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A New Strategy for Treating Frictional Contact
in S heii Structures using Variational Inequalities
Nagi El-Abbasi
A thesis submitted in confomiity with the requirements for the degree of Doctor of Philosophy
Graduate Department of Mechanical and Industrial Engineering University of Toronto
0 Copyright by Nagi El-Abbasi 1999
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A New Strategy for Treating Frictional Contact in Shell Structures
using Variational Inequalities
Nagi Hosni El-Abbasi, Ph.D., 1999
Graduate Department of Mechanical and Industrial Engineering
University of Toronto
Abstract
Contact plays a fundamental role in the deformation behaviour of shell structures.
Despite their importance, however, contact effects are usually ignored andor
oversimplified in finite element modelling. Existing solution techniques for frictional
contact problems involving shell structures suffer from two main deficiencies. Firstly,
commonly used shell elements involve basic assumptions, which are not appropriate for
contact problems, since they do not: (i) account for variations of displacements and
stresses in the transverse direction, and (ii) ailow for double-sided contact. The second
deficiency is in the modelling of contact. To the author's knowledge none of the existing
techniques are based on the more accurate and mathematically consistent variational
inequalities formulation. Typically, the variational formulations are used which employ
contact elements. These contact elements are dependent on user-defined parameters that
affect the accuracy of solution.
In view of the above, three aspects of the problem are accordingly examined. The fint
is concerned with the development of a reliable thick shell element, which accounts for
the thickness change, the normal stress and strain thiaugh the thickness and
accommodates double-sided contact. An assumed naturd strain formulation is used to
avoid shear locking, and a new director interpolation scheme is utilised to prevent
thickness locking. Large deformations and rotations are accounted for by invoking the
appropnate objective stress and strain measures.
The second aspect of the work is concemed with the development of variational
inequalities formulations for large deformation analysis of fnctional contact in shell
structures. The kinematic contact conditions are expressed in terms of the physical
contacting surfaces of the shell. Lagrange multipliers are used to ensure that the
constraints are accurately satisfied and that the solution is free from user defined
parameters.
Finally, the numerical predictions are verified experimentally, compared with
commercial finite element codes, and with theoretical solutions where available. A
number of case studies involving contact, fiction, large deformations and double-sided
contact are also exarnined. The results reveal that the new higher order shell element is
superior to classical shell elements for thick shell applications, and maintains its high
level of accuracy in thin shell problems. Furthemore, the new frictional contact
formulation is more accurate than traditional variational techniques.
Acknowledgements
1 offer my sincere gratitude to Dr. S.A. Meguid for his expert advice, technical guidance,
and his commitment and suppon throughout the course of my research. 1 aiso wish to
thank the members of the Engineering Mechanics and Design Laboratory; specifically,
Mr. A. Czekanski, Dr. M. Refaat, Mr. J.C. Stranart and Dr. G. Shagal. The financial
support of the Natural Sciences and Engineering Research Council of Canada (NSERC),
the Aluminum Company of Amenca (ALCOA) and the University to Toronto is
gratefull y ac know ledged.
Contents
Abstract .................................................................... i
Acknowledgements ........................................................... üi
Contents ................................................................... iv
... List of Figures .............................................................. viu
. . List of Tables ................................................................ mi
... Notation .................................................................... mi
1 Introduction and Justification ................................................ 1
1.1 Contact in Shell Structures ............................................. 1
.............................................. 1.2 Justification of the Study 4
..................................................... 1.3 Aims of the Study 4
.................................................. 1.4 Method of Approach 5
...................................................... 1.5 Layout of Thesis 5
........................................................... 2 Literahire Review 8
.......................................... 2.1 Modelling of Shell Structures 8
................................ 2.1.1 Kirchhoff-Love Type Shell Elements 9
................................... 2.1.2 Shear-Deformable Shell Elements 9
................................. 2.1.3 Higher Order Thick S hell Elements 12
...................................................... 2.1.4 Patch Tests 12
................................... 2.2 Limitations of Existing Shell Models 13
......................................... 2.3 Classical Theories of Contact. 14
........................................... 2.3.1 Hertz Theory of Contact 14
............................................. 2.3.2 Non-Hertzian Contact 15
..................... 2.4 Techniques Adopted in Modelling Frictional Contact 15
............................................. 2.4.1 Variational Approach 15
.............................................. 2.4.2 Solution Techniques 16
................................................. 2.4.3 Contact Elemcnts 17
...................................... 2.5 Variational Inequalities Approach 18
............................................ 2.6 Contact in Shell Structures 19
.................................... 2.7 Large Deformation Elastic Analysis 21
.................................................. 2.7.1 Finite Rotations 23
.................................. 3 Development of a New Thick Shell Element 24
......................................... 3.1 Existing Thick Shell Elements 24
................................... 3.2 New Continuum Based Shell Mode1 - 2 5
............................................ 3.3 Four-noded Shell Element 29
................................................... 3.4 Thickness Locking 30
........................................ 3.5 Discretization of Shell Element 31
............................................. 3.6 Variational Formulation - 3 3
............................................... 3.6.1 Consistent Loading 35
................................................. 3.7 Numerical Examples 37
..................................................... 3.7.1 Patch Tests - 3 8
............................................. 3.7.2 Flat Cantilever Bearn 38
......................................... 3.7.3 Curved Cantilever Beam -40
.............................................. 3.7.4 Pinched Hemisphere 42
................................................. 3.7.5 Pinched Cylinder 44
.................................... 3.7.6 Clamped-Clamped Thick Bearn 46
..................................... 3.7.7 Sphencal Shell Under Pressure 47
4 Anaiysis of Large Deformation Frictionai Contact in Sheiis using Variational
Inequalities ............................................................... 50
......................................... 4.1 Kinematic Contact Conditions 50
................................. 4.2 Variational Inequalities for Continuum 53
........................................ 4.3 Reduced Variational Inequality 54
............................ 4.4 Variational Inequdities for Shell S tmctures - 5 5
.................................................. 4.5 Solution Technique 56
....................................................... 4.6 Discretization 57
.............................................. 4.6.1 Contact Constraints - 5 7
................................................... 4.6.2 Friction Terms 59
........................................... 4.6.3 Finite Element Solution 59
................................................ 4.7 Verifkation Exarnples 60
.............................................. 4.7.1 Three Beam Contact 61
................................................ 4.7.2 Ring Compression 61
................................................ 4.7.3 Strip Friction Test 67
............................................ 4.7.4 Belt-Pulley Assembly -68
........................................... 4.7.5 Strip Compression Test 73
................................................ 5 Experimentai Investigations -77
........................................................ 5.1 Introduction -77
................................................ 5.2 Details of Rings Used 77
.................................................. 5.3 Photoelastic Studies 80
........................................... 5.4 Strain Gauge Measurements 80
........................................ 5.5 Load Deflection Characteristics 80
...................................................... 6 Results and Discussion 83
6.1 Introduction ......................................................... 83
.................... 6.2 Lateral Compression of a Ring Between Curved Dies 83
.................................... 6.3 Two Cylindncal Shells in Contact -92
..................................... 6.4 Compression of a Spherical Shell - 94
................................... 6.5 Saddle-Supported Pressurr Vessels -97
............................................. 7 Conclusions and Future Work 104
................................... 7.1 Definition of the Problem ....... .. 104
7.2 Objectives ......................................................... 104
7.3 General Conclusions ................................................ 105
7.3.1 Thick Shell Element Accounting for Through-thickness Deformation . . 105
7.3.2 Variational Inequalities Contact Formulations for Shell Structures
Undergoing Large Elastic Deformation ............................. 105
7.3.3 Case Studies Considered ......................................... IO6
7.4 Thesis Contribution ................................................. 107
7.5 Future Work ....................................................... 107
References .................................................................. 109
Appendix A: Shell Element Equations ...................................... -122
A . 1 First Interpolation Scheme . iP 1 ...................................... 122
A.2 Second Interpolation Scheme . IP2 .................................... 125
A.3 Third Interpolation Scheme . IP3 ..................................... 127
Appendix B: Cornputer Implementation ..................................... 129
B . 1 Main Program Module .............................................. 129
B.2 Shell Element Equations ............................................. 131
B.3 Contact Search ..................................................... 133
B.4 Contact and Friction Equations ....................................... 134
List of Figures
........................... 1.1 Typical shell structures in engineering applications 2
................................................. 1.2 Contact in shell structures 3
....................................................... 1.3 Method of approach 6
.............................................. 2.1 Schematic of a shell structure 9
............................................... 2.2 Patch test for shell elements 13
................................................. 2.3 A typical contact element 18
..................................................... 2.4 Two shells in contact 20
............................ 3.1 Geometry and degrees of freedom of shell mode1 25
..................... 3.2 Mode of deformation cornsponding to: (a) a3 and (b) 26
........................................... 3.3 Geometry of new shell element -30
3.4 Nonnalised thickness distribution for: (a) sphencal shell and @) cylindrical
................................................................... shell 31
3.5 Extemal shell forces (a) schematic of force system. and (b) location of extemal
................................. forces corresponding to 5-parameter mode1 36
3.6 Cantilever problem: (a) mesh and deformed geometry. and (b) normalised load-
......................................................... deflection curve 39
......................... 3.7 Curved bearn: (a) mesh. and @) convergence results 41
................... 3.8 Pinched hemisphere: (a) mode1 and @) deformed geometry 43
................ 3.9 Deformation at points A and B using Pl and IP3 interpolation 43
......... 3.10 Pinched cylinder: (a) model, (b) uniform mesh. and (c) distorted mesh 45
...................... 3.1 1 Normalised deflection under load for a pinched cylinder 46
................. Clamped-clamped thick bearn: mesh and deformed geometry 46
Clamped-clamped thick bearn: (a) variation of shell thickness. and @) variation
.............................................. of quadratic displacement q - 4 8
................... Spherical shell subjected to intemal and external pressures -49
Location of contact points in four noded shell element ....................... 51
Kinematic contact constraint for shell surfaces .............................. 51
............. Three-beam contact problem: (a) geometry. and (b) contact stages 62
Effective stiffness for three contacting beams ............................... 63
......................... Deformed geometry for three-bearn contact problem 63
Ring contact problem: (a) schematic of loading arrangement. and (b) defomed
............................................................. geometries - 6 4
Variation of contact pressure dong contact distance for a 32.5% reduction in
radius ........................ .. ....................................... - 6 5
Contact pressure distribution for different ring reduction ratios ................ 67
Strip friction test: (a) finite element model. and (b) contact pressure distribution . 68
Finite element mode1 of belt-pulley assembly .............................. - 6 9
Effect of rotation 8 on the contact stress distribution of belt: (a) normal contact
........................................... stress. and (b) tangentid stress - 7 1
(a) Variation of stick-slip angles. ai and Q2. with rotation 0. and @)
cornparison between theoretical and FE stress distributions in belt for 8 = 0.6'. . 72
........................................ Schematic of strip compression test 74
.................................. Deformation stages for strip compression - 7 5
.................................... Effect of friction on the pullsut force -76
...................................................... Expenmental setup 78
Influence of contact and bending stresses on the ring and dies ................. 79
..................................................... Photoeiasticity setup 81
Strain gauge location for (a) thick (t/R= 0.5). and (b) thin rings (tlR = 0.1). .... 81
Finite element mesh of rings .............................................. 86
FE mode1 of ring and curved die ........................................... 86
Photoelastic (left) and finite element (right) maximum shear stress contours
developed in a photoelastic die: (a) Wt = 2 (P = 370 N) and (b) R/t = 4
(P = 50 N) .............................................................. 87
Photoelastic (left) and finite element (right) maximum shear stress contours
developed in a photoelastic die (Rh= 1 O): (a) P = 300 N. (b) P = 500 N and (c)
P = 900 N .............................................................. 88
Variation of normalised circurnferential strain dong inner ring radius (R/t = 10) . 90
Load deflection characteristics for a ring with R/t = 10 ....................... 90
Contact pressure distribution for different ring thicknesses . Lefi hand scale is for
al1 thicknesses except t = 0.43. For t = 0.43. the right hand scale applies ........ 91
........................................... Mode1 of two-ring compression 92
Modes of deformation resulting from contact between two rings .............. -93
............................ Force-deflection characteristics for the two rings 95
.............................. Model of sphericd shell compression problem 95
Deformed geometry of spherical cap: (a) Hertzian contact. (b) edge-dominant
...................................... contact. and (c) post-buckling contact 96
........................ Normalised load-deflection curve for spherical shell -97
......................... A schematic of pressure vesse1 and saddle supports -98
........................... FE mode1 of pressure vesse1 and saddle supports -99
6.16 Effect of saddle to pressure vesse1 radius ratio Rs/Rp on the hoop stresses at the
................................................................ support 101
6.17 Effect of saddle plate extension on the hoop stresses at the support ............ 101
6.18 Effect of saddle placement Le on the longitudinal stresses at 0 = 0'. .......... 102
6.19 Effect of saddle placement Le on the hoop stresses at the support ............. 102
B . 1 Flow chart for main program module ...................................... 130
............................. 8.2 Flow chart for calculation of element equations 132
..................................... B.3 Flow chart for contact search module 133
B.4 Flow chart for evaluation of contact and friction equations ................... 134
List of Tables
Key to analysis options used in numerical simulations ........................ 37
Vertical displacement at tip of beam corresponding to small deformation
................................................................ analysis 38
Vertical displacement at tip of bearn corresponding to large deformation
................................................................ analysis 40
......................... Horizontal displacement under load for curved bearn 42
Displacement at points A and B in pinched hemisphere corresponding to small
deformation analysis and using an 8x8 mesh ................................ 44
Nomalised change in shell thickness a3 l t for Pm= 1000 ...................... 49
Variation of radial stress OR 1 PH through shell thickness for Pm=lOOO . . . . . . . . . 49
Details of geometry and material properties for tested rings .................. -79
..... Details of geometry and material properties of pressure vesse1 and supports 99
xii
Notation
Global contact geometry matrix
Strain displacement matrix
Matrix of active contact constraints
Constitutive rnatrix
Contact traction
Young's Modulus
Linear component of strain tensor Q,
Assembled vector of gap values
Covariant and contravariant vectors at time t =O
Covariant and contravariant vectors at current tirne
Gap function
S hell thickness
Discretized nodal forces
Deformation gradient
Body forces
Identi ty matrix
Moment of area
Quadratic strain displacement matrix
Length of shell
S tifhess matrix
Space of admissible displacements
Unit normal vector
Two-dimensional isoparametric shape functions
Local contact geometry matrix
Rotational component of defonnation gradient
Radius
Second Piola-Kirchhoff stress tensor
Externai traction
Global displacement vector
Stretch component of deformation gradient
Displacement vector
Director vector connecting top and bottom shell surfaces
Unit vector in direction of Vt3
Unit vectors perpendicular to V3
Position vector
Rotationai degrees of freedom
Change in shell thickness
Transverse displacemen t gradient
Green-Lagrange s train tensor
Regularisation parameter
Contact angle
Nonlinear component of strain tensor ~ i j
Second intrinsic variable
Shear correction factor
Vector of Lagrange mu1 tiplien
Lamé constants
Coefficient of Coulomb friction
Poisson's ratio
Cauchy stress tensor
First intrinsic variable
Third intrinsic variable
xiv
Subscripts and Superscripts:
Assumed strain
Bottom surface
Contact
Direct strain
Friction
Linear
Middle surface
Normal component
Quadratic
Tangentid component
Top surface
time
Chapter 1
Introduction and Justification
1.1 Contact in Shell Structures
In almost al1 mechanical and structural engineering systems, there exists a situation in
which one body is in contact with another. This is, in essence, how loads are delivered to
and transmitted from systems. Contact stresses play an important role in determining the
structural integrity and ultimately the resulting failure mode of the contacting bodies.
Figure 1.1 illustrates three typical examples of shell structures: (a) automotive body
panels, (b) fuselage of an aircraft, and (c) space satellites. Fig. 1.2 shows three cases in
which contact govems the mode of deformation and/or failure of the shell
stmcturelcomponent. Fig. 1.2(a) shows the buckling of a spherical shell compressed
between flat plates, Fig. l.Z(b) shows the raceways of an axial thrust bearing, and Fig.
1.2(c) depicts the failure of a pressure vesse1 at the saddle supports due to the presence of
highly localised contact stresses.
In spite of the important and fundamental role played by contact stresses in general
and in shell structures in particular, contact effects are generally ignored or
oversirnplified. This may be due to mathematical and computational difîiculties posed by
modelling contact. With the application of loads to the bodies in contact, the actual
surface on which these bodies meet and the stress at the interface are generdly unknown
and complex to determine.
In addition, the shell elements used in contact formulations do not account for the
variations of displacements and stresses in the transverse normal direction. Existing
elements are also incapable of treating shell structures experiencing double-sided contact.
(a) An automotive body panel [l]
(b) A fuselage of an aeroplane [2]
(c) A space satefite [3]
Figure 1.1 Typical sheil structures in engineering applications.
(a) Buckling o f a spherical shell[4]
@) Raceway of roller bearings [SI
(c) Support of pressure vessels [q
Figure 1.2 Contact in sheli structures.
1.2 Justification of the Study
Analytical solution for contact problems have ken developed dating back to the classical
work of Hertz [7,8]. However, they are restricted to simplified geometries and small
deformations. In order to overcome these limitations, most contact problems are currently
king treated using computationai techniques, with the finite element method being the
most appealing.
In this regard, contact problems can be most accurately modelled as variational
inequalities (VI). Whilst significant developments have ken made in applying variational
inequality fomulations to continuum problems, their applications to thin structures has
been very scarce. Instead, traditionai variational techniques are commonly used. In
cornparison with variational inequalities, the traditional variational methods lack in
mathematical rigor, especially when accounting for hictional effects. These formulations
usually rely on the use of contact elements which involve user-defined parameters that
cause a deterioration in the accuracy of the solution.
Furthemore, commonly used shell elements involve basic assumptions, which are not
appropriate for contact problems. Typicaily, they do not: (i) account for variations of
displacements and stresses in the transverse direction, and (ii) allow for double-sided
contact. These restrictions severely affect the accuracy of the results and lirnit their
application to thin shell structures.
1.3 Aims of the Study
It is therefore the objective of this work to:
(i) develop a new thick shell element which can accommodate variation in the
thickness, normal stresses and aiso ailow simultaneous double-sided contact,
(ii) develop a new variational inequality-based formulation capable of descnbing
frictional contact in thicklthin shell structures,
(iii) employ the newly developed sheil element into the VI formulations for 3D large
deformation problems, and
(iv) to apply the developed algonthms to different case studies involving contact,
fiction, large deformations and double-sided contact.
1.4 Method of Approach
Figure 1.3 shows an outline of the method of approach adopted to achieve the above
stated objectives. To develop the new thick shell element, a new continuum based thick
shell model is first developed. The model accounts for through-thickness deformation,
stresses and strains. Shear locking is avoided using an assumed natural strain
interpolation. Furthemore, an enhanced director field is developed to prevent thickness
locking. The second Piola-Kirchhoff and the Green-Lagrange strains are used as objective
stress and strain measures.
To develop the variational inequalities for contact in shells undergoing large
deformations, the solid continuum variational inequalities are used together with double-
sided kinematic contact constraints. The second Piola-Kirchhoff and the Green-Lagrange
strains are utilised. The resulting fictional terms are regularised to obtain a differentiable
variational form amiable to the finite element implementation. The new shell element and
VI contact formulation are consistently linearized and a solution technique based on
Lagrange Multipliers is adopted. Special attention is devoted to the efficient numerical
implementation of the shell element and the VI contact formulations. The resulting
formulations are verified and used to analyse a number of interesting engineering
problems, where contact plays a critical role in determining the performance of the
studied componentls ystem.
1.5 Layout of Thesis
This thesis contains seven chapters. Following this introductory chapter, chapter 2
provides a critical and careful assessrnent of the literature in two main areas: contact
mechanics and shell element formulations. In chapter 3, we provide a detailed account of
the newly developed thick shell element. The formulations account for large elastic
defonnations and rotations. The chapter also includes several element verification
Continuum thick
2" Piola-Kirchhoff stress Green-Lagninge strain k
Enhanced director interpolation I-
Assumed strain interpolation k
Four-noded thick sheîï element
2& Piola-Kirchhoff stress
Frictional reguiarisation I
YI for sheîï stnictures (large defomations) I
VI contact formuiation I
Solution techniques
Verifkation & applications I Figure 1.3 Method of approach.
problems. In chapter 4, we outline the methodology adopted and the resulting variational
inequalities fomulations for fictional contact in thkWthin shell structures. In chapter 5,
we summarise the photoelastic technique, and the load-displacement experiments as well
as the strain gauge measurements used to validate some of the problem cases examined.
Chapter 6 is devoted to case studies involving contact, friction, large deformations and
double-sided contact. Finally, in chapter 7 we conclude the thesis and outline directions
for future work.
Chapter 2
Literature Review
This thesis is concemed with the development of a new strategy for treating frictional
contact in shell structures. Three areas of scientific research are of direct relevance to the
work examined in this thesis. These are: (i) finite element modelling of shell structures,
(ii) contact mechanics, and (iii) large deformation analysis involving shells. In the
following, we provide a brief overview of the issues pertinent to the current work.
2.1 Modelling of Shell Structures
ShelIs are structural elements in which one of the dimensions is much smaller than the
other two. This leads to the possibility of describing the shell using its midsurface and a
director vector connecting the top and bottom physical shell surfaces (Fig. 2.1). In order
to simplify the modelling of shell structures, three levels of simpliQing assumptions are
commonly imposed [9,10]:
(i) the shell normal remains straight after defornation,
(ii) the shell normal remains normal after deformation, and
(iii) the shell is inextensible in the thickness direction.
The first assumption implies that the director vector connecting the top and bottom shell
surfaces remains straight after deformation. Accordingly, the change in the orientation of
the director vector can be represented by only two independent rotations. This assumption
simplifies the shell formulations and decreases the number of degrees of freedom
involved. The second assumption further restricts the orientation of the director vector.
By assuming that the director vector remains normal to the midsurface, al1 components of
transverse shear deformation are discarded. The third assumption implies that the length
of the director vector is constant. Accordingly, the deformation of the director can be
fûlly expressed in terms of the two rotational variables stated above. Relaxing this
assumption, requires one or more thickness related degrees of Freedom per node.
,A Sheii mid-surface
Figure 2.1 Schematic of a shell structure
2.1.1 Kirchhoff-Love Type Shell Elements
Applying dl three assumptions Ieads to Kirchhoff-Love type shell elements [ 1 1 - 151.
These elements, which require C' continuity, are best suited for thin shell applications. As
a result of the imposed assumptions al1 shear deformations are neglected. The
development of conforming first-order continuous elernents requires a large number of
nodes per element side, e.g. [11,12]. This has motivated the development of non-
conforming elements [l3,14]. This class of shell elements has been extended to large
geometncally nonlinear deformation problems [ 151.
In addition to the large number of nodes required per element, the use of C'
continuous elements for large deformation problems is not favourable. This is specially
m e for "non-smooth" shells as well as elasto-plastic problems, where the development of
plasticity in one part of the mesh induces secondary npple effects over a large part of the
shell 116, 171.
2.1.2 Shear-Deformable Shell Elements
Relaxing the third assumption leads to shear-defonnable elements c o n f i n g to the
Reissner [18] or Mindlin theories [19]. These elements require CO continuity, Lagrangian
interpolation, and involve independent interpolation of displacements and rotations. Most
shell elements belong to this classification 120-321. They can be developed based on
degeneration of continuum models [20-271 or using specific shell theories [28-321. The
main difference between the two is in the discretization [IO]. When using shell theories,
such as those of Sanders 1331, Koiters [34], or Flügge 1351, the thickness reduction is an
integral part of the selected shell theory. For degenerate shell elements, both analytical or
numericai thickness integration are possible. If additional numerical simplifications are
imposed on the shell mode1 and some of the higher order thickness integration terms are
neglected, analytical integration becomes an appealing alternative, which leads to stress
resultant-based elements [21]. The different "levels" of assumptions involved in different
explicit thickness integration schemes are discussed in Ref. [IO]. However, when higher
order terms are included, numerical integration is simpler to perform and is more
computationally efficient 1171. Furthemore, for nonlinear strain or stress relationships
resulting from large deformations andor plasticity, exact analytical through-thickness
integration becomes an even more complex task [36].
This class of elements is highly susceptible to different forms of locking and special
provisions are always necessary to ensure locking-free behaviour. Locking occurs when
the shell is unable to represent a state of pure bending without parasitic shear or
membrane terms. Due to the high shear/membrana to bending stiffnesses, such parasitic
terms dominate the deformation of the shell leading to locking. Numerous research efforts
have been directed to the study of the tocking phenornena [9,37-411. The simplest
technique is that of reduced integration [37,38]. Better results can be obtained by using
selective reduced integration 138,391, where the shear and membrane terms are under-
integrated while full integration is employed for the bending terms. Using reduced
integration, however, introduces zerosrder modes. These are modes of deformation with
zero energy (known as rigid body modes). For 9 and 16-noded Lagrangian elements,
using selective reduced integration results in a limited number of zero order modes most
of which are incommunicable between elements 19,391. Stabilisation techniques, such as
hourglass control, can be used to eliminate these modes 12 1,401.
An alternative method to avoid locking, which does not involve reduced integration,
is the assumed strain interpolation technique first developed by MacNeal [41]. A lower
order shear and/or membrane strain distribution is assumed. These strains are evaluated at
appropriately selected sampling or tying points 120,24273. Many of these lock-preventing
measures were initially developed in the fonn of "numencal tricks". However, later on,
they were proven to be based on generalised variational principles, such as the two-field
Hellinger-Reissner or the three-field Hu-Washizu variational principles [42].
More recently, an enhanced assumed strain method involving independent
interpolation of strain variables, which are condensed at element level, has been used to
avoid shear and membrane locking [43,44]. This leads to improved solution accuracy and
less sensitivity to mesh distortion. However, the additional element degrees of freedom
increase the computational tirne and the problem size.
Shear-deformable shell elements have been enhanced to account for geometric and
material nonlinearities f l6,17,23,26,29]. Some of the difficulties encountered in these
large deformation formulations are related to the correct imposition of the plane stress
assumption, obtaining the correct constitutive relationship and accounting for the change
in shell thickness. The most comrnon procedure for updating the shell thickness is to
partially relax the inextensibility assumption and update the thickness, in the post-
solution stage, by imposing either the plane stress condition [17,45] or volumetric
incompressibility [46,47]. This thickness update is oniy useful for nonlinear problems
involving large membrane strains. It does not enhance the performance of the element in
thick shell applications.
Both Kirchhoff-Love and Mindlin-Reissner type shell element require five degrees of
freedom per node; three mid-surface translations and two rotations of the director. The
axes of the two rotational degrees of fkedom are perpendicular to the current orientation
of the director. Hence, they Vary for different elements and are also a function of the shell
deformation. However, many finite element implementations favour a representation
involving three rotations about the global Cartesian axes. In this case, a fictitious degree
of freedom involving rotation about the shell director (drilling) is usually added to the
shell element, see, e.g. 121,251. This dnlling degree of freedom must be accompanied by
an ad-hoc stifhess value to prevent singularities in the stiffhess matrix. The dnlling
degree of *dom is, however, useful in cases involving non-smooth shells [2 11.
2.1.3 Higher Order Thick Sheli Elements
Several higher order beam and plate theories [48,49], which do not impose the
inextensibility assumption, have been used to formulate beam and plate elements [49,50].
More recently, similar shell elements, which do not impose the inextensibility
assumption, have ben developed [Sl-541. These elements account explicitly for the
thickness change as an additional degree of freedom and account for the through-
thickness stresses and strains. One of the advantages of this approach is that the 3D
constitutive relationships can be directly applied without imposing the plane stress
assumption. However, it is still advantageous to retain the shear correction factor. The
enhanced assumed strain method has also enabled the use of standard 3D solid continuum
elements for modelling shell structures [55,56]. However, the performance of these
elements quickly deteriorates as the shell thickness decreases [56]. Accordingly, when
developing such thick shell elements, it is always necessary to ensure that in the thin shell
lirnit there is no significant decrease in accuracy and that the element does not lock.
In addition to the two types of locking stated previously, the thick shell formulations
will also be susceptible to "thickness locking" resulting from parasitic through-thickness
defortnation. A detailed analysis of thickness locking and techniques for avoiding it is
provided in chapter 3.
2.1.4 Patch Tests
Patch tests have been widely used as a test for shell element convergence [9,57]. The
most commonly used patch test involves a rectangular plate which is discretized using 5-
irregular shell elements (Fig. 2.2). The plate is subjected to different loading conditions
simulating pure bending, membrane, in-plane shear and transverse shear deformation. In
al1 cases, a minimum number of degrees of freedom are constrained to prevent rigid body
motion. The resulting stress field, at a given plane, should be constant in spite of the
element distortion.
1 1 Shear 1
Figure 2.2 Patch test for shell elements.
Passing the patch tests does not guarantee convergence in al1 shell problems.
However, elements which do not pass this test such as the one described in Ref. [58],
should not be used. Accordingly, severai benchmark tests have been proposed to further
assess the convergence of shell elements. These tests have been performed on the newly
proposed shell element in section 3.7.
2.2 Limitations of Existing Shell Models
While being sufficiently accurate for most engineering shell problems, the traditional
Kirchhoff and Mindlin type shells are not accurate for contact problems. The thickness
variation as well as the normal component of the stress and the strain fields are
fundamental to most shell contact problems. Neglecting the influence of these factors
deteriorates the accuracy of solution. Furthemore, traditionai shells are incapable of
modelling double-sided shell contact. This aspezt is discussed in chapter 4.
It is therefore reasonable to postulate that with the exception of those few shell
elements which explicitly account for the thickness degrees of freedom [SI-541, most
existing shell elements are inappropriate for treating contact problems. None of the higher
order thick shell elements listed above have been explicitly used to mode1 frictional
contact problems. However, instead of using one of these higher order elements, an
alternative new shell element is developed in this thesis, which is more suited for thick
and thin shell contact problems. The advantages of the newly developed element
formulation over existing thick shell elements are discussed in section 3.1.
2.3 Classical Theories of Contact
The publication of the pioneenng work of Hertz in 1882 [7,8] can be argued to be the
birth event of contact mechanics. Most analytical solutions of contact problems are based
on the so-called Hertz theory. In these solutions, several simplifjhg assumptions
conceming the size of the contact zone and the contact pressure distribution are imposed.
Friction is neglected and the contacting bodies are usually assumed to be elastic half-
spaces. For thin structures, another class of analytical closed-form solutions can be
obtained by assuming specific beam, plate or shell theories rather than the half-space
idealisation. By using these simplified theories, less restrictions on the size and form of
the contact pressure distribution are warranted. None of these classical theories, however,
c m be used for practical shell problems, since they involve excessive simplifications.
2.3.1 Hertz Theory of Contact
Hertz's theory of contact was developed for elastic smooth frictionless bodies with a
contact region that is small compared with the dimensions of the bodies [7,8]. Hertz
fonnulated the contact conditions which must be satisfied by the normal displacement
field in the two contacting bodies. In order to obtain expressions for the size of the
contact zone and the specific form of the contact pressure distribution, several simplifying
assumptions were imposed.
Based on these assumptions there have been several contributions, most notably the
work of Boussinesq who utilised integral expressions and the half-space formulation to
denve the equilibnum conditions for a number of contact problems 181. There are many
references on the classical theory of Hertz including several texts on mechanics of solids
which have devoted some chapters to the subject, see for example [59,60].
2.3.2 Non-Hertzian Contact
A wide class of simplified contact problems involving thin structures have analytical
solutions that are not founded on Hertz theory. In this class of problems, the analytical
solutions are based on specific bearn. plate or shell theories, such as inextensional
elastica [6 1-63], Kirchhoff-Love type bearnlshell theories 164-661 and Mindlin-type
beam/shell theories [67,68]. These solutions are valid only for the specific beam, plate or
shell theory. Furthemore, the contact length is required to be much larger than the
thickness of the structure. In chapter 4, some of these approximate theoretical solutions
are used to veriS some aspects of the finite element predictions.
Some attempts were also made to utilise plane elasticity solutions expressed in terms
of Fourier transfoms. The elasticity solutions were superimposed on a 2D Bemoulli-
Euler type beam solution [69.70]. While some of these techniques are valid for large
deformation problems [61-63,661, they are restricted to simplified geometnes, loading
and elastic defoxmation. Furthennore, the body in contact with the bearn, plate or shell,
typically refemd to as the indentor, is always assumed ngid.
2.4 Techniques Adopted in Modelling Frictional Contact
Most shell contact problems cannot be approximated to one of the above mentioned
idealised cases. Indeed, numerical solution techniques provide a very powemil
alternative. With the rapid development in the capabilities of digital computers, more
accurate solutions of realistic shell problems are now possible.
2.4.1 Variational Approach
The exact variational representation of fictional contact problems results in a variational
inequality. However, most finite element solutions of contact problems are based on
standard variational principles which involve integrals over unknown contact surfaces
171-791. Chaudhary and Bathe used Lagrange Multipliers to solve the 3D frictional
contact problem [76]. Wriggers and Simo developed consistently linearized penalty-based
contact formulations for 2D problems [72]. Parisch developed consistent tangent stiffness
matrices for treating 3D large defonnation problems [74]. Heege and Alart accounted for
strongly curved rigid contact surfaces using parametric polynomial surface patches [79].
The use of the variational method to formulate contact problems lacks in
mathematical rigour, especially when frictional effects are taken into account. This is
primarily related to the non-differentiability of Coulomb's friction law, which is not
properly addressed in the variational formulations [80-821. Furthemore, it usually results
in the introduction of user defined parameters which influence the accuracy of the
solution and the rate of convergence [80,83,84].
2.4.2 Solution Techniques
The finite element formulation of contact problems can be expressed as a constrained
minimisation problem. For linear fnctionless cases, the minimisation functional takes the
following form:
1 X(U)=-U~KU-F'U subjectto A U I G 2
where U is the required solution, K is the stiffness matrix, F is the vector of externally
applied loads, A is a contact geometry matrix, and G is the vector of gap functions. Most
solution techniques for this minimisation problem are based on either the penalty or
Lagrange Multipliers methods. In the penalty method, the constrained optimisation is
transformed to an unconstrained one by penalising the inter-penetration [Ml:
The penalty method is simple and does not introduce any additional degrees of freedom.
However, it leads to the introduction of user-defined normal and shear stiffness
parameters. The selection of small values for the stiffhess parameters leads to excessive
penebation and slippage, while very high values result in illconditioning of the stifiess
matrix. In addition, in the case of shell structures, the inter-penetration can easily be of
comparable order of magnitude to the shell thickness which is unacceptable.
The constraint equations can be exactly enforced using Lagrange Multipliers [86], viz:
However, new degrees of freedom (the Multipliers fiT) are added to the problem. These
also result in zero-diagonal elements, which requins special precautions in the solver.
The perturbed [77] and the augmented Lagrangian [73,75] methods offer alternative
solution techniques that combine both penalty and Lagrange Multipliers formulations.
Other mathematical programming techniques for constrained optimisation problems were
also applied to fnctional contact problems. The quadratic programming [87,88] and linear
complementarity [89,90] methods are the most cornmon.
2.4.3 Contact Elements
Most commercial finite element software, such as ANSYS [91] and MARC [92], use
contact elements to enforce inter-penetration between the contacting bodies using
penalty-type formulation (Fig. 2.3). Each contact element connects a node on one body to
a node or a surface on another body. The main advantage of contact elements is their
simplicity. However, in addition to the disadvantages of using penalty formulations stated
previously, the use of contact elements significantly increases the size of the problem.
This is especially true if no pnor information about the exact location of the contact zone
is known, which is generally the case in large deformation problems. Accordingly, each
node from one body has to be connected, through a contact element, to al1 the extemal
element surfaces on al1 neighbouring bodies. Furthermore, for thin shell structures, the
inter-penetration can be of the sarne order of magnitude as the shell thickness, thus further
decreasing the solution accuracy.
>Y-
;*:. ; : ','.
I S . . . . a . . .'. * * I . : : ; -,
X . ' : . . Contact .' 8 , i '--. J element . a ,
0 . . ' , . , . . . . .
Target surface and nodes
Figure 2.3 A typical contact element
2.5 Variational Inequalities Approach
Variationai inequalities can be considered as an alternative mathematicai description of
physical problems which proved to be useful in cases involving unilateral constraints. The
theory of variational inequaiities is a relatively young mathematicai discipline. One of the
main bases for its development was the work of Fischera [93] on the solution of the
Signonni problem. Later on, Stampacchia laid the foundation of the theory itself [94].
The variational inequalities approach has not gained popularity because most of the
work in variational inequalities has appeared in the mathematical literature. The focus of
the work was to examine the mathematicai properties of the resulting variational
inequalities [81,95-981. This ngorous treatment enabled the study of existence and
uniqueness of the solution provided by VI formulations of contact problems. In addition,
most of these developments have been documented in Italian and French literature. Only
in the last two decades have interesting results appeared in the English literature, such as
references [95-981.
However, an extensive literature search indicates that little has been carried out to
develop suitable computational techniques to make use of these theoretical results. In this
regard, elastic contact for small deformation was presented by Kikuchi and Oden [8 1,99-
1011. They devoted their efforts to the mathematical questions conceming existence and
uniqueness of the variational inequalities representing different contact problems. They
also presented a solution technique based on the use of the penalty function method and
regularisation technique. Unfortunately, the resulting solution algorithm suffers from the
same disadvantages as those outlined in the traditional penalty approach. They also,
developed a total Lagrangian formulation for the solution of elasto-plastic problems,
which was also treated using the penalty and regularisation rnethods.
Refaat and Meguid 182-84,1021 developed new variational inequaiities for large
deformation elasto-plastic problems based on an updated Lagrangian formulation. They
also developed new solution techniques based on Quadratic Programming as well as non-
differentiable optimisation. These solution strategies did not involve inter-penetration and
were free from user defined parameters. A few other publications, however, have devoted
attention to the practicai implementation of variational inequalities in contact problems
[103,104]. This limited number of contributions is believed to be attributed to the
difficulties encountered by the engineering comrnunity when dealing with the complex
mathematical concepts posed by variational inequality formulations.
2.6 Contact in Shell Structures
Contact plays a fundamental role in the deformation behaviour of shell structures
(Fig. 2.4). Despite their importance, however, contact effects are usually ignored andfor
oversimplified in finite element modelling. Commonly used shell elements involve basic
assumptions, which are not appropriate for contact problems, since they do not:
(i) account for variations of displacements and stresses in the transverse direction, and
(ii) allow for double-sided contact. These restrictions severely affect the accuracy of the
results; especially, for moderately thick plate and shell structures [67]. By neglecting the
variation of displacements in the transverse direction, contact stresses cannot be evaluated
accurately. In addition, double-sided contact plays a significant role in many applications,
such as space satellites, automated manufacturing, sheet metal forming and in the
biomedical field. In such problerns, continuum three dimensional contact formulations
can be used [71,105]. However, they generally result in excessive degrees of freedom
with the necessity for large computational requirements and may also Iead to an ill-
conditioned stiffness matrix 191.
Figure 2.4 Two shells in contact.
Most existing formulations are based on classical variationai methods. In this regard,
Stein and Wnggers developed a contact algorithm for thin Kirchhoff-Love type elastic
shells undergoing frictionless contact [ 1061. Johnson and Quigley developed contact
formulations for thin elastica undergoing large deformations [87]. In addition, efficient
contact search algorithms for shell structures was developed by Benson and Hallquist
assuming single surface contact [107]. Several computational issues related to contact in
shells, such as the local contact search and the master-slave technique, were addressed by
Zhong [78].
The use of variational inequalities for modelling contact in thin structures, however,
has not been given its due attention. Only a limited number of attempts have been made.
These include the work of Ohtake et al. [IO81 which is based upon von Karman plate
theory and is concemed with the developrnent of variational inequalities to treat contact
in plate elements.
2.7 Large Deformation Elastic Analysis
Most practical shell problems involve large deformations even in the elastic range. This
necessitates the use of an objective large deformation formulation which is independent
of ngid body rotations. Furthemore, since shells have rotational degrees of fieedom, it is
essential to accurately account for the non-vectorial nature of these rotations and the
resulting nonlinear displacement terms. The nonlinear analysis can generally be based on
a total or updated Lagrangian formulation. In the total Lagrangian formulation the initial
configuration of the structure is used as the reference in the variational formulations. On
the other hand, in an updated Lagrangian formuiation the reference frarne is the cumnt
configuration. If the appropriate stress and strain measures are used, and the constitutive
relationships are transformed correctly, both formulations would give identical results and
the selection of one or the other becomes a matter of preference [log]. For shell
structures, the constitutive relationships are expressed in terms of the local coordinates
which are only accurately known in the original configuration. Hence, the use of a total
Lagrangian formulation is preferable, see for example Refs. [20,24-26,29-3 11 for further
details on the subject.
Several rneasures of strain an available for large deformation analysis. In order to
maintain objectivity these saain measures should be independent of rigid body rotations.
By decomposing the total deformation gradient F
into a pure stretch U and pure rotational component R
a general class of strain measures based on U can be expressed as follows [110]:
1 e=-(u"-1) for m*O m
E = in(^) for m = O
where different values of m result in different objective strain measures. The four most
commonly used strain tensors are GreenLagrange (m = 2), Biot (m = 1). logarithmic
Hencky (m=O) and the Almansi (m=-2) strain tensors. For small-strain large-
deformation large-rotation analyses, involving compressible materials, the four strain
measures yield sirnilar results. However, the Green-Lagrange tensor involves the least
computations and is therefore the most commonly used strain tensor. This is partially
attributed to the fact that an explicit decomposition of the deformation gradient,
according to Eqn. (2.5), is not necessary, since
Energetically conjugate to the Green-Lagrange strain tensor is the second Piola-
Kirchhoff stress [109,110]. This means that a variational formulation expressed in terms
of the true Cauchy stress, the infinitesimal strain tensors and integrated over the current
domain is equivalent to that expressed in terms of the second Piola-Kirchhoff stress and
the Green-Lagrange strain tensors in the initial configuration. viz:
Shell structures undergoing large deformations also commonly experience non-
conservative deformation dependent loading which may lead to an additional non-
symrnetric stiffhess matrix contribution. These non-conservative forces have been
accurately treated in the literature, sec e.g., Refs. [111,112]. However, for most practicai
engineering problems, the computational effort and storage requirements associated with
solving non-symmetric stifhess matrices far outweighs their benefit. Accordingly, they
are most often neglected in shell formulations.
2.7.1 Finite Rotations
In addition to large deformations, shell elements involve rotational degrees of freedom
which require special treatment in updating the shell configuration. One needs to work
with finite rotations which, unlike infinitesimal rotations, do not possess vector
properties. Procedures for the director update, which account for the large rotation effect,
include those based on Euler or Cardan angles [IO] and rotational vectors [10,1 131. Many
shell formulations, however, can only treat small incremental rotations [114]. An accurate
large rotations formulation leads to additional terms in the consistent linearization of the
director update procedure, see, e.g., Refs. [17,28,115] for further details.
Chapter 3
Development of a New Thick Shell Element
In this chapter, we provide a detailed account of a newly developed thick shell element
which is suitable for modelling large deformation frictional contact problems. It is
essential that this shell element should: (i) explicitly account for the normal contact
stresses, (ii) account for the thickness change as an independent degree of freedom, (iii)
accommodate double-sided shell contact, and (iv) demonstrate accurate locking-free
performance for both thick and thin shell structures. Furthemore, the selected element
should preferably (i) use 3D constitutive equations without any simpliQing assumptions,
and (ii) avoid the singular dnlling degree of freedom. It is worth noting, however, that the
assumptions imposed on the classical Kirchhoff-Love and Reissner-Mindlin type shell
elements Iead to an inaccurate description of shell contact problems.
3.1 Existing Thick Shell Elements
Since normal shell stresses are important in contact problems, it is advantageous to use a
shell formulation which does not impose the inextensibility condition. Simo et al. [SI]
developed one of the first such elements, which accounted for a uniform thickness
stretch. However, the plane stress condition was still imposed for the bending
deformation. In order to avoid this restriction, it is necessary to add a minimum of two
degrees of freedom per node to obtain linearly varying stress and strain fields through the
thickness. In this case, the 3D constitutive relationships can be directly applied without
imposing the plane stress assumption. However, it is still advantageous to retain the shear
correction factor.
Parisch [52] presented a shell formulation using seven degrees of f'reedom per shell
node. In spite of king simple, their formulations used only translational degrees of
freedom, and did not make provisions to maintain a director field of constant magnitude.
Without such a uniform director field, the shell is unable to represent a state of pure
bending without superimposed thickness strains. This leads to thickness locking which is
most critical for thin shells. Büchter et al. [53] developed an alternative formulation
which accounts for a linear variation of strains through the shell thickness, by adding a
thickness degree of Freedom and a Linear strain terni based on an enhanced assumed strain
formulation. A sirnilar shell formulation was developed by Betsch et al. [54]. An
alternative approach to treat thick shells, not used in this work, is based on hierarchical
finite element models of plate and shell structures, e.g. [116,117].
3.2 New Continuum Based Shell Mode1
Consider the shell element shown in Fig. 3.1 in which a pair of points xT and xB, that
make up the top and bottom faces of the element, are connected through a director vector
V', [118]. The geometry of the element can be expressed in ternis of the mid-surface
nodal coordinates, the director VI3, and a quadratic function q as follows:
Figure 3.1 Geometry and degrees of fkeedom of shell rnodel.
The quadratic term 'q, which is initidly zero, is necessary to descnbe a complete linearly
varying strain field through the shell thickness [53]. The incremental displacement field
for the shell element undergoing large deformations can be expressed as:
The above formulation results in seven degrees of freedom per rnid-surface node
(Fig. 3.1). The shell mid-surface invoives three incremental translation components in the
Cartesian coordinates:
The shell director also involves three degrees of fkedom. These are two incremental
rotations al and % about axes VI and V2 (perpendicular to Vg ), and % the change in
shell thickness in the direction of V3. The last degree of freedom represents the change
in the quadratic transverse displacement function 'q in the direction of V3. Fig. 3.2
illustrates the deformation modes comesponding to a3 and a.
M Shell mid-surface /
Figure 3.2 Mode of deformation corresponding to: (a) q and (b) 04.
In order to avoid ill-conditioning in the thin shell limit, it is essentid to decouple the
rotational and the extensional components of the director deformation [SI]. This is
achieved by representing the shell director as the product of a thickness scalar and the
unit clirector vector:
The unit director is updated based only on the rotational degrees of freedom al and q:
(3.5) '+& V3 = IR(' a, ,'a,) 'V3
R is an orthogonal matrix for finite rotations [IO, 1 131:
where
and
's = O t - t l t V , 3 - t a 2 t V a,'VI2+'a2'V,
t a, tV,3+ta2tV, O - t a ~ t V 1 1 - t a 2 t V 2 1 Ja, ' V,2-ta2t Vz 'a, 'V,,+'a,'V2, O
Argyris 11 131 demonstrated that R can be expanded into the following senes:
1 ' 2 1 t 3 ' ~ ( ' a , , ' u , ) = ~ + 'S+- S +- S + ......+( Higherorder terms) 2! 3!
The linear and quadratic terms in the above relationship are important in the consistent
linearization of the resulting variational formulation. Finally, a3 is simply added to the
shell thickness and to the quadratic displacement function q, viz.:
Including the quadratic displacement function q enables the use of 3D continuum
constitutive relationships, without imposing the plane stress condition commonly applied
to shells. The use of a shear correction factor (K= 1.2) is still desirable in order to correct
for the error in stiffbess caused by transverse shear strains, which are constant through the
thickness. Without q the stress-strain relationship in bending has to be modifed,
othenvise an error of the order of v2 would result [53]. This quadratic term can either be
continuous or discontinuous between elements [52]. In the first case, & is included in the
global stiffhess matrix. For a discontinuous quadratic tenn, a c m be condensed at
element level, thus effectively yielding a six parameter shell element. The effect of each
of the two additional degrees of freedorn as well as the condensation of are
investigated in section 3.7 using numerical examples.
Based on a total Lagrangian Framework and the Green-Lagrange strain tensor, the
covariant strain components 5 can be written as [log]:
where gi and Gi are the respective covariant base vectors associated with coordinate 4 at
times t and 0:
aOx Gi =- atx , gi=-- -Gi +- atu agi % i agi
The incremental direct strain is:
It is also necessary to express the elastic stress-strain relationship in covariant
coordinates. For a hyperelastic St. Venant-Kirchhoff type material, the following
relationship is used [53,54]:
Alternatively, a simpler lower order form can be utilised for thin shell elements by
neglecting terms involving G ' ~ and G? This can be more conveniently expressed in
matrix fonn, as follows:
The Cartesian components of the second Piola-Kirchhoff stress and the Green-
Lagrange strain tenson can be evaluated as follows:
Since the 3D constitutive relationships are used without modification, Eqn. (3.12) or
Eqn. (3.13) cm be directly replaced by other compressible hyperelastic constitutive laws.
3.3 Four-noded SheU Element
In this work, a four noded shell element is developed based on the proposed shell model.
Fig. 3.3 shows the pertinent features of the element. The element utilises standard
interpolation for the membrane and bending ternis.
In order to avoid shear locking, the assumed natural strain formulation is used
[2O,26 1. The assumed transverse shear strain fields &: and are related to the direct
strain components iqF at the sampling points as follows:
Mid-surface node
O Integration point
A Shear strain sampling point
Figure 3.3 Geometry of new shell element.
The locations of the sarnpling points A-D are shown in Fig. 3.3.
3.4 Thickness Locking
The shear locking problem for the proposed element has been treated based on an
assumed natural strain formulation (Eqn. (3.16)). Furthemore, this 4-noded element is
not susceptible to membrane locking. However, the proposed 7-parameter shell is
susceptible to another form of locking resulting from the inability of the basic shell
discretization to represent a director field of constant magnitude. In bending-dominated
problems, this introduces unrealistic thickness strains which cm lead to thickness
locking. Therefore, interpolation schemes that preserve the magnitude of the unit director
field should be employed. Like shear locking, thickness locking is most severe in thin
shells, since the ratio of the thickness to bending stiflhesses is proportional to ml4. The thickness error is not significant in the classical 5-parameter shell elements,
where the loss of accuracy associated with a non-unifonn director is not detrimental,
since there are no stresses through the shell thickness. In this case, the thichess error
usually results in a lower estimate of the shell thickness, and hence leads to a more
flexible structure [ 1 O].
3.5 Discretization of Shell EIement
The simplest interpolation fom for the new shell geornetry is as follows:
where the left superscript t denoting time has been ornitted for clarity. This interpolation
will be referred to in subsequent sections as [Pl. It is most commonly used in 5-
parameter shell elements (without the quadratic terni) [21,22,109]. However, using this
interpolation, the shell thickness is not constant except when the curvature is zero
(director vectors are identical at al1 nodes). Fig. 3.4 illustrates the extent and distribution
of this thickness error for two different FE meshes involving a spherical and a cylindrical
shell. Even though the error is zero at the nodes, it can be significant at the integration
points, where the stiffness calculations are performed.
Figure 3.4 Normalised thickness distribution for: (a) spherical shell and @) cylindncal
shell.
There are several ways to enforce the constant director field. One alternative is to
directiy interpolate the rotation variables, e.g. [10,119]. The resulting formulation gives
good results, however, the evaluation of a tangent stiffness matrix is computationally
demanding. In this work, a new approach using only polynomial interpolation is
developed. The error in the magnitude of the director at any point can be eliminated by
normalisation by its magnitude:
One possible interpolation scheme based on the above normalisation is:
This form will be refemd to in subsequent sections as I n . An alternative approach is to
interpolate al1 the pertinent geometric quantities separately:
and use the following interpolation for the shell geometry (IP3).
Note that alI three proposed interpolation schemes do not deviate from the continuum
shell mode1 (Eqn. (3.1)) in which the magnitude of the director is unity by definition.
Regardless of the selected interpolation scheme, the displacement field will not be
linear with respect to the degrees of frezdom. In order to maintain quadratic convergence
it is necessary to retain ail linear and quadratic terms. In this case, the displacement field
cm generally be expressed in terms of the nodal degrees of freedom in the following
form:
where N,"' is a vector of shape functions of size 7xn, and L$" is a square matrix of size
7x11, including the quadratic terms of the displacement interpolation for Cartesian
component i. AU is the vector of nodal degrees of freedom (Fig. 3.1):
The explicit form of N and L depends on the selected interpolation scheme. The detailed
equations and some denvations are provided in Appendix A. Although it is necessary to
include al1 quadratic terms in order to maintain the highest rate of convergence, the
cornputational requirements for some of these terms outweigh their benefit. Appendix A
provides guidelines for determining which terms are more significant than others. Our
numerical tests indicate that on average a 10% increase in the number of iterations and up
to 40% reduction in computational time per iteration is obtained by selecting the
appropriate terms. However, it is essential to account for al1 contributions in the linear
strain term. Saleeb et al. [17] provide more details on the effect of the quadratic
displacement terms on convergence for a 5-parameter shell element. A sirnilar efTect is
present in nonlinear beam formulations [ 1 15,1201.
3.6 Variational FormuMion
The total Lagrangian variational formulation can be expressed using the second Piola-
Kirchhoff stress and the Green-Lagrange strain tenson in covariant form, as follows:
where t + h ~ E 5 t t is the virtual work done by the extemal forces at time t+At. The following
decomposition of the stress and strain components is used 11091:
where eu and qq are the linear and nonlinear strain components. Due to the nonlinear
nature of the displacement field for the shell mode1 used, it is necessary to further
linearize the incremental shains to account for both the linear and the quaciratic
displacement terms of Eqn. (3.22); viz.:
This decomposition results in the following incremental variational formulation:
where
i a t w atsuLo =-(-
2 agi gj +-• gi)ztB, 6U
36,
1 at6uQ a t h Q = -(- gj+-Ogi)=Su T t L, (ij) AU 2 agi aC j
Appendix A details the steps involved in evaluating Bi, B2 and Li matrices for IPl
interpolation.
The assumed field for the transverse shear strain can be expressed as:
where N?) is the shape function for sampling point a and nij is the number of sampling
points for strain component ij. The resulting incremental variationai fomulation can be
expressed in matrix fom as:
where S, is the a vector form of the stress tensor Si' :
Details of the cornputer implementation of the shell element formulations are provided in
Appendix B.
3.6.1 Consistent Loading
Assuming conservative loading, and neglecting the effect of quadratic displacernent ternis
on the extemai loading term results in the following load vector:
The discretized matrix fom is as follows:
Contrary to the 5-parameter shell models, the exact location of the extemal forces (top,
rniddle or bottom shell surface) now plays a more significant role in the consistent
loading of the sheU structure (Fig. 3.5(a)). Note that applying a load to the midsurface of
the shell results in a contribution in the consistent load vector comsponding to q. A
loading pattern that involves only midsurface contributions (similar to the 5-parameter
shell model) can be generated by dividing the external force equdly dong the top and
bottom shell surfaces (Fig. 3.5(b)):
(a) @)
Figure 3.5 Extemal shell forces (a) schematic of force system, and (b) location of extemal
forces correspondhg to 5-parmeter model.
A point load applied to the top, rniddle or bottom shell surfaces will result the following
loading vectors:
1 O 0 O O O 'ha 'Vj:
O O 1 O O O 'ha 'V,", MIDDLE
3.7 Numerical Examples
A number of numerical examples were considered to assess the performance of the newly
developed element. The objectives of the examples were to: (i) demonstrate the enhanced
accuracy of the newly developed element for treating thick shell problems, (ii) evaluate
the performance of the element in thin shell applications, (iii) compare the three different
interpolation schemes as well as the effect of condensing the 7h degree of freedom at
element level, and (iv) demonstrate the extended applicability to new class of shell
problems involving contact. The selected exarnples involve linear and non-linear
analyses. thick and thin shells, bending and membrane dominated deformations, unifonn
and distorted elements, and a wide variety of loading conditions. Whenever possible, the
results are compared with theoretical values and/or sirnilar published numencal results.
Table 3.1 provides a key for the different analysis options that are used in the forthcoming
numerical examples.
7 DOF
6 DOF
5 DOF
Condense
Explanation
Interpolation schemes Pl, IP2 and IP3 (see section 3.5).
7-parameter shell element
6-parameter shell (neglec ting quadratic term)
Classical 5-parameter shell element
Static condensation of 7& degree of freedom
Table 3.1 Key to analysis options used in numencal simulations.
37
3.7.1 Patch Tests
The comrnonly used Selement patch test of Fig. 2.2 is performed [%,3 11. Loading
conditions were imposed to simulate pure bending, tension, in-plane shear, transverse
shear and transverse tension. The element gives a constant state of stress for al1 tests and
interpolation schemes.
3.7.2 Hat Cantilever Beam
A cantilever beam under a point load is modelled using 16 Cnoded elements (Fig. 3.6(a)).
Following Ref. [52], the following properties were selected: E=10~10~, v=0.3, L=lO.O,
w=1 .O, h =0.1 and an incremental force F = 4 0 x h3. The tip displacement was
monitored in both small and large deformation cases.
For the small deformation problem, table 3.2 shows the results obtained using the 7-
parameter element. The normalised tip displacement, according to linear beam theory, is
1.600. The three interpolation schemes give identical results since the shell is initially flat
and the unity of the director field is not violated. Condensing Q at element level gives
slightly better results as it leads to a more accurate imposition of the plane stress
thickness condition that govems this thin shell problem. Neglecting the quadratic term q
reduces the accuracy of solution. This error is directly related to Poisson's ratio V.
Table 3.2 Vertical displacement at tip of beam comsponding to small deformation
analysis.
Interpolation scheme
P I , IP2, IP3
More insight can be gained by looking at the large deformation solution of this
problem. A constant incremental load of F--40000 x h3 was applied for 10 load steps. The
vertical tip deformation after 10 load steps is show in table 3.3. A theoretical solution
based on inextensional elastica is also provided 1611. Fig. 3.6(b) shows the variation of tip
7 DOF
-1.575 1
7 Condense
- 1 .5765
6 DOF
-1.4170
5 DOF
- 1 .5765
O O. 1 0.2 0.3 0.4 OS 0.6 0.7 0.8
Vertical tip displacement ( 6v 1 L )
Figure 3.6 Cantilever problem: (a) mesh and defomed geornetry, and (b) normalised
load-deflection curve.
displacement with load. As anticipated, the results reveal that large deformation analysis
leads to Iarger errors. This highlights the importance of using an appropriate
interpolation. The most accurate 7-parameter shell element results are obtained using IP3
with static condensation. The results obtained without condensation are not significantly
different. Using IP1 (with or without condensation) gives good results only in the first
two load steps, and an increasing error for larger deformation. This is due to thickness
locking which is proportional to the change in curvature. Neglecting the quadratic
displacement term altogether (6 DOF) results in a nearly constant emr throughout the
deformation.
Note that the relative erron in small defomation analysis are higher than those
resulting from large deformation. This can be attributed to the different nature of loading
encountered in both. The small deformation problem involves only bending and
transverse shear stresses which are sensitive to errors in the thickness interpolation. The
large defomation analysis, on the other hand, involves membrane stresses which are
insensitive to errors in the quadratic through-thickness displacement terms.
Table 3.3 Vertical displacement at tip of bearn corresponding to large deformation
analysis.
3.7.3 Curved Cantilever Beam
Interpolation
scheme
Pl
IP2
IP3
O thers
A horizontal tip force is applied to a 90' curved bearn shown in Fig. 3.7(a). The following
properties were selected [52]: ~=200x10~, ~ 4 . 3 , R=20/x, w=1 .O, h=0.1 and the applied
7 DOF
6.0872
7.03 16
7.0460
Theory [6 1 ]
7.0629
5 DOF
7.05 1 1
7 .O3 64 I
7.0497
-
-
7 Condense
6.0872
7.0364
7.0497
Simo et al. [5 11
7.3053
6 DOF
5.9304
6.8 134
6.8242
Parisch [52]
7.083
F hl-
Nodes per side
(W
Figure 3.7 Curved beam: (a) mesh, and (b) convergence results.
force F=40000 x h3. The tip deflection was rnonitored for the linear problem (analytical
solution is 6-0.08106). The results are surnmarised in table 3.4. Contrary to the previous
example, the results differ for the 3 interpolations schemes, with IP3 king the rnost
accurate. Fig. 3.7(b) show the convergence results for different mesh densities. For the 7-
parameter shell element, the results converge to the exact solution, with IP3 (and IP2)
converging much faster than IPl. Neglecting the quadratic term leads to an incorrect
converged solution.
Table 3.4 Horizontal displacement under load for curved beam.
3.7.4 Pinched Hemisphere
Interpolation
scheme
PI
rP2
IP3
The 18' pinched hemisphere shown in Fig. 3.8(a) was modelled with the following
properties: ~=6.825~10', v=0.3, R=10.0, td.04 and a unit load was applied at points A
and B [17,3 1.32.521. This problem is dominated by bending stresses and is an excellent
test of the ability of the element to handle finite 3D rotations. The displacement under the
two loads was monitored. A limiting theoretical solution 0.093 is reported for the small
defmation problem 1311. Table 3.5 shows the resulting deformation of points A and B
for an 8x8 mesh. The results reveal that the IP3 (and IP2) with condensation gives results
closest to the 5-parameter shell.
A large deformation analysis was also perfonned for this problem. Fig. 3.8(b) shows
the deformed geometry, while Fig. 3.9 shows the resuiting deformation at points A and B
for interpolation schemes P l and IP3 with the condensation option. The displacement
7 DOF
-0.0794 1
-0.08039
-0.08052
7 Condense
-0.07943
-0.08045
-0.08054
6 DOF
-0.07 195
-0.07282
-0,0729 1
5 DOF
-0.08060
-0.08045
-0.08054
(a) (b)
Figure 3.8 Pinched hernisphere: (a) mode1 and @) deformed geometry.
0.0 0.1 0.2 0.3 0.4 0 5 0.6
Deflection (6 / R)
Figure 3.9 Deformation at points A and B using IPI and IP3 interpolation (see Table 3.1).
values agree with similar published numerical results [52,119]. Larger deformation at
points A and B is predicted using a 16x16 mesh.
- -- -- - -- - -
Table 3.5 Displacement at points A and B in pinched hemisphere corresponding to small
deformation analysis and using an 8x8 mesh.
Interpolation
scheme
P l
IP2
IP3 I
Others
3.7.5 Pinched Cylinder
A cylinder supported dong the edges with a rigid diaphragrn is loaded by a compressive
point load as shown in Fig. 3.10(a). This is a membrane dominated problem. The
analyticai solution, assuming small deformations, was reported by Lindberg et al. [121].
The following material properties were selected [ 17,261: E=~O.OX 106, v=0.3, R= 100.0,
t= 1 .O and an incrementai force of PO= l82.66/2 is applied for 10 load steps. An 8x8 and a
16x 16 mesh of uniform and distorted elements were used (Fig. 3.10(b)-(c)). Fig. 3.1 1
I I 0.0930 1 0.0939 1 0.09247 1
7 DOF
0.07832
0.09209
0.09276
Theory [3 11
shows the nonnalised deflection under the load in large deformation analysis. Al1 results
are based on IP3 interpolation. The results reveal that the large deformation solution is
not highly sensitive to element distortion, especially for the finer mesh. Furthemore, the
displacement values (in small and large deformations) are in agreement with similar
published work [l7,2 l,26,3 1,321.
7 Condense
0.07837
0.09230
0.09294
Simo et al. [3 11
5 DOF
0.09363
0.0923 1
0.09294 l
Betsch et al. [54]
(cl
Figure 3.10 Pinched cylinder: (a) mode], (b) unifom mesh. and (c) distorted mesh.
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
Deflection (5 I R)
Figure 3.1 1 Normalised deflection under load for a pinched cylinder.
3.7.6 Clamped-Clamped Thick Beam
A thick bearn is clamped at both ends and a concentrated force is applied at the rniddle as
shown in Fig. 3.12. The objective of this example is to examine the effect of the location
of the applied load on the deformation of the beam. The load is applied at the top, bottom
and middle shell surfaces. In addition, the load is also equally divided between the upper
and lower shell surfaces to create the consistent load vector described in Eqn. (3.34). The
following material properties were selected: E= 10x 106, v=0.3, k 10.0, w= 1 .O, h=0.625
and an incremental force F=40000 was applied in each load step. The large deformation
problem involves both membrane and bending deformations.
Figure 3.12 Clamped-clamped thick bearn: Mesh and deformed geometry.
While the vertical deflection is not significantly affected by the location of the load,
the shell thickness h and the quadratic function q are affected. Fig. 3.13(a) shows the
thickness change at different load levels. Using the mid-surface and the combination of
top and bottom surfaces results in a change in thickness, which is mainly attributed to
Poisson's effect. By placing the load on the top or bottom surfaces, an additional
thickness effect is imposed from the direct thickness compression/tension. Fig. 3.13@)
shows the variation of the quaùratic displacement q. The figure reveals that the use of the
top, bottom or combined loading results in similar variation of q. However, applying the
load at the mid-surface results in a different quadratic displacement distribution. Hence, it
is obvious that for thick shells the location of the load (through the thickness) affects the
defonnation behaviour.
3.7.7 Spherical Sheil Under Pressure
A thick spherical shell is subjected to intemal and extemal pressure (Fig. 3.14). One
eighth of the shell was modelled due to symmetry using 192 shell elements. The
following geometric and material properties were selected: ~=6.825x 1 o', v=0.3, R= 10.0,
t=0.625, and an intemal pressure Pm=lûûû. The extemal pressure Povr was varied from O
to 1000. The 5-parameter shell elements predict only the membrane deformation. In
addition, the 7-parameter element is also able to predict the thickness deformation. Table
3.6 shows the change in shell thickness predicted using the 5- and 7-parameter shells, as
well as the 5-parameter shell with post-solution thickness update. Theoretical results
based on elasticity theory are provided [122]. For equal values of inner and outer pressure
a state of hydrostatic loading is obtained, where al1 normal stresses are equd (a = 1000).
In this case, the thickness change is related to the bulk modulus of the matenal. Table 3.7
shows the resulting radial stress at the imer and outer shell surfaces. Theoretically these
stresses should be equal tc the applied pressure. The srnall discrepancy is mainly due to
the discretization and nodal extrapolation emors. The improved accuracy obtained using
the 7-parameter mode1 is a result of the two extra degrees of fieedom, as well as
accounting for the exact location of the load with respect to the shell thickness.
Change in thickness (6 1 t)
(a)
Figure 3.13 Clamped-clamped thick barn: (a) variation of shell thickness, and (b)
variation of quadratic displacement q.
Pm = Constant Po, = Variablc
Figure 3.14 Spherical shell subjected to intemal and extemal pressures.
Degrees of
Freedom
POUT
O I 500 I 1000
7 DOF
Table 3.6 Nomalised change in shell thickness a3 / t for Pm=lOOO.
5 DOF
5 DOF
w ith thic k-update
Theory [122]
-7.282~10'~
-
Table 3.7 Variation of radial stress OR /PH through sheU thickness for Pm=lOOO.
O
-6.957~ 101'
-7.305~ 10"
Shell Surface POUT
-3.934~ i O-'
O L
Inner Surface -0.9408
Outer Surface 0.0525
-5.857~ 1
O
-3 .478x 1 o - ~
-3.945~ 10"
O I
O
-5.8608~ loa
500
-0.9747
-0.4777
1000
-0.10086
-0.10079
Chapter
Analy sis
4
of Large Deformation
Frictional Contact in Shells using
Variational Inequalities
4.1 Kinematic Contact Conditions
The contact formulations are govemed by two constraints: (i) the magnitude of the
normal contact stress must be less than or equal to zero, and (ii) the displacements of the
contacting surfaces must satisQ a kinematic contact condition, so as to avoid penetration.
In addition, the tangential forces and displacements dong the contact surface are assumed
to be govemed by Coulomb's fiction law.
For shell elements, there are two potential contact surfaces: the upper and the lower
physical boundaries of the shell [ 123-1251. This means that for each point on the shell
reference surface, there are two possible offset contact locations (Fig. 4.1)
Based on a master-slave technique, for every point on the master contact surface ï, a
comsponding closest point on the slave surfaces is determined from the kinematics of the
deformation [78]. This is defined for x é Tm as king
where the constraint prevents improper contact between master and slave surfaces. The
gap function for the two shell contact surfaces can be defined as (Fig. 4.2):
Mid-surface node
Q Contact node
O Integration point
A Shear strain tying point
Figure 4.1 Location of contact points in four noded shell element
-T t+d x) - t+~ t x ~ ] . t + ~ t+AtgT(x)=ly ( N~ 2 0
and
t+at g~ (X) = l y * ~ ( t + b X ) - t+~ t X~ let+& NB 10
Figure 4.2 Kinematic contact constraint for shell surfaces.
where 'N can either be the unit outer normal to the master contact surface or the unit
inward normal to the slave contact surface. The gap functions defined above are generally
nonlinear. However, for incremental finite element analysis, a linearized tangential form
yields:
.['uT(x)- 'u(y7)]- ' g T ( x ) ~ o
and
'NB . [<uB(x)- ' ~ ( y * ~ ) ] - 'gB(x)< O
The above inequality constraint equations allow for simultaneous double-sided contact, if
the two contact inequalities are active. To further illustrate this point, the displacement of
the top and bottom shell surfaces are expressed in terms of nodal quantities, and the
contact normal is assumed to be coincident with the shell director for simplicity. In this
case, the following two inequalities are obtained:
where UN is the displacement of the shell rnid-surface in the direction of the normal.
Compared to the classical shell elements assumptions where the change in shell thickness
a3 is neglected, the two constraints cannot be simultaneously satisfied, and hence only
single-sided contact is generally achieved [ 1241.
The second contact constraint is related to the magnitude of the normal stresses which
are compressive. This constraint is expressed in terrns of surface tractions, as follows:
The contact stresses are decomposed into normal and tangential components, viz.:
Coulomb's law of friction, which involves distinct sticking and sliding modes of
deformation is used. Accordingly, the relationship between the normal and tangential
stresses can be expressed as foilows:
where p is the coefficient of friction and  is a positive constant. The tangential
component of displacement for general3D problems c m be expressed as:
with 1 being a 3x3 identity matrix.
4.2 Variational Inequalities for Continuum
The general variational inequality frictional contact formulation can be expressed, in total
Lagrangian framework, in terms of the contravariant second Piola-Kirchhoff stress tensor
Si' and the covariant Green-Lagrange strain tensor qj, as:
where
In the above expression, u is the equilibnum configuration and v represents any
admissible displacement field. The a(u,v-u) term represents the vimial work of the elastic
resistance of deformation from configuration u to v. The f(v-u) term represents the virtual
work done by the external forces, while j(u,v) - j(u,u) is the contribution of the frictional
forces. K is the space of al1 displacements for the points in the domain which satisQ the
kinematic contact and boundary conditions.
4.3 Reduced Variational Inequality
Solution techniques for the VI fnctional contact formulation are based on a reduced VI
mode1 [81-83,991. By assuming that the normal stress within each time step is
independent of the displacement field u a reduced variationai inequdity is obtained [8 11:
where
This assumption will eventually lead to a symmetric form for the tangent stifhess maeix,
thus enabling the use of standard symmetric solvers and significantly decreasing the
computational requirements of the resulting system of equations. In order to solve the VI
of Eqn. (4.1 l), the fnctional term j(v) is replaced with a regularized differentiable form
1961 :
The following form is often used for the regularîsation function [8 1,961:
Consequently, the regularized fnctional term can be replaced by its directional derivative:
The regularised variational inequaiity takes the foiiowing form [125]:
which is still an inequality formulation due to the kinematic contact constraints included
in the space of functions K. The solution of the regularized problem converges to that of
the original unregularized problem as the regularization parameter E tends to zero. Some
insight into the convergence and uniqueness issues related to this sub-problem are
provided in reference [8 11.
An alternative solution technique involving two steps is comrnonly applied to
continuum problems [€Il]. In the first step, the tangentid forces are prescnbed and a full
contact search is performed to evaluate the contact surface and normal contact forces. In
the second step, the contact surface is assumed known and the field variables and
frictional forces are evaluated. Enforcing the contact constraints in step 2 is optional. This
technique was tested for various shell problems. However, Our results indicate that the
solution frequently diverges in the second step, if the contact constraints are not imposed.
This is due to the sensitivity of thin structures to variations in the forces normal to their
mid-surface. When the constraints are enforced, convergence is achieved, but the total
number of iterations increases significantly. Therefore, in d l subsequent analyses, only
the single step solution (Eqn. (4.15)) will be used.
4.4 Variational Inequalities for Shell Structures
A consistent linearization of the general variational inequality is necessary for developing
finite element based solution techniques. Specifically, for the tems involving interna1
energy and friction. This can be achieved using an incremental total Lagrangian
formulation, where the following decomposition of the stress and strain components is
used 1201:
where e, and 11, are the linear and nonlinear s W n components. However, due to the
nonlinear nature of the displacement field for the shell mode1 used, it is necessary to
further linearize the incremental strains to account for both linear and the quadratic
displacement ternis as detailed in section 3.6:
Based on the above linearization, the intemal energy and the residual tems can be
expanded as follows (1 24,1251:
where R,(v) and &(u,v) include al1 ternis that will contribute to the load vector and the
stiffhess matrix respectively. Subscript w is the total displacement vector, which will, for
the sake of clarity, be omitted in the following denvations.
The regularized fiction terni is expressed in tems of the linearized incremental
dis placements:
where q is the deflection relative to the configuration corresponding to sticking friction
and M is a 3x3 matrix that isolates the tangentid displacement, based on Eqn. (4.9).
Finally, the incrementai regularized VI takes the following form:
where u* is the configuration corresponding to sticking fiction.
4.5 Solution Technique
The contact constraints in the VI of Eqn. (4.21) are enforced using Lagrange multipliers
[l24]:
The overbar resembles virtual parameters and K2 is a set of admissible Lagrange
multipliers or contact forces, which is govemed by Eqn. (4.6). The advantage of using
Lagrange multipliers over penalty based methods is that the constraints are satisfied
exactly without any inter-penetration. This inter-penetration could be detrimental to the
accuracy of the solution, since it can be of a comparable order of magnitude to the shell
thickness.
4.6 Discretization
In this section, the contact constraints and the frictional contribution are discretized and
presented in a matrix form suitable for finite element implementation. Using the
discretized shell element equations, derived in Section 3.5, the complete variationai
inequalities frictional contact formulations are expressed in a discrete form. Several
aspects of the solution strategy are then detailed.
4.6.1 Contact Constraints
Each discretized contact constraint can be represented as:
where Ga is the gap, AUa is a vector containing the degrees of freedom of the master
contact node and the target element:
where L is the number of nodes per contacting segment on the slave contact surface. The
A, matrix represents the contact constraint, which is based on the difference between the
displacement of the master and slave surfaces. For each master contact node a, the
general form of the A.. sub-ma& is:
where 6: is the contribution of each of the slave nodes to the normal displacement at the
target point on the slave contact surface and it is determined based on the local contact
search. The Q-matrix is geometry dependent and relates the extemal surface
displacements to the mid-surface degrees of freedom. For the proposed seven parameter
shell element, it takes the following form:
where ys is a constant which equals +1, 0, -1 for the top, rniddle and bottom shell
surfaces, respectively. Note that the seventh degree of freedom does not contribute to the
Q-matrix and therefore it can be condensed at the element level without affecting the
accuracy of solution. Note that the use of the shell mid-surface for contact neglects the
effect of the rotational and extensional degrees of freedorn on the contact constraint, thus
deteriorating the accuracy of the results and preventing double-sided contact [124].
The general form of the A and Q matrices in Eqns. (4.25)-(4.26) can be simplified for
specific contact conditions. If the master or target nodes are represented by 3D solid
elements, the Q-matrix becomes a square unity matrix, formed from the first three
columns of Eqn. (4.26). For a classical five parameter shell elements, the 1st two column
of the Q-matrix can be excluded. If the target is a ngid surface, then L = 0, and only the
first Q sub-matrix in Eqn. (4.25) is retained.
Finally, the assembly of al1 contact constraints, yields a set of inequalities of the form:
where G is the global vector of the gap hinctions, AU is the assembled global
displacement vector and the A-matru represents the standard finite element assembly of
al1 individual A, constraint matrices.
4.6.2 Friction Terms
The fictional term can be discretized as follows:
The fonn of the frictional stiffness component kF(q) depends on the state of fiction:
1 M M ~ O ~ ~ M -- for l%l> e kF(q)= lqTl 1qTr
M - fOrkTI SE E
Our results show that the quadratic term in the above equation is indeed significant,
especially for problems involving large regions of slip. The tangential frictional force
contribution in the residual term is also based on the regularization parameter, and is
evaluated as follows:
where q~ is the displacement relative to the sticking fnction configuration.
Based on the above discretization, the VI formulation for the frictional contact
problem in shells can be expressed in a discretized form as:
4.6.3 Finite Element Solution
The Lagrange multipliers solution to this VI can be expressed in a ma& fonn as:
where the contributions to the stifiess matrix result from the linear, nonlinear geometnc
stifhess, and the quadratic displacernent tems, as well as the fictional tems of
Eqn. (4.28). The C matrix is a subset of the A matrix of Eqn. (4.27) including only the
active contact constraints. This active constraint set is modifîed after each iteration step
and a full contact search is performed. Details of the cornputer implementation of this
solution algorithm are provided in Appendix B.
Equations (4.32) are solved iteratively for 'AU and 'A until convergence is reached.
The resulting displacements and contact forces are used to update the coordinates of the
shell surfaces, the contact surface, the prescribed normal stresses in the fnctional term
and the fnction state (stick-slip). In addition to the energy andor displacement based
convergence criteria necessary for non-linear problems, other convergence cnteria
pertaining to the stability of contact conditions are necessary. This is achieved when al1
master and slave nodes in the active set of contact set remain constant between iterations,
and when the frictional state does not change for al1 contacting nodes.
4.7 Verification Examples
Five exarnples have been selected to demonstrate the flexibility and the accuracy of the
newly developed approach. The first concerns the contact behaviour of three bearns. In
the second, we examine the problem of a ring compressed between two Bat rigid dies. In
the thûd. we focus our attention to a fnction test problem. The fourth example involves a
belt-pulley assembly. Finally, in the fifth example, we examine a flat metallic strip
compressed between two curved dies. The selection of these exarnples was govemed by
our desire to show that the developed formulations and algorithms are capable of
simulating double-sided shell contact and can accommodate friction as well as contact
stresses associated with large deformation in shell structures. In al1 problems, extensive
convergence tests were performed to obtain the optimum mesh density beyond which
there was no significant change in the results.
4.7.1 Three Beam Contact
The first problem involves three layered beams fixed at one end and the top one is loaded
with a unifonn line load at a distance 0.6L from the support, as show in Fig. 4.3(a). The
length of the beams L is 1.0, the width is 0.3, the thickness of each is 0.05 and the gap
between bearns is set to 0.015. Both small and large deformation analyses are performed.
The purpose of the small deformation analysis is to compare with theoretical solutions.
Each beam was modelled using 40 four-noded shell elements of the type detailed
above. The use of this element is necessary in this example to capture the double-sided
contact experienced by the rniddle beam (Fig. 4.3(b)). Initially, no contact is observed.
However, as the load increases the top two beams touch at the edge (stage l) , then al1
three beams contact at the edge (stage 2). As the load further increases, contact spreads
towards the point of application of the load; fint for the top two beams (stage 3) and later
on for both contact locations (stage 4).
Figure 4.4 shows the variation in the transverse stiffhess at the point of application of
the load for both the small and large deformation problems. The stiffbess is nomalised by
the initial transverse stiffhess and the displacement is normalised by the initial gap
between the bearns. An analyticai solution evaluated on the basis of linear beam theory is
also shown for cornparison. The results show a sudden jump in stiffness at the start of
each contact stage. Furthermore, for contact stages three and four, there is a gradual
increase in stiffhess as the load increases due to the change in the contact length.
For the large deformation problem, the sarne four contact stages are expenenced.
Figure 4.5 shows the deformed geometry. This example reveals that the above
formulations are capable of an accurate prediction of double-sided contact, which cm be
very useful in modelling more complex problems such as sheet metal forming.
4.7.2 Ring Compression
The second problem examined in this thesis is that of a cylindncai shell compressed
between two ngid flat plates, Fig. 4.6(a). In view of symmetry of the structure and its
4.3 Three-barn conwt pwblern: (a) geomew. and (b) contgt stages.
It Stage 2
1 - Analyticai solution 1
Figure 4.4 Effective stiffness for three contacting beams.
1 Stage 2
Stage 4
Figure 4.5 Deformed geometry for three-beam contact problem.
Element Face in Contact
Figure 4.6 Ring contact problem: (a) schematic of loading arrangement, and (b) deformed
geometries.
loading, one quarter was modelled using four-noded shell elements. In this example, it is
necessary to include large deformations, since the cylinder undergoes a significant change
in geometry. If one considers only small deformation analysis. a large unredistic
separation between the ring and the ngid plates is predicted at the centre of the contact
region. Fig. 4.6(b) shows the deformed geometry resulting from the newly proposed
formulation, where contact is indicated by the highlighted elements.
In order to validate the formulations and the developed algorithms, the current
technique was compared with traditional FE predictions employing penalty-based contact
elements and single surface shell contact. Fig. 4.7 shows the variation in contact pressure
when the ngid dies reduce the diameter of the ring by 32.546. The vertical axis
corresponds to the nomalised contact pressure and the horizontal axis represents the ratio
of contact length to radius.
---- &&=2xl0' - /KO=~X i o3
------- ICJK,,=~X~O~ . 2C Current 1 ------ 4 - Current 2
Contact distance (x/R)
Figure 4.7 Variation of contact pressure dong contact distance for a 32.5% reduction in
radius.
Three of the curves correspond to the traditional five parameter shell element using a
penalty-based contact formulation with different values of the nomal contact stiffhess
KN, obtained from a commercial FEA code. The contact stiffnesses were normalised by
the initial stiffness (Ko) of the ring in the direction of the applied load. Two curves were
obtained using the current formulations; in the first (current 1), a five degrees of freedom
shell element similar to that employed in the commercial code was used while for the
second curve (current 2), the newly developed seven parameter element was used and the
full poiential of the developed formulations was evaluated. The thin shell solution by
Frisch-Fay [61] for the aven geometry and loading is a contact region of normalised
width 0.091 with the contact load localised at the edges of the contact zone. Due to the
inclusion of the shear deformation terni, however, a continuous pressure distribution over a
the contact region was computed in the present solution. The same effect has been
reported for plates [64] and spherical shells [68].
The results reveal the dependence of the traditional contact element solution upon the
contact stiffnesses, where at low stiffnesses KN excessive inter-penetration is observed,
while at large KN values the program does not converge as a result of an ill-conditioned
stiffhess rnatrix. Furthemore, it shows that even with a five degrees of freedom shell
element, the developed formulations provide accurate results. Finally, using the new
higher order shell element influences the results even without double-sided contact. A
wider contact region was predicted which can be attributed to the newly added shell
flexibility in the thickness direction. This conclusion is in agreement with resuits
presented by Essenburg 1671, where it was shown that the use of a higher order theory for
beams results in the prediction of a wider contact zone and a lowei peak stress level. No
such results were previously presented for cylindrical shells.
The development of the contact area and the corresponding contact pressure
distributions for a thick ring with a radius to thickness ratio of 12.5 are shown in Fig. 4.8
for various levels of ring compression. The length of the contact zone is normalised with
respect to the radius. The results show that the form of the contact pressure changes from
parabolic (Hertzian) to an edge-dominant form as the ring deformation increases.
Furthemore, the contact area initially grows at a slow rate which increases only after the
pressure distribution ceases to be Hertzian. Sirnilar results for thicker rings are presented
in [75], using solid elements with several elements through the ring thickness.
Cornparison with this earlier work reveals that the newly developed shell contact
formulation gives good results.
+ 10 % Reduction + 25 % Rcduction -+ 30 % Rcduction ++ 35 96 Reduction + 40 9% Reduction
O 0.05 0.1 0.15 0.2 0.25 0.3 0.35
Nonnalized contact length (x/R)
Figure 4.8 Contact pressure distribution for different ring reduction ratios.
4.7.3 Strip Friction Test
In this example, a thin strip is wrapped around a cylindrical rigid body and a tensile load
is applied to the fiee end (Fig. 4.9(a)). This test is cornmonly used to evaluate the
coefficient of fnction in metal forming applications [126]. Through elementary
calculations, it can be shown that the ratio of the tensile forces Ti to T2 depends on the
wrap angle <p and the coefficient of fiction p, Le.
Furthermore, d l nodes should be in sliding contact and the contact pressure variation
should comply with the following relation:
Figure 4.9 Strip friction test: (a) finite element model, and (b) contact pressure
distribution.
A strip of dimensions R= 10, t=0.1 and w = 1.0 was modelled using 60 four-noded
elements and the cylinder was modelled as a rigid surface. Fig. 4.9(b) shows the predicted
contact pressure variation for p = 0.5, and TI = 20x10~. A comparison between the
numencal and analytical solutions shows that the normal and tangentid stress
distributions are accurate to within 3% of the theoretical predictions.
4.7.4 Belt-Pulley Assembly
In this problem, the contact behaviour of a belt wrapped 180' around a pulley is examined
(Fig. 4.10). The belt is modelled using 80 shell elements and the pulley as a rigid
cylindrical surface. The following dimensionless geometrical and mechanical properties
of the assembly were assumed: w = 10.0 mm, t = 2.0 mm, R = 120 mm and E = 100x lo9 Pa
A coefficient of friction of 0.4 was assumed between the pulley and the belt, together
with a fnctional regularisation parameter E = 5 ~ 1 0 - ~ mm. Selecting smaller regularisation
values does not significantly affect the results, however, it leads to more iterations.
Initially, a tensile pre-stress To is applied to both ends of the belt. A counter-clockwise
incrementai anplar deformation 0 = 0.1' is then applied per load step. This is equivalent
to applying an increasing torque at the centre of the pulley.
Stick
Figure 4.10 Finite element mode1 of belt- pulley assembly.
A theoretical solution is developed by obtaining the goveming differentiai equations
of the system, based on membrane theory. As a result of contact and fnctional constraints,
the domain is divided into three regions: a central stick region where (DI < @ < m2, and
wo slip regions where @ < mi and > 0 2 . The slip region is govemed by the following
di fferential equation:
where Ti is the tension in the belt The relative angular displacement can be expressed as:
which accounts for the initiai load To as well as the prescribed angular deformation 8. A
similar expression can be obtained for the tension T3 and the relative displacement u3 of
the right-hand side slip region. In the stick region, the equation governing the system is
reduced to:
In this case, the relative angular displacement field is constant:
Boundary and continuity conditions between the three regions were used to evaiuate the
unknown constants as well as the stick-slip regions.
Figure 4. I l(a) shows the numencally predicted contact stress distribution for values
of 0 between O and 0.9, while Fig. 4.1 1(b) shows the corresponding variation for the
frictional stress. In both figures, the stresses are normaiised with respect to the initial
contact stress a0 = Td(Rxw). Let us now examine the mechanics of the system. Initially,
there is no resultant torque and the contact pressure distribution is constant. Furthemore,
d l the nodes experience sticking, with no fictional forces developing. For values of 8 >
O.go, global slipping occurs and the pulley continues to rotate without affecting the
deformation behaviour of the belt. At this stage, the resulting contact stress distribution
becomes exponential.
Figure 4.12(a) compares the theoretical and finite element predictions of the critical
angles QI and Q2. The results indicate a non-symmetric deformation pattern. The slip
region, which is initially non-existent, grows from the edges (O = 9 0 4 towards the
centre. The final point to reach sliding for the examined geometric and material properties
is at @ = 16'. The numerical results are accurate for both values selected for the
regularisation parameter e.
Angular position Q>
-90 60 -30 O 30 60 90
Angular position <O
(b)
Figure 4.1 1 Effect of rotation 0 on the contact stress distribution of belt: (a) normal
contact stress, and (b) tangentid stress.
Rotation angle 8
-90 -60 -30 O 30 60 90
Angular position Q,
(b)
Figure 4.12 (a) Variation of stick-slip angles, <Pl and m2, with rotation 0, and
(b) cornparison between theoreticai and FE stress distributions in belt for 8 = 0.6'.
Figure 4.12(b) provides a comparison between the theoretical and numerical
predictions of the normal and shear stresses, at 8 = 0.6*, for two different regularisation
parameters. The numerical formulations correctly predict the distribution of the contact
and fi-ictional forces, as well as the regions of stick and slip. Note that the normal and
fnctional forces are higher for > O, whereas the slip region is wider for c O. One
notable difference in results is the sudden drop in the frictional forces at the start of the
stick zone based on the theoretical solution. This is not achieved in the numerical
simulation due to the size of the element. The smaller of the two regularisation
parameters E = 5x10-) mm gives results that are closer to the theoretical solution. Selecting
smaller regularisation values does not significantly affect the results, however, it leads to
more iterations.
4.7.5 Strip Compression Test
This example examines the compression of a shell between curved dies (Fig. 4.13). This
is an elaborate double-sided frictional shell contact problem, where most of the shell is in
direct contact with the dies. The test simulates the compression of the strip between the
dies followed by the pull-out of the shell strip. The geometry of the shell and the dies in
this example resemble a metal forming draw-bead application. However, in this case, the
material is assumed elastic and our focus was to demonstrate the versatility of the new
formulations in problems involving double-sided shell contact and multiple contacting
surfaces. A 300 mm long and 2 mm thick sheet was modelled using 120 four noded
elements, and the three dies were modelled as rigid surfaces. The compression stage was
performed in 10 steps, and the pull-out in 26 steps. No constraints were applied to the
right end of the sheet. The left end was fixed during the closing stage and was later given
an incremental leftward displacement to pull out the sheet. The simulation was performed
both with and without friction. For the frictional case, a friction coefficient of 0.05 was
selected together with a regularisation parameter of 0.5 mm.
Upper die u Metal strip
Lower die
Figure 4.13 Schematic of strip compression test.
Figure 4.14 shows the deformed geometry and the contact forces at various stages of
the simulation. They demonstrate clearly the progression of contact as the dies close
together, and during pull-out. When the die is closed, a nearly constant pressure
distribution develops in the flat regions between the die and the blank-holder. This is
coupled with a higher, concentrated force couple at the locations where there is a change
in curvature. The magnitude of the unifonniy distributed load on the flat regions is a
function of the total load applied to the blank-holder, while the magnitude of the force
couple is related to the radius of curvature of the dies.
The deformation mode is similar with and without friction, however, the pull-out
force is significantly higher when frictional effects are taken into account. Fig. 4.15
shows the variation of the pull-out force dong the left edge of the sheet strip. In the bead
closing stage (steps 1-10) the force is negligible for both cases. However, in the pull-out
stage, the resisting force is much higher for the fictional case. This is due to the sliding
of the shell over the rigid surfaces which is resisted mainly by friction.
During load steps 11 to 13, the shell is in full double-sided contact on both sides of
the die and the pull-out force is maximum. This is followed by a linear decrease in the
pull-out force as the shell slides past the right half of the die (steps 14 to 20). From steps
21 to 33 the shell slides p s t the elevated central part of the die. Finally, a constant pull-
out force is reached when the shell is flat and subjected to a simple state of biaxial
Loading (steps 34 to 36). It is worth noting that in a metd forming draw-bead simulation
the resisting force is not solely provided by friction. A simcant contribution is provided
Figure 4.14 Deformation stages for saip compression.
by the plastic loading and unloading of the sheet material as it passes over the bead and
around the fillets [ 1 261.
I Compressim 1
Stage I v
--t- Frictioniess - p = 0.05
1 RiII-out Stage
Load Step
Figure 4.15 Effect of friction on the pull-out force.
Chapter 5
Experimental Investigations
5.1 Introduction
This chapter presents the details of the experimental investigations used to venfy the
newly developed numencai predictions. The shell shucture used in these tests is that of a
ring subjected to lateral compressive loading, as depicted in Fig. 5.1. This problem was
selected due to the importance of ring and tubular structures to many engineering
applications. These include: aerospace satellite assemblies, energy absorption devices,
bal1 bearing technology, pressure vessels, hydraulic and pneumatic devices to support
contact loads. In ail these engineering applications, contact loads play an important role
and can result in the deterioration of the mechanical integrity of these structures. An
extensive review of the available literature on lateral ring compression reveals that most
of the efforts in this area have focused on determining the load deflection behaviour of
the rings under static and dynamic loads [127-1311. However, in this work, the focus is
on the contact problem of both thin and thick ring structures [132]. Details of the ring
samples used and the experimental work are provided below. The results are presented in
chapter 6.
5.2 Details of Rings Used
Photoelastic and duminium nngs of varied radius-to-thickness ratios were
manufactured to a tight tolerance of 76x10') mm. The photoelastic nngs were used in
order to obtain the maximum shear stress contours and thus enable the comparison with
the finite element predictions of the stress field. The aluminium rings were used not only
to characterise the load deflection behaviour but also to measure the strain at the inner
radius of the exarnined rings under different loading conditions. The details of the
geometry and material properties of the different rings an provided in Table 5.1.
t Applied load
Alignment bars -
Die --u Sliding A
brackets
Figure 5.1 Experimental setup.
The radius of curvature of the loading dies was made 10% larger than the radius of
the rings, thus allowing for a wide range of applied loads and contact zones, while
maintainhg elastic deformation. It is worth observing that in the compression of very thin
rings, bending stresses would dorninate the stress field (Fig. 5.2). The loading dies, on the
other hand, would experience a generalised biaxial stress field as a result of the contact
stresses. Accordingly, better visualisation of the contact stresses can be achieved by using
a photoelastic die. Conversely, in the case of thick rings, contact stresses will induce a
generalised biaxial state of stress in both the ring and the loading dies. In this case, better
results can be obtained by using a photoelastic ring and stiffer aluminium dies.
; CD* =B
,, Aluminuni Ring
Figure 5.2 Influence of contact and bending stresses on the ring and dies.
The elastic and optical properties of the photoelastic material were evaiuated using
simple tension, four-point bending and disc compression tests.
Ring Material Aluminium Photoelastic
Outer radius (mm) 43.18 43.18
1 Thickness (mm) 1 4.32,2.62, 1.0 1 21.59, 10.8,4.32 1 Width (mm)
Materiai
-- -- --
Table 5.1 Details of geometry and material properties for tested rings.
Young's modulus @Pa)
Poisson's ratio
Material fringe value, kPal(fnnge1m)
6
Aluminium 606 1
6
Epoxy (PSM-5)
70
0.33
-
2.7
0.36
10.5
5.3 Photoelastic Studies
Photoelastic images were taken with a traditional diffuse light transmission polariscope
system (Fig. 5.3). A digital analysis system and its peripherals, which includes a Hitachi
VK-C360 Camera with 50 mm macrolens, imaging board, image processor and a PC
were used to analyse the isochromatic and isoclinic fnnge patterns. The CCD carnera is
used to scan the chosen photoelastic area, then the image is divided into 512x480 picture
points (pixels). The video signal of the incoming image is converted into a digital signal
with a 24-bit resolution. The photoelastic material used was PSM-5. Young's Modulus
for this materid was 2.7 GPa, which was evaluated using a standard tension specimen
machined from a photoelastic plate. The material fringe value fa was determined, with the
aid of a diarnetrally loaded solid disc, to be 10.5 kPa/fringe/m. An accurate estimate of
fringe fractions was obtained using a Soleil-Babinet null-baiancing compensator [133].
5.4 Strain Gauge Measurements
A single element 3.175rnm (1/8") strain gauge was carefully attached to the inner radius
of the tested rings to measure the circumferential strain at different angular positions
(Fig. 5.4). The different angular positions were obtained by carefully rotating the ring
incrementally with respect to the normal axis using a reference €rame. The strain gauge,
with a gauge factor of 2.1, was thermdly compensated using a quarter Wheatstone bridge.
Furthemore, the strain measurements were taken in a thermally controlled environment.
The strain gauge was connected to a commercial direct-reading strain indicator, which
provided the output directly in tenns of strains.
5.5 Load Deflection Characteristics
Figure 5.1 shows the experimental setup used to obtain the load-deflection diagram of
rings of varying thicknesses subjected to diametrd load between two curved dies. The test
rig was designed and built to a tight tolerance to maintain lateral and horizontal alignrnent
of the rings and the dies to avoid bending effects. Symmetry during loading was
Quarter Quarter wave
Polariser plate plate Analyser
Figure 5.3 Photoelasticity setup.
(a) @)
Figure 5.4 Strain gauge location for (a) thick (t/R= OS), and (b) thin rings (t/R = 0.1).
maintained using a spherical seating arrangement. The contact regions between the ring
and the loading dies were lubricated to minimise fnctional effects. Diametrai deflections
of the rings was measured using a very accurate dia1 gauge with a minimum resolution of
1 0 p . The diametral loading was incrementally applied using dead weights and the
magnitude of the vertical displacement was recorded.
Chapter 6
Results and Discussion
6.1 Introduction
In this chapter, we provide four interesting case studies that utilise the formulations and
solution techniques developed in this work. The selection of these case studies was
motivated by Our desire to examine the main characteristics of the developed shell
element and fictional contact formulations. The first case study deais with the lateral
compression of a ring between curved dies. Specifically, we examine the effect of ring
thickness and loading conditions on the resulting contact region and contact stress
contours. The second case study, involving two cylindrical shells in contact, examines the
large defornation aspects of the newly developed contact strategy. In this case, the mode
of deformation influences the size and location of the contact zones drarnatically. The
latter stages of the deformation involve double-sided shell contact. In the third example, a
sphencal shell is compressed between two Bat platens. In this case, the shell experiences
three different contact stages including both Hertzian and non-Hertzian contact. Finally,
in the fourth case study, we provide design guidelines for saddle supported pressure
vessels.
6.2 Lateral Compression of a Ring Between Curved Dies
The theoretical models developed in chapters 3 and 4 were extensively validated using
the expenmental work detailed in chapter 5. Three different tests were conducted:
(i) photoelastic image analysis to ver@ the mode1 predictions of the stress field,
(ii) strain gauge measurements to validate the finite element predictions of the strain
field, and
(iii) load deflection response to validate the finite element predictions of the
deformation behaviour of the different rings examined.
Prior to analysing the expenmental findings, the existing analytical solutions which are
available for two extreme cases are surnrnarised. The first solution is for a solid disk
compressed between two rigid dies [134]. This is an extension of the classical contact
formulation developed by Hertz [7,8]. For an applied concentrated load P, the size of the
contact zone c is:
where R* is the relative radius of curvature between the ring and the die. E* is the
composite modulus of the system, which is a function of Young's Modulus and Poisson's
ratio of the ring and die. The expression for the contact stress p(x) at a point x dong the
contact length is:
A more accurate analytical solution was reported by Gladwell 11351. However, for the
geometries analysed in this research the results are very similar to the predictions of
Eqns. (6.1-6.2). Both solutions are only valid for small deformations. Photoelastic images
related to Hertzian contact problems cm be found in Refs. [a, 1361.
The other available theoretical solution is for very thin rings and is based on
inextensional elastica [61]. Initially, contact is localised at two points dong the top and
bottom contact surfaces of the ring. When the load P increases beyond a cntical load Po,
the size of the contact zones increases and the contact forces change to two concentrated
loads at the edges of each contact zone. In this case, one can establish that the cntical load
is proportional to EYR~, where the proportionality constant is dictated by the relative
radius of curvahue between ring and the loading die. The proportionality constant was
evaluated to be 0.3, and therefore:
This solution is neither limited to small deformation nor small contact area. However, the
limitation is in the modelling of the ring structure. By assurning an inextensional elastica
only bending type deformation is possible. Membrane, shear and direct contact effects
and their coupling interaction are neglected. This renders the solution accurate only for
very thin rings.
Let us now focus Our attention on the photoelastic validation tests. The thick rings
(Wte 10) were modelled using two dimensional plane stress elements (Fig. 6.1) and
contact was accounted for using the continuum variationai inequalities foxmulations of
Refaat and Meguid 182-84,1021. In view of symmetry of loading and geometry, one
quarter of each ring was discretized using eight-noded elements (Fig. 6.2). The loading
dies were modelled using the appropriate material properties. The thinner rings were
discretized using degenerate shell elements. The solution was obtained using the newly
developed variational inequalities contact formulation.
Figure 6.3 shows the photoelastic isochromatic fnnge patterns and the corresponding
maximum shear stress contours predicted by the current variational inequalities contact
model. The numbers in the figure indicate the actual stress values corresponding to the
different fringes. Figure 6.3(a) corresponds to a ring with Wt = 2, while Figure 6.3(b)
corresponds to the case where R/t = 4. In these figures, the lefi hand side corresponds to
the photoelastic results, while the right hand side corresponds to the finite element
predictions. The experimental and numencal results, which are in close agreement, reveal
the following: (i) for Rh = 2, the maximum shearing stresses are located at the inner
surface at both the horizontal and vertical orientations, and (ii) for Eüt = 4, the maximum
shearing stresses are located at the inner and outer ring radii at the vertical position. For
thinner rings, the maximum stresses are also at the inner and outer ring radii at the
vertical position.
Figure 6.4 shows the photoelastic images of the curved dies and the corresponding
maximum shear stress contours predicted fkom the finite elemeat results for three
different loading levels for Wt = 10. Again, there is good agreement between the
Figure 6.1 Finite element rnesh of rings.
Figure 6.2 FE mode1 of ring and curved die.
Photoelastic Finite Elements
Photoelastic Finite Elements
(b)
Figure 6.3 Photoelastic (left) and finite element (right) maximum shear stress contours
developed in a photoelastic die: (a) Wt = 2 (P = 370 N) and (b) R/t = 4 (P = 50 N).
Photoelastic Finite Elements
(a)
Photoelastic Finite Elements
Photoelastic Finite Elements
Figure 6.4 Photoelastic (left) and finite element (right) maximum shear stress con
developed in a photoelastic die (R/t=lO): (a) P = 300 N, (b) P = 500 N and (c) P = 900 N.
88
numerical and experimental predictions. The figure also reveals that for small loads,
where the size of the contact region is small, the stress distribution is similar to that
resulting from Hertzian type contact [8]. As the load increases, the contact region grows
and the form of the contact pressure distribution gradually changes from a centrally
dominated distribution to one where the edges cany most of the applied load. The critical
load at which this change occurs is closely related to Po of Eqn. (6.3). For P = 900N, there
is some discrepancy in the maximum shear stress contours predicted from finite element
analysis and photoelasticity close to the central axis of syrnrnetry. This rnay be attributed
to frictional effects resulting from imperfect lubrication.
We were hirther interested in verifying the strain distribution at the inner surface of
the thin photoelastic rings. Figure 6.5 shows the angular variation of normalised
circumferential strain (de) for R/t = 10 at two different loads. The angle 8 is measured
counter-clockwise from the horizontal and is normaiised by the expenmental value of the
circumferential strain & at O = 90'. Cornparison between the strain gauge measurements
and the finite element predictions shows a maximum discrepancy of 7 % at 8 = 67' for
P= 6N. For the two load levels shown, the strain distributions are significantly different.
In the case of the smaller load (PcPo), the strain distribution is similar to that induced by
diametrd loading. The point of maximum strain is at the vertical position, where the load
is applied. At the higher load (P>Po), the strain distribution changes significantly-
especially in the contact zone. The maximum strain shifts to the horizontal position. This
means that the location of highest stminlstress and hence the potential failure site is a
function of the contact conditions.
We now tum Our attention to the deformation behaviour of the rings examined.
Figure 6.6 shows the load deflection curve for a thin photoelastic ring (Rit= 10) as
obtained from the finite element predictions and the experimental measurements. The
load deflection cuve indicates that the stiffness of the ring remains relatively constant for
small loads. However, as the load increases and the size of the contact zone increases, the
linearity of load deflection response no longer holds as a result of the stiffening of the
ring structure. Similar observations are noted for thinner rings. For 2.5 mm diametrai
Figure 6.5 Variation of normaiised circumferential strain dong inner ring radius
O 0.5 1 1.5 2 2.5 3 3.5
Diametrd Deflection (mm)
Figure 6.6 Load deflection characteristics for a ring with R/t = 10.
deflection there was a discrepancy of 5 % between the experimentally measured load and
the finite element predicted value, while at 3.5 mm the discrepancy was 20 9%. The figure
also depicts that the finite element shell mode1 predictions are always stiffer than the real
ring structure. This may be attributed to an error in the measurement of the material
parameters or over-stiffhess due to shell element used in modelling a relatively thick
structure.
Figure 6.7 shows the finite element prediction of the contact stress distribution for
different ring thicknesses. The rings were loaded up to a constant contact angle of 20'.
The contact stress GN is normalised by po the average contact pressure resulting from the
applied load. The results reveal that the form of the contact stress distribution changes
from the Hertzian to the edge dominant form as the ring thickness decreases. As the
thickness decreases considerably, the contact stress distribution approaches a point load at
the edge of the contact zone, which is in agreement with theoretical predictions based on
inextensional elastica. Furthemore, for each of the tested rings, the form of the contact
stress distribution changes from Hertzian to edge dominant form as a hinction of the
extemally applied load.
5 10 15
Angle of contact, degrees
Figure 6.7 Contact pressure distribution for different ring thicknesses. Left hand scaie is
for di thicknesses except t = 0.43. For t = 0.43, the right hand scale applies.
Sirnilar results regarding the shift in contact stress distribution have been reported for
plates [64] and sphencal shells [68]. In capturing this contact behaviour, it is essential to
use a thick shell formulation such as the one detailed above.
6.3 Two Cylindrical Shells in Contact
In the first numencal example, the non-linear elastic contact behaviour of two cylinâricai
shells of different radii is examined (Fig. 6.8). This example involves three simultaneous
contact zones, ngid and flexible contact surfaces, and double-sided shell contact. The
ratio of the radii of the shells used was taken as R 2 R i = 1.5. In view of symmetry, one-
half of the contacting cylinders was modelled using the four noded shell element. The top
ngid plate was given an incnmental downward displacement, until the distance between
the rings was reduced to 13% of its original value.
Figure 6.8 Mode1 of two-ring compression.
Figures 6.9(a)-(f) show the deformed shape at different stages of deformation. The
figures clearly show that this problem involves six distinct contact stages. Ioitially,
contact commences dong three lines at the top, middle and bottom ngid surfaces and
Figure 6.9 Modes of deformation resulthg fiom contact between two rings.
between the two rings. Then contact progresses to become an area of contact in the lower
contact zone (stage II), the middle zone (stage Iil) and the top contact zone (stage IV)
respectively. In stage V, contact is initiated between the top and bottom faces of the lower
ring which introduces double-sided contact conditions. Finally in stage VI, an area of
double-sided contacting shells is formed. The forces and deformation compare well with
a theoretical solution of Wu and Plunkett based on inextensional elastica 1621. However,
their analysis fails to predict the occurrence of contact stage VI, since their treatment
mode1 does not account for double-sided shell contact.
Figure 6.10 shows the load-deflection curve for the two rings, where the displacement
is normalised by the initial distance between the two ngid plates, H. The figure shows a
sudden jump in stiffness corresponding to the start of the fifth contact stage. The figure
also shows the locations where the shift between stages occurs. These values are within
2% of the theoretical predictions of Ref. 1621. Due to the small thickness of the rings and
the predorninance of bending stresses, friction did not affect the results, and was,
therefore, excluded to achieve faster convergence.
6.4 Compression of a Spherical Shell
This exarnple involves a spherical shell compressed between two rigid flat platens, as
depicted in Fig. 6.1 1. This problem is important in measuring the intraocular pressure in
the comea of an eye as it contacts an applanation tonometer 11371. A simplified
theoretical analysis is available in Refs. 166,681. One eighth of the spherical shell was
modelled due to syrnmetry and an incremental downward displacement is applied to the
plate. The following material and geometric properties were assumed: R=100.0 mm,
t=3.333 mm, E= 100x 10' Pa and v=0.3.
Figure 6.12(a) illustrates the Hertzian type contact initially experienced by the shell.
As the deformation progresses, an edge dominant contact loading with a flat contact area
develops (Fig. 6.12@)). Similar deformation behaviour has k e n reported for beams and
plates in Refs. 164,671. Beyond a critical load, the central region of the shell curves
inwards to fom an axisymmetric dimple (snap-through) and the contact forces are carried
Stage VI
P
Stage III Stage II
Stage I , I
O 0.2 0.4 0.6 0.8
Deflection (6 /H )
Figure 6.10 Force-deflection characteristics for the two rings.
Figure 6.1 1 Mode1 of spherical sheli compression problem.
Figure 6.12 Deformed geometry of spherical cap: (a) HertUan contact, (b) edge-dominant
contact, and (c) pst-buckling contact.
by an expanding circular line of contact (Fig. 6.12(c)). The angular variation of the
contact pressure and contact length (not shown) reveal that the developed formulations
are insensitive to the distortion present in this mesh.
Figure 6.13 shows the normalised load-deflection characteristics of the shell. The
shell experiences a gradually increasing stiffhess in the flat contact region, which is
followed by a sudden reduction in stiffhess at the onset of snap-through. When the
compression ratio exceeds 5.5, the stiffhess of the spherical shell increases again. Similar
trends have been observed expenmentally 14,681.
O 1 2 3 4 5 6 7
Height Reduction (6 1 t)
Figure 6.13 Normalised load-deflection curve for spherical shell.
6.5 Saddle-Supported Pressure Vessels
Saddle supports are cornmonly used to hold pressure vessels (Fig. 6.14). The design of
saddle supports is inexpensive, and provides an efficient rnethod of carrying the vessel.
The pressure vessel can either be freely standing on the saddle supports or they can be
welded together. In this work, the former case is analysed. The interaction between the
saddle supports and the vessel body is one of the major problem areas in pressure vessel
design, since it involves highly localised contact stresses. The highest stresses are usually
located at the upper-most position of the saddle, called the saddle hom. One of the
commonly adopted design modifications involves increasing the radius of the saddle.
This introduces a gap between the support and the unloaded vessel, which permits the
loaded vessel to deform radially without restra.int. Consequentiy, the pinching effect of
the support at the saddie hom is reduced. The saddle support should also be flexible in the
longitudinal direction to avoid creating high localised stresses at its edges. Accordingly, a
wide saddle plate is usually welded to a thinner base, as depicted in Fig. 6.14.
Figure 6.14 A schematic of pressure vessel and saddle supports.
The ASME Boiler and Pressure Vessels Code [138] does not provide sufficient
details for the design of saddle supports [139]. Instead, a few references are listed which
provide some guidance. The most popular references [140,141] propose a semi-empirical
analysis technique based on bearn theory and assuming that the vessel cross-section
remains round under load. However, more accurate analyses based on cylindrical shell
theory and double Fourier senes expansion are available [139,142,143]. A numericai
study accounting for unilateral contact conditions by formulating a linear
cornplementarity problem was presented by Bisbos et al. [144]. The solution of the
complementarity problem was also obtained using a double Fourier series expansion.
Several attempts have been devoted to the fuiite element analysis of these supports to
obtain more accurate results. Most such analyses are based on simplified shell elements
and contact formulations, see, e.g. [l&, 1461.
In this example, a detailed and more accurate analysis of the saddle support of
pressure vessels is provided for various vessel and sadciie geometries (Table 6.1). Due to
symmetry, a quarter of the pressure vessel and saddle support were modelled (Fig. 6.15).
The newly developed shell element was used for the pressure vessel and the saddle,
which is thicker and stiffer, was modelled using solid elements. Frictionai effects were
accounted for (p=0.2) and were found to have a negligible influence on the solution.
Attention was devoted to studying the effect of the following parameters: (i) the saddle
radius Rs, (ii) the saddle plate extension Bp, and (iii) the overhang LE. The anaiysis
focused on the hoop stresses near the saddle support, because of their importance to the
mechanical integrity of the vessel.
Figure 6.15 FE mode1 of pressure vessel and saddle supports.
Vesse1 radius Rp
Vesse1 length Lp
Saddle location Ls
Table 6.1 Details of geometry and material properties of pressure vessel and supports.
2.0 m
Sadde angle
Saddle radius Rs
Fluid
40-0 m
4 m - 12 m
Vesse1 thickness
150"
2.0 m - 2.1 m
Water
25.0 mm
Vesse1 material
Sadde width
S tee1
1.0 m
Saddle plate thickness
Plate extension
Fluid level
50.0 mm
O" - 15"
Full
Figure 6.16 shows the hoop stresses at the outer surface of the vessel for four saddle
ratios. A support of the same radius as the pressure vessel (Rs/Rp= 1) results in high
compressive hoop stresses at the saddle hom and a smaller tensile region directly above
that hom. Increasing the support radius leads to a reduction in the compressive stresses
and an increases in the tensile stresses. An excessively large saddle radius (Rs/Rp = 1.05)
results in a smaller support area, leading to high tensile stresses over the saddle horn. A
saddle ratio of 1.02 provides the least hoop stresses in the sacidle region. These results are
in agreement with expenmentally measured stress values 11471.
Saddle plate extensions of 0°, SO, 10' and 15" were examined, for a sacidle to pressure
vessel radius ratio of 1.02. The resulting hoop stresses are shown in Fig. 6.17. The plate
extension reduces the pinching effect at the saddle hom which consequently leads to a
reduction in the maximum hoop stress. However, a long unsupported plate extension
suffers from high localised stresses at its base. Since the saddle extension geometry
resembles a curved edge-loaded cantilever bearn, the stress concentration at its root
should Vary with the cube of the length. This localised bending stress in the plate exceeds
the hoop stress in the vessel for the case where 0=15O. Accordingly, a plate extension of
5' - 10' is preferable for the selected geometry.
Finally, we examined the effect of the overhang ratio L&p. According to Ref. 11481
this ratio should not exceed 0.25. Based on beam theory, an overhang of 0.195, which
minimises the longitudinal bending moments, was suggested in Ref. [144]. Fig. 6.18
shows the longitudinal stresses at 0 = O* for different support locations for Rs/Rp = 1.02.
The results indicate a preferable range for LE = 4-6 m which corresponds to L& = 0.1-
0.15. The resulting longitudinal stresses (and bending moments) are significantly different
from the simplified calculations based on bearn theory. The effect of the saddle location
on the hoop stresses is shown in Fig. 6.19. Similar values for the maximum hoop stresses
are obtained for LE =4 m, 6 m and 8 m, while LE 2 10 m leads to higher stresses. This is due
to the pinching effect of the vessel on the saddle support, caused by excessive vessel
deformation. Since the mid-section is less stiff than the ends, then locating the saddles
close to the centre of the vessel subjects them to greater deformation.
200 1 I
- 100 -
400 - Saddle support
4
-500 1 1 1
O 45 90 135 180
Angle, 0
Figure 6.16 Effect of saddle to pressure vesse1 radius ratio R a p on the hoop stresses at
the support.
Figure 6.17 Effect of saddle plate extension on the hoop stresses at the support.
Figure 6.18 Effect of saddle placement Le on the longitudinal stresses at 8 = O*.
Figure 6.19 Effect of saddle placement LE on the hwp stresses at the support.
The previously discussed case studies demonstrate the versatility and accuracy of the
newly developed formulations. The issues examined in these case studies include: contact
stresses associated with large defonnation problerns, the effect of fiction, and double-
sided shell contact.
Chapter 7
Conclusions and Future Work
7.1 Definition of the Problem
Contact stresses play an important role in determining the structural integrity and
ultimately the resulting Mure mode of the contacting bodies. In spite of the important
and fundamental role played by contact stresses in shell structures, contact effects are not
generally taken into account. The reason is that the modelling of contact poses
mathematical and computational difficulties.
Furthemore, commonly adopted shell elements involve basic assumptions, which are
not appropnate for contact problems, since they do not: (i) account for variations of
displacements and stresses in the transverse direction, and (ii) allow for double-sided
contact. These restrictions severely influence the accuracy of the results in cases
involving moderately thick plate or shell structures.
7.2 Objectives
It was therefore the main aim of the current study to develop accurate techniques for
modelling frictional contact in shell structures. To achieve this objective the following
tasks had to be undertaken:
(i) develop new thick shell elements which account for the normal stresses and strain
through the thickness,
(ii) develop variational inequality formulations for shell structures which account for
double-sided contact,
(iii) develop a solution technique which is free of user defined parameters, and
(iv) apply the newly developed shell element and variational inequalities formulation
to treat practical engineering problems involving large elastic deformation.
7.3 General Conclusions
7.3.1 Thick Sheil Element Accounting for Through-thickness
Deformation
A new 7-parameter shell mode1 is presented for thick shell applications. The element
accounts explicitly for the thickness change in the shell, as well as the normal stress and
strain fields through the shell thickness. Large deformations are accounted for by using
the second Piola-Kirchhoff stress and the Green-Lagrange strain tensors. An assumed
transverse shear strain interpolation is used to avoid shear locking. Two new interpolation
schemes for the shell director are developed to avoid thickness locking. These
interpolations are implemented and their consistent linearization is derived. Guidelines
are developed for neglecting some of the quadratic tems in the consistent stifiess matrix
to minimise computational time. The thick shell element performance is tested to show
that the higher order tems result in improved accuracy. It also demonstrates that for thin
shells, there is no significant detenoration in accuracy, compared with traditional 5-
parameter shell elements.
7.3.2 Variational Inequalities Contact Formulations for Shell
Structures Undergoing Large Elastic Deformation
A new variational inequality based formulation is presented for the large deformation
analysis of frictional contact in elastic shell structures. The formulation accounts for the
normal contact stress through the shell thickness and accommodates double-sided shell
contact. The kinematic contact conditions are derived based on the physical contacting
surfaces of the shell. Lagrange multipliers are used to ensure that the kinematic contact
constraints are accurately satisfied and that the solution is free From user intervention.
7.3.3 Case Studies Considered
Several simulations were conducted to demonstrate the utility and flexibility of the
developed formulations. The different problems were selected to highlight some of the
key features of the new solution strategy. These include: element performance, contact,
friction, large deformations and double-sided contact. The following general conclusions
can be drawn from the exarnined cases.
Ring Compression Between Curved Dies
In this case study, thick and thin rings were compressed between curved dies. Both
numericd and experimental results were presented. Photoelastic, strain gauge and
displacement measurements were carried out for a wide range of ring geometries. The
numerical results agree with expenmental measurements and provide some new insight
into the form of the contact pressure distribution.
Two Rings in Contact
This case study was concemed with the prediction of the deformation mode of two thin
rings compressed between flat ngid dies. The problem involved large elastic
deformations and rotations. The developed solution strategy enabled the accurate
evaluation of the six modes of deformation experienced by the rings. The last two stages
involve double-sided contact, which cannot be predicted using traditionai analysis
techniques.
Sphericai Shell Compression
In this thesis, we also devoted attention to the case of a sphencai shell which is
compressed between rigid flat plates. The deformation mode was dictated by the contact
conditions and was divided into three distinct stages. Initially, Hertzian contact is
obtained with a centrally dominant pressure distribution. For higher loads, an edge
contact deformation mode is reached, similar to that noted for rings. However, in this
case, a further increase in the load leads to the formation of an intemal dimple in the
sphere and contact becomes concentrated dong a circula Line. The new shell element and
contact formulations correctly predict the onset of each of the three deformation stages as
well as the contact stress distribution during each stage.
Saddle Supported Pressure Vessels
The contact behaviour of saddle supported pressure vessels was examined. The effect of
the saddle location, radius, and plate extension on the contact stresses was investigated.
Optimum values of these parameters are provided for the selected vesse1 geometry.
7.4 Thesis Contribution
The main contribution of the current work can be summarised as follows:
(i) the development of a novel thick shell element which accounts for the variation of
the displacement, stress and strain fields through the thickness, and is not
susceptible to shear, membrane and thickness locking,
(ii) the development of new variational inequality formulations for the frictional
contact anaiysis of large deformation elastic shell problems accounting for double-
sided contact,
(iii) the implementation of Lagrange multiplier solution techniques for 3D problems,
which are free of user intervention, and
(iv) the application of the new formulations to a number of engineering applications.
The results obtained provide a new insight into the effect of contact on these
s ystems.
7.5 Future Work
The following areas are worthy of future research:
(i) development of variational inequality formulations for the frictional contact
problem accounting for elasto-plasticity,
(ii) implementation of dBerent constitutive laws, including viscoelas tic and
incompressible materials,
(iii) implementation of non-local and nonlinear friction laws, and smooth contact
surface approximation, and
(iv) introduction of dynamic and associated strain rate effects in the variational
inequalities formulation of contact problems.
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Appendix A: Shell
In this section, the displacement fields
are detailed. The lengthy nature of
Element
corresponding to
Equations
the three interpolation schemes
the resulting expressions results in excessive
computational requirements. Therefore, guidelines are presented for neglecting some of
the terms in order to minimise computational efficiency without sacrificing accuracy.
Regardless of the interpolation scheme, there will be nonlinear displacernent terms
involved. One sources of this nonlinearity results from the linearization of the rotational
degrees of freedom of the shell director according to Eqn. (3.5). This nonlinear
contribution is also present in the classicai 5-parameter shell elements [12,20,23]. The
linearized fonn of Eqn. (3.5) including al1 linear and quadratic terms is:
Another source of displacement nonlinearity result from dividing by the director
magnitude (Eqn. (3.18)). Finaily, there are nonlinear displacement terms resulting from
the multiplicative decomposition involving products of the individually interpolated
functions, e.g. Eqn. (3.4).
The strain-displacement matrices for the new shell element were derived in section
3.6 in tems of the Bi, BI, LI and matrices. The explicit fonn of these matrices
depends on the selected interpolation function. Three interpolation functions were
selected: IP 1, IP2 and IP3.
A.l First Interpolation Scheme - IP1
For the IP1 interpolation (Eqn. (3.17)) the following linear and quadratic displacement
terms are obtained:
k k k uL = ~~u~ - F F ( ~ ) N J ~ h a, + F ~ ( ~ ) N , v : ~ ' ~ : + F;'(~)N,V,'~: k k k + F, (I;)N,V3 h a,
In order to evaluate the Bi and B2 matrices of Eqns. (3.28b) and (3.28d), it is necessary to
obtain the spatial derivative with respect to the local coordinates, e,q, and c:
auL -- k k k k k k
36 - - F,V~)N,*~ V, h a, + F;(oN,,c V, h a, + ~:(c)Nk*~ va:
(A-4)
miL -- k k k k k k
mi - N,*,u~ - F ~ ( Q N ~ . ~ V, h a, + F:(I;)Nk,, VI h a, + F:(L;)N,*, vja:
( A 3
auL k k k k k k -=-F:(~)N,V, h a, +F;(~)N,V, h a, +F:(~)N,v~u: ac
k k k +F4(ONkV,h a,
where the functions Fi@ - F 4 ) are defined as:
The cornputationd overhead associated with the cdculation of the quadratic
displacement term is not excessive so it is advisable to retain al1 terms in Eqn. (A.3).
However, the single most effective tenn in this expression is the doubly underlined one,
which involves the product of the two rotations. Note that al1 the quadratic terms involve
two degrees of freedom at the same node. Hence the L$" matrices in Eqn. (3.2%) have a
block diagonal structure which leads to a significant reduction in computational time.
Expressing the relationships of Eqn. (A.4-6) in a matrix form, results in the Bz matrix
of Eqn. (3.28d):
The BI matrix is related to B2 as follows:
B, = G B,
where
[ d l 0 0 d 2 0 0 d 3
(A. 10)
Similarly, the spatial denvatives of the quadratic displacement terms in Eqn. (A.3) are
used to generate matnx ~ ( , ~ j ) of Eqn. (3.28~):
(A. 11)
where each sub-matrix takes the following form:
(A. 12)
The different entries in Eqn. (A.12) are directly based on the local denvatives of
Eqn. (A.3) based on Eqn. (3.28~).
A.2 Second Interpolation Scheme - IP2
For IP2 interpolation (Eqn. (3.19)) the linear displacement field is interpolated as follows:
The spatial derivative with respect to the local coordinate 6 is as follows:
A similar expression is obtained by replacing 6 with q. For the thickness variable
following relationship is obtained:
The quadratic displacement field for the sarne interpolation scheme is as follows:
F3 (0 k k k - 7 ~ k ~ , k ( ~ 3 - V ~ ) c x ~ +(V3 T~)CL;]V~ h a, lv3 I -
Obviously, the computational requirement associated with this equation is excessive.
The terms with a single underline have the srnaIlest magnitude. These ternis are
proportional to the square of the cwature of their element, which makes them
insignificant. The terms with triple underline involve the quadratic degree of freedom 04,
which is much smailer than the rotational or extensionai degrees of freedom, and hence
these too can be neglected. Finally, the tems with a double underline are lineariy
proportional to the curvature, and should not bc neglected. With the exception of the first
seven ternis in Eqn. (A.16), the b'" matrices (Eqn. (3.18~)) resulting from this
interpolation scheme are not sparse.
The quadratic terni which was labelled as the single most effective term in the
interpolation scheme P l , is marked here as being a highly insignificant term. This term
represents the change in the length of the director caused by the large incremental
rotations, and in this interpolation scheme, normalising by the length of the director
diminishes its effect.
A.3 Third Interpolation Scheme - IP3
Finally, for the IP3 interpolation (Eqn. (3.21)) the linear and quadratic displacement fields
are interpolated as follows:
(A. 17)
1 F, (0 (A. 18) --- (C' N ,v ;~ (a~a: +a:a:)+--v, h (V, -v,L)N,(~:~: +a:a:) ' lv3 I Iv3 I
F (0 k k k --N,N,[-(V, -Vr)a(n+(V3 *~ ; )a2 ]v ,h a, IV3 I
Obtaining the spatial denvatives of Eqn. (A.17) and the relative significance of the
quadratic tenns in Eqn. (A. 18) closely follows section A.2.
Appendix B: Computer Implementation
The newly developed shell element and the variational inequalities fnctional contact
formulations outlined in chapters 3 and 4 were implemented in a speciaily developed
computer code using the C-language. The code includes the standard routines needed to
calculate the displacements, strains and stresses. Fig. B.1 provides a flow chart of the
main modules in this software.
B.l Main Program Module
The first step of the main program involves reading the input file. This includes nodal
coordinates, element connectivity, material properties, details of geometry, extemal loads,
boundary conditions, convergence tolerances and other control parameters. Then, the
necessary initialisations and memory reservations are performed.
In order to speed the contact search process, the extemal nodes and elernents are
determined. For problems where prior knowledge about the approximate location of the
contact regions is available, only the nodes and elements belonging to those regions
should be accounted for in this module. The extemal loads are then applied incrementally.
This is followed by a local contact search, based on the master-slave strategy, to
determine the potential contacting nodes and surfaces. Details of this module are
provided in section B.2.
The next step involves calculating the linear and nonlinear components of the
stifiess matrix for al1 elements. The element contribution to the right-hand-side load
vector is also evaluated. These element vectors and matrices are based on a total
Lagrangian formulation employing the second Piola-Kirchhoff stress and the Green-
Lagrange strain tenson. Details of the procedures involved are provided in section B.3.
The contact contribution to the stifhess matrbc and load vector is then evaluated. This
includes the Lagrange multipliers and the fnctional stiffhess resulting ftom the
regularisation process. Details of this module are provided in section B.4.
Determine extemai nodes and elements I
Generate and assemble element equations
- - - --
Genenite contact and fiction equations
1
Reduce stifhess matrix
1 Solve for displacements and contact forces
1 Update shell mid-dace coordinates and director vectors
I Get reactions, stresses and strains -
1 Update contact and fiction s t a t u 1 .---- Check for convergence: enagy, displacement and contact 1
1 End 1
Figure B. 1 Row chart for main program module.
The stiffness matrix is then reduced by imposing the boundary conditions. The
resulting equations are solved using Gaussian elirnination. The order of equations is
changecl, if necessary, to avoid zero-diagonal elements caused by the Lagrange
multipliers. The displacement and rotations are used to calculate the current shell
configuration, as well as the new director vectors based on the large rotation equations of
section 3.2. The reaction forces are then obtained, together with the strains and stresses at
the integration points. These quantities are then extrapolated to the nodes and averaged.
The contact status is then re-evaluated by checking for tensile contact forces and for
nodes exceeding their target surface. The frictionai state is evaluated based on the relative
tangential displacement and the normal force. We then check for convergence based on
an energy norm and/or a displacement norm as well as any change in the contact status. A
change in status, such as a stick to slip transition or a new node initiating contact, requires
an extra iteration to ensure solution accuracy.
When convergence is reached, the displacements, stresses, strains and reaction forces
are stored in the output files. Al1 contact and friction related information are aiso stored.
The procedure is then repeated for al1 loading increments.
B.2 Sheii Element Equations
Figure B.2 shows the procedure involved in calculating the shell element equations. The
detailed denvation of these equations is provided in Appendix A. The first step involves
evaluating the BI, B2 and Li matrices of Eqn. (3.28) at the four sarnpling points. Then for
each integration point, the same three matrices are re-evaluated, and then the assumed
strain form is computed according to Eqn. (3.29). The Jacobian is calculated and used
together with the D and Bi matrices to determine the linear stiffhess matrix according to
Eqn. (3.30). This is followed by the evaluation of the nonlinear stifhess matrix, with
terms resulting from the Green-Lagrange strain as well as the large shell rotations, and the
normalisation of the shell director (see, Eqns. (3.22) and (3.25)). The right-hand-side load
vector contribution resulting from the incremental loads and from any applied pressure
loads is also evaluated. The linear and nonlinear stiffness matrices are added to the total
Evaluate B,, B, and L, matrices at 4 sampling points 1 1
r----- + Evaluate B,, 8, end L, matrices at integration point
1 r CI CI
Obtain assumed strain matrices: B,, L, and %
Evaluate jacobian J J Evaluate linear a e s s matrix K, & Evaluate nonlinear stifniess matrix IC, and IZ, .
m
Evaluate interna1 and e x t d load vectors Fm F, I
Condense 7& degree of kedom ifnecessary
l
Figure 8.2 Flow chart for computation of element equations.
.-------- Add to global stifFness matrix and load vectors
stiffness matrix of the element. M e r al1 the integration points are evaluated, the seventh
quadratic degree of fkedom is condensed if this option is enabled (see, section 3.2).
Finally, the resulting stifhess matrix and load vector are assembled in their global
counterparts.
B.3 Contact Search
Figure B.3 outlines the steps involved in the contact search. For each potential contact
node the closest target surface is located. Different procedures are employed for contact
with ngid surfaces and for contact with other elements. The normal vector and the gap are
then evaluated. The selected surface should not violate the constraint on the shell normal
vectors, as defined in Eqn. (4.2). The local coordinates of the target contact point are then
calculated. Finally, the local coordinates corresponding to sticlcing friction are evaluated.
These coordinates are identical to the target contact point, unless the contacting node was
in stick condition in the previous loading step. In this case, the old stick location is
maintained.
- -
Check for inappropriate shell contact
Search for closest element and surface
53 1 E aa
I BI Calculate local coordinates for contact point I
m I
Calculate normal vector and gap
Figure B.3 Flow chart for contact search module.
B.4 Contact and Friction Equations
Figure B.4 outlines the steps involved in evaluating the equations resulting from contact
and friction. For each active contact constraint the Q and C matrices of Eqn. (4.32) are
fint evaluated. Then the Lagrange multiplier 2. is created. The contribution of the normal
force to the right-hand-side vector is then calculated, followed by the fictional terms.
These include the frictional stiffness and the load vector resulting from the regularisation
of the variational inequality formulation (Eqn. (4.28)). The equations used for fnction
evaluation depend on the stick-slip state of the contact node-target segment involved.
Finally, the constraint is assembled in the global stiffness matrix and the procedure is
repeated for al1 contact nodes.
*--.---- I !
Calculate contact constraint maüix C 1 Generate Lagrangian multiplier 1 k
1
Calculate contribution to internai load vector F,
I
Caicuiate fkictional load vector F, 1 I
Figure B.4 Flow chart for evaluation of contact and fnction equations.
I
I '----- Assemble constraint in global stifniess matrix