A New Strategy for Treating Frictional Contact in Sheii 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
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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 gratefully 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 ...... -25 3.3 Four-noded Shell Element ...... 29 3.4 Thickness Locking ...... 30 3.5 Discretization of Shell Element ...... 31 3.6 Variational Formulation ...... -33 3.6.1 Consistent Loading ...... 35 3.7 Numerical Examples ...... 37 3.7.1 Patch Tests ...... -38 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 Stmctures ...... -55 4.5 Solution Technique ...... 56 4.6 Discretization ...... 57 4.6.1 Contact Constraints ...... -57 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) a3and (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 ...... -48
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...... -64
Variation of contact pressure dong contact distance for a 32.5%reduction in radius ...... -65
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 ...... -69
Effect of rotation 8 on the contact stress distribution of belt: (a) normal contact stress. and (b) tangentid stress...... -71
(a) Variation of stick-slip angles. aiand 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 ...... -75
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 OR1 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 Shell thickness Discretized nodal forces Deformation gradient Body forces Identi ty matrix Moment of area Quadratic strain displacement matrix Length of shell Stifhess 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 displacement gradient Green-Lagrange strain tensor Regularisation parameter Contact angle
Nonlinear component of strain tensor ~ij 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 ofa 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 componentlsystem.
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 VI contact formuiation I
2" Piola-Kirchhoff stress 2& Piola-Kirchhoff stress Green-Lagninge strain k
Enhanced director interpolation I-
Assumed strain Frictional interpolation k reguiarisation I
Four-noded YI for sheîï stnictures thick sheîï element (large defomations) 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.
,ASheii 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 me 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 confing 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, cm 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 AUIG 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. ;*:. ;: ','. I S. . . .a.'. **I.. . .':: : ;.. -, Contact X .' 8 .a,, i '--. J element >Y- >Y- .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 aland % 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:
'+& V3= IR(' a,,'a,) 'V3 (3.5)
R is an orthogonal matrix for finite rotations [IO, 1 131:
where
and
O -tltV,3-ta2tV t a,'VI2+'a2'V,
's= t a,tV,3+ta2tV, O -ta~tV11-ta2tV21 Ja, ' V,2-ta2tVz 'a,'V,,+'a,'V2, O
Argyris 11 131 demonstrated that R can be expanded into the following senes:
'~('a,,'u,)=~+'S+- 1' S 2 +-1t S 3 + ...... +(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 cm 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 atx Gi =- , 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 [ 1O].
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 In. 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~E5tt 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 atw atsuLo =-(- gj +-• gi)ztB,6U 2 agi 36,
1 at6uQ athQ = -(- 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.
Explanation Interpolation schemes Pl, IP2 and IP3 (see section 3.5). 7 DOF 7-parameter shell element 6 DOF 6-parameter shell (neglec ting quadratic term) 5 DOF Classical 5-parameter shell element Condense 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=40x 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.
Interpolation scheme 7 DOF 7 Condense 6 DOF 5 DOF PI,IP2, IP3 -1.575 1 - 1.5765 -1.4170 - 1.5765
Table 3.2 Vertical displacement at tip of beam comsponding to small deformation analysis.
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 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.
Interpolation 7 DOF 7 Condense 6 DOF 5 DOF scheme Pl 6.0872 6.0872 5.9304 7.05 11 IP2 7.03 16 7.0364 6.8 134 7 .O364
I IP3 7.0460 7.0497 6.8242 7.0497 Theory [61 ] Simo et al. [5 11 Parisch [52] -
Others 7.0629 7.3053 7.083 -
Table 3.3 Vertical displacement at tip of bearn corresponding to large deformation analysis.
3.7.3 Curved Cantilever Beam
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 hl-F
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.
Interpolation 7 DOF 7 Condense 6 DOF 5 DOF scheme PI -0.0794 1 -0.07943 -0.07 195 -0.08060 rP2 -0.08039 -0.08045 -0.07282 -0.08045 IP3 -0.08052 -0.08054 -0,0729 1 -0.08054
Table 3.4 Horizontal displacement under load for curved beam.
3.7.4 Pinched Hemisphere
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 Pl and IP3 with the condensation option. The displacement (a) (b)
Figure 3.8 Pinched hernisphere: (a) mode1 and @) deformed geometry.
0.0 0.1 0.2 0.3 0.4 05 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.
Interpolation 7 DOF 7 Condense 5 DOF scheme Pl 0.07832 0.07837 0.09363 IP2 0.09209 0.09230 0.0923 1 IP3 0.09276 0.09294 0.09294
I l Others Theory [3 11 Simo et al. [3 11 Betsch et al. [54]
I I 0.0930 1 0.0939 1 0.09247 1 ------
Table 3.5 Displacement at points A and B in pinched hemisphere corresponding to small deformation analysis and using an 8x8 mesh.
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.OX106, 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 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. (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: ~=6.825x1o', 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 POUT Freedom O I 500 I 1000 7 DOF -7.282~10'~ -3.934~i O-' -5.857~1 5 DOF O O O
I 5 DOF -6.957~101' -3.478x 1 o-~ O w ith thick-update Theory [122] -7.305~10" -3.945~10" -5.8608~loa
Table 3.6 Nomalised change in shell thickness a3 / t for Pm=lOOO.
Shell Surface POUT O 500 1000
L Inner Surface -0.9408 -0.9747 -0.10086 Outer Surface 0.0525 -0.4777 -0.10079
-
Table 3.7 Variation of radial stress OR /PH through sheU thickness for Pm=lOOO. Chapter 4 Analysis 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+AtgT(x)=ly-T ( t+dx)-t+~t x~].t+~N~ 20 and
t+at g~ (X) = ly*~(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)]- 'gT(x)~o and
'NB. [ 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 cmbe 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 dl 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 sWn 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 (124,1251: 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: --M M~O~~Mfor l%l> e kF(q)= lqTl 1qTr -M fOrkTISE 1 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 solution1 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=~Xi o3 ------ICJK,,=~X~O~ . 2C Current 1 ------Current 2 4 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 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 lo9Pa A coefficient of friction of 0.4 was assumed between the pulley and the belt, together with a fnctional regularisation parameter E = 5~10-~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 woslip regions where @ < mi and > 02. 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 T3and 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, dl 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 = 904 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 (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, 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. uUpper die 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 pst 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 [ 1261. --t- Frictioniess -p = 0.05 I Compressim 1 Stage I v 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) 6 6 Materiai Aluminium 606 1 Epoxy (PSM-5) Young's modulus @Pa) 70 2.7 Poisson's ratio 0.33 0.36 Material fringe value, kPal(fnnge1m) - 10.5 ------Table 5.1 Details of geometry and material properties for tested rings. 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 10p. 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 R2Ri = 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 ken 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 2.0 m Vesse1 thickness 25.0 mm Vesse1 length Lp 40-0 m Vesse1 material Stee1 Saddle location Ls 4 m - 12 m Sadde width 1.0 m Sadde angle 150" Saddle plate thickness 50.0 mm Saddle radius Rs 2.0 m - 2.1 m Plate extension O" - 15" Fluid Water Fluid level Full Table 6.1 Details of geometry and material properties of pressure vessel and supports. 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 LE2 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 Rapon 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 LEon 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. 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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, LIand 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: uL = ~~u~ - FF(~)NJ~kkk h a, + F~(~)N,v:~'~:+ F;'(~)N,V,'~: + F, (I;)N,V3kkk h a, In order to evaluate the Bi and B2matrices 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 -- kkk kkk - - F,V~)N,*~V, h a,+ F;(oN,,c V, h a, + ~:(c)Nk*~va: 36 (A-4) miL -- - N,*,u~- F~(QN~.~V, kkkh a, + F:(I;)Nk,, VI kkkh a, + F:(L;)N,*, vja: mi (A3 auL kkk -=-F:(~)N,V, h a, +F;(~)N,V,kkk h a, +F:(~)N,v~u: ac +F4(ONkV,hkkk a, where the functions Fi@ - F4)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 B2as follows: B, = G B, where [dl 0 0 d2 0 0 d3 (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 kkk -7~k~,k(~3lv3 I -V~)cx~ +(V3 T~)CL;]V~-h a, 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 Pl, 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) (A. 18) ---' (C' N,v;~ (a~a:+a:a:)+--v, 1 F, (0 h (V, -v,L)N,(~:~:+a:a:) lv3 I Iv3 I --N,N,[-(V,F (0 -Vr)a(n+(V3*~;)a2]v,h kkk 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 statu 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 Limatrices 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 % JEvaluate jacobian J &Evaluate linear aessmatrix K, . Evaluate nonlinear stifniess matrix IC, and IZ, m Evaluate interna1 and extdload vectors Fm F, I Condense 7°ree of kedom ifnecessary l .------Add to global stifFness matrix and load vectors Figure 8.2 Flow chart for computation of element equations. stiffness matrix of the element. Mer 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. --Search for closest element and surface m I 53 Calculate normal vector and gap 1 Eaa Check for inappropriate shell contact Calculate local coordinates for contact point I BI I 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 k1 Calculate contribution to internai load vector F, I Caicuiate fkictional load vector F, 1 I I I '----- Assemble constraint in global stifniess matrix Figure B.4 Flow chart for evaluation of contact and fnction equations.