study is the response of a long heavy elastic strip on a horizontal
frictional foundation under the action of compressive loads. An analytical description of the
static problem is first presented. The dynamics of the localized ruck is then analyzed. Based
on Vella et al. (2009); O’Keefe and Gooch (2015), the dissipation mechanism, arising from the
effect of friction, is also explained.
Static analysis
Let us consider a heavy elastica of length L resting on a horizontal rigid foundation under the
action of a pair of equal and opposite compressive forces. If the elastica is considered to be
“long” in the sense of Wang (1984b), its buckling is accompanied by a formation of a ruck of
length Lr  L, while the rest remains in contact with the rigid foundation, as shown in Figure
6.2 a).
124
a)
rL
/ 2fL / 2fL/ 2∆
Y X
Ruck
/ 2∆
X
( )rX L L= −∆
D
Y
rL
( )SΘ
b)
QQ
QQ
Figure 6.2: Problem description a) heavy elastica of length L, b) ruck formation of length Lr
The potential energy of the “heavy” elastica can be expressed as follows
Π =
∫ L
0
{
1
2
EI
(
dΘ(S)
dS
)2
+ wY (S) +Q (cos Θ(S)− 1)
}
dS (6.3)
where E is the Young modulus, I is the moment of inertia, Θ(S) is the inclination angle, w is the
weight per length of the elastica and Q is the applied horizontal force. The energy functional
involves the bending energy, the gravitational potential energy and the external work of the
applied force Q, respectively.
By vanishing the first variation of the functional (6.3), the governing equation of the elastica
is obtained EId2Θ/dS2 + Q sin Θ − w (L/2− S) cos Θ = 0. The elastica is then divided into
the buckled shape, also called ruck, with length Lr  L and the flat-lying segment of length
Lf = L− Lr . The solution of the unstressed section is trivial, thus the problem is reduced to
the analysis of the ruck. In particular, due to symmetry the left-half part of the ruck is only
analyzed, as shown in Figure 6.2 b).
Scaling of the problem follows from adopting lg = (EI/w)
1/3 as length scale, fg = EI/`2g =(
EIw2
)1/3 as force scale and mg = EI/`g = ([EI]2 w)1/3 as moment scale (i.e., weight based
scaling law (WSL)) Wang (1981b)
s =
S
lg
, x =
X
lg
, ` =
L
lg
, y =
Y
lg
, d =
D
lg
, δ =
∆
lg
,Q = Q
fg
which leads to the governing equation of the ruck
d2θ
ds2
+Q sin θ −
(
`r
2
− s
)
cos θ = 0 (6.4)
with boundary conditions θ(0) = θ′(0) = θ (`r/2) = 0.
If we assume small displacements and apply the boundary conditions, we get tan
(
`r
√Q/2) =
`r
√Q/2 with minimum root `r
√Q/2 = α where α = 4.49341.
125
Nevertheless, the product form of `r
√Q makes the explicit solution of the parameters (e.g.,
the end-shortening δ) a difficult task. Instead, an intermediate scaling is imposed to overcome
this complexity. Starting from the original equation of the ruck and assuming a length scale
lf = (EI/Q)
1/2 (i.e., force based scaling law (FSL)), the equation of the ruck becomes
d2θ
ds¯2
+ sin θ − ε
( ¯`
r
2
− s¯
)
cos θ = 0 (6.5)
where ¯`r = Lr/lf is the length of the ruck. For small displacements θ  1, which also implies
sin θ = θ − θ3/6 and cos θ = 1 − θ2/2, (6.5), shows that the parameter ε = w (EI/Q3)1/2
is necessarily small. Based on this assumption, an asymptotic analysis can be applied with
expansion θ = εϑ(s¯) +O(ε2), which is substituted into (6.5) to get the first-order term
O (ε) : d
2ϑ
ds2
+ ϑ =
( ¯`
r
2
− s¯
)
(6.6)
with boundary conditions ϑ(0) = ϑ′(0) = ϑ
(
¯`
r/2
)
= 0 . Its solution is ϑ(s¯) = sin s¯− ¯`r cos ¯s/2+
¯`
r/2− s¯, where ¯`r = 2α and α = 4.49341.
The cartesian parametrization of the elastica x¯(s¯) = X(S)/lf , y¯(s¯) = Y (S)/lf is derived by
applying the inextensibility-unshearability conditions
x¯(s¯) =
∫ s¯
0
cos θdξ, y¯(s¯) =
∫ s¯
0
sin θdξ (6.7)
Taking into account the asymptotic expansion of the inclination θ, consistent expansions of the
cartesian components are assumed x¯(s¯) = s¯ − ε2ζ (s¯), y¯(s¯) = εη(s¯) with dη(s¯)/ds¯ = ϑ(s¯) and
dζ(s¯)/ds¯ = ϑ2(s¯)/2. Considering that x¯(0) = η(0) = 0 and y¯(0) = ζ (0) = 0, we get
η (s¯) = 1− cos s¯− s¯ sin s¯+ s¯
¯`
r
2
− s¯
2
2
(6.8)
ζ(s¯) = −1
2
{ ¯`
r
4
cos 2s¯+
1
4
( ¯`2
r
4
− 1
)
sin 2s¯+ ¯`r s¯ sin s¯+ 2s¯ cos s¯
−2
( ¯`2
r
4
+ 1
)
sin s¯+ s¯2
(
s¯
3
−
¯`
r
2
)
+
1
2
(
3¯`2r
4
+ 1
)
s¯−
¯`
r
4
}
(6.9)
Using the values ζ
(
¯`
r/2
)
= 37.80215 and η
(
¯`
r/2
)
= 15.6987, we can evaluate the involved
parameters, namely the maximum vertical component d, the end-shortening δ, the length of
the deformed segment `r and the maximum bending moment m. In particular, we combine
the two distinct scaling laws to express all terms with respect to the horizontal force Q, or the
end-shortening δ = 2ε2ζ
(
¯`
r/2
)
lf/lg = 75.6043Q−7/2Q = 3.441δ2/7
126
d = Y (Lr/2)l
−1
g = εη
(
¯`
r/2
)
lf l
−1
g = 15.6987Q−2, or d = 1.326δ4/7 (6.10)
`r = Lrl
−1
g =
¯`
rlf l
−1
g = 2αQ−1/2, or `r = 4.844δ1/7 (6.11)
m = EIεm−1g l
−1
f
dϑ
(
¯`
r/2
)
ds¯
= −5.6033Q−1, or m = −1.6284δ2/7 (6.12)
The equations (6.10)-(6.12) are then used to calculate the energy terms included in (6.3). Taking
into account the energy scale lgfg, the bending energy of the ruck is given by
Ub = 2
lgfg
[∫ Lr/2
0
1
2
EI
(
dθ(S)
dS
)2
dS
]
' α
∫ ¯`
r/2
0
(
dϑ(s¯)
ds¯
)2
ds¯ (6.13)
where α = EIε2/lf lgfg = 3.441−5/2δ5/7 and
∫ ¯`
r/2
0
(dϑ(s¯)/ds¯)
2
ds¯ = 45.065, leading to Ub =
2.06δ5/7. The gravitational potential energy is similarly obtained
Uw = 2
lgfg
[∫ Lr/2
0
wY (S)dS
]
' 2β
∫ ¯`
r/2
0
η (s¯) ds¯ (6.14)
where β = wl2fε/lgfg = 3.441
−5/2δ5/7 and
∫ ¯`
r/2
0
η (s¯) ds¯ = 30.242, which gives Uw = 2.75δ5/7.
The external work of the horizontal force can be approximated by substituting Q = 3.441δ−2/7
and evaluating the integral with respect to the change of the end-shorteningWR =
∫
3.441δ−2/7dδ =
4.82δ5/7 (O’Keefe and Gooch, 2015). We can easily verify that WR = Ub + Uw.
Dynamics
When the motion of the “dynamic” ruck is analyzed, the translational and rotational kinetic
energies should be also included, which in dimensionless form (i.e, WSL scaling law) become
Ek = 1
2
∫ `r
0
(
x˙2 + y˙2
)
ds+
1
2
α
∫ `r
0
θ˙2ds (6.15)
where α = glg/(E/ρ) and tg = (lg/g)
1/2 is the time scale. However the velocity of the elastica is
much slower than the speed of the sound α 1. This implies that the rotational kinetic energy
is negliglible when compared to the translational kinetic energy and therefore it is omitted
(Dichmann et al., 1996; Dichmann and Maddocks, 2000).
By vanishing the first variation of the total energy functional (6.13-6.15), the governing
equations of the ruck are given by
∂R¯x
∂s
=
∂2x
∂t2
,
∂R¯y
∂s
=
∂2y
∂t2
(6.16)
127
∂2θ
∂s2
= R¯x sin θ − R¯y cos θ (6.17)
where (6.16), (6.17) are the force and moment balances and R¯x, R¯y are the dimensionless
horizontal and vertical components, respectively.
Nevertheless, Vella et al. (2009) deduced by experiments that the ruck moves at a constant
speed and its shape remains almost identical, except at the early and final stages of its dynamical
motion. Hence, Vella et al. (2009) proposed to transform the PDE system (6.16-6.17) to an
ODE, assuming steady motion. More specifically, Vella et al. (2009) introduced a new variable
η ≡ s − vrt where vr is the constant speed of the ruck which leads to R¯x(s, t) ≡ −
(Q+ v2r),
R¯y(s, t) ≡
(
¯`
r/2− η
)
and a single equation for θ(s, t) = θ¯(η):
∂2θ¯
∂η2
= − (Q+ v2r) sin θ¯ + (`r2 − η
)
cos θ¯ (6.18)
which is the same with the governing equation of the “static” ruck (6.4) if we replace the “static”
horizontal forceQ by the “effective” termQ+v2r and modify the boundary conditions by imposing
the change of variables. This similarity also implies that the shapes of a steadily moving ruck
and a static ruck (see Section 6.2.1) are identical for a given δ if small displacements are assumed.
Inspired by Vella et al. (2009), O’Keefe and Gooch (2015) suggested a simple procedure to
evaluate the kinetic energy of the ruck. The ruck can be divided into n identical elements of
length `e = `r/n, which are moving with constant speed vr. For given length `e and speed vr,
the time step dt is then derived by dt = `e/vr. In this manner, the horizontal displacement of
the ruck at the time step dt is exactly equal to the length of the element `e.
To clarify the procedure, we assume that the ruck is divided into 8 elements (see Figure 6.3).
Since the motion is under constant speed, the number and the length of the elements remain
unchanged. At time t the first and the last nodes [1], [9] which are in contact with the rigid
foundation are replaced by the nodes [2], [10] at time t+ dt.
Let us now follow the path of a specific node for a steadily moving ruck. For instance, the
displacement of the node [7] is denoted as u[7] in Figure 6.3. Considering that the horizontal
motion is prescribed by the constant speed vr and the horizontal displacement is equal to the
length of the element `e = `7, we can approximately evaluate the displacement of the node [7]
u2[7] = 2`
2
7 (1− cos θ7) by applying the law of cosines for the corresponding triangle, as shown in
Figure 6.3.
Hence, in general if the set of nodes is It = {[1] , ..., [n+ 1]} for n elements at time t, the
new set of nodes at time t+ dt becomes It+dt = {[2] , ..., [n+ 2]}. In addition, the displacement
of a node [i] can be approximated as u2[i] = 2`
2
i (1− cos θi). If we assume that the velocity of a
node [i] is given by v[i] ≈ u[i]/dt, which defines the speed of the element i, the kinetic energy
128
can be approximated as follows
Ek = 1
2
n∑
i=1
v2[i] =
1
2
n∑
i=1
(u[i]
dt
)2
=
=
(
`e
dt
)2 n∑
i=1
(1− cos θi) = v2rδ (6.19)
where for the first node u[1] = 0. Consequently the total energy of the ruck can be expressed in
terms of the end-shortening and the speed of the ruck as follows
E = Ub + Uw + Ek = 4.82δ5/7 + v2rδ (6.20)
Time step: t+dt
[7]u
10
9
8
7
6
5
4
3
2 9
8
7
6
5
4
3
2
1
7
Time step: t
`e
`7 = `e
Figure 6.3: Steady motion
Effect of friction
The frictional effect between the elastica and the rigid foundation on the motion of the ruck is
investigated in the following Section. We consider that the left end of the elastica is fixed, while
the right end is free to move. Following the experiments we can define the initial condition of the
ruck by predefing its shape based on a given end-shortening δ and assigning an initial “steady”
velocity of the ruck vsr .
129
F y
dx
x
d
rv
`f
`f
`r/2`r/2
(`r  )/2
R¯xR¯x
Figure 6.4: Problem description
Let us now define a system of coordinates (x, y) with origin at the right extremity of the
elastica in its unstressed state, as shown in Figure 6.4. We denote as xd the coordinate of the
x-position of the ruck of the maximum height d and F the frictional force between the elastica
and its foundation, which is derived by integration of the corresponding length of elastica (Lee
et al., 2015).
Based on experimental observations (O’Keefe and Gooch, 2015; Vella et al., 2009), the ruck
is initially moving with constant velocity vr and its shape remains unchanged until the onset of
slipping. In particular, we can assume that the shape of the traveling ruck is identical to the
static case of the ruck for a given end-shortening δ, while the “dynamic” horizontal force can be
approximated as follows
R¯x = Q− v2r = 3.441δ−2/7 − v2r (6.21)
where vr is the speed of the ruck. More details for the derivation of (6.21) can be found in
Sections (6.2.1-6.2.1).
On the other hand, the friction force can be approximated as follows
F =
(
`f +
`r
2
)
µ =
(
−xd − δ
2
)
µ
where µ is the friction coefficient. If the friction force F is larger than the applied horizontal force
F > R¯x, the flat-lying segments does not move and the end-shortening remains unchanged. At
this stage, the shape of the ruck does not change and its speed vr = vsr . This condition is called
sticking behavior. The slipping process initiates when the friction force between the flat-lying
segment and the rigid foundation becomes lower than the “dynamic” horizontal force.
Past the “slipping point”, the flat-lying segment starts accelerating. This leads to a gradual
reduction of the end-shortening δ, which affects the “static” horizontal force Q = 3.441δ−2/7 and
the shape of the ruck. More specifically, the height of the ruck starts decreasing, while the ruck
130
is moving with higher speed (Vella et al., 2009). The latter effect can be explained as follows;
since a ruck of lower height d has less potential energy and requires less kinetic energy to travel
at its current velocity, the excess energy is transferred to the kinetic energy of the “reduced”
ruck, which starts accelerating (O’Keefe, 2015). Eventhough the velocity of the ruck increases
and a steady motion is not present, we can assume that (6.21) continues to be valid with updated
values of horizontal force and speed (O’Keefe, 2015). This approximation is considered to be
consistent with experimental results, which show that the shape of the ruck remains almost
symmetrical, implying uniform motion of the reduced ruck.
Nevertheless, this gradual increase of the velocity of the ruck may lead to the vanishing value
of the horizontal force R¯x when vr = (3.441)1/2 δ−1/7 . This is called “self-sustaining” velocity
vs. Past this point, the “dynamic” horizontal force R¯x becomes tensile. Eventhough the force
R¯x opposes the motion of the flat-lying segment, it continues moving because of its inertia with
a resultant force equal to the sum of the “dynamic” horizontal and the friction forces. This
causes the flat-lying section to reduce in velocity and cease slipping. At this stage, the ruck
attains a constant height, which is called residual condition with a “residual” velocity vrr . This
final condition is only met when the length of the heavy elastica is sufficiently long.
The whole slipping process is presented in Figure 6.5. A simple algorithm following this
approximate mathematical process can be found in O’Keefe (2015).
s
rv r
rv
rd
Onset of slipping
sd
Constant
r
rv
sv
r
rvHeight
R¯xR¯xR¯x R¯x
Figure 6.5: Slipping process
6.2.2 “Short” heavy elastica on a horizontal foundation
When the heavy elastica is short in the sense of Wang 1984b, the elastica is not in contact with
the horizontal wall except at its extremities. The governing equation of the elastica is given by
EId2θ/dS2+Q sin θ−W (L/2− S) cos θ = 0 with boundary conditions θ(0) = 0 and θ(L/2) = 0.
Scaling of this problem follows from adopting lb = L as length scale and fb = EI/L2 as force
scale (i.e., bending based scaling law (BSL)), which leads to
d2θ
ds2
+Q sin θ − w
(
1
2
− s
)
cos θ = 0 (6.22)
131
Assuming small displacements θ  1 we can deduce from (6.22) that the dimensionless weight
w is necessarily small. Considering w = ε3, an asymptotic analysis is applied with expansions
for both the axial load Q = Qo + εQ1 + ε2Q2 and the inclination angle θ = εϑo + ε2ϑ1 + ε3ϑ2,
respectively. If we also expand sin θ = θ− θ3/6 and cos θ = 1− θ2/2, (6.22) is reformulated with
boundary conditions θ(0) = 0 and θ(1/2) = 0. Accordingly, the first three order terms can be
obtained
O(ε) : d
2ϑo
ds2
+Qoϑo = 0 (6.23)
O(ε2) : d
2ϑ1
ds2
+Qoϑ1 = −Q1ϑo (6.24)
O(ε3) : d
2ϑ2
ds2
+Qoϑ2 = −Q1ϑo −Q1ϑ1 + Qoϑ
3
o
6
+
(
1
2
− s
)
(6.25)
The first order solution ϑo(s) is given by ϑo(s) = Θ sin
(√Qos) with Qo = 4pi2 and Θ the
inclination angle at the inflection point at s = 1/4. The next order solution ϑ1(s) can be
expressed as follows
ϑ1(s) = − Q1Θ¯
2
√Qo
s sin
(√
Qos
)
(6.26)
which satisfies the boundary conditions only when Q1 = 0. Hence, ϑ1(s) = 0. Lastly, the terms
ϑ2(s) and Q2 are given by
ϑ2(s) =
1− 2s
2Qo +
1
4Qo
(QoΘ3
8
−Q2Θ
)
(1− 2s) cos
√
Qos+ Θ
3
192
sin 3
√
Qos (6.27)
and Q2 = QoΘ2/8+Q−1/2o Θ−1. For the critical buckling load, we seek the minimum of equation
dQ2/dΘ = 0 that leads to Q2 = 3 · 2−5/3. Hence, the final expression of the horizontal force
becomes Q = 4pi2 + 3 · 2−5/3w2/3.
The vertical and horizontal coordinates are derived by the inextensibility constraints as
dx/ds = cos θ, dy/ds = sin θ. Taking into account the expansions of the trigonometric identities,
we deduce that their asymptotic expansions are given by x ≈ xo + ε2x1 and y ≈ εyo where
xo(s) = s, x1(s) = Θ2
(
s− sin 2√Qos/2
√Qo
)
/4 and yo(s) = Θ
(
1− sin√Qos
)
/
√Qo. The
vertical displacement at s(1/2), the horizontal coordinate x(1) and the end-shortening δ are
then approximated as y(1/2) = Θpi−1w1/3, x(1) = 1 − (Θ2/4)w2/3 and δ = (Θ2/4)w2/3,
respectively. In a similar manner, the bending moment dθ/ds can be easily evaluated.
6.2.3 Optimal control formulation
The buckling response of a heavy elastica constrained by a rigid horizontal, or inclined foundation
is analyzed by applying the optimal control method. Utilizing a bending based scaling law
(BSL)), the nonlinear optimal control problem P for clamped-clamped boundary conditions can
132
be expressed as follows
Minimize J = 12
∫ 1
0
u2(s)ds+
∫ 1
0
w [y(s) cosβ + x(s) sinβ] ds
subject to
y′(s) = sin θ(s)
x′(s) = cos θ(s)
θ′(s) = u(s)
x(0) = y(0) = y(1) = θ(0) = θ(1) = 0
x(1) = 1− δ
−y(s) ≤ 0
(6.28)
where β is the inclination of the foundation, θ is the inclination angle with respect to the x-axis
and x, y are the cartesian coordinates. The pure-state constraints is g(y) = y with first and
second time derivatives given by g1(θ) = − sin θ and g2(y, u) = −u cos θ. This implies that the
pure state constraint is of second-order. In the case of a closed contact, we have g1u(y) = 0 and
g2u(y, u) = − cos θ.
Based on the optimal control problem P, the simple Hamiltonian is given by
H(λ,X,u) =
1
2
u2(s) + w (y(s) cosβ + x(s) sinβ) + λx cos θ + λy sin θ + λθu (6.29)
where λ = {λy, λθ} ∈ R2 are the covectors of the dynamics f = {sin θ, u} ∈ R2. The set of the
boundary conditions is described by the scalar function
Φ = ν1y(0) + ν2y(1) + ν3θ(0) + ν4θ(1) + ν5x(0) + ν6x(1) (6.30)
for clamped-clamped boundary conditions.
The Lagrangian of the Hamiltonian, or the augmented Hamiltonian additionally includes the
pure-state constraint, namely H¯ (µ,λ,X,u) = H(λ,X,u) + µ (−y(s)), where the complemen-
tarity condition µ⊥g is satisfied. The minimization of the augmented Hamiltonian with respect
to the control variable is then expressed as
∂H¯
∂u
=
∂H
∂u
= 0⇒ u+ λθ = 0 (6.31)
Eliminating the control variable in (6.30), the lower Hamiltonian is derived
H = w (y(s) cosβ + x(s) sinβ) + λy sin θ − 1
2
λ2θ + µ (−y(s))
which is constant since it does not depend explicitly on s.
133
The adjoint equations are given by
−λ′x =
∂H¯
∂x
= w sinβ (6.32)
−λ′y =
∂H¯
∂y
=
{
w cosβ if unconstrained
w cosβ − µ if y(τc) = 0
(6.33)
−λ′θ =
∂H¯
∂θ
= −λx sin θ + λy cos θ (6.34)
with initial and terminal transversality conditions λx(0) = −ν5, λy(0) = −ν1, λθ(0) = −ν3 and
λy(1) = ν2, λθ(1) = ν4, λx(1) = ν6. Based on (6.32), the horizontal force is a linear function
λx(s) = −ν5 − w sinβ and satisfies the terminal condition λx(1) = ν6. A similar expression for
the shear force is derived λy(s) = −ν1 − w cosβs if the unconstrained case is investigated. In
this particular case, if we consider (6.34) and make use of (6.31), we get
θ′′ = −λx sin θ + λy cos θ (6.35)
with boundary conditions θ(0) = θ(1) = 0 for a clamped elastica.
Let us now assume that we have a single contact point ζc, where y(ζc) = 0 is satisfied.
In this case, the shear force is λy(s) = −ν1 − w cosβs, for s ∈ [0, ζc), λy (ζ−c ) − λy (ζ+c ) = η
(i.e., η1 =
∫ ζc+ε
ζc−ε µ1ds) at the contact point ζc which leads to λy(s) = −ν1 − w cosβs − η1 =
ν2, for s ∈ (ζc, 1]. On the other hand, λθ (τ−c )−λθ (τ+c ) = 0, because the pure state constraint is
independent of θ. This also implies that λθ = −u = −θ′ is continuous while its spatial derivative
−λ′θ = −u′ is discontinuous at the contact points (6.34) since it depends on the discontinuous
shear force λy(s). In the case of a continuous segment, the only difference is the vanishing
control variable u(ζ) = 0 for ζ ∈ [ζen, ζex], where ζen and ζex denote the entry and exit points
of a continuous contact.
Results
Based on the optimal control formulation, we investigate the post-buckling behavior of a heavy
elastica with different values of self-weight w and different foundation gradients β. To establish
the correctness of the model we first analyze the well-known post-buckling response of a heavy
elastica on a horizontal foundation β = 0 (Wang, 1984b; Domokos et al., 2003; Santillan et al.,
2006; Chen and Lin, 2013).
In this class of problem, displacement-controlled conditions are assumed. Load-controlled
conditions cannot be applied, because the load initially decreases until a minimum value and
then starts increasing, as shown in Figure 6.6 a). The bifurcation graph illustrates the evolution
of the “buckling” load Q, derived indirectly by the adjoint variable λx, with respect to the
134
end-shortening δ for distinct values of self-weight w. From Figure 6.6 a), the branch of zero
self-weight w can be considered as an “envelope” since all other branches of non-zero values of
self-weight w approach asymptotically this line. Figure 6.6 b) indicates that the height of the
localized ruck attains a maximum value h ∼ 0.4 for a weightless elastica, which slightly decreases
as the self-weight increases.
Let us now investigate the transition behavior from a symmetric to an asymmetric configu-
ration, also known as the symmetry breaking bifurcation, which was first observed by Domokos
et al. (2003). In Figure 6.7, we indicate this transition condition by a solid line (above the solid
line, asymmetric configurations are obtained). The graph also shows that the transition occurs
at a lower value of end-shortening δ as the self-weight of the elastica increases. Based on this
analysis, Domokos et al.’ conclusion is confirmed, which states that the symmetry-breaking
bifurcation occurs before a self-contact is established. Nevertheless, this result is valid only if
the self-weight is w ≥ 130. For a self-weight w < 130, the buckling behavior of the elastica
is similar to a weightless elastica, where only a self-contact can occur (no development of an
asymmetric configuration).
The corresponding bifurcation diagrams for different foundation gradients β are similar to
Figure 6.6. A representative example (i.e., self-weight w = 200 and foundation gradient β = pi/3)
is only shown in Figure 6.8.
135
0 50 100 150 200
0.0
0.2
0.4
0.6
0.8
1.0
w=0
w=100
w=50
δ
w=10
Q
0 . 0 0 . 2 0 . 4 0 . 6 0 . 80 . 0
0 . 1
0 . 2
0 . 3
0 . 4
0 . 5
w = 1 0 0
h h
eigh
t
δ
w = 0
Figure 6.6: a) Bifurcation diagram for a clamped-clamped elastica for different values of self-
weight w with one-sided horizontal foundation and b) height of the formed ruck for two distinct
values of self-weight w = 0, 100
136
0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0
0 . 4
0 . 5
0 . 6
0 . 7
0 . 8
0 . 9
δ
w
Figure 6.7: ”Symmetry-breaking bifurcation” for a horizontal foundation
δ= 0 . 0 3δ= 0 . 2
δ= 0 . 5
δ= 0 . 7 β
Figure 6.8: Deformation shapes for different values of end-shortening δ for self-weight w = 200
and foundation slope β = pi/3
6.3 Heavy elastica constrained by rigid horizontal walls
In the following Section, the buckling problem of a heavy elastic sheet with one-sided foundation
is extended to the stability analysis of a heavy elastica constrained by two rigid symmetrically,
or asymmetrically located walls, as shown in Figure 6.9.
137
Applying a bending-based scaling law (BSL), a planar unit-length elastica with a self-weight
w is considered. The left end of the elastica is fixed, while the right end is constrained to move
horizontally. The extremities of the elastica are either clamped at zero inclination or pinned,
i.e., free to rotate.
We define a system of coordinates (x, y) with origin at the left end of the elastica and with
the x-axis pointing towards the elastica right end. To describe a deformation shape of the
elastica, we introduce the arc-length coordinate s with its origin s = 0 at x = 0 and y = 0,
and the local inclination angle θ(s) of the elastica with respect to the x-axis. Applying the
inextensibility-unshearability constraints, its deformed configuration x(s), y(s) is fully defined
by the function θ(s) and the fixed end at s = 0. The unilateral constraints associated with the
presence of the two walls is represented by the inequality condition −cl ≤ y(s) ≤ cu where cl, cu
are the clearances of the lower and upper walls, respectively.
The initial configuration of the elastica depends on its self-weight w and the clearance cl
of the lower wall. In particular, if the clearance cl is zero, the elastica rests on the lower wall
and its initial buckling behavior is identical to the case of a “long” heavy elastica with one-sided
foundation, see Section 6.2.1. On the other hand, if the lower wall is located at y = −cl with
cl 6= 0, two distinct cases are obtained; the elastica is in static equilibrium due to its self-weight
either without, or with a point/continuous contact with the lower wall. In the former case,
the initial response of the heavy elastica is similar to the post-buckling behavior of a weightless
elastica with initial imperfection, while in the latter case the effect of the weight is dominant at
the initial state.
We seek to determine the (post)-buckling behavior of the elastica in response to the ap-
plication of a displacement δ = 1 − x(1) at the right end that tends to shorten the distance
between the two ends. Let us ignore the tensile load, present at the initial state of the elastica.
In other words, we restrict our attention to the evolution of the compressive load Q. Hence,
for a vanishing compressive load (starting point of our study), the end-shortening δ is always
non-zero when the self-weight is non-negligible.
The same problem can be also slightly modified in order to solve the constrained buckling
problem of a heavy elastica of variable length, which is the “insertion problem”. In this particular
problem, the distance between the supports remains constant, while the length 1 + δ¯ of the
inserted elastica gradually increases. In its initial state, we assume that the length of the elastica
L is identical to the horizontal distance between the supports, meaning that a tensile load is
present when the self-weight is non-negligible. Hence, the same bending scaling law (BSL) can
be also applied here. If we ignore the tensile force, obtained at the initial state, the presence of
a compressive load Q implies that the inserted length is always 1 + δ¯ > 1 for a non-negligible
self-weight w of the elastica. The evolution of the compressive load Q with respect to the change
of the inserted length δ¯ is our main concern below.
Using optimal control, the formulation of the classical problem is first presented. A detailed
138
analysis for the insertion problem is omitted due to its similarity. Then we illustrate some
representative results for the buckling problem of the elastica constrained by two rigid walls
with clearances cl = 0 and cr > 0 ( see Figure 6.9a)). The same analysis is then performed
for symmetrically located walls (see Figure 6.9b)). To illustrate the effect of the weight in the
buckling behavior of the constrained elastica, the above examples are calculated for both a
weightless and a heavy elastica. The effects of the boundary conditions and the clearances of
the walls are also investigated in combination with the presence of the self-weight w.
w(s)
w(s)
P4P3
P2
δ
P1
w(s)
y x
θ(s)
Q
w(s)
w(s)
P3
P2
δ
P1
w(s)
y
x
θ(s)
Q
Figure 6.9: Buckling problem of elastica constrained by a) asymmetrically and b) symmetrically
located rigid walls
6.3.1 Optimal control formulation of constrained heavy elastica
The constrained buckling problem of a heavy elastica is solved by applying the optimal con-
trol method. In particular, in Section 6.3.1, the classical problem is first demonstrated for
clamped-clamped boundary conditions. The framework is generic in order to demonstrate the
theoretical aspects of the problem. The optimal control formulation for the insertion problem
is also presented in Section 6.3.1.
Classical problem
A constant length elastica is constrained by two rigid horizontal walls. A bending-based length
scale (BSL) is applied with length scale, the length of the elastica. Accordingly, the nonlinear
optimal control problem P for displacement-controlled conditions can be expressed as
139

Minimize J = 12
∫ 1
0
u2(s)ds+
∫ 1
0
wy(s)ds
subject to
y′(s) = sin θ(s)
x′(s) = cos θ(s)
θ′(s) = u(s)
x(0) = y(0) = y(1) = θ(0) = θ(1) = 0
x(1) = 1− δ
y(s)− cu ≤ 0
−y(s) + cl ≤ 0
(6.36)
where θ is the inclination angle with respect to the x-axis, x, y are the cartesian coordinates,
δ is the end-shortening and cu, cl are the clearances of the upper and lower horizontal walls,
respectively. Clamped-clamped boundary conditions are assumed in (6.36). The pure-state
constraints read
g(y) =
{
y − cu
−y + cl
}
:∈ RN2 (6.37)
The first and second time derivatives of (6.37) are given by
g1(θ) =
{
sin θ
− sin θ
}
g2(y, u) =
{
u cos θ
−u cos θ
}
(6.38)
where g1u(y) = 0 and g2u(y, u) = {cos θ,− cos θ}> at contact points, or continuous contacts,
which implies that the pure state constraints (6.37) are of second-order.
The simple Hamiltonian is given by
H(λ,X,u) =
1
2
u2(s) + wy(s) + λx cos θ + λy sin θ + λθu (6.39)
where λ = {λy, λθ} ∈ R2 are the covectors of the dynamics f = {sin θ, u} ∈ R2. The boundary
conditions are described by the following scalar function
Φ = ν1y(0) + ν2y(1) + ν3θ(0) + ν4θ(1) + ν5x(0) + ν6x(1) (6.40)
for clamped-clamped boundary conditions.
The augmented Hamiltonian also includes the inequality constraints
H¯ (µ,λ,X,u) = H(λ,X,u) + µ1 (y(s)− cu) + µ2 (−y(s)− cl) (6.41)
where the complementarity condition µ⊥g should be satisfied.
The minimization of the augmented Hamiltonian with respect to the control variable leads
140
to u = −λθ. Eliminating the control variable in (6.40), the lower Hamiltonian is then derived
H = wy(s) + λy sin θ − 1
2
λ2θ + µ1 (y(s)− cu) + µ2 (−y(s) + cl)
which remains constant for every s ∈ [0, 1] as it does not depend explicitly on s. Accordingly,
the adjoint equations are obtained
−λ′x =
∂H¯
∂x
= 0 (6.42)
−λ′y =
∂H¯
∂y
=

w if unconstrained
w + µ1 if y(τc) = cu
w − µ2 if y(τc) = −cl
(6.43)
−λ′θ =
∂H¯
∂θ
= −λx sin θ + λy cos θ (6.44)
with initial and terminal transversality conditions λx(0) = −ν5, λy(0) = −ν1, λθ(0) = −ν3 and
λy(1) = ν2, λθ(1) = ν4, λx(1) = ν6. Based on (6.42), the horizontal force is uniform, meaning
that λx(s) = −ν5 = ν6 and it is equal to the applied load Q.
For the unconstrained case the shear force is a linear function λy(s) = −ν1 − ws, which
satisfies at the right end λy(1) = −ν1−w = v2. If we consider (6.44) and make use of u = −λθ,
we get
θ′′ = −λx sin θ + λy cos θ (6.45)
with boundary conditions θ(0) = θ(1) = 0.
Let us now assume that we have a single contact point ζc (i.e., y(ζc) = cu). In this case, the
shear force is λy(s) = −ν1 − ws, for s ∈ [0, ζc), λy (ζ−c ) − λy (ζ+c ) = η1 (i.e., η1 =
∫ ζc+ε
ζc−ε µ1ds)
at the contact point ζc which leads to λy(s) = −ν1 − ws − η1 = ν2, for s ∈ (ζc, 1]. On the
other hand, λθ (ζ−c )− λθ (ζ+c ) = 0 as the pure state constraint is independent of θ. This implies
that the curvature, or the bending momentλθ = −u = −θ′ is continuous. However −λ′θ = −u′
is discontinuous at the contact points (6.44) since it depends on the shear force λy(s). In the
case of a continuous segment, the only difference is the vanishing control variable u(ζ) = 0 for
ζ ∈ [ζen, ζex].
The numerical procedure of the constrained optimization problem is not described here. A
detailed description can be found in Chapter 4. Nevertheless, the effect of the loading conditions
is shortly explained. For displacement-controlled conditions, a predefined value of the end-
shortening δ is assigned, while the applied force Q is derived indirectly by the covector λx. The
same problem can be also formulated for force-controlled conditions if we include the external
work of an imposed applied load Q in the energy functional J . In this case, the end-shortening
δ becomes the unknown variable, which is computed by the solution of the problem. However
141
force-controlled conditions are not considered here.
Insertion problem
The nonlinear optimal control problem P¯ of the insertion problem becomes
Minimize J = 12
∫ 1+δ¯
0
u2(s¯)ds¯+
∫ 1+δ¯
0
wy(s)ds¯
subject to
y′(s) = sin θ(s)
x′(s) = cos θ(s)
θ′(s) = u(s)
x(0) = y(0) = y(1 + δ¯) = θ(0) = θ(1 + δ¯)) = 0
x(1 + δ¯)) = 1
y − cu ≤ 0
−y − cl ≤ 0
(6.46)
for clamped-clamped boundary conditions. Further analysis is not included since it is identical
to the “classical problem”, see Section 6.3.1.
6.3.2 Results
The buckling response of a heavy elastica with a self-weight w constrained by two rigid horizontal
walls is analyzed by applying an optimal control method. Several effects, such as the clearance,
the boundary conditions and the value of the self-weight are analyzed.
Asymmetric walls
The post-buckling behavior of a constant length elastica constrained by two asymmetrically
located walls with clearances cl = 0 and cu = 0.1 is investigated. To understand the effect of
the self-weight, two distinct values are applied, namely w = 0 and w = 100, as shown in Figures
6.10 and 6.11.
The initial response of the constrained weightless elastica is not comparable with the case of
the heavy elastica. In particular, the weightless elastica remains in its unstressed configuration
with δ = 0 on the rigid foundation until it reaches the first (symmetrical) buckling mode of
clamped-clamped elastica with applied force Q/pi2 = 4. A nearly straight line is then followed
until it comes in contact with the upper wall at the midpoint position at C2. Past C2, an abrupt
growth of the stiffness Q/δ is detected due to the remarkable increase of the horizontal force Q
when compared to the slight progression of the end-shortening δ. Accordingly C3 corresponds
to the vanishing of the moment at the discrete contact. This leads to the onset of the continuous
contact.
142
Even though the contact pattern 3 − 4 of a clamped-clamped elastica constrained by two
symmetrically located walls is unstable, this is not the case when the lower wall has a clearance
cl = 0. This implies that, the lower wall functions as an additional stabilization to the clamped
supports. The length of this continuous contact gradually increases until the initiation of a
secondary buckling at C4. Past C4, a free-standing fold is formed with configuration 4 − 5
(see (Pocheau and Roman, 2004; Roman and Pocheau, 1999)). The maximum height of this
segment progressively increases until a new discrete contact is detected at C5 with a symmetrical
configuration 5− 6. This new discrete contact turns gradually into a continuous contact at C6.
In a similar manner, a new secondary buckling occurs at C7, which is a new form of a free-
standing fold with shape 7−8. At C8, a new discrete contact occurs, leading to the symmetrical
configuration 8− 9.
The buckling behavior of the heavy elastica with self-weight w = 100 is entirely different
at the initial stage. The heavy elastica remains unstressed on the rigid horizontal wall and an
infinite horizontal force is required for the formation of a ruck, as explained in Section 6.2.1. The
height of this ruck gradually increases and the horizontal force initially decreases with increasing
the end-shortening. If the upper wall is not present, the horizontal force reaches a minimum
value and then starts increasing, while the height of the ruck reaches a plateau (maximum value).
If the value of the maximum height is lower than the clearance of the upper wall, a discrete
contact at the midpoint position is met, which is also the case in this example at C2. Comparing
with the weightless elastica, which is not in contact with the lower wall except at its end-points
(see shape 2 − 3 in Figure 6.10), a small part of the heavy elastica, the ruck, is unconstrained
(see shape 2−3 in Figure 6.11). Nevertheless, the length of the constrained ruck increases, while
a formation of a continuous contact is not possible. Instead, the flat-lying segment buckles as
the first buckling mode of a clamped-clamped elastica, leading to a second ruck at C3. This
leads to the asymmetrical shape 3− 4 with a softening response. When the new ruck comes in
contact with the upper wall, a symmetrical shape 4−5 appears at C4. Beyond C4, the sequence
of the equilibrium configurations is identical to the weightless elastica. However the values of
Rx, δ are dissimilar (see Table 6.1).
143
a) Weightless Elastica
Points δ Q/pi2
2 0.0251 4.05
3 0.033 18.05
4 0.047 33.96
5 0.105 16.92
6 0.139 62.65
7 0.171 81.16
8 0.266 41.71
b) Elastica with
self-weight w = 100
Points δ Q/pi2
2 0.035 12.54
3 0.055 39.82
4 0.106 19.10
5 0.131 54.62
6 0.182 85.55
7 0.269 46.66
Table 6.1: Points of evolution of the contact patterns for the bifurcation diagram for a con-
strained clamped-clamped elastica with self-weight w = 0, 100 and clearances cl = 0 and cu = 0.1
for the classical problem
144
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
0
20
40
60
80
100 9
8
7
6
5
4
3
2
δ
1
Q/
⇡
2
2 - 3
3 - 4
8 - 97 - 8
6 - 75 - 6
4 - 5
1 - 2
Figure 6.10: Bifurcation diagram for a constrained clamped-clamped elastica with self-weight
w = 0 and clearances cl = 0 and cu = 0.1 for the classical problem
145
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
0
20
40
60
80
100
δ
5
1
2
3
4
6
7
8
Q/
⇡
2
7 - 8
6 - 75 - 6
4 - 53 - 4
2 - 31 - 2
Figure 6.11: Bifurcation diagram for a constrained clamped-clamped elastica with self-weight
w = 100 and clearances cl = 0 and cu = 0.1 for the classical problem
The same problem is also solved as an insertion problem considering an elastica of variable
length 1 + δ¯ with constant distance between the supports x¯(1 + δ¯) = 1 (see Section 6.3.1). The
bifurcation diagrams of the applied force Q/pi2 with respect to the change of the length of the
elastica δ¯ for w = 0, 100 are shown in Figures 6.12 and 6.13, respectively. One main difference is
that the post-buckling behavior of the weightless elastica buckles α via a sub-critical bifurcation
Q/pi2 < 4 rather than a subcritical bifurcation Q/pi2 > 4, which is the case in the classical
146
problem. Nevertheless, the sequence of the contact patterns and the complete diagram are
similar to the case of the classical problem. The transition points, where the contact patterns
evolve are summarized in Table 6.2, while the corresponding contact patterns are shown in
Figures 6.12 and 6.13.
Points w = 0 w = 100
δ¯ Rx/pi2 Q/pi2 δ¯ Rx/pi2 Q/pi2
2 0.021 3.88 3.68 0.035 12.64 12.64
3 0.031 16.14 16.14 0.049 32.04 32.04
4 0.044 29.9 29.9 0.095 16.74 15.05
5 0.095 14.04 11.68 0.111 41.26 41.26
6 0.12 51.22 51.22 0.144 63.24 63.24
7 0.14 64.3 64.3 0.206 28.52 20.31
8 0.204 27.33 18.35 0.233 68.52 68.02
Table 6.2: Transition points of contact patterns for w = 0, 100 consistent with Figures 6.12 and
6.13
147
0 . 0 0 0 . 0 5 0 . 1 0 0 . 1 5 0 . 2 0 0 . 2 5
0
1 0
2 0
3 0
4 0
5 0
6 0
7 0
8 0
9 , 9 '
8
8 '
7 , 7 '
6 , 6 '
5
5 '
4 , 4 '
3 , 3 '
2 , 2 '


/pi2 ,

x/pi2
δ
1 , 1 '
2 - 3
3 - 4
8 - 97 - 8
6 - 75 - 6
4 - 5
1 - 2
Figure 6.12: Bifurcation diagram for a constrained clamped-clamped elastica with self-weight
w = 0 and clearances cl = 0 and cu = 0.1 for the insertion problem; (i) solid line denotes the
evolution of the internal horizontal force Rx/pi2 and (ii) dashed line denotes the evolution of the
applied load Q/pi2
148
0 . 0 0 0 . 0 5 0 . 1 0 0 . 1 5 0 . 2 0 0 . 2 5
0
1 0
2 0
3 0
4 0
5 0
6 0
7 0
8 0 1 , 1 '
8 , 8 '
7
7 '
6 , 6 '
5 , 5 '
4
4 '
3 , 3 '
2 , 2 '


/pi2 ,

x/pi2
δ
7 - 8
6 - 75 - 6
4 - 53 - 4
2 - 31 - 2
Figure 6.13: Bifurcation diagram for a constrained clamped-clamped elastica with self-weight
w = 100 and clearances cl = 0 and cu = 0.1 for the insertion problem; (i) solid line denotes the
evolution of the internal horizontal force Rx/pi2 and (ii) dashed line denotes the evolution of the
applied load Q/pi2
149
Symmetrical walls
The post-buckling behavior of the constrained clamped-clamped elastica constrained by sym-
metrically located walls with clearance cu = cl = 0.1 is investigated. Two disitinct values of
self-weight w = 0 and w = 100 are examined, as shown in Figure 6.14.
In the case of a weightless elastica, the equilibrium configurations with negative y coordinate
(see Figure 6.14) can be also replaced by the same contact patterns with positive sign without
affecting the bifurcation diagram. For instance, the branch 7 − 8 can be described either by
the shape 7′ − 8′, or 7′′ − 8′′. This symmetry is broken whenever the self-weight is w 6= 0. In
particular, the branches 7′ − 8′ and 7′′ − 8′′ for a heavy elastica with self-weight w = 100 are
entirely different responses (see Figure 6.14), while the branch 7 − 8 of the weightless elastica
lies approximately in the middle of them.
Let us now analyze the behavior of the weightless elastica in more detail. For low values of
end-shortening δ, the first buckling mode of a clamped-clamped elastica is obtained without any
contact. Due to symmetry the elastica comes in contact with the wall at the midpoint position
at C2. The position of the discrete contact remains unchanged until C4, which corresponds to
the vanishing of the moment at the discrete contact. Nevertheless, it does not continue to a
continuous contact, while a simultaneous change of the sign of the moments of both ends is also
not feasible. Hence, beyond point 4, the sign of the moment changes only at one end, leading
to the asymmetrical configuration 4 − 5 with one discrete contact. Along Branch 4 − 5, the
horizontal load decreases with increasing δ (i.e., softening branch) until it reaches C5. Past C5,
a symmetrical configuration 5 − 7 with two discrete contacts is obtained. At C7, the bending
moments at the discrete contacts vanish simultaneously for symmetrical reasons. Beyond this
point, a new softening branch 7−8 is detected until a third discrete contact occurs at C8. Lastly,
past this point we get a symmetrical configuration 8− 9 with increasing horizontal force as the
end-shortening further increases.
a) Weightless Elastica
Points δ Q/pi2
2 0.0251 4.051
4 0.0305 15.51
5 0.113 8.84
7 0.171 27.4
8 0.295 17.95
b) Elastica with w = 100
Points δ Q/pi2
3 0.0268 0
4 0.0519 36.7
5 0.11831 12.28
6 0.14 21.4
7′′ 0.19 27.33
8′′ 0.29 16.78
Table 6.3: Points of evolution of the contact patterns of the bifurcation diagram for a constrained
clamped-clamped elastica with self-weights w = 0, 100 and clearances cl = cu = 0.1
In the case of a heavy elastica with self-weight w = 100, the initial configuration of the
elastica is modified. More specifically, at C1, the weight effect is dominant and a symmetrical
150
configuration 3− 4 with a continuous contact is detected. Increasing the end-shortening δ, the
length of the continuous contact and the horizontal force Q gradually increases. At C4, an
asymmetrical configuration 4− 5 with a softening response is derived with one discrete contact.
When a second discrete contact occurs at C5, a symmetrical symmetrical configuration 5 − 6
is obtained. At C6, the discrete contact with the lower wall turns into a continuous contact.
This deformation shape remains until C7”, where a softening branch 7′′ − 8′′ is obtained. If we
further increase the end-shortening δ, a symmetrical configuration with three discrete contacts
is obtained starting at C8”. The branches 7′ − 8′ and 8′ − 9′ are also stable branches but their
minimized energy cost is higher than the corresponding branches 7′′−8′′ and 8′′−9′′, respectively.
Hence, in diplacement-controlled loading conditions, the branches 7′′ − 8′′ and 8′′ − 9′′ are the
most favorable.
151
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
0
5
10
15
20
25
30
35
40
6
9''
9'
w=0
3
9
8'
8''
8
7',7''7
4
5
5
4
2
δ
1
w=100
Q/
⇡
2
8 ' ' - 9 ' '8 ' - 9 '
7 ' ' - 8 ' '7 ' - 8 '
6 - 7 ' , 7 ' '5 - 6 , 7
4 - 53 - 4
2 - 31 - 2
Figure 6.14: Bifurcation diagram for a constrained clamped-clamped elastica with self-weights
w = 0, 100 and clearances cl = cu = 0.1
Let us now investigate the effect of the self-weight, keeping the same clearance cu = cl = 0.1
and assigning a self-weight w = 10. As shown in Figure 6.15, the sequence of the contact
patterns remains unchanged except at the initial stage. In particular, the effect of the self-
weight is not dominant, while a clear post-buckling behavior is also not possible. The initial
response is similar to a post-buckling behavior of a weightless elastica with initial imperfection.
152
The effect of the clearance of the symmetrically located walls is also examined. Keeping
a constant value of self-weight w = 10, a lower value of clearance cu = cl = 0.05 is assigned.
Comparing Figures 6.16 and 6.15, we can observe that the sequence of contact patterns is
identical. Nevertheless, for lower value of clearance, higher values of applied load Q and lower
values of end-shortening δ are obtained. This implies that the stiffness Q/δ of all branches is
steeper in this example.
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
0
5
10
15
20
25
30
δ
1
2
3
4
5
6
7
8
Q/
⇡
2
1 - 2 2 - 3
3 - 4
5 - 6 6 - 7
7 - 8
4 - 5
Figure 6.15: Bifurcation diagram for a constrained clamped-clamped elastica with self-weight
w = 10 and clearances cl = cu = 0.1
153
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07
0
10
20
30
40
7
5
4
2
δ
1
3
6
8
Q/
⇡
2
1 - 2 2 - 3
3 - 4
5 - 6 6 - 7
7 - 8
4 - 5
Figure 6.16: Bifurcation diagram for a constrained clamped-clamped elastica with self-weight
w = 10 clearances cl = cu = 0.05
Now, we investigate the effect of the boundary conditions in the buckling response of a con-
strained heavy elastica. Consider a pinned-pinned elastica with self-weight w = 10 constrained
by symmetrically located walls with clearances cl = cu = 0.05. If we compare Figure 6.16 with
Figure 6.17, an entirely different sequence of contact patterns is observed. Even though the
initial configuration of a clamped-clamped elastica is unconstrained, this is not the case in a
154
pinned-pinned elastica. Its initial deformation shape is consistent with the first buckling mode of
a pinned-pinned elastica with a discrete contact at the midpoint. Increasing the end-shortening,
the discrete contact gradually becomes a continuous contact. The length of the continuous con-
tact then increases until it buckles as the first buckling mode of a clamped elastica at C3. Beyond
this point, a free standing fold is obtained with a softening response. When C4 is reached, a
new discrete contact with the upper wall at the midpoint is detected. This shape 4− 5 remains
unchanged until one of the discrete contacts with the lower wall turn into a continuous contact.
Similarly, the subsequent configurations of the elastica are obtained, as shown in Figure 6.17.
From this example, we can observe that, for the same range of end-shortening δ, the number
of discrete contacts is higher here when compared to the clamped-clamped elastica. This also
leads to higher values of applied force Q.
Elastica with w = 10
Points δ Q/pi2
3 0.017 17.9
4 0.025 5
6 0.096 55.7
Table 6.4: Points of evolution of the contact patterns of the bifurcation diagram for a constrained
pinned-pinned elastica with self-weights w = 10 and clearances cl = cu = 0.05
155
0.00 0.04 0.08 0.12 0.16 0.20
0
10
20
30
40
50
60 8
5
1
7
6
4
3
2
δ
Q/
⇡
2
1 - 2 2 - 3
3 - 4 4 - 5
5 - 6 6 - 7
7 - 8
Figure 6.17: Bifurcation diagram for a constrained pinned-pinned elastica with self-weight w =
10 and clearance cl = cu = 0.05
6.3.3 Summary
The effect of the self-weight w in the buckling response of the elastica constrained by rigid walls
is investigated in this Chapter. A short review of previous studies on the loop tests and the
buckling problem of a heavy elastica lying on a horizontal foundation is included. The latter
problem is then solved by applying optimal control. In this manner, we first confirm the validity
of the optimal control method. The “symmetry breaking bifurcation” is also investigated and the
156
results are shown to be in agreement with Domokos et al. (2003)’ analysis. The same problem
is also investigated for inclined foundations.
Extension of the one-sided foundation to two-sided foundations is then provided. By applying
an optimal control method, two distinct classes of problems are analyzed; (1) the classical
buckling problem and (ii) the insertion buckling problem. In the first class of problems, the
elastica is of constant length and the right end is free to move towards the fixed left end under
displacement-controlled conditions. On the other hand, in the second class of problems, the
heavy elastica is of variable length while the distance of the two supports is unchanged.
The buckling response of the heavy elastica constrained by two asymmetrically located walls
with cl = 0 is first analyzed. At the initial state, the response of the elastica is identical to the
case of an elastica resting on a horizontal foundation. Nevertheless, the additional constraint of
the upper wall leads to a different sequence of deformation shapes if a sufficient development of
the end-shortening δ for the classical problem, or the inserted length of the elastica 1 + δ¯ for the
insertion problem occurs.
The buckling analysis of the heavy elastica constrained by two symmetrically located walls is
then analyzed. If we assign different values of self-weight w for the same clearance and boundary
conditions, the derived bifurcation diagrams are similar, except at the initial state, where three
distinct cases are met; (i) if the elastica is weightless, a classical post-buckling response is
observed, (ii) if the self-weight w is relatively small, a post-buckling behavior of a weightless-like
elastica with initial imperfection is detected and (iii) if the self-weight w is sufficiently large, an
initial configuration with a discrete, or a continuous contact is met. Several other effects, such
as the clearances and the boundary conditions are also investigated. All results are compared
with the buckling response of a weightless elastica in order to clearly illustrate the effect of the
self-weight.
Chapter 7
Insertion problem of spatial elastica
7.1 Introduction
The aim of the Chapter is the study of the constrained buckling problem of a variable-length
spatial elastica. The main focus is placed on the derivation of the configurational or Eshelby-like
force, which is generated at the insertion point of the sliding sleeve (see (Bigoni et al., 2015)
for the planar case). The reason behind the development of this configuration force is two fold;
(1) the change of the length of the elastica as it is gradually inserted inside the conduit and (2)
the isoperimetric constraint of a fixed distance between the extremities of the inserted elastica.
This effect leads to a reduction of the applied axial force at the point of insertion.
Based on the calculus of variations, we first calculate the configuration force. Then, a
representative example is solved by applying the optimal control method. The effect of the
configurational force is used to compute the applied force.
7.2 Derivation of Eshelby-like force
A weightless spatial (isotropic) elastica of total length L¯t of uniform bending EI and torsional
GJ rigidity is constrained by a cylindrical wall of clearance C and it is initially unstressed. The
elastica is clamped, or pinned at the left end and it is inserted through a fixed frictionless sliding
sleeve that functions as a clamped end (see Figure 7.1). Hence, the distance L between the
fixed end of the elastica and the insertion point is constant, while the inserted length L¯ of the
elastica varies (i.e., inserted length increment ∆¯ = L¯−L). We seek to analyze the post-buckling
response of the spatial elastica under the action of a compressive load Q.
A system of coordinates (X,Y, Z) is defined with origin at the left extremity of the elastica
and with the X-axis pointing towards the elastica right end. A cylindrical wall with clearance
157
158
C with respect to the centerline is also present, which can be described by Y 2 + Z2 = C2.
We introduce the arc-length coordinate S with its origin S = 0 at X = 0, Y = 0, Z = 0, while
three Euler angles Θ(S), Ψ(S),Φ(S) are used to define the rotation matrix with respect to the
reference configuration (see Chapter 5 for a detailed derivation). In view of the inextensibity and
unshearability constraints characterizing the elastica, its deformed configuration Xˆ(S), Yˆ (S),
Zˆ(S) is fully defined by
Xˆ ′ = cos Θ cosΨ
Yˆ ′ = cos Θ sinΨ
Zˆ ′ = − sinΘ
(7.1)
with boundary conditions Xˆ(0) = Yˆ (0) = Zˆ(0) = 0, while Yˆ (L¯) = Zˆ(L¯) = 0 and Xˆ(L¯) = L
in accordance with the imposed boundary condition ∆¯ = L¯−X(L¯). The unilateral constraint
associated with the presence of the cylindrical wall is characterized by the inequality condition
Yˆ 2(S) + Zˆ2(S) ≤ C2 with S ∈ [0, L¯].
The internal force transmitted by the elastica is denoted by R(S) with components Rx, Ry, Rz
along the corresponding axes. The functions Ry and Rz become piecewise uniform whenever
there is a discrete or a continuous contact with the cylindrical wall. In addition the applied
compressive force Q does not coincide with the axial force Rx. More specifically, at the insertion
point, a configurational force is generated, which leads to a reduced value of the applied load
Q when compared to the axial force. This effect was first verified by Bosi et al. (2015) for the
insertion problem of an unconstrained planar elastica. Here, we derive this effect for the spatial
elastica by applying the calculus of variations.
Figure 7.1: A spatial elastica constrained by a cylindrical wall with clearance C located with
respect to the centerline with the left end fixed, while the elastica is gradually inserted through
a fixed frictionless sleeve at the right end.
159
7.2.1 Governing equations of spatial elastica
Scaling of this problem follows from adopting L as length scale and EI/L2 as force scale
s =
S
L
, x =
X
L
, y =
Y
L
, z =
Z
L
, δ¯ =
∆¯
L
, c =
C
L
, R = RL
2
EI
, Q = QL
2
EI
(7.2)
where δ¯ is the change of inserted length of the elastica. The inserted length is denoted by
ρ = 1 + δ¯ and the total length of the elastica is ρt ≥ ρ.
The cartesian parametrization of the elastica, xˆ(s), yˆ(s), zˆ(s) is derived by applying the
inextensibility and unshearability constraints (7.1)
xˆ(s) =
∫ s
0
cos θ cosψds
yˆ(s) =
∫ s
0
cos θ sinψds
zˆ(s) = − ∫ s
0
sin θ
(7.3)
where the boundary conditions xˆ(0) = yˆ(0) = zˆ(0) = 0 and the Euler angles {ψ, θ, φ} have been
taken into account.
7.2.2 The calculus of variations
The potential energy of the spatial elastica (see Chapter 5) can be expressed as
Π = Us −WQ = 1
2
∫ ρ
0
(
κ22(s) + κ
2
3(s)
)
ds+
α
2
∫ ρ
0
κ21(s)ds−Q
(
ρt −
∫ ρt
0
cos θ cosψds
)
(7.4)
where Us is the elastic strain energy, WQ is the external work of the axial compressive load Q,
α is the ratio of torsional to bending moduli, κ1 = τ is the torsion, κ2 and κ3 are the curvatures
about the director axes
κ1 = φ
′ − ψ′ sin θ
κ2 = ψ
′ sinφ cos θ + θ′ cosφ
κ3 = ψ
′ cosφ cos θ − θ′ sinφ
(7.5)
and Q (ρt − ∫ ρt0 cos θ cosψds) is the external work of the axial compressive force Q under the
presence of the equality and inequality constraints(∫ s
0
cos θ sinψdξ
)2
+
(∫ s
0
− sin θdξ)2 − c2 ≤ 0, s ∈ [0, ρ]∫ ρ
0
cos θ cosψds = 1∫ ρ
0
cos θ sinψds = 0∫ ρ
0
− sin θds = 0
(7.6)
160
and the boundary conditions, which are ψ(0) = ψ(ρ) = θ(0) = θ(ρ) = φ(0) = φ(ρ) = 0 for
clamped-clamped boundary conditions. If displacement-controlled conditions are assumed, the
applied force Q becomes the unknown Lagrange multiplier associated with the imposed bilateral
constraint ρt −
∫ ρt
0
cos θ cosψds = δ¯.
Let us investigate an equilibrium configuration of the constrained elastica with a single
discrete contact at a position s = `1. In a similar manner, the derivation can be accordingly
extended to a multiple number of contact points. Based on this assumption, the potential energy
can be expressed as
Π = Us + Ui −WQ −WF
=
1
2
∫ ρ
0
(
ψ′2 cos2 θ + θ′2
)
ds+
γ
2
∫ ρ
0
(φ′ − ψ′ sin θ)2 ds−Q
(
ρt −
∫ ρt
0
cos θ cosψds
)
− λx
(
1−
∫ ρ
0
cos θ cosψds
)
+ λy
∫ ρ
0
cos θ sinψds+ λz
∫ ρ
0
− sin θds
+ F
(∫ `l
0
cos θ sinψdξ
)2
+
(∫ `l
0
− sin θdξ
)2
− c2
 (7.7)
where Ui is the energy produced by the isoperimetric constraints (7.6)ii)-iv) with associated
Lagrange multipliers λx, λy, λz and WF is the external work produced by a closed unilateral
constraint (7.6)i) at s = `l with associated Lagrange multiplier F and `l is the length of the
segment from the origin (i.e., s = 0) to the discrete contact.
To derive the first variation of the potential energy, variations of the involved variables are
assumed
θ(s, ε) = θˆ(s) + εη(s)
ψ(s, ε) = ψˆ(s) + εζ(s)
φ(s, ε) = φˆ(s) + εχ(s)
(7.8)
ρ(ε) = ρˆ+ εγ
`l (ε) = ˆ`l + εγl
(7.9)
where θˆ(s) , ψˆ(s), φˆ(s),ρˆ, ˆ`l are the extremal values and η(s), ζ(s), χ(s), γ , γl are their
corresponding variations, while ε  1 is a positive parameter. Based on these variations and
taking into account the boundary conditions θ(ρ) = θˆ(ρˆ) = 0, ψ(ρ) = ψˆ(ρˆ) = 0, φ(ρ) = φˆ(ρˆ) = 0
and θ (`l) = θˆ
(
ˆ`
l
)
= 0, ψ (`l) = ψˆ
(
ˆ`
l
)
= 0, φ (`l) = φˆ
(
ˆ`
l
)
= 0, the following compatibility
equations are obtained;
γθˆ′(ρˆ) + η(ρˆ) = 0
γψˆ′(ρˆ) + ζ(ρˆ) = 0
γφˆ′(ρˆ) + χ(ρˆ) = 0
(7.10)
161
γθˆ′(ˆ`l) + η(ˆ`l) = 0
γψˆ′(ˆ`l) + ζ(ˆ`l) = 0
γφˆ′(ˆ`l) + χ(ˆ`l) = 0
(7.11)
The first variation of the potential energy is then deduced by applying the following derivation
δΠ(θ, θ
′
, ψ, ψ
′
, φ, φ
′
) =ε→0
∣∣∣∣dΠdε = δUs + δUi − δWQ − δWF (7.12)
which leads to
δUs =
∫ ρˆ
0
(
ψˆ′ζ ′ cos2 θˆ −
(
ψˆ′2 sin 2θˆ/2
)
η + θˆ′η′
)
ds
+ α
∫ ρˆ
0
(
φˆ′χ′ + ψˆ′ζ ′ sin2 θˆ +
(
ψˆ′2 sin 2θˆ/2
)
η
)
ds
− α
∫ ρˆ
0
(
sin θˆ
(
ψˆ′χ′ + φˆ′ζ ′
)
+ φˆ′ψˆ′ cos θˆη
)
ds
+
γ
2
(
ψˆ′2 (ρˆ) cos2 θˆ (ρˆ) + θˆ′2 (ρˆ)
)
+
αγ
2
(
φˆ′2 (ρˆ) + ψˆ′2 (ρˆ) sin2 θˆ (ρˆ)− 2φˆ′ (ρˆ) ψˆ′ (ρˆ) sin θˆ (ρˆ)
)
(7.13)
δUi = −λx
(∫ ρˆ
0
sin θˆ cos ψˆηds+
∫ ρˆ
0
cos θˆ sin ψˆζds
)
+ γλx cos θˆ (ρˆ) cos ψˆ (ρˆ)
+ λy
(
−
∫ ρˆ
0
sin θˆ sin ψˆηds+
∫ ρˆ
0
cos θ cos ψˆζds
)
+ γλy cos θˆ (ρˆ) sin ψˆ (ρˆ)
+ λz
∫ ρˆ
0
− cos θˆηds− γλz sin θˆ (ρˆ) (7.14)
δWQ = Q
(∫ ρˆ
0
sin θˆ cos ψˆηds+
∫ ρˆ
0
cos θˆ sin ψˆζds
)
(7.15)
δWF = −2F
(∫ ˆ`
l
0
cos θˆ sin ψˆdξ
)[
−
∫ ˆ`
l
0
sin θˆ sin ψˆηds+
∫ ˆ`
l
0
cos θ cos ψˆζds+ γl cos θˆ
(
ˆ`
l
)
sin ψˆ
(
ˆ`
l
)]
− 2F
(∫ ˆ`
l
0
− sin θˆdξ
)[∫ ˆ`
l
0
− cos θˆηds− γl sin θˆ
(
ˆ`
l
)]
(7.16)
All terms in (7.13) that involve η′(s), ζ ′(s), χ′(s), are transformed to η(s), ζ(s), χ(s) by applying
integration by parts in combination with the compatibility equations (7.10). In this manner, we
162
arrive at the final form
δΠ =
∫ ρˆ
0
[
−θˆ′′ + ψˆ′2 sin 2θˆ (α− 1) /2− φˆ′ψˆ′ cos θˆ − (Q+ λx) sin θˆ cos ψˆ
]
ηds
+
∫ ρˆ
0
[(
−λy + FyH
(
s− ˆ`l
))
sin θˆ sin ψˆ +
(
−λz + FzH
(
s− ˆ`l
))
cos θˆ
]
ηds
+
∫ ρˆ
0
[
−ψˆ′′
(
cos2 θˆ + α sin2 θˆ
)
+ ψˆ′ sin 2θˆ (1− α) + α
(
φˆ′′ sin θˆ + φˆ′ cos θˆ
)]
ζds
+
∫ ρˆ
0
[
− (Q+ λx) cos θˆ sin ψˆ +
(
λy − FyH
(
s− ˆ`l
))
cos θˆ cos ψˆ
]
ζds
+ α
∫ ρˆ
0
[
ψˆ′′ sin θˆ + ψˆ′ cos θˆ
]
χds
+ γ
[
λx −
(
θˆ′ (ρˆ)2 + ψˆ′ (ρˆ)2 + αφˆ′ (ρˆ)2
)
/2
]
(7.17)
where Fy = 2F
∫ ˆ`
l
0
cos θˆ sin ψˆdξ, Fz = −2F
∫ ˆ`
l
0
sin θˆdξ and H(s) is the Heaviside function. By
vanishing (7.17) for the three independent variations η(s), ζ(s), χ(s), the system of the governing
equations of the elastica is accordingly obtained
−θ′′ + ψ′2 sin 2θ (α− 1) /2− φ′ψ′ cos θ −Rx sin θ cosψ
−Rx sin θ cosψ −Ry sin θ sinψ −Rz cos θ = 0 (7.18)
−ψ′′ (cos2 θ + α sin2 θ)+ ψ′ sin 2θ (1− α) + α (φ′′ sin θ + φ′ cos θ)
−Rx cos θ sinψ +Ry cos θ cosψ = 0 (7.19)
ψ′′ sin θ + ψ′ cos θ = 0 (7.20)
where ˆ is omitted, Rx = Q + λx, Ry = λy − FyH (s− `l) and Rz = λz − FzH (s− `l) are
the internal forces and s ∈ [0, ρ]. From this derivation, it is deduced that the applied axial
force Q does not coincide with internal axial force Rx. Their difference is the configurational
or Eshelby-like force P = Rx − Q =
(
θ′ (ρ)2 + ψ′ (ρ)2 + αφ′ (ρ)2
)
/2, which develops at the
insertion point of the sliding sleeve.
7.3 Optimal control
The analytical derivation of the potential energy of a Kirchhoff rod model is explained in Chapter
5. The optimal control formulation for the constained buckling problem of a variable length
elastica can be written as
163

Minimize J [x(·),u(·)] = 12
∫ 1+δ¯
0
(
κ22(s) + κ
2
3(s)
)
ds+ α2
∫ 1+δ¯
0
κ21(s)ds
subject to
x′ = cos θ(s) cosψ(s)
y′ = cos θ(s) sinψ(s)
z′ = − sin θ(s)
ψ′ = u1
θ′ = u2
φ′ = u3
x(0) = y(0) = z(0) = y(1 + δ¯) = z(1 + δ¯) = 0
ψ(0) = ψ(1 + δ¯) = θ(0) = θ(1 + δ¯) = φ(0) = φ(1 + δ¯) = 0
x(1 + δ¯) = 1
y2(s) + z2(s) ≤ c2
(7.21)
where the state variables are the cartesian coordinates {x, y, z} of the elastica and the Euler
angles {ψ, θ, φ}, the control variables are the spatial derivatives of the Euler angles {ψ′, θ′, φ′}
and α is the ratio of the torsional to the bending moduli, while the rotational strains are given
by (7.5).
Further analysis is omitted due to its similarity with the classical problem (see Chapter
5). For displacement-controlled loading conditions, the inserted length of the elastica gradually
increases, while the applied load Q is given by
Q = Rx − P (7.22)
where the internal horizontal force Rx is indirectly derived by the corresponding adjoint variable
and the Eshelby-like force P is obtained by
P =
(
θ′
(
1 + δ¯
)2
+ ψ′
(
1 + δ¯
)2
+ αφ′
(
1 + δ¯
)2)
2
(7.23)
as shown in Section 7.2.
7.3.1 Example
In this Section, we investigate a clamped-clamped spatial elastica of variable length constrained
by a cylindrical wall of c = 0.1 and α = 5/7. The inserted length of the elastica gradually
increases (increase of δ¯), while the applied load is indirectly obtained, as shown in Section 7.3.
The elastica remains unstressed until it buckles according to the first buckling mode of a
clamped-clamped planar elastica at an applied load Q/pi2 = 4. If we further increase the inserted
164
length, the applied load Q/pi2 and the internal force Rx/pi2 decrease until C2, where a discrete
contact at the midpoint position occurs. This is a planar configuration of the elastica. At C3,
the torsion starts increasing, leading to a spatial deformation with identical contact conditions.
When C4 is reached, a second discrete contact appears. In particular, for symmetrical reasons,
these discrete contacts lie in symmetrical positions next to the midpoint position. Then, at C5, a
third discrete contact at the midpoint position takes place, which rapidly turns into a continuous
contact when C6 is achieved. After further increase of the inserted length, a single continuous
segment is obtained at C7, associated with a gradual reduction of the load (softening branch).
Past C8, the single continuous segment is transformed to three shorter continuous segments. In
Table 7.1, the points of evolution of the contact patterns are summarized, while the complete
bifurcation diagram is shown in Figure 7.2.
Insertion Problem
Points 2, 2′ 3, 3′ 4, 4′ 5, 5′ 6, 6′ 7, 7′ 8, 8′
δ 0.024 0.028 0.054 0.112 0.12 0.30 0.43
Rx/pi2 3.86 7.59 7.32 8.45 8.62 10.35 6.32
Q/pi2 3.68 7.45 6.85 7.44 7.55 8.46 5.15
Table 7.1: Points of evolution of the contact patterns for the bifurcation diagram of a constrained
clamped-clamped spatial elastica of variable length with clearance c = 0.1
165
0.0 0.1 0.2 0.3 0.4 0.5
0
2
4
6
8
10
12
9'
9
8
8'
L-L-L
7'
7
6'5'
65
4'
4
3
3'
2
2'
L
P-L-P
P-P-P
P-P
P
P
1,1'
Q/
⇡
2
,R
x
/⇡
2
¯
Figure 7.2: Bifurcation diagram for a constrained clamped-clamped elastica of variable length
with clearance c = 0.1; (1-2) Planar one point (P), (3-4) Spatial one point, (4-5) Two discrete
contacts (P-P), (5-6) Three discrete contacts (P-P-P), (6-7) Point-Line-Point (P-L-P), (7-8) Line
contact (L) and (8-9) Line-Line-Line (L-L-L) (Same holds for the corresponding points denoted
by ′). The black and grey lines represent the evolution of the internal force Rx/pi2 and the
applied load Q/pi2 with respect to the change of the inserted length δ¯, respectively.
7.4 Conclusions
This Chapter presents the generation of a configurational or Eshelby-like force at the insertion
point of the sliding sleeve as the elastica is progressively inserted inside the conduit. This causes
a reduction in the applied load. Its effect is also illustrated through a representative example of
a clamped-clamped elastica constrained by a cylindrical wall of clearance c = 0.1.
166
Chapter 8
Conclusions
8.1 Summary
Slender elastic structures subjected to inside or outside constraints under various loading and
boundary conditions forms a class of problems that has attracted a great deal of attention. For
instance, the stability behavior of drillstrings inside a rigid conduit is of major concern in the oil
and gas industry. In the medical field, the insertion of guidewire and catheters for interventional
radiology operations is also a crucial problem and requires a real-time simulation rather than a
stability analysis.
Motivated by the buckling problem in oil wells, our main focus is placed on the stability
analysis of a constrained elastica, which is an inextensible-unshearable elastic rod. This is an
appropriate choice in our case due to the large length/clearance ratio. Two distinct problems
are mainly analyzed: (1) the classical buckling problem and (2) the insertion buckling problem.
The bifurcation diagram is obtained in both cases. In the former case, the bifurcation diagram
is expressed in terms of the axial force-end-shortening (i.e., distance between the two supports of
the elastica), while in the latter case, the end-shortening is replaced by the change of the length
of the elastica. Several alternatives in terms of loading, boundary conditions and clearances of
the walls are investigated.
The analysis of these problems involves two steps; (i) the derivation of an equilibrium state of
the elastica under particular loading and boundary conditions and (ii) the stability analysis of (i).
Due to the presence of the unilateral constraints several numerical difficulties are encountered.
Some remarkable numerical techniques can be found in literature (Lazarus et al., 2015; Ro et al.,
2010; Manning and Bulman, 2005). One way is to transform the infinite dimensional problem
to a finite dimensional system of equations by applying some discretization method and then
derive the stability of an equilibrium configuration by computing the spectrum of the Hessian
matrix (Lazarus et al., 2015). Nevertheless the potential existence of weakly active constraints
167
168
and the abrupt changes of the structure of the Hessian matrix, when the contact patterns evolve
discontinuously, makes the analysis very sensitive to numerical errors. In our situation, an
additional numerical issue is met. Even if a deformation shape is stable (i.e., positive-definite
Hessian matrix), the condition number of the Hessian matrix can be remarkable high (i.e., at
least one eigenvalue close to zero). This is true when the derived axial loads are very close to the
Euler-buckling loads. For instance, this is a common issue when we analyze the post-buckling
response of a planar weightless elastica with symmetrically located walls. Hence, if we apply a
classical continuation method, an extremely small step length is required and even in this case
the accuracy of the results is questionable.
For these reasons the majority of the research works preserves the infinite dimensionality
of the problem. The most notable studies have been performed by (Domokos et al., 1997; Ro
et al., 2010; Manning and Bulman, 2005). Domokos et al. (1997) solved the problem using
two different methods; (1) an analytical technique using elliptic integrals and (2) a numerical
method applying a classical linear programming method, the simplex method (Holmes et al.,
1999). The problem was only partially solved, because the stability of the derived equibrium
states was not investigated. Manning and Bulman (2005) reformulated Domokos et al. (1997)’s
problem by replacing the unilateral constraint with a smooth potential and accordingly applied
a conjugate point theory on the unconstrained problem. Even though the stability analysis
is complete, the approach remains approximate due to the assumption of smooth contact. In
addition, the method is not generic since it should be adjusted whenever Neumann-Neumann
boundary conditions are present, as explained by Manning (2009).
On the other hand, Ro et al. (2010) provided an Eulerian description of the problem in
order to take into account the potential variation of the contact points between the elastica
and the rigid obstacles during vibration. Then a shooting method was applied, which can
provide information regarding the stability of the problem. This method also involves numerical
difficulties. First of all, it is known that the shooting method is always sensitive to initial guesses
of the involved variables. This becomes very evident as the number of the unknown variables
increases. Secondly, the derivation of the mathematical formulation of every distinct problem
requires a lot of effort.
Based on above observations we tried to solve this constrained stability problem, taking
into account two prerequisites; (i) there are no further assumptions in the formulation of the
problem (e.g., the assumption of a smooth constraint) and (ii) the methodology is as simple and
generic as possible. First, in Chapter 2, we show why the use of the calculus of variations is
not recommended. Accordingly, two novel approaches are suggested, namely a geometry-based
method that provides an equilibrium map of the problem and the optimal control method, which
completely solves the constrained stability problem. These two contributions are summarized
next.
The post-buckling response of the planar elastica constrained by rigid horizontal walls relies
169
on the proposed geometry-based method. We first define a specific sequence of deformation
shapes of the elastica for a given buckling mode. For a known configuration the phase portait
of the elastica (i.e., inclination θ- curvature θ′) is then determined. Based on it, the elastica is
divided into distinct segments consistent with a single canonical segment with clamped-pinned
boundary conditions. The complete solution is accordingly calculated by solving all distinct sub-
problems and satisfying the continuity properties at the common extremities of these segments.
The solution of these subproblems is obtained by applying an asymptotic analysis, or by using
elliptic integrals.
In Chapter 3, the analysis is presented for both classes of problem; (1) the classical problem
(some supplementary material can be also found in Appendix B) and (2) the insertion problem.
In the latter case, a configurational force is generated at the insertion point that leads to a
reduced value of the axial applied force, which is not present in the classical problem. This
effect is also illustrated through concrete examples.
The same approach can be also adopted for the analysis of different equilibrium problems
of inflectional elastica as far as the phase portrait of the elastica can be determined (known
positions of zero inflection points θ′ = 0 and zero inclination angles θ = 0). An alternative
option is to discretize the elastica by applying a finite-difference scheme, leading to the so-called
“discrete elastica” (Domokos and Holmes, 1993). This idea in combination with an asymptotic
analysis was first suggested by Challamel et al. (2015) for the post-buckling analysis of an
unconstrained elastica. It can be also adopted herein for the solution of the subproblems.
Another geometry-based contribution of the present work is a stability rule for the planar
case, which is based on the number of the inflection points and the number and nature of the
contact conditions. Its verification was achieved by using previous research works (Ro et al.,
2010; Manning and Bulman, 2005) and applying the optimal control method. However, this rule
has not been formally demonstrated and therefore remains a conjecture. As explained in Chap-
ter 3, we can deduce about the stability of a given configuration of the planar elastica purely by
its geometry without the need of any futher analysis. This stability criterion, however, cannot
be adopted for a zero value of clearance, because the lower wall functions as an additional sta-
bilization factor in this case and different contact patterns are obtained. The latter observation
is explained in Chapter 6.
The application of the optimal control to the stability analysis of elastic structures is not very
well-known yet. Maurer and Mittelmann (1991) were the first to derive preliminary results for
the stability behavior of a nonlinear beam constrained by two symmetrically located longitudinal
obstacles using an optimal control method. Maurer and Mittelmann (1991)’s innovative idea
remained quasi unknown for years. Even though the equivalence between a calculus of variations
and an optimal control problem was proved as early as in 1960, see for instance (Berkovitz, 1961,
1962), only recently the optimal control has started being used for the stability analysis of an
elastic structure (Liberzon, 2012; Sachkov, 2008b,a; O ’ Reilly and Peters, 2011).
170
In short, the optimal control formulation for our constrained stability problem involves the
following parts; (1) the functional, which is the Lagrangian of the problem (i.e., the potential
energy of the elastica), (2) the spatial dynamics, which is deduced from the inextensibility-
unshearability constraints, (3) the control variables, which are the spatial derivatives of the
Euler angles (e.g., the curvature for the planar elastica), (4) the unilateral constraints (i.e, rigid
obstacles) and (5) the boundary conditions. Based on Hamiltonian mechanics, the optimal-
ity conditions, which constitute the Pontryagin’s minimum principle, are then derived. These
necessary optimality conditions consist of the minimization of the Hamiltonian with respect to
the control, the canonical equations of the (augmented) Hamiltonian and the satisfaction of the
boundary conditions. Detailed derivation of the Pontryagin’s theorem can be found in Chapter
4.
The main advantage of the optimal control method when compared to calculus of variations
is the assumption of strong rather than weak variation of the involved variables. This leads to the
additional Weierstrass necessary condition, which provides the minimization of the Hamiltonian
with respect to the control, see Chapter 4. Based on it, we obtain the “optimal” equilibrium
state that minimizes the total energy of a system rather than just an equilibrium configuration,
as in calculus of variations. Hence, we can deduce about the stability behavior of an equilibrium
state even if we just apply the first-order optimality conditions. The inclusion of the unilateral
constraints is also an additional reason to choose an optimal control method. In our constrained
stability problem, the unilateral constraints are pure-state constraints of second order, which
implies that set-valued functions of the adjoint equations are present. Physically, this means
that discontinuous shear forces are obtained whenever a contact with the wall is present. Even
in this case, the use of optimal control is straightforward while in calculus of variations several
assumptions are required, such as the assumption of smooth constraints, or the addition of slack
variables. For all these reasons, the optimal control method is a more efficient technique when
compared to the classical calculus of variations.
A variety of examples is analyzed in Chapters 5 , 6 and 7 using the optimal control method.
These involve the classical and insertion stability problems while several factors such as the
gravity, the clearances of the walls and the boundary conditions are also investigated. Before
referring to some remarkable results, it is important to point out that every distinct stability
problem can be formulated without any special effort. For instance, let us explain how we can
transform the classical to the insertion stability problem for displacement-controlled conditions.
In the classical problem, we predefine the distance between the extremities of the elastica by
keeping fixed the length of the elastica, while in the insertion problem exactly the opposite holds.
This is the only difference between these problems, which shows how easy it is to switch from
one formulation to another. In addition there is no need for any special treatment regarding
the boundary conditions, or the types of the constraints. All above indicate that the optimal
control method is simple and generic at the same time.
171
Based on the optimal control method some notable results were obtained. One important
conclusion is that the classical and the insertion stability problems lead to the same sequence of
contact patterns if the elastica is weightless. Some differences are observed when the self-weight
is non-negligible. Another interesting result is the comparison between the stability behavior
of the planar elastica and the spatial elastica with or without the unilateral constraints. In
particular, the stability behavior of the unconstrained spatial elastica coincides with the planar
elastica until the onset of out-of-plane deformation, which depends on the ratio of the torsional
over the bending moduli. Past this point, the buckling response of the spatial elastica is entirely
different. Similar conclusions are reached by solving the constrained problem, see Chapters 5
and 7.
The effect of the weight has been also investigated using the optimal control method. Only
partial analysis of this effect can be found in the literature (Wicks et al., 2008; Gao and Huang,
2015). Based on the self-weight and the clearances of the walls, two possible initial configurations
of the elastica can be obtained; (1) an unconstrained shape, or (2) a constrained shape with a
discrete, or continuous contact with the lower wall. An unconstrained, or a constrained shape
is obtained if the maximum height derived by buckling of the elastica under its own weight is
higher, or lower than the clearance of the lower wall, respectively.
Another important observation is the case of one-sided foundation with a high value of self-
weight and low or zero value of clearance. Initially, the elastica remains unstressed on the rigid
foundation and an infinite axial force is required for the formation of a “ruck” (see Chapter
6). After the onset of buckling, the height of this ruck increases while the axial force initially
decreases with increasing the end-shortening until it reaches a minimum value. Beyond this
point, the axial force starts increasing and the height of the ruck gradually reaches a plateau. If
an upper wall is also present and its clearance is lower than this plateau, a sequence of contact
patterns is obtained. Otherwise, its behavior is identical to the stability problem with one-sided
foundation. Several other interesting results are also available in Chapters 5, 6 and 7.
To summarize, the geometry-based method is only applicable to the planar case and for
particular kinds of problems when compared to the optimal control problem whose use is more
generic. The major contributions of the present work are summarized below.
8.2 Major contributions
The most notable contributions of this thesis are:
• Development of a geometry-based technique for the analysis of the classical stability
problem: A geometry-based segmentation technique is applied to derive the bifurcation
diagram, including the sequence of contact patterns. A simple geometry-based stability
criterion based on the number of the inflection points and the nature of contact conditions
172
is also conjectured.
• Analysis of the insertion problem: The constrained buckling problem of a variable
length elastica is solved. Based on the calculus of variations, we show that an Eshelby-
like or configuration force is generated at the insertion point of the sliding sleeve, which
was ignored in previous studies (Ro et al., 2010). Its effect for the constrained buckling
problem of a spatial elastica is also illustrated.
• Application of optimal control in the stability analysis of constrained thin
elastic structures: Optimization-based stability analysis of a planar, or a spatial elastica
constrained by rigid horizontal walls is performed. Derivation of bifurcation diagrams for
different loading and boundary conditions is achieved for both the classical and insertion
stability problems.
• Weight considerations in the stability analysis of constrained elastica: To the
best of our knowledge, the weight effect has not been investigated before. The problem is
only partially solved as in the oil-drilling community, it is assumed that the drillstring is
always in contact with the borehole.
8.3 Future work
Further analysis could also include second-order sufficient conditions. This would require an
indirect method rather than a direct approach. In particular, after solving the first order op-
timality conditions of the problem, the second-order optimality conditions can be derived by
applying a sensitivity analysis of the shooting mapping. This mapping should include a regular
structure with a finite number of contact points and the strict complementarity conditions has
to be also satisfied such that the implicit function theorem is applicable, as explained in detail
in (Malanowski and Maurer, 2001; Maurer and Oberle, 2002). This could give a more com-
plete solution of our problem. However, the comparison of our results with other studies has
shown that the first-order optimality conditions are sufficient enough in order to derive accurate
bifurcation diagrams without any exception.
The optimal control method can be also used to analyze the quasi-static, or buckling response
of an elastic rod under complex loading and boundary conditions, including a non-uniform
density and/or discontinuities along its arclength (e.g., kinks) and in the presence of bilateral
and/or unilateral constraints. It can be also adopted for the stability analysis of any other
elastic structure, such as shells and plates. Hence, we anticipate that this novel approach (see
(Maurer and Mittelmann, 1991; Liberzon, 2012)) will gain more attention in the future.
Except for its application to the analysis of a stability problem, we can also use it to optimize
the design (i.e., optimization of material or mechanical properties) of an elastic structure, such
173
that any catastrophic failure (i.e., buckling) is avoided or its optimal performance is achieved
(Bendsoe and Sigmund, 2003; Huang and Xie, 2010). For instance, this analysis is essential in
medical and engineering applications, such as the insertion of a guide-wire (Tang et al., 2012)
or in oil-drilling operations (Wicks et al., 2008). A comprehensive literature review about the
latter application is included in Appendix A.
The main limitation of the current study is that the effect of friction is ignored (frictionless
rigid walls are only studied here). The effect of friction in a constrained buckling problem can
be better understood if a dynamic stability analysis is performed, as explained by (Su et al.,
2013). A classical stability analysis can only partially solve this problem (Gao and Miska, 2009a).
Some preliminary study about it (particularly in oil-drilling operations) can be found in (Liu
and Chen, 2013; Liu et al., 2015) (see also Appendix A). As explained in (Liu et al., 2015), the
effect of friction in the buckling response of an elastic rod is complex as it can be beneficial
or catastrophic, depending on the initial, boundary and loading conditions under consideration.
Hence, further numerical and experimental analysis is necessary. This analysis, however, requires
a different approach rather than the optimal control method as a history-dependent response of
the elastic rod is expected.
Bibliography
Alderliesten, T., M. K. Konings, and W. J. Niessen (2007, January). Modeling Friction, Intrin-
sic Curvature, and Rotation of Guide Wires for Simulation of Minimally Invasive Vascular
Interventions. IEEE Transactions on Biomedical Engineering 54 (1), 29–38.
Alfutov, N. A. (2000). Stability of Elastic Structures. Foundations of Engineering Mechanics.
Berlin, Heidelberg: Springer Berlin Heidelberg.
Antman, S. S. (2005). Nonlinear problems of elasticity. New York, NY: Springer.
Arslan, M., M. E. Ozbayoglu, S. Z. Miska, M. Yu, N. Takach, and R. F. Mitchell (2012). Buckling
of Bouyancy Assisted Tubular. Society of Petroleum Engineers.
Atanackovic, T. (1997). Stability Theory of Elastic Rods. Stability, Vibration and Control of
Systems, Series A.
Athisakul, C., T. Monprapussorn, and S. Chucheepsakul (2011, mar). A variational formulation
for three-dimensional analysis of extensible marine riser transporting fluid. Ocean Engineer-
ing 38 (4), 609–620.
Augustin, D. and H. Maurer (2001). Computational sensitivity analysis for state constrained
optimal control problems. Annals of Operations Research 101 (1/4), 75–99.
Barclay, A., P. E. Gill, and J. B. Rosen (1998). SQP methods and their application to numerical
optimal control. In Variational Calculus, Optimal Control and Applications, pp. 207–222.
Birkhauser Basel.
Barquins, M. (1985, August). Sliding friction of rubber and Schallamach waves - a: review.
Materials Science and Engineering 73, 45–63.
Bendsoe, M. P. and O. Sigmund (2003). Topology Optimization. Springer-Verlag Berlin Heidel-
berg New York.
174
175
Benson, D. A., G. T. Huntington, T. P. Thorvaldsen, and A. V. Rao (2006, nov). Direct
trajectory optimization and costate estimation via an orthogonal collocation method. Journal
of Guidance, Control, and Dynamics 29 (6), 1435–1440.
Berkovitz, L. D. (1961, aug). Variational methods in problems of control and programming.
Journal of Mathematical Analysis and Applications 3 (1), 145–169.
Berkovitz, L. D. (1962, dec). On control problems with bounded state variables. Journal of
Mathematical Analysis and Applications 5 (3), 488–498.
Berkovitz, L. D. and N. G. Medhin (2012). Nonlinear Optimal Control Theory (Chapman &
Hall/CRC Applied Mathematics & Nonlinear Science). Chapman and Hall/CRC.
Bernoulli, D. (1742). The 26th letter to euler. in correspondence mathematique et physique,
volume 2. p. h. fuss,.
Bernoulli, J. (pp. 223-227, 1692). Die Werke von Jakob Bernoulli: Bd. 5: Differentialgeometrie
"Quadratura curvae, e cujus evolutione describitur inflexae laminae curvatura". Birkhauser.
Betts, J. T. (1998, mar). Survey of numerical methods for trajectory optimization. Journal of
Guidance, Control, and Dynamics 21 (2), 193–207.
Betts, J. T. (2010, jan). Practical Methods for Optimal Control and Estimation Using Nonlinear
Programming. Society for Industrial and Applied Mathematics.
Biegler, L. and V. Zavala (2009, mar). Large-scale nonlinear programming using IPOPT: An
integrating framework for enterprise-wide dynamic optimization. Computers & Chemical En-
gineering 33 (3), 575–582.
Bigoni, D. (Ed.) (2016). Extremely deformable structures. Number Vol. 562 in Courses and
lectures / International Centre for Mechanical Sciences. Wien: Springer.
Bigoni, D., F. D. Corso, F. Bosi, and D. Misseroni (2015, jan). Eshelby-like forces acting on
elastic structures: Theoretical and experimental proof. Mechanics of Materials 80, 368–374.
Bonnans, J. F. (Ed.) (2006). Numerical optimization: theoretical and practical aspects (2nd ed
ed.). Universitext. Berlin ; New York: Springer.
Bonnans, J. F. and A. Hermant (2007a, jul). No-gap second-order optimality conditions for
optimal control problems with a single state constraint and control. Mathematical Program-
ming 117 (1-2), 21–50.
Bonnans, J. F. and A. Hermant (2008, feb). Stability and sensitivity analysis for optimal control
problems with a first-order state constraint and application to continuation methods. ESAIM:
Control, Optimisation and Calculus of Variations 14 (4), 825–863.
176
Bonnans, J. F. and A. Hermant (2009, feb). Second-order analysis for optimal control problems
with pure state constraints and mixed control-state constraints. Annales de Institut Henri
Poincare (C) Non Linear Analysis 26 (2), 561–598.
Bonnans, J. F. B. and A. Hermant (2007b, jan). Well-posedness of the shooting algorithm
for state constrained optimal control problems with a single constraint and control. SIAM
Journal on Control and Optimization 46 (4), 1398–1430.
Born, M. (1906). Untersuchungen uber die Stabilitaet der elastischen Linie in Ebene und Raum,
under verschiedenen Grenzbedingungen. Ph. D. thesis, University of Goettingen.
Bosi, F., D. Misseroni, F. D. Corso, and D. Bigoni (2015, jul). Self-encapsulation, or the
‘dripping’ of an elastic rod. Proceedings of the Royal Society A: Mathematical, Physical and
Engineering Science 471 (2179), 20150195.
Bosi, F., D. Misseroni, F. Dal Corso, and D. Bigoni (2015, September). Development of config-
urational forces during the injection of an elastic rod. Extreme Mechanics Letters 4, 83–88.
00005.
Bryson, A. (2016). Applied optimal control: Optimization, estimation and control. CRC Press.
Bryson, A. E. and W. F. Denham (1964, jan). Optimal programming problems with inequality
constraints. ii - solution by steepest-ascent. AIAA Journal 2 (1), 25–34.
Bryson, A. E., W. F. Denham, and S. E. Dreyfus (1963, nov). Optimal programming problems
with inequality constraints. AIAA Journal 1 (11), 2544–2550.
Bulirsch, R. and J. Stoer (2010). Introduction to Numerical Analysis. Springer.
Byrd, R. H., N. I. Gould, J. Nocedal, and R. A. Waltz (2003, may). An algorithm for nonlinear
optimization using linear programming and equality constrained subproblems. Mathematical
Programming 100 (1).
Byrd, R. H., J. Nocedal, and R. A. Waltz (2006). Knitro: An Integrated Package for Nonlinear
Optimization, pp. 35–59. Boston, MA: Springer US.
Chai, H. (1998, July). The post-buckling response of a bi-laterally constrained column. 46 (7),
1155–1181.
Challamel, N., A. Kocsis, and C. Wang (2015, December). Discrete and non-local elastica.
International Journal of Non-Linear Mechanics 77, 128–140.
Cheatham, J. B. (1984, August). Helical Postbuckling Configuration of a Weightless Column
Under the Action of m Axial Load. Society of Petroleum Engineers Journal 24 (04), 467–472.
177
Chen, J., J. Wu, M. Luo, and J. Zhang (2014, jun). A knitro-based real-time locomotion method
for imitating the caterpillar-like climbing strategy. 11th IEEE International Conference on
Control & Automation (ICCA).
Chen, J.-S. and J. Fang (2013, October). Deformation sequence of a constrained spatial buckled
beam under edge thrust. International Journal of Non-Linear Mechanics 55, 98–101.
Chen, J.-S. and Y.-C. Lin (2013, April). Vibration and stability of a long heavy elastica on rigid
foundation. International Journal of Non-Linear Mechanics 50, 11–18.
Chen, J.-S., C.-J. Lu, and C.-Y. Lee (2015, November). On the use of energy method with
element splitting to determine the stability of a constrained elastica. International Journal of
Non-Linear Mechanics 76, 77–86.
Chen, J.-S. and H.-H. Wu (2011, January). Deformation and stability of an elastica under a
point force and constrained by a flat surface. 53 (1), 42–50.
Chen, V., V. Lin, and J. Cheatham (1989). An Analysis of Tubing and Casing Buckling in
Horizontal Wells. Offshore Technology Conference.
Chucheepsakul, S., S. Buncharoen, and T. Huang (1995, jul). Elastica of simple variable-arc-
length beam subjected to end moment. Journal of Engineering Mechanics 121 (7), 767–772.
Chucheepsakul, S. and T. Monprapussorn (2000, aug). Divergence pf variable arc-length elastica
pipes transporting fluid. Journal of Fluids and Structures 14 (6), 895–916.
Chucheepsakul, S., T. Monprapussorn, and T. Huang (2003, feb). Large strain formulations of
extensible flexible marine pipes transporting fluid. Journal of Fluids and Structures 17 (2),
185–224.
Chucheepsakul, S. and B. Phungpaigram (2004, jan). Elliptic integral solutions of variable-arc-
length elastica under an inclined follower force. ZAMM 84 (1), 29–38.
Clauvelin, N., B. Audoly, and S. Neukirch (2009, May). Elasticity and Electrostatics of Plec-
tonemic DNA. Biophysical Journal 96 (9), 3716–3723.
Comninou, M. (1978). The interface crack in a shear field. Journal of Applied Mechanics 45 (2),
287.
Cunha, J. (2004, March). Buckling of Tubulars Inside Wellbores: A Review on Recent Theoret-
ical and Experimental Works. SPE Drilling & Completion 19 (01), 13–19.
Davies, M. A. and F. C. Moon (1993, jan). 3-d spatial chaos in the elastica and the spinning top:
Kirchhoff analogy. Chaos: An Interdisciplinary Journal of Nonlinear Science 3 (1), 93–99.
178
Daviet, G., F. Bertails-Descoubes, and L. Boissieux (2011). A hybrid iterative solver for robustly
capturing coulomb friction in hair dynamics. pp. 1. ACM Press.
Dawson, R. (1984, October). Drill Pipe Buckling in Inclined Holes. 36 (10), 1734–1738.
Denoel, V. and E. Detournay (2011, February). Eulerian formulation of constrained elastica.
International Journal of Solids and Structures 48 (3-4), 625–636.
Dichmann, D. J. and J. H. Maddocks (2000). An Impetus-Striction Simulation of the Dynamics
of an Elastica. In Journal of Nonlinear Science (Ed.), Mechanics: From Theory to Computa-
tion, pp. 217–238. New York, NY: Springer New York.
Dichmann, D. J., J. H. Maddocks, and R. L. Pego (1996, December). Hamiltonian dynam-
ics of an elastica and the stability of solitary waves. Archive for Rational Mechanics and
Analysis 135 (4), 357–396.
Domokos, G., W. B. Fraser, and I. Szeberenyi (2003, November). Symmetry-breaking bifurca-
tions of the uplifted elastic strip. Physica D: Nonlinear Phenomena 185 (2), 67–77.
Domokos, G. and P. Holmes (1993, December). Euler’s problem, Euler’s method, and the
standard map; or, the discrete charm of buckling. Journal of Nonlinear Science 3 (1), 109–
151.
Domokos, G., P. Holmes, and B. Royce (1997, June). Constrained euler buckling. Journal of
Nonlinear Science 7 (3), 281–314.
Doraiswamy, S., K. R. Narayanan, and A. R. Srinivasa (2012, January). Finding minimum
energy configurations for constrained beam buckling problems using the Viterbi algorithm.
International Journal of Solids and Structures 49 (2), 289–297.
Duriez, C., S. Cotin, J. Lenoir, and P. Neumann (2006, November). New approaches to catheter
navigation for interventional radiology simulation. Computer Aided Surgery 11 (6), 300–308.
Elnagar, G., M. Kazemi, and M. Razzaghi (1995). The pseudospectral legendre method for
discretizing optimal control problems. IEEE Transactions on Automatic Control 40 (10),
1793–1796.
Euler, L. (1744). Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive
solutio problematis isoperimetrici lattissimo sensu accepti, chapter Additamentum 1. Lausanne
& Geneve, apud Marcum-Michaelem Bousquet & socios.
Falugi P., K. E. and van Wyk (2010). Imperial college london optimal control software user guide
(iclocs). Technical report, Department of Electrical Engineering, Imperial College London,
London, UK.
179
Fang, J., S.-Y. Li, and J.-S. Chen (2013, November). On a compressed spatial elastica con-
strained inside a tube. Acta Mechanica 224 (11), 2635–2647.
Fletcher, R. (1970, mar). A new approach to variable metric algorithms. The Computer Jour-
nal 13 (3), 317–322.
G. Fasano, J. P. (2012). Modeling and Optimization in Space Engineering. Springer-Verlag
GmbH.
Gao, D.-L. and W.-J. Huang (2015, August). A review of down-hole tubular string buckling in
well engineering. Petroleum Science 12 (3), 443–457.
Gao, G., Q. Di, S. Z. Miska, and W. Wang (2011). Stability Analysis of Pipe with Connectors
in Horizontal Wells. Society of Petroleum Engineers.
Gao, G. and S. Miska (2010, September). Dynamic Buckling and Snaking Motion of Rotating
Drilling Pipe in a Horizontal Well. 15 (3), 867–877.
Gao, G. and S. Z. Miska (2009a, December). Effects of Boundary Conditions and Friction on
Static Buckling of Pipe in a Horizontal Well. SPE Journal 14 (4), 782–796.
Gao, G. and S. Z. Miska (2009b). Effects of Friction on Post Buckling Behavior and Axial Load
Transferring of Pipe in a Horizontal Well. In SPE Journal. Society of Petroleum Engineers.
Gao, G. and S. Z. Miska (2009c). Effects of Friction on Post Buckling Behavior and Axial Load
Transferring of Pipe in a Horizontal Well. In SPE Journal. Society of Petroleum Engineers.
Gelfand, I. M. (2000). Calculus of variations. Mineola, N.Y: Dover Publications.
Gill, P. E., W. Murray, and M. A. Saunders (2002, jan). SNOPT: An SQP algorithm for
large-scale constrained optimization. SIAM Journal on Optimization 12 (4), 979–1006.
Goh, C. and K. Teo (1988, jan). Control parametrization: A unified approach to optimal control
problems with general constraints. Automatica 24 (1), 3–18.
Goldfarb, D. (1970, jan). A family of variable-metric methods derived by variational means.
Mathematics of Computation 24 (109), 23–23.
Goldstein, H., C. P. Poole, and J. L. Safko (2002). Classical mechanics. San Francisco: Addison
Wesley.
Goss, V. G. A. (2008, jul). The history of the planar elastica: Insights into mechanics and
scientific method. Science & Education 18 (8), 1057–1082.
180
Goyal, S., N. Perkins, and C. Lee (2005, oct). Nonlinear dynamics and loop formation in kirch-
hoff rods with implications to the mechanics of DNA and cables. Journal of Computational
Physics 209 (1), 371–389.
Hartl, R. F., S. P. Sethi, and R. G. Vickson (1995, jun). A survey of the maximum principles
for optimal control problems with state constraints. SIAM Review 37 (2), 181–218.
He, X. and A. Kyllingstad (1995, March). Helical Buckling and Lock-Up Conditions for Coiled
Tubing in Curved Wells. 10 (1), 10–15.
Hermant, A. (2009a, jun). Homotopy algorithm for optimal control problems with a second-order
state constraint. Applied Mathematics and Optimization 61 (1), 85–127.
Hermant, A. (2009b, jan). Stability analysis of optimal control problems with a second-order
state constraint. SIAM Journal on Optimization 20 (1), 104–129.
Holmes, P., G. Domokos, J. Schmitt, and I. Szeberenyi (1999, March). Constrained Euler
buckling: an interplay of computation and analysis. Computer Methods in Applied Mechanics
and Engineering 170 (3-4), 175–207.
Huang, W. and D. Gao (2014, July). Helical buckling of a thin rod with connectors constrained
in a cylinder. 84, 189–198.
Huang, W., D. Gao, and F. Liu (2015, April). Buckling Analysis of Tubular Strings in Horizontal
Wells. SPE Journal 20 (02), 405–416. 00005.
Huang, W., D. Gao, and Y. Liu (2016, nov). A study of tubular string buckling in vertical wells.
International Journal of Mechanical Sciences 118, 231–253.
Huang, W., D. Gao, and Y. Liu (2017, apr). Inter-helical and intra-helical buckling analyses
of tubular strings with connectors in horizontal wellbores. Journal of Petroleum Science and
Engineering 152, 182–192.
Huang, W., D. Gao, S. Wei, and P. Chen (2015, December). Boundary Conditions: A Key
Factor in Tubular-String Buckling. SPE Journal 20 (06), 1409–1420.
Huang, X. and M. Xie (2010). Evolutionary Topology Optimization of Continuum Structures:
Methods and Applications. John Wiley & Sons.
Humer, A. and H. Irschik (2009, may). Large deflections of beams with an unknown length of
the reference configuration. In 50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural
Dynamics, and Materials Conference. American Institute of Aeronautics and Astronautics.
Humer, A. and H. Irschik (2011, may). Large deformation and stability of an extensible elastica
with an unknown length. International Journal of Solids and Structures 48 (9), 1301–1310.
181
Hunt, G. W. and A. Blackmore (1997, nov). Homoclinic and heteroclinic solutions of upheaval
buckling. Philosophical Transactions of the Royal Society A: Mathematical, Physical and
Engineering Sciences 355 (1732), 2185–2195.
Huntington, G. T. and A. V. Rao (2008, mar). Comparison of global and local collocation
methods for optimal control. Journal of Guidance, Control, and Dynamics 31 (2), 432–436.
Huynen, A., E. Detournay, and V. Denoel (2016). Eulerian formulation of elastic rods. Proceed-
ings of the Royal Society A: Mathematical, physical and engineering sciences 472 (2190).
Jacobson, D., M. Lele, and J. Speyer (1971, aug). New necessary conditions of optimality for
control problems with state-variable inequality constraints. Journal of Mathematical Analysis
and Applications 35 (2), 255–284.
Jaculli, M. A., J. R. P. Mendes, and K. Miura (2017, may). Dynamic buckling with friction
inside directional wells. Journal of Petroleum Science and Engineering 153, 145–156.
Jae Ho Jung, Tae Jin Kang, and Kwan Soo Chung (2004, October). An Analytical Solution
to the Lateral Buckling of an Elastica with Self-Contact. Textile Research Journal 74 (10),
899–904.
Jin, M. and Z. Bao (2008, April). Sufficient conditions for stability of Euler elasticas. Mechanics
Research Communications 35 (3), 193–200.
Jin, M. and Z. Bao (2014, July). A proof of instability of some Euler elasticas. Mechanics
Research Communications 59, 37–41.
Jin, M. and Z. B. Bao (2013, July). Extensibility Effects on Euler Elastica ’ s Stability. Journal
of Elasticity 112 (2), 217–232.
Jorge Nocedal, S. W. (2006). Numerical Optimization. Springer New York.
Kameswaran, S. and L. T. Biegler (2007, nov). Convergence rates for direct transcription of
optimal control problems using collocation at radau points. Computational Optimization and
Applications 41 (1), 81–126.
Kirchhoff, G. (1859). Uber das gleichgewicht und die bewegung eines unendlich dunnen elastis-
chen stabes. Journal fur Mathematik (Crelle) 56, 285–313.
Klarbring, A. (1988, January). On discrete and discretized non-linear elastic structures in
unilateral contact (stability, uniqueness and variational principles). International Journal of
Solids and Structures 24 (5), 459–479.
Kolinski, J. M., P. Aussillous, and L. Mahadevan (2009, October). Shape and Motion of a Ruck
in a Rug. Physical Review Letters 103 (17).
182
Konings, M. K., E. B. Kraats, T. Alderliesten, and W. J. Niessen (2003, November). Analytical
guide wire motion algorithm for simulation of endovascular interventions. Medical & Biological
Engineering & Computing 41 (6), 689–700.
Kopp, C., C. D. Rahn, and F. W. Paul (2000, oct). Measuring deformations of limp fabrics for
material handling. Textile Research Journal 70 (10), 920–932.
Kornienko, I., L. T. Paiva, and M. do Rosário de Pinho (2014). Introducing state constraints in
optimal control for health problems. Procedia Technology 17, 415–422.
Kraft, D. (1985). On converting optimal control problems into nonlinear programming problems.
Computational Mathematical Programming , 261–280.
Kuru, E., A. Martinez, S. Miska, and W. Qiu (2000). The Buckling Behavior of Pipes and Its
Influence on the Axial Force Transfer in Directional Wells. 122 (3), 129.
Kuznetsov, V. and S. Levyakov (2002, September). Complete solution of the stability problem
for elastica of Euler’s column. International Journal of Non-Linear Mechanics 37 (6), 1003–
1009.
Lazarus, A., C. Maurini, and S. Neukirch (2015). Stability of discretized nonlinear elastic
systems. pp. 1–53.
Lazarus, A., J. T. Miller, and P. M. Reis (2013, August). Continuation of equilibria and stability
of slender elastic rods using an asymptotic numerical method. Journal of the Mechanics and
Physics of Solids 61 (8), 1712–1736.
Lee, A. A., C. Le Gouellec, and D. Vella (2015, December). The role of extensibility in the birth
of a ruck in a rug. Extreme Mechanics Letters 5, 81–87.
Lenoir, J., S. Cotin, C. Duriez, and P. Neumann (2006, June). Interactive physically-based
simulation of catheter and guidewire. Computers & Graphics 30 (3), 416–422.
Levenberg, K. (1944, jul). A method for the solution of certain non-linear problems in least
squares. Quarterly of Applied Mathematics 2 (2), 164–168.
Levien, R. (2008). The elastica: a mathematical history. Ph. D. thesis, University of California
at Berkeley.
Levyakov, S. V. and V. V. Kuznetsov (2010, April). Stability analysis of planar equilibrium
configurations of elastic rods subjected to end loads. Acta Mechanica 211 (1-2), 73–87.
Li, S., J. Qin, J. Guo, Y.-P. Chui, and P.-A. Heng (2011a). A Novel FEM-Based Numerical
Solver for Interactive Catheter Simulation in Virtual Catheterization. International Journal
of Biomedical Imaging 2011, 1–8.
183
Li, S., J. Qin, J. Guo, Y.-P. Chui, and P.-A. Heng (2011b). A Novel FEM-Based Numerical
Solver for Interactive Catheter Simulation in Virtual Catheterization. International Journal
of Biomedical Imaging 2011, 1–8.
Liberzon, D. (2012). Calculus of variations and optimal control theory: a concise introduction.
Princeton University Press.
Liu, C.-W. and J.-S. Chen (2013, January). Effect of friction on the planar elastica constrained
inside a circular channel with clearance. 50 (1), 270–278.
Liu, J., T. Su, N. Wicks, J. Pabon, and K. Bertoldi (2015, October). The Instability Mechanism
of a Confined Rod Under Axial Vibrations. Journal of Applied Mechanics 83 (1).
Lloyd, D., W. Shanahan, and M. Konopasek (1978, jan). The folding of heavy fabric sheets.
International Journal of Mechanical Sciences 20 (8), 521–527.
Love, A. E. H. (2013). A treatise on the mathematical theory of elasticity. Cambridge: Cambridge
University Press.
Lu, Z.-H. and J.-S. Chen (2008, may). Deformations of a clamped–clamped elastica inside a
circular channel with clearance. International Journal of Solids and Structures 45 (9), 2470–
2492.
Lubinski, A. (1950, January). A Study of the Buckling of Rotary Drilling Strings.
Lubinski, A. and W. Althouse (1962, June). Helical Buckling of Tubing Sealed in Packers.
14 (6), 655–670.
Maddocks, J. (1984). Stability of nonlinearly elastic rods. Archive for Rational Mechanics and
Analysis 85 (4).
Mahadevan, L. and J. B. Keller (1999, jan). Periodic folding of thin sheets. SIAM Review 41 (1),
115–131.
Majumdar, A. and A. Goriely (2013, jun). Static and dynamic stability results for a class of three-
dimensional configurations of kirchhoff elastic rods. Physica D: Nonlinear Phenomena 253,
91–101.
Majumdar, A., C. Prior, and A. Goriely (2012, September). Stability Estimates for a Twisted
Rod Under Terminal Loads: A Three-dimensional Study. Journal of Elasticity 109 (1), 75–93.
Malanowski, K. (1995, sep). Stability and sensitivity of solutions to nonlinear optimal control
problems. Applied Mathematics & Optimization 32 (2), 111–141.
184
Malanowski, K. (1997, jan). Sufficient optimality conditions for optimal control subject to state
constraints. SIAM Journal on Control and Optimization 35 (1), 205–227.
Malanowski, K. and H. Maurer (2001). Sensitivity analysis for optimal control problems subject
to higher order state constraints. Annals of Operations Research 101 (1/4), 43–73.
Malanowski, K., H. Maurer, and S. Pickenhain (2004, dec). Second-order sufficient conditions
for state-constrained optimal control problems. Journal of Optimization Theory and Applica-
tions 123 (3), 595–617.
Maltby, T. and C. Calladine (1995a, September). An investigation into upheaval buckling of
buried pipelines - ii. theory and analysis of experimental observations. International Journal
of Mechanical Sciences 37 (9), 965–983.
Maltby, T. and C. Calladine (1995b, sep). An investigation into upheaval buckling of buried
pipelines-ii. experimental apparatus and some observations. International Journal of Mechan-
ical Sciences 37 (9), 943–963.
Manning, R. S. (2009, February). Conjugate Points Revisited and Neumann - Neumann Prob-
lems. SIAM Review 51 (1), 193–212.
Manning, R. S. (2014, April). A Catalogue of Stable Equilibria of Planar Extensible or Inextensi-
ble Elastic Rods for All Possible Dirichlet Boundary Conditions. Journal of Elasticity 115 (2),
105–130. 00003.
Manning, R. S. and G. B. Bulman (2005, August). Stability of an elastic rod buckling into
a soft wall. Proceedings of the Royal Society A: Mathematical, Physical and Engineering
Sciences 461 (2060), 2423–2450.
Manning, R. S., K. A. Rogers, and J. H. Maddocks (1998, December). Isoperimetric conjugate
points with application to the stability of DNA minicircles. Proceedings of the Royal Society
A: Mathematical, Physical and Engineering Sciences 454 (1980), 3047–3074.
Martinez, A., S. Miska, E. Kuru, and J. Sorem (2000). Experimental Evaluation of the Lateral
Contact Force in Horizontal Wells. 122 (3), 123.
Maurer, H. (1979, mar). Differential stability in optimal control problems. Applied Mathematics
& Optimization 5 (1), 283–295.
Maurer, H. and K. Malanowski (1998, feb). Sensitivity analysis for state constrained optimal
control problems. Discrete and Continuous Dynamical Systems 4 (2), 241–272.
Maurer, H. and H. D. Mittelmann (1991, jan). The non-linear beam via optimal control with
bounded state variables. Optimal Control Applications and Methods 12 (1), 19–31.
185
Maurer, H. and H. J. Oberle (2002, jan). Second order sufficient conditions for optimal con-
trol problems with free final time: The Riccati approach. SIAM Journal on Control and
Optimization 41 (2), 380–403.
Maurer, H. and S. Pickenhain (1995, sep). Second-order sufficient conditions for control problems
with mixed control-state constraints. Journal of Optimization Theory and Applications 86 (3),
649–667.
McCann, R. and P. Suryanarayana (1994). Experimental Study Of Curvature And Frictional
Effects On Buckling. Offshore Technology Conference.
McCourt, I., T. Truslove, and J. Kubie (2004, September). On the penetration of tubular drill
pipes in horizontal oil wells. Proceedings of the Institution of Mechanical Engineers, Part C:
Journal of Mechanical Engineering Science 218 (9), 1063–1081.
Menand, S., H. Sellami, J. Akowanou, C. Simon, L. P. Y. Macresy, P. Isambourg, and D. C.
Dupuis (2008). How Drillstring Rotation Affects Critical Buckling Load ? Society of
Petroleum Engineers.
Mesterton-Gibbons, M. (2009). A Primer on the Calculus of Variations and Optimal Control
Theory. American Mathematical Society.
Miersemann, E. and H. Mittelmann (1991, June). Stability and continuation of solutions to
obstacle problems. Journal of Computational and Applied Mathematics 35 (1-3), 5–31.
Miersemann, E. and H. D. Mittelmann (1989, June). Continuation for parametrized nonlinear
variational inequalities. Journal of Computational and Applied Mathematics 26 (1-2), 23–34.
Miersemann, E., H. D. Mittelmann, and W. Tornig (1986). A free boundary problem and
stability for the nonlinear beam. Mathematical Methods in the Applied Sciences 8 (1), 516–
532. 00012.
Mikata, Y. (2007, May). Complete solution of elastica for a clamped-hinged beam, and its
applications to a carbon nanotube. Acta Mechanica 190 (1-4), 133–150.
Miller, J., T. Su, E. Dussan V., J. Pabon, N. Wicks, K. Bertoldi, and P. Reis (2015, October).
Buckling-induced lock-up of a slender rod injected into a horizontal cylinder. International
Journal of Solids and Structures 72, 153–164.
Miller, J. T. (2014). Mechanical behavior of elastic rods under constraint. Thesis, Massachusetts
Institute of Technology. Thesis: Ph. D., Massachusetts Institute of Technology, Department
of Civil and Environmental Engineering, 2014.
186
Miska, S. and J. Cunha (1995). An Analysis of Helical Buckling of Tubulars Subjected to Axial
and Torsional Loading in Inclined Wellbores. Society of Petroleum Engineers.
Miska, S., W. Qiu, L. Volk, and J. Cunha (1996). An Improved Analysis of Axial Force Along
Coiled Tubing in Inclined/Horizontal Wellbores. Society of Petroleum Engineers.
Misseroni, D., G. Noselli, D. Zaccaria, and D. Bigoni (2015, September). The deformation of an
elastic rod with a clamp sliding along a smooth and curved profile. International Journal of
Solids and Structures 69-70, 491–497.
Mitchell, R. (1982, October). Buckling Behavior of Well Tubing: The Packer Effect. 22 (5),
616–624.
Mitchell, R. (1986, December). Simple Frictional Analysis of Helical Buckling of Tubing. 1 (6),
457–465.
Mitchell, R. (1988, September). New Concepts for Helical Buckling. 3 (3), 303–310.
Mitchell, R. F. (2005). The Pitch of Helically Buckled Pipe. Society of Petroleum Engineers.
Morse, M. (2013). Introduction to Analysis in the Large. Literary Licensing, LLC.
Nocedal, J. (2006). Numerical optimization (2nd ed ed.). Springer series in operations research.
New York: Springer.
O ’ Reilly, O. M. and D. M. Peters (2011, November). On Stability Analyses of Three Classical
Buckling Problems for the Elastic Strut. Journal of Elasticity 105 (1-2), 117–136.
O Reilly, O. and T. Tresierras (2011, February). On the static equilibria of branched elastic
rods. International Journal of Engineering Science 49 (2), 212–227.
O’Keefe, A. (2015). Determining the feasibility of building Len Lye’s kinetic artwork sun, land
and sea. Ph. D. thesis, University of Kanterbury.
O’Keefe, A. N. and S. D. Gooch (2015, feb). On the properties of a traveling ruck in a flexible
strip. Journal of Applied Mechanics 82 (4).
O’Reilly, O. M. and D. M. Peters (2012, January). Nonlinear stability criteria for tree-like struc-
tures composed of branched elastic rods. Proceedings of the Royal Society A: Mathematical,
Physical and Engineering Sciences 468 (2137), 206–226.
Paiva, L. T. and F. Fontes (2015, apr). Adaptive time–mesh refinement in optimal control
problems with state constraints. Discrete and Continuous Dynamical Systems 35 (9), 4553–
4572.
187
Paslay, P. R. and D. B. Bogy (1964). The Stability of a Circular Rod Laterally Constrained to
Be in Contact With an Inclined Circular Cylinder. 31 (4), 605.
Peterson, K. and R. Manning (2010, March). Ineffective perturbations in a planar elastica.
Involve, a Journal of Mathematics 2 (5), 559–580.
Plaut, R. H. (2015, may). Formulas to determine fabric bending rigidity from simple tests.
Textile Research Journal 85 (8), 884–894.
Pocheau, A. and B. Roman (2004, June). Uniqueness of solutions for constrained Elastica.
Physica D: Nonlinear Phenomena 192 (3-4), 161–186.
Pulngern, T., T. Sudsanguan, C. Athisakul, and S. Chucheepsakul (2013, mar). Elastica of
a variable-arc-length circular curved beam subjected to an end follower force. International
Journal of Non-Linear Mechanics 49, 129–136.
Reddien, G. W. (1979, mar). Collocation at gauss points as a discretization in optimal control.
SIAM Journal on Control and Optimization 17 (2), 298–306.
Ro, W.-C., J.-S. Chen, and S.-Y. Hong (2010, August). Vibration and stability of a constrained
elastica with variable length. International Journal of Solids and Structures 47 (16), 2143–
2154.
Roman, B. and A. Pocheau (1999, June). Buckling cascade of thin plates: Forms, constraints
and similarity. Europhysics Letters (EPL) 46 (5), 602–608.
Ross, M. I. (2015). A Primer on Pontryagin’s Principle in Optimal Control. Collegiate Pub-
lishers.
Sachkov, Y. L. (2007, December). Optimality of Eulerâs elastica. Doklady Mathematics 76 (3),
817–819.
Sachkov, Y. L. (2008a, July). Conjugate Points in the Euler Elastic Problem. Journal of
Dynamical and Control Systems 14 (3), 409–439.
Sachkov, Y. L. (2008b, April). Maxwell strata in the Euler elastic problem. Journal of Dynamical
and Control Systems 14 (2), 169–234.
Santillan, S. T., L. N. Virgin, and R. H. Plaut (2006). Post-buckling and Vibration of Heavy
Beam on Horizontal or Inclined Rigid Foundation. Journal of Applied Mechanics 73 (4), 664.
Schallamach, A. (1971, April). How does rubber slide? Wear 17 (4), 301–312.
188
Schulz, M. and S. Pellegrino (2001). Equilibrium Paths of Mechanical Systems with Unilateral
Constraints I. Theory. In Mathematical, Physical and Engineering Sciences, Volume 456, pp.
2223–2242.
Shanno, D. F. (1970, sep). Conditioning of quasi-newton methods for function minimization.
Mathematics of Computation 24 (111), 647–647.
Sorenson, K. G. and J. B. Cheatham (1986). Post-Buckling Behavior of a Circular Rod Con-
strained Within a Circular Cylinder. 53 (4), 929.
Spillmann, J. and M. Teschner (2007). CoRdE: Cosserat rod elements for the dynamic simulation
of one-dimensional elastic objects. In In Proc. ACM Siggraph/Eurographics Symposium of
computer animation, pp. 63–72.
Stuart, I. M. (1966, September). A loop test for bending length and rigidity. British Journal of
Applied Physics 17 (9), 1215–1220.
Su, T., N. Wicks, J. Pabon, and K. Bertoldi (2013, July). Mechanism by which a frictionally
confined rod loses stability under initial velocity and position perturbations. 50 (14), 2468–
2476.
Sun, Y., Y. Yu, and B. Liu (2015, January). Closed form solutions for predicting static and
dynamic buckling behaviors of a drillstring in a horizontal well. European Journal of Mechanics
- A/Solids 49, 362–372.
Tang, W., T. R. Wan, D. A. Gould, T. How, and N. W. John (2012, August). A Stable and Real-
Time Nonlinear Elastic Approach to Simulating Guidewire and Catheter Insertions Based on
Cosserat Rod. IEEE Transactions on Biomedical Engineering 59 (8), 2211–2218.
Thongyothee, C. and S. Chucheepsakul (2015, jun). Postbuckling of unknown-length nanobeam
considering the effects of nonlocal elasticity and surface stress. International Journal of Ap-
plied Mechanics 07 (03).
Timoshenko, S. (2009). Theory of elastic stability (2nd ed., Dover ed ed.). Mineola, N.Y: Dover
Publications.
Ulbrich, S. (2003, nov). On the superlinear local convergence of a filter-SQP method. Mathe-
matical Programming 100 (1).
Vaillette, D. P. and G. G. Adams (1983). An Elastic Beam Contained in a Frictionless Channel.
Journal of Applied Mechanics 50 (3), 693.
van der Heijden, G., S. Neukirch, V. Goss, and J. Thompson (2003, jan). Instability and self-
contact phenomena in the writhing of clamped rods. International Journal of Mechanical
Sciences 45 (1), 161–196.
189
van der Heijden, G. H. M. (2001, March). The static deformation of a twisted elastic rod
constrained to lie on a cylinder. 457 (2007), 695–715.
Vella, D., A. Boudaoud, and M. Adda-Bedia (2009, October). Statics and Inertial Dynamics of
a Ruck in a Rug. Physical Review Letters 103 (17).
Viswanathan, K., A. Mahato, and S. Chandrasekar (2015, jan). Nucleation and propagation of
solitary schallamach waves. Physical Review E 91 (1).
Vlassenbroeck, J. and R. V. Dooren (1988, apr). A chebyshev technique for solving nonlinear
optimal control problems. IEEE Transactions on Automatic Control 33 (4), 333–340.
von Stryk, O. (1993). Numerical solution of optimal control problems by direct collocation.
ISNM International Series of Numerical Mathematics: Optimal control , 129–143.
Wachter, A. and L. T. Biegler (2005, apr). On the implementation of an interior-point filter line-
search algorithm for large-scale nonlinear programming. Mathematical Programming 106 (1),
25–57.
Waltz, R., J. Morales, J. Nocedal, and D. Orban (2005, nov). An interior algorithm for non-
linear optimization that combines line search and trust region steps. Mathematical Program-
ming 107 (3), 391–408.
Wang, C. (1984a, January). Buckling and postbuckling of the lying sheet. International Journal
of Solids and Structures 20 (4), 351–358.
Wang, C. (1997, November). Post-buckling of a clamped-simply supported elastica. International
Journal of Non-Linear Mechanics 32 (6), 1115–1122.
Wang, C. M., C. M. Wang, C. Y. Wang, and J. N. Reddy (2004). Exact solutions for buckling
of structural members. Number 6 in CRC series in computational mechanics and applied
analysis. Boca Raton, FL: CRC Press.
Wang, C.-Y. (1981a). Folding of Elastica-Similarity Solutions. Journal of Applied Mechan-
ics 48 (1), 199.
Wang, C. Y. (1981b, January). The Ridging of Heavy Elastica. ZAMM - Journal of Applied
Mathematics and Mechanics 61 (2), 125–126.
Wang, C. Y. (1984b, June). On Symmetric Buckling of a Finite Flat-Lying Heavy Sheet. Journal
of Applied Mechanics 51 (2), 278–282.
Wang, C. Y. (1986). A critical review of the heavy elastica. International Journal of Mechanical
Sciences 28 (8), 549–559.
190
Weiss, H. (2002). Dynamics of Geometrically Nonlinear Rods: I. Mechanical Models and Equa-
tions of Motion. Nonlinear Dynamics 30 (4), 357–381.
Wicks, N., B. L. Wardle, and D. Pafitis (2008, March). Horizontal cylinder-in-cylinder buckling
under compression and torsion: Review and application to composite drill pipe. International
Journal of Mechanical Sciences 50 (3), 538–549.
Wu, J. and H. Juvkam-Wold (1995a, mar). Coiled tubing buckling implication in drilling and
completing horizontal wells. SPE Drilling & Completion 10 (01), 16–21.
Wu, J. and H. C. Juvkam-Wold (1993). Helical Buckling of Pipes in Extended Reach and Hori-
zontal Wells Part 2: Frictional Drag Analysis. Journal of Energy Resources Technology 115 (3),
196.
Wu, J. and H. C. Juvkam-Wold (1995b). Buckling and Lockup of Tubulars in Inclined Wellbores.
117 (3), 208.
Wu, J. and H. C. Juvkam-Wold (1995c). The Effect of Wellbore Curvature on Tubular Buckling
and Lockup. Journal of Energy Resources Technology 117 (3), 214.
Wu, J., H. C. Juvkam-Wold, and R. Lu (1993). Helical Buckling of Pipes in Extended Reach
and Horizontal Wells Part 1: Preventing Helical Buckling. 115 (3), 190.
Zhou, X., C. Majidi, and O. M. O’Reilly (2015, sep). Flexing into motion: A locomotion
mechanism for soft robots. International Journal of Non-linear Mechanics 74, 7–17.
Appendix A
Constrained Buckling of Drillstrings
A.1 Introduction
The buckling of columns under compression, first investigated by Euler, is a well researched
problem in structural and mechanical engineering (Timoshenko, 2009). However, the constrained
buckling of a rod, a key problem in petroleum engineering, has not yet been fully explored, as
other factors have to be taken into account when inserting a slender elastic rod, the so-called
drillstring inside a rigid cylindrical conduit, the wellbore.
The main differences between the classical buckling of a column and the buckling of a drill-
string are the constraint of the cylindrical conduit, the dominant role of gravity in the loading,
and the effect of friction. These factors affect the force distribution within the drillstring de-
pending on the geometry of the wellbore.
In a vertical, or steeply inclined well, the weight on the bit (i.e., the axial load transmitted
by the drillstring to the bit) is the dominant factor that can cause buckling of the part of the
drillstring under compression. In horizontal, or moderately inclined borehole, contact between
the drillstring and the wellbore due to gravity leads to significant axial friction forces during
insertion of the drillstring. Friction forces, which oppose the motion of the drillstring, cause a
build-up of the axial force that could lead to buckling of the drillstring. In both cases, the rest
of the length of the drillstring (i.e., away from the bit) is under tension and remains straight.
In early studies (Dawson, 1984; Paslay and Bogy, 1964) the effect of friction was neglected.
Based on this assumption, and since the drillstring has a very large length/diameter ratio,
the first buckling mode is expected to be sinusoidal. Past the critical load, or the sinusoidal
buckling load, there is a transition of the configuration of the rod from a sinusoidal to a helical, or
secondary buckling shape, which is observed due to the increasing contact force between the rod
and the wellbore. This transition region is also called post-buckling region. Its understanding
is crucial in oil-drilling operations (Cheatham, 1984; Sorenson and Cheatham, 1986) since any
191
192
potentiality of failure should be avoided.
Friction is the most important factor that affects the buckling response of drillstring in an
inclined/horizontal well. When friction is under consideration (Gao and Miska, 2009c; Martinez
et al., 2000; McCourt et al., 2004; Miller, 2014; Wu and Juvkam-Wold, 1993), past the first mode
buckling load, the normal contact force between the rod and the wellbore increases significantly,
causing a substantial increase of the friction force. Once secondary buckling is reached, limited
additional insertion is feasible. When further insertion is impossible, a “sticking” behavior be-
tween the rod and the wellbore is recognized. This condition is the so-called ’lock-up failure’
(He and Kyllingstad, 1995; Wicks et al., 2008; Wu and Juvkam-Wold, 1995b).
The Section starts with the description of the buckling response of a drillstring constrained by
a vertical, or steeply inclined borehole. All stability factors are illustrated. Then we present the
stability behavior of a drillstring constrained by a horizontal, or a moderately inclined borehole
which is the main concern of the study. Special care is placed on the effect of friction.
A.1.1 Vertical boreholes
Since the 1950 pioneering work of Lubinski (1950) on the buckling of drillstrings, there have
been various efforts aimed at assessing the critical weight on bit at which in-plane buckling takes
place (Dawson, 1984; Gao and Miska, 2009b; Paslay and Bogy, 1964). The key question that
arises when attempting to solve this problem is the determination of the characteristic length
`∗ that scales the buckling wavelength.
In a vertical, or steeply inclined borehole, the drillstring does not contact the borehole prior
to buckling, and the buckling problem is analogous to that of a beam buckling under its own
weight. The characteristic length and the force scale are given by
`v∗ =
(
EI
w
)1/3
, fv∗ =
(
EIw2
)1/3 (A.1)
where EI is the flexural rigidity of the bottom-hole assembly (BHA) and w is the effective
weight/length of the BHA (after subtracting the weight of the displaced drilling fluid). This
leads to the dimensionless quantities ς = s/`v∗ and Q = F/fv∗ , where s is the arc-length of the
drillstring and F is the axial force. Assigning the origin at the so-called neutral point at which
the axial force vanishes and neglecting the frictional effect, the axial force Q is identical to the
distance ς, or in other words the weight W = ws is identical to the axial force F .
Considering that the drillstring is initially stress-free and increasing gradually the weight
on the bit, Lubinski (1950) calculated the critical weight on bit Wc that causes buckling of the
BHA. In particular, the drillstring first buckles in a planar deformation shape which is consistent
with a first-order buckling mode with critical length ςc = 1.94. Lubinski (1950) also calculated
the secondary critical length ςc = 3.75 for a second-order buckling mode.
193
Later, Lubinski and Althouse (1962) mentioned that the drillstring does not buckle into a
higher-order planar buckling mode. Instead the onset of a spatial deformation is met. Lubinski
and Althouse (1962) assumed that the drillstring is in continuous contact with the borehole and
more particularly a helical deformation shape is predefined. In this paper, based on geometric
considerations, a force-pitch relationship is derived for a weightless pipe as
F =
8pi2EI
p2
(A.2)
where p is a constant pitch (i.e., distance between spirals) throughout and F is the applied
axial compression force at the lower end of the tubing which is above the helical buckling load.
Additionally, the same expression can be applied to non-weightless pipes, where the pitch is not
constant but varies with the distance from the neutral point. Typically, in oil drilling operations,
the distance of the lower end to the neutral point is very large (i.e., several thousands of feet)
leading to a number of spirals around 100 ∼ 200 (Lubinski and Althouse, 1962).
Cheatham (1984) also investigated the case when the axial load starts decreasing from its
helical buckling load for a weightless pipe. During unloading from (A.2), the pitch remains
initially constant until the load reaches the value 4pi2EI/p2. Any further decrease of the axial
load is then accompanied by an increase of the pitch length and the pipe starts losing contact
with the wall. This leads to the bounds of the force in terms of the pitch length 4pi2EI/p2 ≤
F ≤ 8pi2EI/p2, which was later verified by Miska and Cunha (1995).
Recently, Huang et al. (2016) demonstrated a more comprehensive stability analysis of a
drillstring constrained by a vertical wellbore. Huang et al. (2016) provides five distinct deforma-
tion shapes of the drillstring depending on the weight on the bit; (1) the straight configuration,
(2) the planar shape with discrete contacts, (3) the spatial shape with discrete contacts, (4) the
spatial shape with discrete contacts and a single continuous segment, and (5) the helical shape.
These configurations are successively obtained as we gradually increase the weight on the bit.
The sequence of the deformation shapes is shown in Figure A.1.
For the planar and spatial deformation shapes with discrete contacts, a finite number of
distinct segments is obtained. Every configuration is then solved by the beam-column model
(see Section A.2). The complete solution is finally obtained by applying the necessary continuity
conditions at the common extremities. Nevertheless when a continuous segment is also present,
its solution is obtained by the buckling differential equation (see Section A.2).
To clarify the procedure followed by Huang et al. (2016), we need to denote two distinct
positions which are the extremities of the drillstring; (1) ςo ≤ 0 is the length on the hook (i.e.,
upper end) and (2) ςc ≥ 0 is the length on the bit (i.e., lower end), where the origin is assigned
at the neutral point, as before (Lubinski, 1950). In this manner, the critical length on the bit ςc
ranges between two limits; (1) the lower limit ς(l)c is derived by assuming ςo → −∞ and (2) the
upper limit ς(u)c is obtained by setting ςo = 0. These limits are identical to the lower and upper
194
bounds of the axial force ς(l)c ≤ Qc ≤ ς(u)c for a frictionless vertical borehole. Depending on the
weight on the bit, distinct deformation shapes are obtained with different associated bounds.
Let us now describe in more detail the buckling mode transition for a drillstring with weight
constrained by a vertical borehole, as explained by Huang et al. (2016). Initially the drillstring
is unstressed, as shown in Figure A.1(a). When the axial force reaches the value of 1.8546 ≤
Qv2d ≤ 2.6012, the drillstring buckles into a planar deformation shape which is in contact with
the borehole. When the drillstring is weightless a first-order buckling mode is only present with a
single discrete contact. This is not the case when the drillstring is non-weightless. As we increase
the weight on the bit, the drillstring comes in contact with the borehole in more than one discrete
contact implying that higher-order buckling modes can be gradually obtained. The first two-
order buckling modes are illustrated in Figure A.1(b,c). If 3.9494 ≤ Qv3d ≤ 4.3758 is reached,
the planar deformation shape becomes unstable and a spatial deformation shape appears, see
Figure A.1(d). This configuration involves several discrete contact points with the borehole. As
shown in Figure A.1(e), the discrete contact at the middle part of the drillstring is successively
transformed to a line, or continuous segment when 5.2497 ≤ Qvl ≤ 6.1864 is achieved. If we
further increase the weight on the bit, the axial load becomes 8.3954 ≤ Qvh ≤ 9.2453 and one
pitch of helix is developed leading to a helical deformation shape, as indicated in Figure A.1(f).
The analytical derivation can be found in Huang et al. (2016).
195
Figure A.1: Configurations of the drillstring constrained by a vertical well; (a) Initial straight
configuration, (b) first-order planar shape, (c) second-order planar shape, (d) spatial shape, (e)
spatial shape with a continuous segment, and (f) helical shape Huang et al. (2016).
Taking into account the buckling differential equation for a helical deformation shape, the
contact force can be also computed. In particular, using (A.2) Mitchell (1988) derived the
contact force between the drillstring and the vertical borehole for a helical shape when the
frictional effect is omitted;
N = %Q
2
h
4
(A.3)
where N = N/w is the contact force per unit length of the drillstring and % = r/`v∗ is the
dimensionless radial clearance.
Nevertheless if the friction is present, the axial force Q does not coincide with the distance
ς. In this case, the differential equation of the axial force can be expressed as follows
dQ
dς
= 1− µ%
4
Q2 (A.4)
where µ is the friction coefficient (Mitchell, 1986). Integrating (A.4) Mitchell (1986) calculated
196
an analytical expression of the axial force for a helical shape;
Q = 2√
µ%
[
tanh
(
1
2
ς
√
µ%+ c
)]
(A.5)
where c is an integral constant. Recently, Huang et al. (2016) defined this constant c by assigning
the position of the entry point of the line, or continuous contact ςl, leading to the following
analytical formula
Q(ς) =
2√
µ%
tanh
(
1
2
(ς − ςl)√µ%+ arctanh
(
1
2
ςl
√
µ%
))
(A.6)
More details can be found in Huang et al. (2016).
A.1.2 Horizontal/Inclined boreholes
In a horizontal, or moderately inclined borehole, the characteristic length arises from consider-
ation involving the transverse deflection of a horizontal beam under its own weight relative to
the clearance between the drillstring and the borehole wall.
By assuming a sinusoidal deflection of the drillstring of specific mode, the critical buckling
load for a drillstring constrained by a horizontal borehole subject to both compression and
gravity is derived by Paslay and Bogy (1964);
Fs =
EIpi2
λ2
(
4 +
λ4
4pi4
w
EIr
)
(A.7)
where r is the clearance between the drillstring and the borehole and λ is the wavelength. The
critical buckling load (A.7) is equivalent to the buckling load of a beam on an elastic foundation
of stiffness β = w/r.
After reaching the critical buckling load and if we further increase the axial force, the sinu-
soidal deformation shape becomes unstable and a new configuration of helical shape appears.
Chen et al. (1989) derived a formula similar to (A.7) for the initiation of secondary, or helical
buckling in inclined/horizontal wellbores assuming constant axial load along the helical buckling
process;
Fh = EI
pi2
λ2
(
4 +
λ4
pi4
w
2EIr
)
(A.8)
Adopting as length and force scales
`h∗ =
(
EIr
w
)1/4
, fh∗ =
(
EIw
r
)1/2
(A.9)
we get ς = s/`h∗ , σ = λ/`h∗ and Q = F/fh∗ , where s is the arc-length of the drillstring, λ is
the wavelength and F is the axial force. The dimensionless form of the sinusoidal and helical
197
buckling loads can be written as follows
Qs = pi
2
σ2s
(
4 +
σ4s
4pi4
)
, Qh = pi
2
σ2h
(
4 +
σ4h
2pi4
)
For a sufficiently long cylinder (Wicks et al., 2008) the asymptotic length-independent loads are
given by Qs = 2 and Qh = 2
√
2 with wavelengths σs = 2pi and σh = 2
√
2pi for sinusoidal and
helical buckling, respectively. For inclined wells the same formula can be applied if we replace
w by w cosα with α = 0, α = pi/2 for a horizontal and a vertical wellbore, respectively.
Several studies for the derivation of the sinusoidal and helical buckling loads can be found
which are typically based on predefined displacement fields (Wicks et al., 2008; Gao and Huang,
2015; Cunha, 2004; Arslan et al., 2012). Even though the critical buckling loads are well-
established, there is no complete understanding of the transition stage from the sinusoidal
buckling load Qs to the onset of secondary buckling Qh. Nevertheless several suggestions are
available. According to (Wu et al., 1993) the axial force increases linearly from the sinusoidal
buckling load Qs = 2 to the helical buckling load Qh = 2
√
2− 1. Miska et al. (1996) suggested
a linear increase of the axial force for the entire loading process which leads to higher values of
buckling load Qs = 3.75 and Qh = 4
√
2. Similar assumptions can be found in (Gao and Miska,
2009b; Wicks et al., 2008; Gao and Huang, 2015). The difference between these works clearly
indicates that further examination is required.
Effect of boundary conditions
The effect of boundary conditions on the stability analysis of a drillstring constrained by a
horizontal borehole is an important factor, as explained by (Mitchell, 1982; Wu and Juvkam-
Wold, 1995a; Huang et al., 2015; Gao and Miska, 2009a). The main concern is the influence of
the boundary conditions when a sinusoidal, or a helical deformation shape has been developed.
In particular, it is assumed that the configuration of the drillstring is composed by two parts; (1)
a transition region close to the extremities of the drillstring and (2) a fully-developed sinusoidal,
or helical segment in the middle (Huang et al., 2015).
Mitchell (1982) was the first who attempted to involve this transition region in the buckling
analysis of the drillstrings. In particular, Mitchell (1982) assumed that the transition region
involves a single suspended segment which is solved by the beam-column method while the
helical shape is derived by solving the buckling differential equation. A similar idea was also
adopted by Sorenson and Cheatham (1986) and Mitchell (2005). From these studies it was
found that the transition region is smaller than one pitch of helix. Hence, if a drillstring has
at least 3.5 pitches of helix, the transition section can be considered to be negligible (Wu and
Juvkam-Wold, 1995c). The derived lengths of the transition section for fixed and pinned ends
based on different approaches are summarized in Huang et al. (2015).
198
Recently, Huang et al. (2015) classified the boundary conditions based on a virtual work ap-
proach. The first type of boundary conditions corresponds to zero virtual work of the bending
moment and lateral force. In this case, the fully-developed helical deformation shape is unaf-
fected by the boundary conditions if the drillstring is sufficiently long L/`h∗ ≥ 5pi, as explained
in detail by Gao and Miska (2009a,b). This assumption is typically adopted in previous studies
(Wicks et al., 2008; Gao and Miska, 2009a). Nevertheless in the second type of boundary con-
ditions, the virtual work of the bending moment and lateral force is non-negligible and it can
influence the helical configuration. More precisely, the stability of the helical buckling shape
may change depending on the boundary conditions. For instance, the helical buckling direction
can abruptly reverse whenever the virtual work of the bending moment and/or the lateral force
goes across some critical values. These critical conditions are analytically derived by (Huang
et al., 2015,?).
Figure A.2: Configuration of a weightless drillstring constrained in a straight borehole Huang
et al. (2015)
Effect of friction
Most studies on the constrained stability analysis of a drillstring do not consider the frictional
effect (Cunha, 2004; Gao and Huang, 2015). In fact, an energy approach is typically adopted
which provides the critical and secondary buckling loads for the frictionless case (Wicks et al.,
2008). Nevertheless friction alters the buckling response of a confined drillstring. In the presence
of friction, the buckling load of a confined drillstring has been reported to be substantially higher
than the frictionless case (Gao and Miska, 2009b,a; Jaculli et al., 2017). The unloading process
also reveals a hysteresis behavior which cannot be predicted by the classical frictionless theory
(McCann and Suryanarayana, 1994).
While the friction has been shown to enhance the critical buckling load, its effect can be also
catastrophic. After initiation of the buckling, the friction can cause a diminishing transfer of the
axial force which may subsequently lead to a lock-up failure. This phenomenon was first studied
by Wu et al. (1993) who computed an analytical expression of the axial force in horizontal and
inclined wells. Other studies are also available which additionally consider the effect of boundary
199
conditions (Gao and Miska, 2009b; Jaculli et al., 2017).
Recently, Su et al. (2013) studied the buckling behavior of a drillstring constrained by a
horizontal borehole based on a dynamic perspective. Using the buckling differential equation
and applying a Fourier series theory, a second-order ODE is deduced similar to a mass-spring
system. Based on this approach, Su et al. (2013) investigated how friction affects the dynamics
of the system and accordingly the buckling behavior of the confined rod. It was proved that the
friction enhances the critical buckling loads, as expected. Interestingly, the initial conditions
were found to be the key-factor in the buckling response of the confined drillstring when friction
is non-negligible. Liu et al. (2015) extended this study in order to understand the additional
effect of the axial vibration. Liu et al. (2015) found that this factor is essential, but its trend is
very complex.
The Section starts with the demonstration of the frictional effect by performing a static anal-
ysis of a confined drillstring. The critical buckling loads and approximate analytical expressions
of the axial force are provided, including the theoretical derivation of the lock-up failure. Some
methods that improve the axial force transfer are also included. We then present a more realis-
tic methodology to capture the effect of friction which involves the dynamics of the constrained
drillstring. Special focus is placed on the effect of axial vibration which is commonly thought
as a means to further improve the stability behavior of the drillstring.
Static analysis
The effect of friction in horizontal and low inclination wellbores has been shown recently to
be a dominant factor on the stability behavior of drillstrings. On the one hand, friction force
may be a stability factor since it usually delays the onset of the buckling. Nevertheless, after
initiation of buckling the friction force can cause an abrupt reduction in axial force transfer and
eventually “lock up”, as explained by Kuru et al. (2000).
The fruitful effect of the friction in the critical buckling load was first confirmed by exper-
imental results (McCann and Suryanarayana, 1994). Based on an energy approach, Gao and
Miska (2009b) then computed the sinusoidal and helical buckling loads in a horizontal borehole
by applying the Coulomb’s law of sliding friction;
Qfs = Qs
[
1 + 1.233µ2/3
]
(A.10)
Qfh =
√
2Qs
(
1 +
(
30 + 7pi2
)
30pi
µ
)
(A.11)
where the dimensionless axial force is denoted as Qf to indicate the frictional effect. These
theoretical results are found to be consistent with experimental results.
Even though the critical and secondary buckling loads can be easily computed by an energy
200
approach, the derivation of the axial force distribution function requires a different methodology.
The understanding of the axial force transfer is crucial since a lock-up failure can occur under
the action of friction. To investigate the axial force transfer in the presence of the friction, a
new length scale is adopted, the so-called lock-up length;
`f∗ =
1
µ
(
EI
wr
)1/2
(A.12)
which leads to ςf = s/`f∗ while the force scale fh∗ (A.9) remains unchanged.
Wu et al. (1993) was the first who investigated the axial force transmission in horizontal and
inclined wells under the action of the friction force. Wu et al. (1993) distinguished the frictional
drag for an unbuckled and a helically buckled drill-pipe, ignoring the first critical buckling.
Based on Coulomb friction law Wu et al. (1993) derived the following differential equation of
the axial load
dQf
dςf
= (cosα+N )− sinα/µ (A.13)
where µ is the friction coefficient, N = N/w is the contact force per unit length for helically
buckled weightless drillpipes with N ≈ Q2h cosα/4. Based on Wu et al. (1993)’s approach the
axial force distribution function in a horizontal borehole can be written as follows
Qfh(ς) = 2 tan (c− ς) (A.14)
where c is an integral constant. A similar expression can be also derived in an inclined wellbore.
More details can be found in Wu et al. (1993). Quite recently, (Gao and Miska, 2009b) also
calculated the axial force function in a horizontal borehole for sinusoidal and helical buckling,
respectively;
Qfs (ς) = 2
[
−a3 + 1
a2
tan
(
−a1a2
2
ς + c
)]
, Qfh(ς) = Qh
1
1−Qhς (A.15)
where a1 = 0.79 + 0.043µ2/3, a2 = 0.446− 0.17µ2/3, a3 = 1.16− 0.093µ2/3 and c is an integral
constant.
Taking into account the differential equation of the axial load (A.13) and assuming that a
cylinder is pushed into an already existing horizontal wellbore (i.e., with no load on the bit),
the total lock-up failure length can be approximated. In this particular case, the lock-up failure
length is defined as the maximum distance that the drill-pipe can be pushed inside the wellbore
due to the frictional effect (Wicks et al., 2008). Under these conditions, the total lock-up failure
length is approximately ςL = 3
√
2`f in a horizontal borehole.
There is a variety of methods that can improve the transfer of axial force and consequently
201
eliminate the lock-up failure. For instance, the rotation of the drillstring can modify the direction
of the friction force, leading to a substantial reduction of the axial friction force. Nevertheless
the rotation can also affect the critical buckling loads, as explained by Menand et al. (2008). In
particular, Menand et al. (2008) showed that rotation can remarkably reduce the helical buckling
load when compared to a non-rotating case under the presence of friction, while Gao and Miska
(2010) deduced that the critical buckling load is unaffected by the rotation in the frictionless
case. Recently, other techniques are also adopted which can minimize the unfavorable effect
of friction on the transmission of axial force, namely the hydraulic vibration and the use of
connectors (Huang et al., 2017; Huang and Gao, 2014; Gao et al., 2011). More details can be
found in (Gao and Huang, 2015).
Dynamic analysis
Let us now investigate the buckling behavior of a drillstring constrained by a horizontal
borehole including inertia effects. More precisely, we assume that the drillstring is in continuous
contact with the horizontal borehole. Hence, the governing equation of the drillstring is based
on the buckling differential equation (see Section A.2) which can be approximated as follows
EIr
∂4θ
∂s4
+ rF
∂2θ
∂s2
+ wθ +mr
∂2θ
∂t2
+ Ff sign(θ˙) = 0 (A.16)
neglecting higher order terms and assuming a sliding friction force per unit length Ff = µN .
Taking into account the length and force scales (A.9) and a time scale th∗ = (r/g)
1/2, we get the
dimensionless quantities ς = s/`h∗ , τ = t/th∗ , Q = F/fh∗ and Ff = Ff/
(
fh∗ /`
h
∗
)
. In this manner,
(A.16) becomes
∂4ϑ
∂ς4
+Q∂
2ϑ
∂ς2
+
∂2ϑ
∂τ2
+ ϑ+ Ff sign(ϑ˙) = 0 (A.17)
Let us now apply a Fourier series expansion on the angular displacement ϑ(ς, τ) and the sign
function with respect to the angular term ϑ˙(ς, τ)
ϑ(ς, τ) =
∞∑
i=1
Ai(τ) sinωiς, sign
(
ϑ˙
)
=
∞∑
i=1
Bi(τ) sinωiς (A.18)
where Ai, Bi are the time dependent coefficients for the i-mode with frequency ωi = ipi/ςL.
Assuming that one mode k is dominant, the coefficient Bk can be approximated as follows
Bk ≈ sign
(
A˙k
) 2
ςL
∫ ςL
0
|sinωkς| dς = 4
pi
sign
(
A˙k
)
(A.19)
202
Substituting (A.18) and (A.19) into (A.17), we get the second-order ODE
A¨k + ω
2
k
[
ω2k + ω
−2
k −Q
]
Ak +
4Ff
pi
sign
(
A˙k
)
= 0⇒
A¨k +KkAk + F˜f sign
(
A˙k
)
= 0 (A.20)
where F˜f is the “effective” value of the sliding friction force per unit length and Kk describes
the stiffness of the system and is a linear function of the axial force Q.
Let us first investigate the frictionless case Ff = 0. In this case, three distinct cases are
possible; (1) Kk > 0, (2) Kk = 0 and (3) Kk < 0. The first and third cases lead to periodic and
hyperbolic functions, respectively. This implies that the first case is only stable. In the special
case of Kk = 0, the onset of instability is obtained with critical buckling load Qc = ω2k + ω−2k .
The critical load is minimized by setting dQc/dωk = 0. This leads to Qc = 2 which coincides
with the critical, or sinusoidal buckling load Qs, as explained in Section A.1.2.
When the friction is present, the response of the drillstring does not only depend on the
sign of Kk but also the initial conditions Ak, A˙k. In particular, the initial conditions define the
length of the stationarity line and accordingly the area of the stable zone. The stationarity line
represents the so-called “sticking condition” which leads to a dissipation of the system.
Let us first demonstrate the special case of Kk = 0. As Kk → 0 any solution tends to a
stationary line independently of the initial perturbation Ak. This indicates that the case of
Kk = 0 does not provide the onset of instability in the frictional case. Instead, the onset of
instability occurs for Kk < 0. Decreasing Kk the axial Q accordingly increases and becomes
obviously higher than Qs. Even if Kk < 0, a stable zone can be also derived whose size depends
on the initial conditions. Hence, the onset of instability is not uniquely derived but it clearly
depends on the initial perturbation (Su et al., 2013). In addition, the analysis shows that friction
can strongly affect the number of the dominant buckling mode. The higher the friction force Ff
is the higher the buckling mode is obtained.
Recently, Liu et al. (2015) extended the above analysis in order to additionally investigate
the effect of the axial vibration in the stability behavior of the confined rod as it is said to be
capable of delaying buckling and lock-up in the oil field. To incorporate the axial vibration into
the above derivation, an axial vibration force dQ sin τa is included with τa = ωτ . The frequency
ω is chosen in such a way that
√
E/ρ ≥ ωL/2pi (i.e., the period of the vibration force is much
larger than the time for an axial wave to travel through the rod Liu et al. (2015)). The axial
vibration force is added into (A.17), leading to the new form
∂4ϑ
∂ς4
+ [Q+ dQ sin τa] ∂
2ϑ
∂ς2
+
∂2ϑ
∂τ2a
+ ϑ+ F¯f sign(ϑ˙) = 0 (A.21)
with F¯f = µθ (ς, τa)N/
(
fh∗ /`
h
∗
)
a more realistic representation of the sliding friction force per
203
unit length. µθ (ς, τa) is the transverse friction coefficient which depends on the constant friction
coefficient µ and the ratio of shear to the axial velocity components. This is a reasonable
representation of the friction force when we only seek the critical buckling loads, because at
the instant of buckling initiation the shear velocity approaches infinity and the shear friction
becomes accordingly dominant, as explained by (Gao and Miska, 2009b). Its derivation is
shown in Section A.2. In this study, µθ is assumed to be constant and a simple approximation
is contained in (Liu et al., 2015).
After applying a Fourier series expansion (A.18-A.19), (A.21) becomes
A¨k +
[K¯k − P¯k sin τa]Ak + F˜f sign(A˙k) = 0 (A.22)
where ω¯k = ωk/ω, leading to the new stiffness K¯k = ω¯2k
[
ω¯2k + ω¯
−2
k −Q
]
, P¯k = ω¯2kdQ and
F˜f = 4F¯f/pi.
In the case of a frictionless system with dQ 6= 0 , (A.22) is the well-known Mathieu’s equation
for the k-mode. Its stability depends on the two parameters K¯k and P¯k and not on the initial
conditions. We cannot also assume that a dominant mode is present in this case. Hence, the
stability analysis is obviously a more difficult task herein as the derivation of the stable modes
requires the investigation of several combinations of K¯k and P¯k.
From the stability analysis remarkable conclusions can be withdrawn. One main result is
that the negative value of the stiffness K¯k < 0 implies instability of the system, as above. The
stability analysis also shows that by increasing any of the quantities ω, or Q, or dQ, the system
may become unstable as some of the modes fall outside the stable region. More precisely, if we
increase the amplitude of the vibrational force dQ, the stable region shrinks, especially when
dQ reaches the value of 0.5Qc. On the other hand, the effect of the frequency ω is complex. In
particular, for a given value of dQ the range of frequencies for which the rod is stable is unclear.
For instance, there are some frequencies for which the drillstring is stable even for large axial
forces Q while for other frequencies the instability may initiate for very low values of Q.
In the frictional case F¯f 6= 0, the stability behavior of k-mode of the confined drillstring is
controlled by four parameters, namely the coefficients K¯k, P¯k, F¯f and the initial velocity A˙k(0).
Let us first investigate the effect of the friction F¯f . For low values of friction the buckling
response is close to the frictionless case while the stability is enhanced when we increase the
value of F¯f . This also implies that the negative value of K¯k does not imply instability, as
above. Keeping the value of friction constant, the stability behavior of the confined drillstring
is affected by the quantities dQ,ω in a similar manner to the frictionless case. Surprisingly, in
the frictional case the stability region remains unchanged if we increase the axial force Q while
keeping constant the value of friction F¯f . On the other hand, the effect of the initial velocity
A˙k(0) is dominant. In particular, as we decrease the initial velocity A˙k(0) the stable domain is
remarkably enhanced even for negative values of K¯k.
204
A.2 Drilling equations
The stability problem of drillstrings constrained by a straight borehole is analyzed in this Section.
Based on the contact conditions with the borehole two distinct models can be adopted; (1) the
beam-column model and (2) the so-called buckling differential equation. The two-dimensional
beam-column model is applied when there is no contact with the wellbore, or only a few discrete
contact points are present. Nevertheless when a continuous contact is present, it is preferable to
assume a cylindrical coordinate system, meaning that the lateral displacements can be expressed
as u2 = r cos θ and u3 = r sin θ, where θ is the angular displacement and r is the known clearance
of the borehole. In this particular case, the governing equation of the drillstring becomes one-
dimensional (it is only a function of θ). These models are analyzed in short in the following
Sections.
A.2.1 Beam-column model
The clearance of the wellbore is much smaller than the total length of the tubular string. Hence
the linear elastic theory assuming small displacements can be applied in this case. In particular,
the governing equation of an inclined straight wellbore is derived by adopting the beam-column
model (Timoshenko, 2009; Gao and Huang, 2015);
d4u2
ds4
+
d
dS
(
F1 − ws sinα
EI
du2
ds
)
− w cosα
EI
= 0
d4u3
ds4
+
d
ds
(
F1 − ws sinα
EI
du3
ds
)
= 0
where u2 and u3 are the lateral displacements along Y and Z coordinates, respectively; s is the
axial distance; F1 is the axial compressive force at the bottom end; EI is the bending stiffness;
w is the weight per unit length of the tubular string; α = 0 for a horizontal well and α = pi/2
for a vertical well.
A.2.2 Buckling differential equation
When the drillstring is in continuous contact with the wellbore, it is more convenient to express
the governing equations of the drillstring with respect to the angular displacement θ rather
than the two lateral displacements u2,u3 as a constant clearance r is present. Based on this
simplification, two distinct approaches can be adopted to derive the buckling differential equation
(Gao and Huang, 2015); (1) the energy principle (Gao and Miska, 2009a) and (2) the general
theory of elastic rods (Gao and Miska, 2010; Mitchell, 1988). In the former case, the virtual
works of shear force and bending moment at the extremities of the drillstring are neglected
while in the latter case the higher-order terms of the curvature and twist are considered to be
205
negligible. Both methods lead to the same buckling differential equation.
In the next Section, we demostrate the static and dynamic analysis of the drillstring by
applying the general theory of elastic rods (Gao and Miska, 2010). An improved model that
includes higher-order terms can be found in (Huang et al., 2015).
Static analysis
Let us suppose that the reference configuration {e1, e2,e3}, where e1 = {1, 0, 0}, e2 = {0, 1, 0}
and e3 = {0, 0, 1}. In the reference configuration the rod is located at (s, 0,−r), where s is
the arclength and r is the uniform clearance of the wellbore. If we denote θ(s) the angular
displacement and u1(s) the axial linear displacement at a point s, the position vector can be
written as follows
r(s, t) = (s+ u1) e1 + r sin θe2 − r cos θe3 (A.23)
which implies that the continuous contact with the wellbore u2 = r sin θ and u3 = −r cos θ.
The spatial derivative of (A.23) gives the tangential direction of the pipe’s axis is
τ (s) =
∂r
∂s
= e1 + r
∂θ
∂s
eθ (A.24)
where eθ = cos θe2 + sin θe3 while the term ∂us/∂s 1 and it is neglected. The unit vectors in
the normal direction n and binormal direction b are
κn =
∂τ
∂s
= κrer + κθeθ (A.25)
and
κb = τ × kn = −κθer + κreθ (A.26)
where er = sin θe2 − cos θe3, κ =
√
κ2r + κ
2
θ is the curvature of drillpipe and κr = −r (∂θ/∂s)2
and κθ = r∂2θ/∂s2 are the components of κn on er and eθ axes, respectively.
The total displacement can be written as u1(s) = ua(s) + ub(s), where ua(s) is the axial
displacement and ub(s) is the axial displacement due to the lateral buckling. Considering that
the total change of length can be approximated as follows
∆L ≈ −1
2
∫ L
0
[(
∂ry(s, t)
∂s
)2
+
(
∂rz(s, t)
∂s
)2]
ds =
1
2
r2
∫ 1
0
(
∂θ
∂η
)2
dη
the displacement ub of a material element ds is similarly derived
ub(s) = −1
2
r2
∫ s
0
(
∂θ
∂η
)2
dη (A.27)
206
The distributed pressure consists of the weight W and the normal contact pressure N can be
expressed as follows
W = −mg cosαkds = w cosα (cos θer − sin θeθ) ds (A.28)
N = −Nerds (A.29)
Using (A.28-A.29) the total body force becomes
f = (mg cosα cos θ −N) er −mg cosα sin θeθ (A.30)
where α = 0 for a horizontal well and α = pi/2 for a vertical well. In addition the forces F, the
moments M and their spatial derivatives yield to the following expressions
F = F1e1 + Frer + Fθeθ (A.31)
M = Mrer +Mθeθ (A.32)
∂F
∂s
=
∂F1
∂s
e1 +
(
∂Fr
∂s
− Fθ ∂θ
∂s
)
er +
(
∂Fθ
∂s
+ Fr
∂θ
∂s
)
eθ (A.33)
∂M
∂s
=
(
∂Mr
∂s
−Mθ ∂θ
∂s
)
er +
(
∂Mθ
∂s
+Mr
∂θ
∂s
)
eθ (A.34)
The consistutive relations between the forces (moments) with the displacements (angular dis-
placements) are required. In particular, Hooke’s law is adopted to derive the relationship between
the axial deformation and the axial compressive force F1;
F1(s) = −EA∂ua
∂s
= −EA∂u1
∂s
− 1
2
EAr2
(
∂θ
∂s
)2
(A.35)
where E is the Young’s modulus and A is the cross-sectional area. Similarly, we assume that
the bending moment M(s, t) is proportional to the curvature κb;
M(s, t) = −EIκb (A.36)
where EI is the bending stiffness.
The conservation of linear and angular momentum is then obtained;
∂F
∂s
− f = 0 (A.37)
∂M
∂s
+ τ × F = 0 (A.38)
207
which can be decomposed along the three orthogonal directions τ , er and eθ respectively;
∂F1
∂s
= 0 (A.39)
∂Fr
∂s
− Fθ ∂θ
∂s
+N − w cosα cos θ = 0 (A.40)
∂Fθ
∂s
+ Fr
∂θ
∂s
+ w cosα sin θ = 0 (A.41)
∂Mr
∂s
−Mθ ∂θ
∂s
− Fθ + Fsr ∂θ
∂s
= 0 (A.42)
∂Mθ
∂s
+Mr
∂θ
∂s
+ Fr = 0 (A.43)
where (A.39) indicates that F1 is uniform and
Mθ = EIr
(
∂θ
∂s
)2
(A.44)
Mr = EIr
∂2θ
∂s2
(A.45)
Substituting (A.44-A.45) into (A.42) and (A.43), we get
Fθ = EIr
[
∂3θ
∂s3
−
(
∂θ
∂s
)3]
+ F1r
∂θ
∂s
(A.46)
Fr = −3EIr∂θ
∂s
∂2θ
∂s2
(A.47)
(A.46) and (A.47) are then substituted into (A.40) and (A.41), leading to the nonlinear buckling
equations
N = −EIr
[(
∂θ
∂s
)4
− 3
(
∂2θ
∂s2
)2
− 4∂θ
∂s
∂3θ
∂s3
]
+ F1r
(
∂θ
∂s
)2
+ w cosα cos θ (A.48)
and
EIr
[
∂4θ
∂s4
− 6
(
∂θ
∂s
)2
∂2θ
∂s2
]
+ rF1
∂2θ
∂s2
+ w cosα sin θ = 0 (A.49)
208
Dynamic analysis
If we extend the static analysis described in the previous Section, we need to modify the con-
servation of linear and angular momentum to include inertia effects
∂F
∂s
− f +m∂v
∂t
= 0 (A.50)
∂M
∂s
+ τ × F + I ∂ω
∂t
= 0 (A.51)
where v(x, t) is the linear velocity
v(x, t) =
∂r
∂t
=
∂u1
∂t
e1 + r
∂θ
∂t
q (A.52)
and ω(x, t) involves two terms; (1) the angular velocity of the element rotating about the
centerline of the drillstring (ωτ ) with ω the uniform angular velocity of the rod and (2) the
angular velocity produced by the change in the direction of the centerline (∂τ/∂x);
ω(s, t) = ωτ +
∂τ
∂t
= ωe1 + ωrer +
(
ωr
∂θ
∂s
+ ωθ
)
eθ (A.53)
where ωr = −r (∂θ/∂s) (∂θ/∂t) and ωθ = r∂2θ/∂s∂t whose absolute values are much smaller
than ω.
The nonlinear buckling equations (A.48-A.54) are then modified, leading to the dynamic
buckling equations;
+EA
∂u1
∂s
+ EAr2
∂2θ
∂s2
∂θ
∂s
−m∂
2u1
∂t2
= 0 (A.54)
N = −EIr
[(
∂θ
∂s
)4
− 3
(
∂2θ
∂s2
)2
− 4∂θ
∂s
∂3θ
∂s3
]
+ F1r
(
∂θ
∂s
)2
+ Iω
(
∂ωθ
∂s
+ ωr
∂θ
∂s
)
+ w cosα cos θ +mr
(
∂θ
∂t
)2
(A.55)
and
EIr
[
∂4θ
∂s4
− 6
(
∂θ
∂s
)2
∂2θ
∂s2
]
+ r
∂
∂s
(
F1
∂θ
∂s
)
+Iω
(
∂ωr
∂s
− ωθ ∂θ
∂s
)
+ w cosα sin θ +mr
∂2θ
∂t2
= 0 (A.56)
209
Friction
Based on (Gao and Miska, 2009b,a), the friction force can be also included in the above deriva-
tion. The friction force involves two components which are given by
Ff (s, t) = Ff1(s, t)e1 + Ffθ (s, t)eθ (A.57)
where Ff1(s, t), Ffθ (s, t) are the axial and lateral components, respectively;
Ff1(s, t) = −sign(
∂u1
∂t
)µ1Nds, Ffθ (s, t) = −sign(
∂θ
∂t
)µθNds (A.58)
and µ1, µθ are the corresponding friction coefficients on the axial and lateral directions. As
shown in (Gao and Miska, 2009b), the friction coefficients µ1, µθ can be derived by introducing
a constant friction coefficient µ and taking into account that the ratio of µ1 and µθ is proportional
to the ratio of the corresponding speeds;
µ1 = µ
|v1|√
v21 + v
2
θ
, µθ = µ
|vθ|√
v21 + v
2
θ
(A.59)
where v1 = ∂u1/∂t and vθ = r∂θ/∂t. Substituting (A.59) and (A.58) into (A.57), the friction
force can be written as follows
Ff (s, t) = − µNds√(
∂u1
∂t
)2
+ r2
(
∂θ
∂t
)2
[
sign(
∂u1
∂t
)
∣∣∣∣∂u1∂t
∣∣∣∣ e1 + sign(∂θ∂t )
∣∣∣∣r ∂θ∂t
∣∣∣∣ eθ] (A.60)
Based on (A.60), the final form of the dynamic buckling equations is derived;
+EA
∂u1
∂s
+ EAr2
∂2θ
∂s2
∂θ
∂s
−m∂
2u1
∂t2
− µsign(
∂u1
∂t )
∣∣∂u1
∂t
∣∣√(
∂u1
∂t
)2
+ r2
(
∂θ
∂t
)2N = 0 (A.61)
EIr
[
∂4θ
∂s4
− 6
(
∂θ
∂s
)2
∂2θ
∂s2
]
+ r
∂
∂s
(
F1
∂θ
∂s
)
+Iω
(
∂ωr
∂s
− ωθ ∂θ
∂s
)
+ w cosα sin θ +mr
∂2θ
∂t2
− µsign(
∂θ
∂t )
∣∣r ∂θ∂t ∣∣√(
∂u1
∂t
)2
+ r2
(
∂θ
∂t
)2N = 0 (A.62)
N = −EIr
[(
∂θ
∂s
)4
− 3
(
∂2θ
∂s2
)2
− 4∂θ
∂s
∂3θ
∂s3
]
+ F1r
(
∂θ
∂s
)2
210
+ Iω
(
∂ωθ
∂s
+ ωr
∂θ
∂s
)
+ w cosα cos θ +mr
(
∂θ
∂t
)2
(A.63)
Appendix B
Supplementary material to Chapter
3
In this Appendix, we first present the principle of minimum energy for a particular example
of a clamped-clamped variable-length elastica constrained by symmetrically located walls with
clearance c = 0.05 (see Chapter 3 for its complete bifurcation diagram). We then derive the ana-
lytical solution of a clamped-clamped elastica of constant length (classical problem) constrained
by symmetrically located walls. The complete bifurcation diagram for this particular problem
is also computed by applying the geometry-based method, as proposed in Chapter 3.
B.1 Principle of minimum energy
The constrained buckling problem cannot be solved by applying the calculus of variations in
the traditional manner. The reason behind that is the presence of the unilateral constraints,
which affect the space of allowed variations. Hence, a second-order analysis cannot be easily
performed. Even if the contact conditions are assumed to be known (i.e., closed contacts), the
variation of the associated contact points should be included for a second-order analysis. Their
inclusion is particularly crucial for the investigation of the bifurcation points, where a smooth
or a nonsmooth evolution of the contact patterns is present under further loading. These issues
are explained in detail in (Ro et al., 2010; Chen et al., 2015; Manning and Bulman, 2005).
As an alternative, we apply the principle of minimum energy, which is typically used to
identify the most likely equilibrium state (see (Alfutov, 2000; Doraiswamy et al., 2012)). Even
though this criterion does not address the stability problem rigorously, it can be used to deduce
about the most favorable, or optimal equilibrium configuration among other candidates for
particular loading conditions. An optimal solution is also called stable in the current study.
211
212
To illustrate this optimality criterion, we revisit the constrained buckling problem of a
clamped-clamped elastica of variable length for a clearance c = 0.05 (see Chapter 3). Tak-
ing into account that there is no dissipation (i.e., system is conservative), the external work is
given by
W =
∫ δ¯
0
Qdδ¯ (B.1)
is identical to the bending energy U (Antman, 2005). Hence, the principle of minimum energy
can be applied by utilizing either the external work, or the bending energy. Here, the bending
energy is computed in terms of two distinct bifurcations parameters, the applied load Q for
force-controlled conditions or the change of the inserted length of the elastica δ¯ for displacement-
controlled conditions. The corresponding diagrams U(Q), U(δ¯) are shown in Figs. B.1 and B.2,
respectively. Based on them, for a fixed value of a bifurcation parameter, an equilibrium state of
the elastica is then called optimal if its energy is minimum when compared to other candidates.
Consider first displacement-controlled conditions. By increasing the inserted length of the
elastica, a sequence of equilibrium configurations is obtained, indicated with black line in Fig.
B.1. The corresponding contact patterns can be found in Fig. B.1. Special attention is placed
on the bifurcation point 3′, where the bending moment vanishes at the discrete contact and at
the end-points of the elastica for symmetrical reasons. Beyond this point, the bending moments
at the extremities of the elastica could possibly evolve in a symmetrical or anti-symmetrical
manner, leading to the shapes 3 − 6 and 3 − 4, respectively. As shown in Fig. B.1, for a fixed
δ¯, the energy of the asymmetric configuration 3− 4 is lower than the energy of the symmetrical
shape 3−6. In addition the initiation of a continuous contact is not feasible for the same reason.
Hence, the asymmetric configuration 3 − 4 is the optimal solution. The same conclusion has
been deduced by (Ro et al., 2010).
For force-controlled conditions (see Fig. B.2), the main difference when compared to the
displacement-controlled loading condition is that all softening branches are unstable here. In
particular, at points 1′ and 3′, there is no smooth evolution of the contact patterns. Instead,
there is a jump to the closest possible in terms of energy equilibrium configuration of the same
applied load Q/pi2. For instance, at 1′, there is a jump to a point of the branch 2′ − 3′ with the
same applied load Q/pi2 = 4.
213
0.00 0.01 0.02 0.03 0.04 0.05 0.06
0
1
2
3
4
5
6
7
8
0.0000 0.0025 0.0050 0.0075 0.0100
0.0
0.2
0.4
0.6
0.8
1.0
5'
6'
4'
3'2'
1',8'
1',8'
3'
2'
U
U
¯
¯
Figure B.1: Bending energy U with respect to δ¯
0 5 10 15 20 25 30 35
0
1
2
3
4
5
6
7
8
8'
5'
6'
4'
3'2'
1'
Q/⇡2
U
Figure B.2: Bending energy U with respect to Q
214
B.2 Analytical solution of constrained clamped-clamped elas-
tica
The analytical solution of the post-buckling behavior of the clamped-clamped elastica con-
strained by two symmetrically located walls at y = ±c is presented. The derivation is restricted
to the symmetrical configurations of the elastica (odd buckling modes). Because of symmetry
the elastica is divided by 1/4k segments where k is an odd number of buckling mode and thus
the solution of the first segment is only illustrated below.
The anti-symmetrical configurations (even buckling modes) are not analyzed because they
are similar to the clamped-pinned elastica which can be found in (Domokos et al., 1997).
B.2.1 Unconstrained case
The equilibrium equation of the unconstrained clamped-clamped elastica can be written as
θ′2
2
−R cos θ = −R cos θ1/4k (B.2)
with boundary conditions θ (0) = θ′ (1/4k) = 0, R is the compressive load with inclination angle
α = 0 and k = 2%− 1 is the number of buckling mode with % ∈ {1, 2, ...}.
Multiplying (B.2) by dθ and integrating from s = 0 to s = 1/4k, the horizontal load R
becomes √
R = 4kK(k¯) (B.3)
where k¯ = sin
(
θ1/4k/2
)
is the modulus of elliptic integrals.
Accordingly the cartesian component x¯(1) at the right end of the elastica for the k buckling
mode is given by
x¯(1)
4k
=
1√R
[
2E(k¯)−K(k¯)] (B.4)
Combining (B.3) and (B.4) the horizontal component becomes x¯(1) = 2E(k¯)/K(k¯)− 1.
The above expressions are applicable until the elastica touches one of the horizontal walls.
When a contact occurs in the absence of a contact force, the vertical component at s = 1/4k
becomes
y¯(
1
4k
) =
∫ 1
4k
0
sin θdζ =
c
2
(B.5)
Combining (B.3) and (B.5) a relation between the compressive load and the clearance can be
derived
√R = 4 sin
(
θ
(n)
1/4k/2
)
/c.
215
B.2.2 Point contact
Past the first contact event, a contact force appears which implies that α 6= 0. The positions of
the contact points on the elastica remain unchanged and the vertical component at the right end
of the first segment is given by (B.5). In this situation, the governing equation of the elastica
becomes
ψ′2
2
−R cosψ = −R cosψ1/4k (B.6)
with boundary conditions ψ(0) = −α, ψ′(1/4k) = 0 and ψ(1/4k) = θ1/4k − α.
Starting from (B.6) and applying the boundary conditions, the following expression is ob-
tained
F
(
pi/2; k¯
)
+ F
(
φo; k¯
)
=
1
4k
√
R (B.7)
where k¯ = sin
(
ψ1/4k/2
)
and φo = arcsin
[
sin (α/2) / sin
(
ψ1/4k/2
)]
.
For the k buckling mode the horizontal component x¯(1) and the clearance c can be expressed
in terms of ψ, α as follows
x¯(1)
4k
=
∫ 1/4k
0
cos(ψ + α)dξ,
c
2
=
∫ 1/4k
0
sin(ψ + α)dξ (B.8)
where ∫ 1/4k
0
cosψds =
1√R
{
2
[
E(k¯) + E(φo, k¯)
]− [K(k¯) + F (φo, k¯)]} (B.9)
∫ 1/4k
0
sinψds =
2k¯√
R˜
cosφo (B.10)
B.2.3 Onset of continuous contact
For the special case where R passes through the contact point, a line contact (second contact
event) initiates, and a geometrical expression is obtained tanα = tanψ1/4k = 2kc/x¯(1), leading
to
√R = 8kK(k¯).
Further analysis other than the onset of the continuous contact is not required since past
this point the elastica loses stability.
B.3 Geometry-based analysis of the constrained clamped-
clamped Elastica
We consider next the case of a constrained elastica clamped at both ends. There are two
important differences between this case and the pinned-pinned elastica analyzed above. First,
there exist two distinct roots for the first buckling mode of a clamped-clamped elastica, with
the smaller root corresponding to a stable symmetric configuration and the larger root to an
216
unstable anti-symmetric configuration. Second, the first symmetric buckling mode configuration
with continuous contact is unstable, while it was stable for a pinned-pinned elastica. We analyze
separately the symmetric, anti-symmetric, and asymmetric configurations.
B.3.1 First Buckling Mode (Symmetric Case)
Unconstrained Buckling
Referring to Figure B.3, the unconstrained elastica is divided into four identical segments for
the first symmetric buckling mode. More generally, there are m = 4k with np = 2k for the
symmetric k-buckling mode, with the inclination θ1(s) and the horizontal force R1 of the first
segment given by
θ1(s) = θ[2]
{
cos
[
2kpi
(
s− s[2]
)]
+
θ2[2]
192
(
cos
[
2kpi
(
s− s[2]
)]− cos [6kpi (s− s[2])])
}
R1 = 4(kpi)2
[
1 +
θ2[2]
8
]
(B.11)
and θ[2] viewed here as the loading parameter. Using symmetry considerations, the solution is
trivially extended to the other segments.
Constrained Buckling with Discrete Contact
This first contact event is characterized by contact force F[j] = 0, j ∈ Iw and ∆yi = c/2, i ∈ I`
for each segment. Past this first contact event, increasing θ[2] leads to the build-up of contact
force F[j] > 0, j ∈ Iw at the discrete contacts and the progressive decrease of the moment at
the location of the contacts. The derivation is similar to the pinned-pinned case presented in
Section 3.4.4 and is thus omitted.
c 54
32y x
P i n n e dC l a m p e d
1 - δ1
Figure B.3: Symmetrical first buckling mode of clamped-clamped elastica
217
Secondary Buckling
With increasing θ[2], the moment at the discrete contact eventually vanishes. In contrast to
the pinned-pinned elastica, this condition does not mark the onset of continuous contact as this
configuration of the elastica become unstable (n(l)p = 2m
(l)
e ). In addition the same holds for the
secondary buckling described below, for which n(d)p = m
(d)
e + 2. Due to vanishing moments of
both ends, the secondary buckling corresponds to a change in the sign of the moment at the
clamped ends, as reflected in the deformed configuration shown in Fig. B.4.
To construct the solution for the secondary buckling with one discrete contact, only one half
of the elastica needs to be considered. As shown in Fig. B.4, there are two distinct clamped-
clamped segments with an inflection point at the midpoint of each segment. Two distinct
canonical problems have thus to be solved, corresponding to segments 2 (bounded by nodes 2
and 3) and 3 (bounded by nodes 3 and 4), with a combined length `2 + `3 = 1/4. The segments
have relative inclinations ψ2(s) = θ2(s) + α2 and ψ3(s) = θ3(s)− α3, respectively with α2 ≥ 0,
α3 > 0. As the resultant force Ri, i ∈ I` is uniform, αi = αi+1, i ∈ I`. Due to continuity of the
bending moment at node 3, β2 = −β3, and a relation between length `i and angle βi is derived
`i =
1
4pi
arccosβi (B.12)
Hence, Ri = 16pi2, i ∈ I`.
The contact constraints for the segments are ∆y1 = ∆y2 = −γc/2 and ∆y3 = ∆y4 =
c (1 + γ) /2 with 0 ≤ γ < 1. Number γ increases with end-displacement δ, starting at 0 at
the onset of secondary buckling up to 1 when the elastica contacts the opposite wall at two
symmetric points. With three discrete contacts, the segmentation remains the same. However
the relative inclination for the second segment is modified according to ψ2(s) = θ2(s)− α2 and
the transverse components of the force transmitted by the segments are related by
R2 sinα2 = R3 sinα3 + F[3] (B.13)
This means that the inclination angles α2 6= α3 are different. Using the continuity of the bending
moment at node 3 and the uniformity of horizontal horizontal force, the complete solution can
be easily derived.
218
cy x
1−δ
P i n n e dC l a m p e d
8 97
6
5
4
32
1
Figure B.4: Secondary buckling of clamped-clamped elastica.
B.3.2 First Buckling Mode (Anti-symmetric Case)
The anti-symmetric case of the second root differs remarkably from the symmetric one. It is
a double clamped-pinned case, which requires some special consideration. An asymmetrical
deformation shape similar to the anti-symmetric case is also discussed.
Unconstrained Buckling
The second root of the first buckling mode of a clamped-clamped elastica corresponds to the
anti-symmetric configuration (see Fig. B.5). Hence, only one-half of the elastica has to be
solved. Using the segmentation technique, one half of the elastica is divided into three distinct
segments, with segments 1 (bounded by nodes 1 and 2) and 2 (bounded by nodes 2 and 3)
being identical; the combined lengths 2`2 + `3 = 0.5. Considering that the node 4 lies along the
centerline, the sum of the vertical offsets of the segments satisfy
3∑
i=1
∆yi = 0 (B.14)
In the absence of contact, the force in the elastica is uniform. The solution can be obtained by
applying continuity of the bending moment at node 3.
Constrained Buckling with Discrete Contact
The contact event occurs when the contact force F[3] = 0 and ∆y3 = c. The deflection of
the segments is then constrained by 2∆y1 = 2∆y2 = ∆y3 = c and the segment lengths by
2`2 + `3 = 0.5. Further increase of δ leads to F[3] > 0 and thus to a jump in the transverse
component of the force transmitted by segments 2 and 3.
219
7
65
4
32
c
y x
C l a m p e dP i n n e d
1−δ1
Figure B.5: Asymmetrical first buckling mode of clamped-clamped elastica
Asymmetrical configuration (displacement controlled)
For a displacement-controlled loading, an asymmetrical configuration of the elastica is observed
beyond the point of vanishing moment at the discrete contact 3, as shown in Figure B.6. It is
similar to the second root of first buckling mode but in an asymmetrical manner and with a
single discrete contact. With increasing δ an abrupt decrease of the horizontal force Ro occurs.
The position of the discrete contact varies (i.e., s[3] ≤ 0.5), while the vertical displacement of the
two last segments (i.e., ∆y5 + ∆y6) increases until a second discrete contact develops at node 5
.
To derive the solution for the asymmetrical configuration, the elastica should be divided into
three clamped-clamped segments with an inflection point at the midpoint of each segment. Three
canonical problems should be solved, e.g., segments 2 (bounded by nodes 2 and 3), 4 (bounded
by nodes 4 and 5) and 5 (bounded by nodes 5 and 6) with combined length `2+`4+`5 = 1/2. The
segments have relative inclinations ψ2(s) = θ2(s)−α2, ψ4(s) = θ4(s)−α4 and ψ5(s) = θ5(s)+α5,
respectively, assuming positive inclinations. The horizontal applied force Ri cosαi, i ∈ I` is
uniform. In addition, R4 = R5 and thus α4 = α5. Due to continuity of the bending moment at
node 5, β4 = −β5 and a relation between lengths `4, `5 and their resultant force is obtained
`4 + `5 =
pi
R24
(B.15)
Continuity of bending moment at the discrete contact at node 3 has also to be satisfied. The
contact constraints for the segments are ∆y1 = ∆y2 = −c/2 , ∆y3 = ∆y4 = c (1 + γ) /2 and
∆y5 = ∆y6 = γc/2 with 0 ≤ γ < 1. For γ = 1, the deformation configuration is characterized
by two discrete contacts and the elastica should be divided differently, as explained in Section
B.3.2.
220
c P i n n e d
c 76
5
4
3
2
y x1 1−δ C l a m p e d
Figure B.6: Asymmetrical configuration of clamped-clamped elastica
B.3.3 Numerical Results
The calculated post-buckling response of a constrained clamped-clamped elastica for clearance
c = 0.05 is in full agreement with the numerical results of Manning and Bulman (2005). The
corresponding bifurcation diagram is shown in Fig.B.7.
The behavior for the first symmetric buckling mode shares some similarities with the pinned-
pinned case. Due to symmetry the elastica comes in contact with the wall at the midpoint po-
sition at Point 2 (θ[2] = 0.16,δ = 0.0062, Q/pi2 = 4). This contact position remains unchanged
until Point 3 (θ[2] = 0.2,δ = 0.0075, Q/pi2 = 15.9) for both force-controlled and displace-
ment–controlled conditions. The Point 3 corresponds to the vanishing of the moment at the
discrete contact. Nevertheless it does not continue to a line contact, as confirmed by the con-
jectured rule n(l)p = 2m
(l)
e (see Chapter 3). In addition the simultaneous change of the moments
of both ends is not feasible (Branch 3-7), noting also that n(d)p = m
(d)
e + 2.
For displacement controlled, beyond Point 3 the sign of the moment at the right end changes,
leading to an asymmetrical configuration with one discrete contact point (i.e., n(d)p = m
(d)
e + 1).
Along Branch 3-5, the horizontal load decreases with increasing δ (i.e., softening branch). This
branch is present until a second discrete contact occurs at Point 5 (θ[2] = 0.255, δ = 0.026,
Q/pi2 = 8.3), where the configuration becomes anti-symmetric with two discrete contacts. For
force-controlled conditions the softening Branch 3-5 is not possible. Instead the elastica jumps
to a point (θ[2] = 0.266, δ = 0.0285, Q/pi2 = 16) of Branch 5-6.
The second root of the first buckling mode, corresponding to an anti-symmetric configuration
of the elastica is also shown in Figure B.7 (Branch 4-5). The unconstrained case is unstable
(n(u)p > 2) . However, when the elastica comes in contact with the wall at Point 5 (θ[2] =
0.40,δ = 0.026, Q/pi2 = 8.3), the configuration becomes stable (i.e., Branch 5-6), noting that
n
(d)
p = m
(d)
e .
Upon unloading, the response follows Branch 5-6 until one of the contact points vanishes. It
unloads further according to its unconstrained first mode.
221
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07
0
5
10
15
20
25 8
7
6
54
3
2
/π
2
δ
 Stable
 Stable (D.C.)
 Unstable
1
Q
5 - 6
2 - 3
3 - 7 7 - 8
4 - 5
1 - 2
3 - 5
Figure B.7: Bifurcation diagram for a constrained clamped-clamped elastica with clearance
c = 0.05
B.4 Clamped-pinned elastica
The post-buckling response of a constrained clamped-pinned elastica has also been computed.
Results for clearance c = 0.053, chosen to enable comparison with the numerical and experi-
mental results of Domokos et al. (1997) are shown in the bifurcation diagram of Fig. B.8. The
numerical results are not shown since they coincide with the current study.
The elastica is stable for the first buckling mode. The elastica comes in contact with the
222
wall at Point 2 (θ[1] = 0.215,δ = 0.00735, Q/pi2 = 2.05) and the second configuration is is stable
(n(d)p = m
(d)
e ). The first two configurations are actually similar to those corresponding to the
second root of the first buckling mode of a clamped-clamped elastica, except that the length of
the clamped-pinned segment is here the total length of the elastica.
At Point 3 (θ[1] = 0.32,δ = 0.0098, Q/pi2 = 8.8), where the moment at the discrete contact
vanishes, the configuration switches smoothly from 3 to 4 for a displacement-controlled loading.
The switch is associated with a change in the sign of the moment at the clamped support
changes. Along this branch the axial load decreases (i.e., softening) and the right segment
approaches the lower wall. When it reaches Point 4 (θ[1] = 0.387,δ = 0.0293, Q/pi2 = 6.7),
a discrete contact with the opposite wall appears, corresponding to configuration 4-5. For a
force-controlled loading, Branch 3-4 is unstable (i.e., n(d)p = m
(d)
e + 1) and the elastica jumps
from Point 3 to a point (θ[1] = 0.4,δ = 0.03, Q/pi2 = 8.8) of Branch 4-5. This response is in
agreement with experimental results (Domokos et al., 1997).
0.00 0.01 0.02 0.03
0
2
4
6
8
10 5
4
3
2
Stable (D.C.)
/π
2
δ
Experimental
Stable
1
Q
3 - 41 - 2
4 - 52 - 3
Figure B.8: Bifurcation diagram for a constrained clamped-pinned elastica with clearance c =
0.053