UNIVERSITY OF HAWAII COLLEGE OF ENGINEERING DEPARTMENT OF CIVIL AND ENVIRONMENTAL ENGINEERING

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1 UNIVERSITY OF HAWAII COLLEGE OF ENGINEERING DEPARTMENT OF CIVIL AND ENVIRONMENTAL ENGINEERING

2 ACKNOWLEDGMENTS This report consists of the dissertation by Ms. Yan Jane Liu, submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Civil Engineering. Prof. H.Ronald Riggs was her research advisor. Other committee members were Prof. Harold S. Hamada, Prof. Craig M. Newtson, Prof. Ian Robertson, and Prof. Ronald. H. Knapp. The authors appreciate the financial support for this work from NASA Langley Research Center under grant NAG Dr. Alexander Tessler was the NASA Technical Officer. The first author gratefully acknowledges the financial support provided in the form of graduate teaching assistantship from the Department of Civil Engineering of University of Hawaii at Manoa. ii

3 ABSTRACT A general formulation for a family of N-node, higher-order, displacement-compatible, triangular, Reissner/Mindlin shear-deformable plate elements, MIN-N, is presented in this work. The development of MIN-N has been motivated primarily by the success of the 3-node, 9 degree-of-freedom, low-order, constant moment, anisoparametric triangular plate element, MIN3. This element avoids shear locking by using so-called anisoparametric interpolation, which is to use interpolation functions one degree higher for transverse displacement than for bending rotations. The methodology to derive members of the MIN- N family based on the anisoparametric strategy is presented. The family of MIN-N elements possesses complete, fully compatible kinematic fields. In the thin limit, the element must satisfy the Kirchhoff constraints of zero transverse shear strains. General formulas for these constraints are developed. As an example of a higher-order member, the 6-node, 8 degree-of-freedom, triangular element MIN6 is developed. MIN6 has a cubic variation of transverse displacement and quadratic variation of rotational displacements. The element, with its straightforward, pure, displacement-based formulation, is implemented in a finite element program and tested extensively. Numerical results for both isotropic and orthotropic materials show that MIN6 exhibits good performance for both static and dynamic analyses in the linear, elastic regime. Explicit formulas for Kirchhoff constraints in the thin limit, in terms of element degrees-of-freedom, are developed. The results illustrate that the fully-integrated MIN6 element neither locks nor is excessively stiff in the thin limit, even for coarse meshes. iii

4 TABLE OF CONTENTS ACKNOWLEDGMENTS... ii ABSTRACT... iii TABLE OF CONTENTS... iv LIST OF TABLES... vi LIST OF FIGURES... vii CHAPTER : INTRODUCTION.... Overview....2 Scope of Work...3 CHAPTER 2: REVIEW OF PLATE THEORY AND PLATE ELEMENTS General Comments Plate Theories Kirchhoff Plate Theory Reissner/Mindlin Plate Theory Potential Energy Functional Kirchhoff Constraints Mindlin Plate Element Modeling Finite Element Model and Shear Locking Phenomenon Concept of Anisoparametric Interpolations Review of Plate Elements Overview Existing Low-Order Anisoparametric Element, MIN CHAPTER 3: DERIVATION OF THE MIN-N FAMILY Introduction Basic Concepts of MIN-N Element Family Virgin Elements Shear Constraints General Derivation of MIN-N Constraint equations Examples Constrained Interpolations Element Matrices and Stress Resultants of MIN-N Kirchhoff Constraints of MIN-N Application: Element MIN iv

5 3.8. Virgin Interpolations of MIN Constrained Interpolations of MIN Kirchhoff Constraints of MIN Kirchhoff Edge Constraints CHAPTER 4: FORMULATION OF MIN Introduction Shear Constraints Interpolation Functions of Virgin Element Constrained Interpolation Functions of MIN Kirchhoff Constraints of MIN Kirchhoff Element Constaints Kirchhoff Edge Constraints... 5 CHAPTER 5: NUMERICAL RESULTS FOR MIN General Comments Shear Relaxation Coefficient Performance for Thin Plates Test Problems Overview Patch Test for Thin Plate Meshing Strategy Isotropic Thin and Moderately Thick Square Plates (L/t =000 and 0) Thin Circular Plates (2R/t = 00) Twisted Ribbon Tests Cantilever Plates (L/t = 00; L/t = 5) Orthotropic Square Plate (L/t = 30) Free Vibration of Thin and Moderately-Thick Plates Remarks on MIN6...9 CHAPTER 6: CONCLUSIONS AND RECOMMENDATIONS Conclusions Recommendations...93 APPENDIX I: MIN6 Element User Guide...94 APPENDIX II: MIN6 Element Input Data for Test Problems...96 REFERENCES...23 v

6 LIST OF TABLES Page Table 3. Displacement interpolation for virgin and constrained elements Table 5. Location of interior nodes Table 5.2 Exact solutions for center deflection and bending moment of a square plate.. 73 Table 5.3 Exact solutions for center deflection and bending moment of a circular plate 77 Table 5.4 Cantilever plate results using MIN6 elements Table 5.5 Non-dimensional frequencies for simply-supported square plates Table 5.6 Percent error (%) in frequencies for simply-supported square plates... 9 vi

7 LIST OF FIGURES Page Figure 2. Plate variables and sign convention [3]...6 Figure 2.2 Kirchhoff kinematics (x-z plane)...7 Figure 2.3 Kinematics of Reissner/Mindlin plate (x-z plane)...9 Figure 2.4 Nodal configurations for initial and constrained displacement field...7 Figure 3. Virgin and constrained elements nodal configuration. (a) N = 3, Linear. (b) N = 6, Quadratic. (c) N = 0, Cubic. (d) N = 5, Quartic Figure 3.2 Notation for MIN-N element...25 Figure 3.3 Correlation between A mn and polynomial terms...40 Figure 3.4 MIN3 triangular plate element...4 Figure 3.5 Nodal configuration for MIN3 s edge constraints...44 Figure 4. MIN6 triangular plate element...47 Figure 4.2 Nodal configuration for MIN6 s edge constaints...53 Figure 4.3 Nodal configuration for MIN6 s interior Kirchhoff constraints...55 Figure 4.4 MIN6 cross-diagonal pattern...57 Figure 5. Meshes of one quadrant of a thin, symmetric, square plate with a concentrated center load...60 Figure 5.2 Cs for MIN6...6 Figure 5.3 Performance of MIN6 (Cs = 0, /0, /9) and ISOMIN6 with varying L/t ratios (simply supported and center load)...63 Figure 5.4 Performance of MIN6 (Cs = 0, /0, /9) and ISOMIN6 with varying L/t ratios (clamped and center load)...64 vii

8 Figure 5.5 Performance of MIN6 (Cs = 0, /0, /9) and ISOMIN6 with varying L/t ratios (simply supported and uniform load)...65 Figure 5.6 Performance of MIN6 (Cs = 0, /0, /9) and ISOMIN6 with varying L/t ratios (clamped and uniform load)...66 Figure 5.7 Performance of MIN6 (Cs = 0) and ISOMIN6 with varying L/t ratios and three meshes (simply supported)...67 Figure 5.8 Performance of MIN6 (Cs = 0) and ISOMIN6 with varying L/t ratios and three meshes (clamped)...68 Figure 5.9 Patch test for plate: a = 0.2; b = 0.24; t = 0.00; E =.0x0 6 ; and ν = Figure 5.0 The cross-diagonal meshes of MIN6 and MIN3 for plate problems...73 Figure 5. Meshes for one quadrant of doubly-symmetric square plates (dashed lines indicate the MIN3 elements)...73 Figure 5.2 Convergence of center deflection for a thin square plate with center load...74 Figure 5.3 Convergence of center deflection for thin square plate with uniform load...75 Figure 5.4 Convergence of center deflection for moderately thick square plate with uniform load...76 Figure 5.5 MIN6 and SHELL93 meshes for /4 thin circular plate...78 Figure 5.6 MIN3 meshes for /4 thin circular plate...78 Figure 5.7 Convergence of center deflection for thin circular plate (2R/t =00)...79 Figure 5.8 Twisted ribbon examples with corner forces, and moments...80 Figure 5.9 Mesh I for twisted ribbon examples...8 Figure 5.20 Mesh II for twisted ribbon examples...8 Figure 5.2 Twisted ribbon tests for tip deflection (E = 0 7, ν = 0.25, t = 0.05)for Mesh I...82 Figure 5.22 Twisted ribbon tests for tip deflection (E = 0 7, ν = 0.25, t = 0.05) for Mesh II...83 viii

9 Figure 5.23 Cantilever plate: load case and 2; L/t =00; L/t = Figure 5.24 Cantilever plate: numerical load models for case and Figure 5.25 Mesh of cantilever plate with MIN6 elements...85 Figure 5.26 Convergence for center deflection of orthotropic square plate (L/t = 30)...87 Figure 5.27 Convergence of moment Mxy of orthotropic square plate (L/t = 30)...87 Figure 5.28 Convergence of moment Mx and My of orthotropic square plate (L/t = 30)..88 Figure 5.29 Convergence of shear force Qx and Qy of orthotropic square plate (L/t = 30)...89 ix

10 CHAPTER INTRODUCTION. Overview Application of the finite element method to the bending of plates dates to the early 960s. Nevertheless, the subject remains an active area of research because of the importance of plate structures and the difficulty to develop accurate and robust plate finite elements. A very large number of plate bending elements have been developed based on the most commonly used plate theories, those of Kirchhoff and Reissner/Mindlin []. The former theory ignores transverse shear deformation, while the latter includes it. Interest in the shear deformable Reissner/Mindlin plate theory has increased significantly during the past two decades primarily because of the increased use of laminated composite materials. These types of structures, such as adhesively bonded laminated plates, usually exhibit much lower strength in the transverse direction and between the layer interfaces. Therefore, transverse shearing behavior can be significant in many cases. Consequently, the finite element method based on Reissner/Mindlin theory can be a very effective tool to provide more accurate modelling of moderately thick and anisotropic (composite) plates with significant shear deformation. Also, in the finite element model, Reissner/ Mindlin plate theory allows that rotations and the displacement derivatives are not directly coupled. Consequently, rotation fields and the displacement field are independently introduced into the functional of total transverse strain energy. Hence, only C 0 dis-

11 placement and rotation continuity is required for the finite element formulations. This low continuity requirement makes the implementation much easier than elements based on Kirchhoff plate theory, which in theory requires C continuity of displacement. Many displacement-based finite elements have been proposed which are able to model shear deformable behavior in plates [2]. However, many low-order Reissner/Mindlin plate elements become very stiff when used to model thin plates. This phenomenon is called shear-locking. One of the more successful low-order Mindlin plate elements in the literature is Tessler and Hughes MIN3 element [3], which is a 3-node, 9 degree-of-freedom (DOF) triangle. It avoids shear locking by using so-called anisoparametric interpolation. The anisoparametric interpolation strategy is to use interpolation functions one degree higher for transverse displacements than for bending rotations. This strategy avoids shear locking and ill-conditioning in the thin limit for even low-order Reissner/Mindlin plate elements. Therefore, a straightforward, pure displacement-based formulation with full integration of the stiffness matrix is possible. In addition to these favorable characteristics, MIN3 produces accurate numerical results. A more detailed review of the MIN3 element will be given in Chapter 2. The interpolation functions for the MIN3 element have also been used to develop a smoothing element analysis for improved stress recovery in finite element analysis [4-6]. The characteristics of the interpolation functions, when the Kirchhoff zero shear constraints are enforced, leads to a C continuous recovered stress field. The success of the low-order, constant moment, anisoparametric triangular plate ele- 2

12 ment, MIN3, has motivated the development of a family (MIN-N) of higher-order triangular anisoparametric Reissner/Mindlin plate elements. A secondary motivation is the potential to use these higher-order interpolation functions to develop higher-order smoothing elements. A general methodology to derive members of the MIN-N family is developed herein. The transverse displacement is interpolated by a polynomial one order higher than the rotational displacements. The transverse displacement is coupled with the bending rotations by enforcing continuous shear constraints in the element. MIN3, with a quadratic variation of transverse displacement, is the lowest order member of the MIN-N family. As an example of a higher-order member, the 6-node MIN6, with a cubic variation of transverse displacement, is developed herein. MIN6 is a higher-order, displacementcompatible, complete polynomial, fully-integrated, triangular Reissner/Mindlin plate element, with neither shear locking nor excessive stiffness in the thin limit. The element is implemented in a finite element program and tested extensively..2 Scope of Work The primary objective of this study is to develop a general formulation for a family of higher-order, pure displacement-based, triangular Mindlin plate elements; use the general formulation to derive a 6-node element, MIN6, of the MIN-N family; and evaluate the numerical performance of the element for linear elastic problems with isotropic and orthotropic material behavior. The dissertation is organized as follows. In Chapter 2, the two most common plate theories, Kirchhoff and Reissner/Mindlin theories, are reviewed briefly. Because the MIN- 3

13 N element is based on Reissner/Mindlin theory, the general review of previous plate elements focuses on the different type of Mindlin plate elements. The low-order anisoparametric Mindlin beam and plate elements [3, 7, 8] and some discrete Kirchhoff plate elements [9, 0] are emphasized in the review section. The original formulation of the three-node, triangular, anisoparametric plate element MIN3 will be reviewed briefly, as well as the well-known discrete Kirchhoff triangular plate element, DKT. Chapter 3 gives the derivation of the general formulation of the family of higher order, N-node triangular anisoparametric Mindlin plate elements, in which N is also the number of terms in a complete 2-D polynomial (i.e., 3, 6, 0, etc.). As an application of the general MIN-N formulation, Tessler s 3-node MIN3 element [3], which was originally derived with a different strategy, has been verified as a member (N = 3) of the MIN-N element family. In Chapter 4, the formulation of the 6-node, 8 degree-of-freedom (DOF), triangular, anisoparametric plate element (MIN6) is presented in detail. Numerical results with MIN6 for thin and moderately thick plates in the linear, elastic regime for both static and dynamic analyses are presented in Chapter 5, and these results are compared with those for MIN3 and SHELL93, which is an ANSYS 6 to 8-node shell element with transverse shear deformation. Finally, conclusions and recommendations are given in Chapter 6. 4

14 CHAPTER 2 REVIEW OF PLATE THEORY AND PLATE ELEMENTS 2. General Comments This chapter begins with a brief review of two widely used linear plate bending theories, the Kirchhoff (classical) and Reissner/Mindlin plate theories and their finite element formulations. The basic concept of shear locking for Reissner/Mindlin plate elements and different approaches to avoid shear locking in the literature are then reviewed. The finite element formulation for the discrete Kirchhoff triangular plate element, DKT, and Tessler s anisoparametric Reissner/Mindlin plate element, MIN3, are reviewed in detail. The review provides a background for the present work. 2.2 Plate Theories Consider a plate in the x-y plane with thickness t and mid-plane area A. Let uxyz (,, ) and vxyz (,, ) be the plate in-plane displacements of a point ( x, y, z) and wxy (, ) be the transverse displacement of a point ( x, y) on the mid-surface of the plate. Let θ x ( x, y) and θ y ( x, y) be the rotations of the normal to the midplane about the x and y- axes, respectively. See Figure 2. for the sign convention. 5

15 Figure 2. Plate variables and sign convention [3] 6

16 2.2. Kirchhoff Plate Theory The kinematics of Kirchhoff (classical) plate theory are based on the assumption that plane sections remain plane and normal to the deformed midplane, as depicted in Figure 2.2. Therefore, θ y = w, x, θ x = w, y (2.) and u = zw, x, v = zw, y (2.2) A comma is used to denote partial differentiation. From equation (2.2), the in-plane strains are ε = ε x ε y γ xy = w, xx z w, yy 2w, xy (2.3a) z,w γ xz = 0 θ y w, x w, x midplane u(x, y, z) w(x,y) x, u Figure 2.2 Kirchhoff kinematics (x-z plane) 7

17 and the transverse shear strains are γ γ xz w, x + θ y = = = γ yz w, y + θ x 0 (2.3b) The plate bending deformations, i.e., curvatures, are κ κ xx w, xx = κ yy = w, yy κ xy w, xy (2.3c) Given the above strain definitions, the moment curvature relations for an isotropic material are M M x = M y = M xy D w, xx + νw, yy w, yy + νw, xx ( ν)w, xy (2.4) t 3 in which D = E ( ν 2 ) is the bending rigidity; E is the modulus of elasticity; and ν is Poisson s ratio. Because of the assumption of zero shear strain in Kirchhoff plate theory, the transverse shear forces and cannot be obtained from the stress strain relations Q x Q y τ xz = Gγ xz and τ yz = Gγ yz, where G is the shear modulus. Instead, they must be obtained based on equilibrium: Q x M x, x + M xy, y Q = = Q y M y, y + M xy, x (2.5) Refer to Figure 2. for the positive definitions of the plate bending variables. 8

18 2.2.2 Reissner/Mindlin Plate Theory The Reissner/Mindlin plate theory uses a generalization of the Kirchhoff hypothesis, i.e., plane sections remain plane but not necessarily normal to the deformed midplane. Based on this assumption we have γ xz w, x + θ y = γ yz w, y + θ x (2.6) θ x θ y in which and are the rotations of the line that was initially normal to the undeformed midsurface. The displacements are therefore uxyz (,, ) = zθ y ( x, y), vxyz (,, ) = zθ x ( x, y) and wxy (,, z) = wxy (, ) (2.7) as depicted in Figure 2.3. Based on equation (2.7), the in-plane strains are z,w θ y γ xz w, x w, x midplane u(x, y, z) w(x,y) x, u Figure 2.3 Kinematics of Reissner/Mindlin plate (x-z plane) 9

19 ε x ε= ε y = z θ x, y = + γ xy θ y, x θ y, y θ x, x z κ xx κ yy κ xy (2.8a) and the transverse shear strains are γ γ xz w, x + θ y 0 = = = x γ yz w, y + θ x 0 y w θ y θ x (2.8b) The plate bending curvatures are κ κ xx 0 θ y, x x = κ yy = θ x, y = 0 κ xy θ y, y + θ x, x y y x θ y θ x (2.8c) Given the strain definitions in equations (2.8a) and (2.8c), the moment curvature relations for an isotropic material are M M x = M y = M xy D ν 0 ν ν 2 κ xx κ yy κ xy (2.9) and based on equation (2.8b), the shear forces are Q x Q = = Gk 2 t 0 0 Q y γ xz γ yz (2.0) 0

20 in which k 2 is the shear correction factor for non-uniform shear distribution (usually with the value 5/6) Potential Energy Functional The total strain energy of shear deformable plates can be decomposed into membrane energy, U m, bending energy, U b, and transverse shearing energy, U s : U = U m + U b + U s (2.) in which the membrane energy is considered as zero (i.e., assume the in-plane displacements of the midplane are zero and linear kinematics). For a linear elastic, isotropic material, the total transverse strain energy functional, including only the bending and shearing energies, may be written as U = --D ( ν) ( θ 2 x, x + θ y, y ) 2 da θ2 y, x 2νθ y, x θ x, y θ2 x, y λ --- ( w, A x + θ y ) 2 + ( w, y + θ x ) 2 + [ ]] da (2.2) in which λ = 6k 2 ( ν) --- A. t 2 From equation (2.2) the total transverse strain energy functional for Kirchhoff plate can be reduced to: U = --D 2 [( w, xx ) 2 + 2νw, xx w, yy + ( w, yy ) ( ν) ( w, xy ) 2 ] da (2.3) It is clear from equation (2.2) that the rotation fields, and the transverse displacement w are considered as independent variables in the formulation and the highest derivatives of displacements and rotations are of first order, which means that only C 0 - θ x θ y

21 continuity is required in the finite element formulation. This is a great simplification compared to equation (2.3), the Kirchhoff formulation, in which C -continuity is required for the displacement field w Kirchhoff Constraints In equation (2.2), the second integral is referred to as a penalty constraint functional. λ represents a penalty parameter which tends to infinity as the plate thickness, t, approaches zero. In the thin plate regime, this penalty parameter enforces the Kirchhoff constraints: γ xz = 0 and γ yz = 0 (2.4a) or w, x + θ y = 0 and w, y + θ x = 0 (2.4b) The total strain energy given by equation (2.2) reduces to that of the classical (Kirchhoff) plate theory in equation (2.3) when w, x + θ y = 0 and w, y + θ x = 0, i.e., when the transverse shear strains are assumed to be zero. 2.3 Mindlin Plate Element Modeling 2.3. Finite Element Model and Shear Locking Phenomenon The general form of the finite element model can be obtained by expressing the displacements and the rotations in terms of interpolation functions and nodal variables and then minimizing the total transverse strain energy, resulting in (K b + λ K s ) d = F (2.5) 2

22 2.3.2 Concept of Anisoparametric Interpolations To relieve the problems associated with shear locking, different methods have been developed. Relatively successful solutions have involved discrete penalty constraints, reduced-integration procedures, improved penalty-strain interpolations, and penaltyparameter modifications [3]. The selective, reduced integration approach is one of the most popular methods. However, this can result in a rank-deficient stiffness matrix. Another approach is to use different degree of the interpolation functions for displacement and rotation, the so-called anisoparametric interpolation functions [3]. This approach has proven to be effective in avoiding shear locking. To understand why the anisoparametric approach is effective in avoiding locking, let us use a simple numerical model to demonstrate the mathematics of shear locking. Con- K b is the bending stiffness matrix, which results from the first integral in equation (2.2); K s is the shear stiffness matrix, which results from the second integral in equation (2.2); d is the vector of nodal displacements; and F is the vector of equivalent nodal loads. As the plate thickness approaches zero, (i.e., as γ and ), U s = d T xz 0 γ yz 0 --λ K s d must 2 approach zero. However, as the thickness t goes to zero, λ approaches infinity. For the strain energy U s to remain bounded, K s d therefore must approach zero. This means either d must approach zero, which results in shear locking, or K s must be a singular matrix. The latter is achieved in many elements by using selective-reduced integration. Unfortunately, this solution may result in the introduction of spurious zero energy modes, which require an additional stabilization procedure. 3

23 sider a 3-node isoparametric triangular Reissner/Mindlin element. The transverse displacement, w, and rotational displacements, and, with linear variations are given by the following equations: θ x θ y w = 3 i = ξ i w i (2.6a) θ x 3 = ξ i θ xi and θ y = ξ i θ yi, i =, 2, 3 (2.6b) i = 3 i = in which the ξ i are both area-parametric coordinates and linear interpolation functions, and w i, θ xi and θ yi are nodal DOFs. The interpolation functions in terms of Cartesian coordinates are ξ i = ( c 2A i + b i x + a i y) (2.6c) A is the area of the triangle, and the coefficients a i, b i, and c i are given by a i = x k x j b i = y j y k c i = x j y k x k y j (2.6d) with a cyclic permutation of the indices (i =, 2, 3; j = 2, 3, ; k = 3,, 2). and are the nodal coordinates of node i. The transverse shear strain field is x i y i γ xz = w, x + θ y (2.7) Substitution of equations (2.6a) and (2.6b) into (2.7) yields the shear strain in the Cartesian coordinates x, y as 3 γ xz = b 2A [( i w i + c i θ yi ) + b i θ yi x + a i θ yi y] i = (2.8a) 4

24 Equation (2.8a) may be rewritten as γ xz = A 0 ( w i, θ yi ) + A ( θ yi )x + A 2 ( θ yi )y (2.8b) In the thin plate limit, γ xz = 0 is imposed to satisfy the Kirchhoff constraint. Thus, the coefficients A 0, A and A 2 must go to zero, which gives three Kirchhoff constraint equations. A 0 = 0 establishes a linear relation between the nodal rotations and the nodal transverse displacements. However, from A = 0 and A 2 = 0 we obtain 3 b i θ yi = 0 and a i θ yi = 0 (2.9) i = i = These equations imply that the nodal rotations 3 have to be either zero or constant values (note that a i = 0 and b i = 0 ). As a result, the bending energy in equation i = (2.2) vanishes and the element locks. 3 i = The idea of the anisoparamatric approach is that the interpolation for w must be one degree higher than that for θ x and θ y. Therefore, the deflection slopes w,x and w,y are represented by the same complete polynomials as and (i.e. linear variation), and it θ yi is therefore possible to represent γ xy = w, x + θ y = 0. 3 θ x θ y 2.4 Review of Plate Elements 2.4. Overview Recent research on displacement-based thin plate elements has focused on relaxing the assumptions of Kirchhoff thin-plate theory. The elements produced by relaxing these assumptions are called discrete Kirchhoff elements. The most successful of these elements is the DKT (Discrete Kirchhoff Triangle) plate element which was reexamined in 980 by 5

25 Batoz et al. [9] based on the original elements QQ3 and KC published by Stricklin et al. and Dhatt in 969. An explicit formulation has been presented by Batoz [0] also. The DKT element only enforces the Kirchhoff condition of zero shear at discrete points in the element. This 3-node, 9 degree-of-freedom DKT element is one of the best triangular plate elements known []. It combines the advantages of both theories: shear locking free analysis of thin plate (Kirchhoff) and C 0 -continuity requirements for independently interpolated displacements and rotations (Reissner/Mindlin). Reissner/Mindlin theory of moderately thick plates has C 0 -continuity requirements for the displacement assumption, which is an important advantage compared to the Kirchhoff theory, as discussed in the previous sections. Many successful C 0 elements have been developed by Hughes et al. [2], Hughes and Tezduyar [3], Pugh et al. [4], MacNeal, [5], Crisfield [6], Belytschko, [7], and Tessler and Hughes [3, 8]. Tessler s MIN3 element will be presented in detail subsequently. Mixed elements can be derived for Kirchhoff and Reissner/Mindlin theories. In mixed elements, both displacements and stresses are approximated. The highest degree of derivatives of displacement and moments occurring in the functionals is, which means that C 0 -continuous shape functions are sufficient. This is a great advantage, and many successful elements are available based on this approach. However, nodal variables are displacements and moments, which make these elements difficult to implement. The system of equations has zeros on the main diagonal, and therefore special equation solvers are required. So-called hybrid elements use special procedures to eliminate the nodal moment components in the interior mixed field. Therefore, hybrid elements have only nodal dis- 6

26 placement components. Hybrid elements are successfully used in the practical analysis of thin plates Existing Low-Order Anisoparametric Element, MIN3 In this section the basic concept and formulation of MIN3 are reviewed. More details of the derivation can be found in [3]. As mentioned in [2], many types of shear deformable plate elements have been developed because of their advantages over elements based on the thin plate theory of Kirchhoff. However, numerical results indicate that many of the elements become excessively stiff, and sometimes rigid, for very thin plates. This phenomenon is known as shear locking because the excessive stiffness results from the zero transverse shear deformation constraint which occurs in the thin limit. MIN3 uses anisoparametric interpolation functions and a penalty parameter modification to avoid shear locking and excessive stiffness problems, respectively Figure 2.4 Nodal configurations for initial and constrained displacement field 7

27 The initial interpolation functions for MIN3 are given in terms of area-parametric coordinates for the 6-node triangle shown in Figure 2.4. The quadratic deflection is interpolated as w = Nw v (2.20a) where the interpolation functions and nodal DOFs are N T = { N i }, w v = { w i }, i =, 2,..., 6 (2.20b) with N i = ξ i ( 2ξ i ), N i + 3 = 4ξ i ξ k, i =, 2, 3; k = 2, 3, (2.20c) These are the standard Lagrange type interpolation functions for a 6-node triangle. The linear rotation fields are θ x = ξθ x and θ y = ξθ y (2.2a) where the linear interpolation functions and nodal DOFs are ξ = { ξ i }, θ x = { θ xi } and θ y = { θ yi }, i =, 2, 3 (2.2b) Note that in this initial, or virgin, element, 6 nodes have transverse displacement DOFs and only the 3 vertex nodes have rotational DOFs. This in inconvenient, and to eliminate the 3 transverse displacements at the mid-edge nodes, the shear strain along the element edges are required to be constant, i.e., γ sz,s = ( w, s + θ n ), s 0, i =, 2, 3 (2.22) ξ i = 0 where s denotes the edge coordinate and θ n is the tangential edge rotation (refer to Figure 2.). Enforcing these constraints leads to a constrained deflection field in terms of vertex 8

28 DOFs. The displacements and rotations become w = N N 2 N 3 d θ x = N θ x θ y = N θ y (2.23a) The shape functions N T = { ξ i } NT 2 = { N 2i } NT 3 = { N 3i } i =, 2, 3 (2.23b) are given in terms of area-parametric coordinates as [3]: ξ i = ( c 2A i + b i x + a i y) ξ i = 3 i = N 2i = -- ( b 2 k ξ i ξ j b j ξ k ξ i ) N 2i = 0 3 i = N 3i = -- ( a 2 j ξ k ξ i a k ξ i ξ j ) N 3i = 0 3 i = (2.23c) (2.23d) (2.23e) with a cyclic permutation of the indices (i =, 2, 3; j = 2, 3, ; k = 3,, 2). The element stiffness matrix, K, may be expressed in terms of its bending, K b, and transverse shear, K s, components as K = K b + K s =, (2.24a) B T T b D b B b da + B s G s B s da A A where 0 0 N, x B b = 0N, y 0 (2.24b) 0N, x N, y 9

29 N B, x N 2, x N 3, x + N s = N, y N 2, y + N N 3, y (2.24c) The bending and transverse shear rigidity matrices are, respectively, D D 2 D 6 G D b = D 22 D 26 G G s = 2 sym. G 22 sym. D 66 (2.24d) where D ij -----h 3 2 = C 2 ij G ij = k ijhc6 i, 6 j with C ij denoting the plane stress elastic moduli and k 2 ij the shear correction factors. (2.24e) The consistent load vector due to the distributed normal load, q, as well as bending moments, M xx, M yy, and transverse shear force, Q, prescribed on the portion Γ σ of the element boundary Γ may be written as T T F = q N N 2 N 3 da + 0 M yy N M xx N dγ + Γ σ A T Q N N 2 N 3 dγ Γ σ (2.25) The element bending moment and transverse shear stress resultants may be determined from the relations M xx M yy M xy = D b B b d (2.26) 20

30 Q x Q y = G s B s d (2.27) It is noted that full integration is used to evaluate the element matrices in equations (2.24a) and (2.25). The stiffness matrix therefore has full rank. Tessler [3] defined the shear relaxation factors k 2 ij in equation (2.24e) to incorporate both the classical shear correction factor (taken as 5/6), k 2 ij k ij, and a finite element relaxation factor, φ 2. That is, 2 = φ 2 k 2 ij, where subscripts i and j are used to denote a general anisotropic material case. The element stiffness matrix K may be written as K = K b + K s = K b + φ 2 K s (2.28a) where K s is the unrelaxed element shear stiffness that includes the classical shear correction factor. The introduction of the element-appropriate shear relaxation factor, φ 2, can be interpreted as follows. In the thin regime, the numerical values in K s can be very large relative to those in K b. Therefore a numerically ill-conditioned matrix and subsequent loss of accuracy in finite precision arithmetic can result. Hence, φ 2 is introduced to avoid this problem by appropriately relaxing the shear stiffness. Tessler [6] defined φ 2 = C s α φ 2 as (2.28b) where α = 9 k sii i = k bii i = 4 (2.28c) 2

31 The parameter α is the ratio of the sum of the diagonal shear and bending stiffness coefficients associated with the rotational DOFs, and C s is a constant which was determined numerically. Note that the factor φ 2 tends to zero as the plate thickness approaches zero, and that it is closer to for thick plates. Another advantage of the additional relaxation introduced by φ 2 is that the excessive stiffness caused by coarse meshes is reduced. The success of MIN3 is due to the anisoparametric interpolation functions, which eliminate shear locking, and the element-appropriate shear relaxation factor, which results in a well-conditioned element stiffness matrix over the entire range of element thicknessto-area ratios. Consequently, the element has no deficiencies in either the classical (thin) or shear-deformable (thick) regimes and at the same time produces rapidly convergent solutions. MIN3 is an excellent element, based upon numerical testing on linear, elastostatic problems. 22

32 CHAPTER 3 DERIVATION OF THE MIN-N FAMILY 3. Introduction As mentioned in Chapter, the success of the existing low-order anisoparametric triangular element MIN3 has motivated the development of a family of arbitrary, triangular, anisoparametric Mindlin plate elements, MIN-N. In this chapter, the basic concepts and the derivation of MIN-N are presented in detail. More detailed information of the existing low-order, displacement-compatible beam, plate and shell elements that are based on anisoparametric interpolations can be found in other sources [3, 7, 8, 9-22]. 3.2 Basic Concepts of MIN-N Element Family Based on the anisoparametric strategy, the desired triangular N-node and 3N-DOF elements, MIN-N, must have a complete polynomial interpolation for the transverse displacement, w, which is one degree higher than the interpolation of the normal rotations, and. θ y The first step in the derivation is to generate a family of virgin elements that are based on independent interpolations of the kinematic variables (transverse displacement and normal rotations). Lagrange-type interpolations for the rotations are used with the standard triangular nodal configuration, as shown in Figure 3.. Hierarchical interpolation functions are employed for the higher-order terms of the transverse displacement field. From the virgin elements, a family of constrained elements, MIN-N, can be developed by θ x 23

33 enforcing a continuous shear constraint along any line L on the N-node element (Figure 3.2). As a result of imposing these constraints, the hierarchical DOFs are eliminated and the transverse displacement of the constrained element is then coupled with the bending rotations. The constrained element then has the same nodal configuration, with the same nodal DOF, as the virgin element. The deflection field for virgin and constrained elements is illustrated in Table 3.. Once the interpolation functions for constrained elements are developed, formulating element stiffness, consistent mass matrices, and consistent load vectors follows the straightforward, standard procedure for displacement-based finite elements. The derivation of the MIN-N interpolation functions is discussed in the following sections (a) (b) (c) (d) Figure 3. Virgin and constrained elements nodal configuration. (a) N = 3, Linear. (b) N = 6, Quadratic. (c) N = 0, Cubic. (d) N = 5, Quartic. 24

34 z, w y θ y s θ s L θ x x α n θ n Figure 3.2 Notation for MIN-N element Table 3. Displacement interpolation for virgin and constrained elements Element type Virgin element Constrained element Degree of interpolation fields (p + ); p (p + ); p wxy (, ); θ x, θ y Degree of shear strain p p - γ sz = w, s + θ n Number of nodes N N N = ( p+ ) ( p + 2) 2 Number of nodal DOF 3N 3N Number of hierarchical parameters p Number of constraint equations 0 p +2 25

35 3.3 Virgin Elements Let w represent the transverse displacement field, and and the rotations of midsurface-normals about the x and y axes, respectively (Figure 3.2). The nodal displacements of the N-node element are w i, θ xi and θ yi, i =, 2... N, and θ xi θ yi w = { w i } θ x = { θ xi } θ y = { θ yi } (3.a) For the virgin elements, independent interpolation is used for w,, and. Lagrange-type interpolation is used for the rotations, such that θ x θ y θ x = Nθ x θ y = Nθ y (3.b) and N is the x N vector of interpolation functions: N T = { N i } i =, 2,... N (3.c) These are the standard Lagrange interpolation functions, used extensively for plane triangular elements, that can be found in many finite element texts []. These interpolation functions have degree p = ( + 8N 3) 2, such that p = when N = 3, p = 2 when N = 6, p = 3 when N = 0, etc. The transverse displacement is interpolated with functions of degree p+. The terms of degree up to p come from the Lagrange interpolation functions N i ; hierarchical functions are used for the p+2 higher-order terms. The nodeless degrees-of-freedom associated with the hierarchical terms are represented by a = { a j } j =, 2...(p+2) (3.d) The displacement can then be expressed as 26

36 w = N00N a d v (3.2a) in which N a is the x (p+2) vector of hierarchical interpolation functions N a T = { N aj } j =, 2...(p+2) (3.2b) and d v = w θ x θ y a (3.2c) Equations (3.b), (3.2a) and (3.2c) can be combined to obtain u = N v d v (3.3a) in which u = w θ x θ y (3.3b) and N v = N 0 0 N a 0 N N 0 (3.3c) Note that the virgin elements have 3N nodal degrees-of-freedom and p+2 nodeless degrees-of-freedom. 27

37 3.4 Shear Constraints γ sz The transverse shear along any arbitrary line L that makes an angle α with the x- axis (Figure 3.2) is defined as γ sz = w, s + θ n (3.4a) in which θ n is the midsurface-normal rotation as depicted. Hence γ xz and γ yz are defined as γ xz = w, x + θ y ; γ yz = w, y + θ x (3.4b) γ sz can be expressed in terms of γ xz and γ yz as follows. Based on the directional derivative, w, s = w x, cosα + w y sin, α (3.4c) The relation θ n is θ n = θ y cosα + θ x sinα (3.4d) Substitution of equations (3.4c) and (3.4d) into (3.4a) results in γ sz = γ xz cosα+ γ yz sinα (3.4e) Our desired element, MIN-N, is obtained from the virgin element by eliminating the p+2 nodeless degrees-of-freedom, a. The MIN-N elements have therefore 3N nodal degrees-of-freedom only (displacement and 2 rotations at each node). As a result of the elimination of a, the transverse displacement will be coupled with the rotational degreesof-freedom. p+2 constraints are needed to eliminate a. The p+2 constraint equations are obtained from 28

38 γ sz, ss s p 0 (3.5) Equation (3.5) says that the p-th partial derivative of the shear strain γ sz along any direction is required to be zero. For example, for a 3-node element, p = and the first partial derivative has to be zero; for a 6-node element, p = 2 and the second partial derivative has to be zero; for a 0-node element, p = 3 and the third partial derivative has to be zero; and for an N-node element, the p-th order partial derivative should be zero. Note that this constraint reduces the degree of completeness of the shear strain polynomial by one. The explicit derivation of the p + 2 shear constraint equations is given in Section General Derivation of MIN-N 3.5. Constraint equations From equations (3.5) and (3.4e), we have p γ sz s p 0 (3.6a) where 29

39 p γ sz s p = sinα + cosα γ x y sz p = sinα + cosα p ( γ x y xz cosα + γ yz sinα) = = = p m = 0 C p m p C m p m = 0 p m = ( cosα) p m ( cosα) p m C m + p ( ) m ( γ xz cosα + γ yz sinα) x p m y m sinα + sinα p p p + γ yz (3.6b) ( ) m γ xz x p m y m ( cos α ) p m ( sinα) m x p m y m p γ xz x p ( m+ ) y m + p γ yz C + m p x p m y m ( cosα) p m ( sinα) m + in which C m p! p = m! ( p m)! and C 0 p =, C p = C p + p = 0. To satisfy equation (3.6a), because angle α is arbitrary, each term in equation (3.6b) must be zero: C m + p p γ xz x p ( m+ ) y m + p γ yz C + m p x p m y m = 0 m =, 0,, p (3.7) The p+2 equations represented by equation (3.7) can be simplified as p γ xz = 0 x p p γ xz γ yz ( p m) x p ( m+ ) + ( m + ) y m + x p m = 0 y m p γ yz y p = 0 p m = 0,, 2,...(p-) (3.8) 30

40 The constraint equations (3.8) can be used to eliminate the p+2 hierarchical degrees-offreedom Examples From the general equation (3.8) the constraint equations for different elements in the MIN-N family are easily obtained.. MIN3 is the first element in the family. It is a 3-node triangle with p =. The element has a quadratic displacement field and linear rotation fields. The virgin element has 9 degrees-of-freedom, with three nodeless degrees-of-freedom. The three constraint equations are γ xz, x = 0 + = 0 γ xz, y γ yz, x γ yz, y = 0 (3.9a) 2. MIN6 is the second element in the family. It is a 6-node triangle with p = 2. The element has a cubic displacement field and quadratic rotation fields. The virgin element has 8 nodal degrees-of-freedom and four nodeless degrees-of-freedom. The four constraint equations are γ xz, xx = 0 + = 0 2γ xz, xy γ yz, yy γ xz, yy 2γ yz, xy γ yz, yy = 0 + = 0 (3.9b) 3. MIN0 is the third element in the family. It is a 0-node triangle with p = 3. The element has a quartic displacement field and cubic rotation fields. The virgin element 3

41 has 30 nodal degrees-of-freedom and five nodeless degrees-of-freedom. The five constraint equations are γ xz, xxx = 0 + = 0 3γ xz, xxy γ yz, xxx 2γ xz, xyy 2γ yz, xxy γ xz, yyy 3γ yz, xyy γ yz, yyy = 0 + = 0 + = 0 (3.9c) 4. MIN5 is the fourth element in the family. It is a 5-node triangle with p = 4. The element has a quintic displacement field and quartic rotation fields. The virgin element has 45 nodal degrees-of-freedom and six nodeless degrees-of-freedom. The six constraint equations are γ xz, xxxx = 0 + = 0 4γ xz, xxxy γ yz, xxxx 3γ xz, xxyy 2γ yz, xxxy 2γ xz, xyyy 3γ yz, xxyy γ xz, yyyy 4γ yz, xyyy γ yz, yyyy = 0 + = 0 + = 0 + = 0 (3.9d) Constrained Interpolations γ xz γ yz The definitions of and in terms of displacement can be substituted in the constraint equation (3.8) to obtain the constraints in terms of the displacement variables: 32

42 ( p + ) p + x p + p + x p m y m + p + y p + ( m + ) 0 p x p m y m p y p ( p m) p x p p + x p ( m+ ) y m + 0 w θ x θ y = 0 (3.0a) where m = 0,, 2,...(p - ). Let [ ] = ( p + ) p + x p + p + x p m y m + p + y p + ( m + ) 0 p x p m y m p y p ( p m) p x p p + x p ( m+ ) y m + 0 (3.0b) in which [ ] is a (p+2) x 3 matrix. Substitution of equation (3.3a) and (3.0b) into equation (3.0a) results in [ ]N v d v = 0 (3.a) or [ ] Let N 0 0 N a 0 N N 0 w θ x = 0 θ y a (3.b) 33

43 B c [ ] N 0 0 N a 0 N N 0 0B θx B θy B a = = ( p + 2) ( 3N + p+ 2) (3.2a) so that equation (3.b) can be written as 0B θx B θy B a w θ x θ y a = 0 (3.2b) Based on equation (3.2b), the nodeless degrees-of-freedom a can be written in terms of the nodal degrees-of-freedom as w a B a 0B θx B θ = θy x θ y (3.3) Then d v I w θ = x = B a 0B θx B θy θ y Td (3.4a) where I is a 3N x 3N identity matrix, the transformation matrix T is (3N + p + 2) x 3N, T = B a 0B θx B θy I (3.4b) and 34

44 d = w θ x θ y (3.4c) d v With = Td, u = N v Td, we obtain the constrained interpolation functions N c = N v T (3.5a) in which N c = NLM 0 N N (3.5b) such that u = N c d (3.5c) N c is the 3 x 3N matrix of constrained interpolation functions for MIN-N elements. Notice that as a result of enforcing the continuous shear constraints along any line in the element, the transverse displacements are now coupled to the bending rotations and can be expressed as w = NLMd (3.6a) in which L T = { L i } M T = { M i } i =, 2,...N (3.6b) L and M are x N vectors and have degree p+, one order higher than N. That is, the transverse displacement is no longer interpolated independently of the rotations, as it is for the virgin elements. 35

45 To guarantee that the rigid body motions are satisfied, the interpolation functions N, L, and M for MIN-N elements must satisfy the conditions n N i, L i 0, M i 0 (3.7) i = n i = n i = The restriction on N i is clearly satisfied because these are the standard Lagrange-type interpolation functions. It will be shown later that the other two conditions are satisfied for specific elements. One can argue, however, that they will be satisfied in general. The virgin interpolation functions certainly can represent rigid body motions. The constrained interpolation functions are obtained from the virgin functions by restricting the variation of the shear strains (equation (3.6a)). Because rigid body motions involve zero shear, the constraints do not affect the ability of the constrained functions to represent rigid body motion, and hence equation (3.7) will be satisfied. 3.6 Element Matrices and Stress Resultants of MIN-N Once the interpolation functions of MIN-N are developed, formulating the element stiffness, consistent mass matrices and consistent load vectors follows the straightforward procedure in the displacement-based finite element formulation. Stiffness, consistent mass matrices and consistent load vectors are integrated with full integration. Using the displacement-based finite element model based on Mindlin plate theory, the element stiffness matrix, K, can be expressed as the sum of the bending component, K b, and transverse shear component, K s : 36

46 K = K b + K s = B T T b D b B b da + B s G s B s da A A (3.8a) in which 0 0 N, x B b = 0N, y 0 (3.8b) 0N, x N, y N, B x L, x M, x + N s = N, y L, y + N M, y (3.8c) The bending and transverse shear rigidity matrices are, respectively, D D 2 D 6 G D b = D 22 D 26 G G s = 2 sym. G 22 sym. D 66 (3.8d) in which D ij -----h 3 2 = C 2 ij G ij = k ijhc6 i, 6 j with C ij are the plane stress elastic moduli and k 2 ij (3.8e) are the shear relaxation factors (the same as used for MIN3 [3]). The shear relaxation factors are introduced here in the MIN- N formulas to verify their effects for the higher-order elements. When considering inertia forces, the element s consistent mass matrix is M = N T c mn c da A (3.9a) in which 37

47 m= m 0 0 sym. mt mt (3.9b) where m is the mass per unit area of the plate and mt 2 2 is the rotary inertia. The consistent load vector for the distributed normal load, q, as well as applied bending moments, M xx, M yy, and transverse shear force, Q, prescribed on the portion Γ σ of the element boundary Γ may be written as A F = q NLM T da + + Γ σ Q NLM T dγ Γ σ 0 M yy N M xx N T dγ (3.20) The element bending moment and transverse shear stress resultants may be determined from the relations M xx M yy M xy = D b B b d (3.2) Q x Q y = G s B s d (3.22) 3.7 Kirchhoff Constraints of MIN-N The p+2 shear constraints in equation (3.8) for MIN-N limit the variation of the γ xz γ yz shear strains. The shear strains and are in the form 38

48 p γ xz = A mn x n m n = 0 n m = 0 y m + p m = A mp x p m y m (3.23a) p γ yz = B mn x n m n = 0 n m = 0 y m + p m = 0 B mp x p m y m (3.23b) to satisfy the first and last shear constraints in equation (3.8). Figure 3.3 gives the relations between the polynomial terms and the corresponding coefficient A mn and B mn in equation (3.23a). The relations for B mn are similar. To satisfy the other p constraint equations p γ xz γ yz ( p m) x p ( m+ ) + ( m + ) y m + x p m = 0 y m p m = 0,, 2,...(p-) (3.24) we obtain the p relations A + mp B 0 ( p m)p = m =, 2,...p (3.25) In the thin plate regime as L t, the Kirchhoff constraints γ xz 0 and γ yz 0 (3.26) are enforced over the entire element domain. Therefore, with (3.23a), (3.23b) and (3.25) the Kirchhoff constraints are A mn,( A lp = B ( p l)p ), B mn 0 m = 0,,...n; n = 0,,...p-; l =, 2,...p. (3.27) These are the p(p+2) Kirchhoff constraint equations for MIN-N in the thin plate limit. 39

49 A 00 x y A 0 A x 2 xy y 2 x 3 x 2 y xy 2 y A 02 A 2 A 22 A 03 A 3 A 23 A x p- x p-2 y xy p-2 y p- A 0(p-) A (p-) A (p-2)(p-) A (p-)(p-) x p x p- y xy p- y p A 0p A p A (p-)p A pp Figure 3.3 Correlation between A mn and polynomial terms 3.8 Application: Element MIN3 As stated previously, the success of the 3-node, triangular, constant-moment MIN3 element presented by [3] has motivated this development of the MIN-N family, which includes higher-order elements. It is verified in this section that Tessler s MIN3 element is the lowest order member in the MIN-N family, even though his derivation and the derivation of MIN-N are somewhat different. The detailed derivation of the constrained interpolation functions for MIN3 is now presented. The nodal configuration for MIN3 is shown in Figure

50 3 y 2 x Figure 3.4 MIN3 triangular plate element 3.8. Virgin Interpolations of MIN3 The interpolation functions for MIN3 s virgin element are given in terms of areaparametric coordinates as: N T = { N i }, NT a = { N ai } (3.28a) where N i = ξ i, i =, 2, 3 and N ai = ξ i ξ k, i =, 2, 3; k = 2, 3,. (3.28b) The N i are also used as shape functions, and therefore the linear relation between Cartesian and area coordinates is ξ i = ( c 2A i + b i x + a i y) (3.28c) As before, A is the area of the triangle, and the coefficients a i, b i, and c i are 4

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