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dissertation entitled
A 3-D Laminated Plate Finite Element
with Zig-Zag Sublaminate Approximations

for Composite and Sandwich Panels

presented by

Yuen Cheong Yip

has been accepted towards fulfillment
of the requirements for

Ph.D. Mechanics

degree in

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A 3-D LAMINATED PLATE FINITE ELEMENT
WITH ZIG-ZAG SUBLAMINATE APPROXIMATIONS
FOR COMPOSITE AND SANDWICH PANELS
By

Yuen Cheong Yip

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Materials Science and Mechanics

1996

ABSTRACT
A 3-D LAMINATED PLATE FINITE ELEMENT
WITH ZIG-ZAG SUBLAMINATE APPROXIMATIONS
FOR COMPOSITE AND SANDWICH PANELS
By .

Yuen Cheong Yip

This dissertation describes the development of a new zig-zag theory and an eight-node
brick finite element using zig-zag sub-laminate approximations which are suitable for the
analysis of both thick and thin laminated composite plates and sandwich panels. The zig-
zag theory employs the sublaminate concept, in which each computational layer (or
sublaminate) contains several, even many, physical layers. The new element derived from
this new theory has the topology of a three-dimensional (3-D) continuum-type element (an
eight-node brick) although it is based on plate kinematics. This eight-node brick topology
permits the stacking of elements through-the-thickness of a laminate, which is not
convenient in conventional plate/shell elements whose kinematics are usually with respect
to the reference mid-surface. This allows sub-layering for through-the-thickness
refinement if higher accuracy is desired.

This zig-zag sublaminate approach has the following desirable properties: (i) satisfies
interlaminar transverse shear stress continuity; (ii) satisfies transverse shear tractions at the
top and bottom surfaces of the plate exactly; (iii) tends to the correct prescribed load value
for the transverse normal tractions at the top and bottom with refinement in the thickness

direction; (iv) has a small and fixed number of degrees-of-freedorn per sublaminate; (v)

uses traditional engineering degrees-of-freedom (displacements and rotations); and (vi)
permits the use of adaptive techniques for through-the-thickness discretization when used
with the eight-node brick topology.

In thin plate applications, the parasitic shear locking phenomenon has plagued shear
deformable elements Since their inception. This new eight-node brick finite element using
zig-zag sub-laminate approximations utilizes the assumed strain field approach to extend
the element to thin plate applications. The element shear strain fields meet both field-
consistency and edge-consistency requirements.

Numerical results demonstrate that this new element is robust, accurate and
computationally efficient. The element also passes the membrane and Kirchhoff patch
tests for plates. The current models Show excellent promise for efficient and accurate
analysis of thick laminated composites and sandwich panels. Furthermore, the eight-node
brick topology will permit the use of adaptive techniques for through-the-thickness
discretization and will allow the coupling of the new elements to conventional 3D

continuum-based elements.

to my beloved wife,
Regina Wai Cheng Hooi
and daughter,

Michelle Shi Yun Yip.

iv

ACKNOWLEDGENIENT S

I would like to thank NASA Langley Research Center for their extensive support for
this research under Grant NAG-l-159l. Dr. Jerry Housner and Dr. Alex Tessler were the
grant monitors. A special thanks to Dr. Alex Tessler for his assistance in some of the work.

I would like to acknowledge my doctoral committee members, Professors Thomas J.
Pence, Dahsin Liu, Nicholas J. Alterio and Zhengfang Zhou for their guidance and
suggestions during the course of the research.

I must also thank the Computational Mechanics Research Group (CMRG) for their
helpful discussions and contributions to my work, with special thanks to my colleagues
Venkateshwar Rao Aitharaju, Yong-Bae Cho and Lorenzo Marco Smith for their
friendship and useful discussions this past year. I am grateful to my wife, Regina for her
understanding and support throughout the years that I have known her and to Professor
Anthony Waas for if not for him, all this would not be possible.

Finally, I must acknowledge my advisor, Professor Ronald C. Averill. In the
preparation of this dissertation and throughout the whole period of my studies at Michigan
State, I have had the good fortune to benefit from his guidance, advice and
encouragement. I will always be grateful and indebted to him for giving me the

opportunity as well as the financial support to pursue this study.

TABLE OF CONTENTS

LIST OF TABLES ....................................................................................................... viii

LIST OF FIGURES ..................................................................................................... ix
CHAPTER 1

INTRODUCTION AND BACKGROUND ................................................................. 1
1.1 Introduction ................................................................................................... 1
1.2 Background: Laminated Plate/Shell Theories .............................................. 4
1.3 Background: Laminated Plate/Shell Finite Element Models ........................ 9
1.4 The Present Study ......................................................................................... 11
1.5 Interpolation Theory: The C-Concepts ......................................................... 12
1.6 C0 versus C Finite Elements ..................................................................... 14
1.7 Interdependent Interpolation ......................................................................... 14
1.8 Organization of the Dissertation ................................................................... 15
CHAPTER 2

LAYERWISE BEAM ELEMENT ............................................................................... 17
2.1 Introduction ................................................................................................... 17
2.2 Theory Formulation ...................................................................................... 18
2.3 Finite Element Model ................................................................................... 24
2.4 Numerical Studies ......................................................................................... 27
2.5 Summary ....................................................................................................... 31
CHAPTER 3

MIN DLIN PLATE ELEMENT .................................................................................... 3S
3. 1 Introduction ................................................................................................... 35
3.2 Interdependent Interpolation ......................................................................... 38
3.3 Finite Element Model ................................................................................... 43
3.4 Consistent Shear Strain Fields ...................................................................... 45
3.5 Edge Consistency .......................................................................................... 47
3.6 Numerical Studies ......................................................................................... 50
3.7 Summary ....................................................................................................... 63
CHAPTER 4

LAYERWISE PLATE ELEMENT - THICK PANELS ............................................... 65
4.1 Introduction ................................................................................................... 65

vi

4.2 Theory Formulation ...................................................................................... 66

4.3 Theory and Finite Element Model ................................................................ 72
4.4 Numerical Studies ......................................................................................... 80
4.5 Summary ....................................................................................................... 87
CHAPTER 5

LAYERWISE PLATE ELEMENT - THIN PANELS .................................................. 100
5. 1 Introduction ................................................................................................... 100
5.2 Formulation ................................................................................................... 102
5.3 Interdependent Interpolation ......................................................................... 102
5.4 Field Consistent Shear Strain Fields ............................................................. 103
5.5 Consistent Penalty Constraints ..................................................................... 109
5.6 Consistent Transverse Normal Strain Field .................................................. 111
5.7 Edge Consistency .......................................................................................... 113
5.8 Numerical Studies ......................................................................................... 117
5.9 Summary ....................................................................................................... 124
CHAPTER 6

NON-LINEAR MODEL .............................................................................................. 141
6. 1 Introduction ................................................................................................... 141
6.2 Theory Formulation ...................................................................................... 142
6.3 Newton-Raphson Iteration Method ............................................................... 147
6.4 Numerical Results ......................................................................................... 150
6.5 Summary ....................................................................................................... 152
CHAPTER 7

CONCLUSIONS .......................................................................................................... 157
7.1 Review .......................................................................................................... 157
7.2 Area of Future Research ............................................................................... 160
BIBLIOGRAPHY .................................................................................................. 162
APPENDICES
APPENDIX A - Inplane Displacement Field Constants (Layerwise Beam) ......... 171
APPENDIX B - Interdependent Interpolation ....................................................... 173
APPENDD( C - Shape Functions .......................................................................... 180
APPENDIX D - Transformed Shape Functions .................................................... 185
APPENDIX E - Interpretation of Stress and Strain Tensors in the Case of

Small Strains/Large Rotations ...................................................... 186

APPENDIX F - Element Stiffness Matrix (von Karman strains) .......................... 190
APPENDIX G - Direct Stiffness Matrix (von Karman strains) ............................. 192
APPENDIX H - Tangent Stiffness Matrix (von Karman strains) .......................... 201

vii

Table 1

Table 2

Table 3

Table 4

Table 5

Table 6

Table 7

Table 8

Table 9

Table 10

Table 11

Table 12

Table 13

Table 14

LIST OF TABLES

Material Properties Used in the Numerical Analyses ............................ 28
Description of Laminate Sequence ........................................................ 29
Straight Cantilever Beam ....................................................................... 52
Curved Cantilever Beam ........................................................................ 5 3
Long Cantilever Beam Test .................................................................... 54
Results for Rectangular Plate, Simple Supports: Uniform Load ........... 55
Results for Rectangular Plate, Clamped Supports: Center Load ........... 56
Normalized Center Deflection: Uniform load ........................................ 63
Lamination Scheme for Random Five-Layer Panel ............................... 82
Material Properties for Random Five-Layer and Sandwich Panel ......... 82
Lamination Scheme for Sandwich Panel ............................................... 83
Material Properties for TACOM’S Composite Armored Vehicle

(CAV) Panel ........................................................................................... 84
Results for Square Plate, Simple Supports: Sinusoidal Load ................ 86
Results for Square Plate, Simple Supports: Sinusoidal Load ................ 122

viii

Figure 1 -

Figure 2 -

Figure 3 —

Figure 4 -

Figure 5 -

Figure 6 -

Figure 7 -

Figure 8 -

Figure 9 -

Figure 10 -

Figure 11-

Figure 12 -

Figure 13 -

LIST OF FIGURES

Schematic of Sublaminate and Layer Divisions in the Laminate

Theory .................................................................................................... 18
Nodal Topology of Unconstrained Element (Top) and Constrained

Element (Bottom) .................................................................................. 27
Schematic of Loading and Support Conditions ..................................... 27

Predicted Normalized Maximum Deflection and Normalized
Maximum Inplane Normal Stress versus Number of Elements in the
Mesh ....................................................................................................... 32

Normalized Center Deflection versus Span-to—Thickness Ratio of a
Five-Layer Simply-Supported Beam Subjected to a Sinusoidal Load .. 32

Axial Displacement versus Normalized Thickness Coordinate at the
End of a Simply-Supported Beam Subjected to a Sinusoidal Load ....... 33

Axial Stress versus Normalized Thickness Coordinate at the Midspan
of a Simply-Supported Beam Subjected to a Sinusoidal Load .............. 33

Transverse Shear Stress versus Normalized Thickness Coordinate at
x=L/10 of a Simply-Supported Beam Subjected to a Sinusoidal Load.. 34

Transverse Deflection versus Normalized Thickness Coordinate at
Midspan of a Simply-Supported Beam Subjected to a Sinusoidal

Load ....................................................................................................... 34
Sign Convention for Rotations and Stress Resultants ............................ 40
Topology of the Finite Element Model .................................................. 44
Quadrilateral Element Coordinate Description ...................................... 44
Kirchhoff Patch Test for Plates .............................................................. 51

ix

Figure 14 -
Figure 15 -
Figure 16 -

Figure 17 -

Figure 18 -

Figure 19 -

Figure 20 -

Figure 21 -

Figure 22 -

Figure 23 -

Figure 24 -

Figure 25 -

Figure 26 -

Figure 27 -

Figure 28 -

Figure 29 -

Figure 30 -

Straight Cantilever Beam ....................................................................... 51
Curved Cantilever Beam ........................................................................ 53
Long Cantilever Beam ........................................................................... 53

Geometry, Material Properties, Boundary Conditions and Theoretical
Solutions for the Rectangular Plate Tests .............................................. 55

Normalized Center Deflection versus Span-to-Thickness Ratio of a
Simply-Supported Square Plate Under Sinusoidal Loading (a/b = 1)... 57

Normalized Center Deflection versus Span-to-Thickness Ratio of a
Simply-Supported Square Plate Under Sinusoidal Loading (a/b = 5)... 57

Normalized Maximum Normal Stress versus Span-to-Thickness
Ratio of a Simply-Supported Square Plate Under Sinusoidal
Loading (a/b = 1) ................................................................................... 58

Normalized Maximum Normal Stress versus Span-to—Thickness

Ratio of a Simply-Supported Square Plate Under Sinusoidal

Loading (a/b = 5) ................................................................................... 5 8
Clamped Circular Plate (Quadrant) ........................................................ 59

Normalized Deflection (W = w 161tD/ (Pa4)) Along Line of Symmetry
(y=0) of a Thin (a/h=50) Clamped Circular Plate Under Center Load . 60

Normalized Deflection (W = w161tD/(Pa4)) Along Line of Symmetry
(y=0) of a Thick (a/h=2.5) Clamped Circular Plate Under Center Load 60

Normalized Deflection (W = w64D/(qa2)) Along Line of Symmetry
(y=0) of a Thin (a/h=50) Clamped Circular Plate Under Uniform Load 61

Normalized Deflection (W = w640/(qa2)) Along Line of Symmetry
(y=0) of a Thick (a/h=2.5) Clamped Circular Plate Under Uniform

Load ....................................................................................................... 61
Morley’s Acute Skew Plate .................................................................... 62
Schematic of Sublaminate and Layer Divisions .................................... 67
Topology of the LZZ3 Finite Element Model ........................................ 80

Schematic of Loading and Boundary Conditions for Example
Problems ................................................................................................ 81

Figure 31 -

Figure 32 —

Figure 33 -

Figure 34 -

Figure 35 -

Figure 36 -

Figure 37 -

Figure 38 -

Figure 39 -

Figure 40 -

Figure 41 -

Axial Displacement versus Normalized Thickness Coordinate of a
Simply-Supported Square Plate (Random 5-Layer) Subjected to
Sinusoidal Load ..................................................................................... 89

Transverse Deflection versus Normalized Thickness Coordinate of a
Simply-Supported Square Plate (Random 5-Layer) Subjected to
Sinusoidal Load ..................................................................................... 89

Inplane Normal Stress versus Normalized Thickness Coordinate of a
Simply-Supported Square Plate (Random 5-Layer) Subjected to
Sinusoidal Load ..................................................................................... 90

Transverse Shear Stress versus Normalized Thickness Coordinate
of a Simply-Supported Square Plate (Random 5-Layer) Subjected to
Sinusoidal Load ..................................................................................... 90

Inplane Shear Stress versus Normalized Thickness Coordinate of a
Simply-Supported Square Plate (Random 5-Layer) Subjected to
Sinusoidal Load ..................................................................................... 91

Transverse Normal Strain versus Normalized Thickness Coordinate
of a Simply-Supported Square Plate (Random 5-Layer) Subjected to
Sinusoidal Load ..................................................................................... 91

Axial Displacement versus Normalized Thickness Coordinate of a
Simply-Supported Square Plate (Sandwich) Subjected to Sinusoidal
Load ....................................................................................................... 92

Transverse Deflection versus Normalized Thickness Coordinate of a
Simply—Supported Square Plate (Sandwich) Subjected to Sinusoidal
Load ....................................................................................................... 92

Inplane Normal Stress versus Normalized Thickness Coordinate of a
Simply-Supported Square Plate (Sandwich) Subjected to Sinusoidal
Load ....................................................................................................... 93

Inplane Normal Stress versus Normalized Thickness Coordinate of a
Simply-Supported Square Plate (Sandwich) Subjected to Sinusoidal
Load - Top Face Sheet ........................................................................... 93

Transverse Shear Stress versus Normalized Thickness Coordinate
of a Simply-Supported Square Plate (Sandwich) Subjected to
Sinusoidal Load ..................................................................................... 94

xi

Figure 42 -

Figure 43 -

Figure 44 -

Figure 45 -

Figure 46 -

Figure 47 -

Figure 48 -

Figure 49 -

Figure 50 -

Figure 51 -

Figure 52 -
Figure 53 -

Figure 54 -

Figure 55 -

Inplane Shear Stress versus Normalized Thickness Coordinate of a
Simply-Supported Square Plate (Sandwich) Subjected to
Sinusoidal Load ..................................................................................... 94

Inplane Shear Stress versus Normalized Thickness Coordinate of a
Simply-Supported Square Plate (Sandwich) Subjected to
Sinusoidal Load - Top Face Sheet .......................................................... 95

Transverse Normal Strain versus Normalized Thickness Coordinate
of a Simply-Supported Square Plate (Sandwich) Subjected to
Sinusoidal Load ..................................................................................... 95

Axial Displacement versus Normalized Thickness Coordinate of a
Simply~Supported Square Plate (CAV) Subjected to Sinusoidal Load . 96

Transverse Deflection versus Normalized Thickness Coordinate of a
Simply-Supported Square Plate (CAV) Subjected to Sinusoidal Load . 96

Inplane Normal Stress versus Normalized Thickness Coordinate of a
Simply-Supported Square Plate (CAV) Subjected to Sinusoidal Load . 97

Inplane Normal Stress versus Normalized Thickness Coordinate of a
Simply-Supported Square Plate (CAV) Subjected to Sinusoidal Load -
Outer Core .............................................................................................. 97

Transverse Shear Stress versus Normalized Thickness Coordinate
of a Simply-Supported Square Plate (CAV) Subjected to Sinusoidal
Load ....................................................................................................... 98

Inplane Shear Stress versus Normalized Thickness Coordinate of a
Simply-Supported Square Plate (CAV) Subjected to Sinusoidal Load . 98

Transverse Normal Strain versus Normalized Thickness Coordinate
of a Simply-Supported Square Plate (CAV) Subjected to Sinusoidal

Load ....................................................................................................... 99

Quadrilateral Element Coordinate Description ...................................... 103
Patch Test for Plates ............................................................................... 1 18
Clamped Circular Plate (Quadrant) ........................................................ 119

Normalized Deflection (W = wl61tD/ (Pa4)) Along Line of Symmetry
(y=0) of a Clamped Circular Plate Under Center Load ......................... 120

xii

Figure 56 -

Figure 57 -

Figure 58 -

Figure 59 -

Figure 60 -

Figure 61 -

Figure 62 -

Figure 63 -

Figure 64 -

Figure 65 -

Figure 66 -

Figure 67 -

Normalized Deflection (W = w640/(qa2)) Along Line of Symmetry
(y=0) of a Clamped Circular Plate Under Uniform ............................... 120

Normalized Center Deflection versus Span-to—Thickness Ratio of a
Simply-Supported Square Plate (Isotropic) Subjected to Sinusoidal
Load ....................................................................................................... 125

Normalized Center Deflection versus Span-to—Thickness Ratio of a
Simply-Supported Square Plate (Sandwich) Subjected to Sinusoidal
Load ....................................................................................................... 125

Normalized Center Deflection versus Span-to-Thickness Ratio of a
Simply-Supported Square Plate (CAV) Subjected to Sinusoidal Load . 126

Axial Displacement versus Normalized Thickness Coordinate of a
Simply-Supported Square Plate (Sandwich) Subjected to
Sinusoidal Load ..................................................................................... 126

Transverse Deflection versus Normalized Thickness Coordinate
of a Simply-Supported Square Plate (Sandwich) Subjected to
Sinusoidal Load ..................................................................................... 127

Inplane Normal Stress versus Normalized Thickness Coordinate
of a Simply-Supported Square Plate (Sandwich) Subjected to
Sinusoidal Load ..................................................................................... 127

Inplane Normal Stress versus Normalized Thickness Coordinate
of a Simply-Supported Square Plate (Sandwich) Subjected to
Sinusoidal Load - Top Face Sheet .......................................................... 128

Transverse Shear Stress versus Normalized Thickness Coordinate
of a Simply-Supported Square Plate (Sandwich) Subjected to
Sinusoidal Load ..................................................................................... 128

Inplane Shear Stress versus Normalized Thickness Coordinate of a
Simply-Supported Square Plate (Sandwich) Subjected to
Sinusoidal Load ..................................................................................... 129

Inplane Shear Stress versus Normalized Thickness Coordinate of a
Simply-Supported Square Plate (Sandwich) Subjected to
Sinusoidal Load - Top Face Sheet .......................................................... 129

Transverse Normal Strain versus Normalized Thickness Coordinate
of a Simply-Supported Square Plate (Sandwich) Subjected to
Sinusoidal Load ..................................................................................... 130

xiii

Figure 68 -

Figure 69 -

Figure 70 -

Figure 71 -

Figure 72 -

Figure 73 -

Figure 74 -

Figure 75 -

Figure 76 -

Figure 77 -

Figure 78 -

Axial Displacement versus N orrnalized Thickness Coordinate of a
Simply-Supported Square Plate (CAV) Subjected to Sinusoidal
Load .......................................................................................................

Transverse Deflection versus Normalized Thickness Coordinate
of a Simply-Supported Square Plate (CAV) Subjected to Sinusoidal
Load .......................................................................................................

Inplane Normal Stress versus Normalized Thickness Coordinate
of a Simply-Supported Square Plate (CAV) Subjected to Sinusoidal
Load .......................................................................................................

Inplane Normal Stress versus Normalized Thickness Coordinate
of a Simply-Supported Square Plate (CAV) Subjected to Sinusoidal
Load - Outer Shell ..................................................................................

Transverse Shear Stress versus Normalized Thickness Coordinate
of a Simply-Supported Square Plate (CAV) Subjected to Sinusoidal
Load .......................................................................................................

Inplane Shear Stress versus Normalized Thickness Coordinate of a
Simply-Supported Square Plate (CAV) Subjected to Sinusoidal
Load .......................................................................................................

Transverse Normal Strain versus Normalized Thickness Coordinate
of a Simply-Supported Square Plate (CAV) Subjected to
Sinusoidal Load .....................................................................................

Axial Displacement versus Normalized Thickness Coordinate of a
Simply-Supported Square Plate (Sandwich) Subjected to
Sinusoidal Load .....................................................................................

Transverse Deflection versus Normalized Thickness Coordinate
of a Simply-Supported Square Plate (Sandwich) Subjected to
Sinusoidal Load .....................................................................................

Inplane Normal Stress versus Normalized Thickness Coordinate
of a Simply-Supported Square Plate (Sandwich) Subjected to
Sinusoidal Load .....................................................................................

Inplane Normal Stress versus Normalized Thickness Coordinate

of a Simply-Supported Square Plate (Sandwich) Subjected to
Sinusoidal Load - Top Face Sheet ..........................................................

xiv

130

131

131

132

132

133

133

134

134

135

135

Figure 79 -

Figure 80 -

Figure 81 -

Figure 82 -

Figure 83 -

Figure 84 -

Figure 85 -

Figure 86 -

Figure 87 -

Figure 88 -

Figure 89 -

Transverse Shear Stress versus Normalized Thickness Coordinate
of a Simply-Supported Square Plate (Sandwich) Subjected to
Sinusoidal Load .....................................................................................

Inplane Shear Stress versus Normalized Thickness Coordinate of a
Simply-Supported Square Plate (Sandwich) Subjected to
Sinusoidal Load .....................................................................................

Transverse Normal Strain versus Normalized Thickness Coordinate
of a Simply-Supported Square Plate (Sandwich) Subjected to
Sinusoidal Load .....................................................................................

Axial Displacement versus Normalized Thickness Coordinate of a
Simply-Supported Square Plate (CAV) Subjected to Sinusoidal
Load .......................................................................................................

Transverse Deflection versus Normalized Thickness Coordinate of a
Simply-Supported Square Plate (CAV) Subjected to Sinusoidal
Load .......................................................................................................

Inplane Normal Stress versus Normalized Thickness Coordinate
of a Simply-Supported Square Plate (CAV) Subjected to Sinusoidal
Load .......................................................................................................

Inplane Normal Stress versus Normalized Thickness Coordinate
of a Simply-Supported Square Plate (CAV) Subjected to Sinusoidal
Load - Outer Shell ..................................................................................

Transverse Shear Stress versus Normalized Thickness Coordinate
of a Simply-Supported Square Plate (CAV) Subjected to Sinusoidal
Load .......................................................................................................

Inplane Shear Stress versus Normalized Thickness Coordinate of a
Simply-Supported Square Plate (CAV) Subjected to Sinusoidal
Load .......................................................................................................

Transverse Normal Strain versus Normalized Thickness Coordinate
of a Simply-Supported Square Plate (CAV) Subjected to Sinusoidal
Load .......................................................................................................

Load-Deflection Curves for the Bending of a Simply-Supported,

Square, Orthotropic Plate under Uniform Transverse Loading - Effect
of Inplane Mesh Refinement ..................................................................

XV

136

136

137

137

138

138

139

139

140

140

154

Figure 90 -

Figure 91 -

Figure 92 -

Figure 93 -

Figure 94 -

Figure B.1-

Load-Deflection Curves for the Bending of a Simply-Supported,
Square, Orthotropic Plate under Uniform Transverse Loading - Effect
of Through-Thickness Mesh Refinement ...............................................

Load-Deflection Curves for the Bending of a Simply-Supported,
Square, Orthotropic Plate under Uniform Transverse Loading -
Comparison Among Different Theories and Experimental Results ......

Load-Deflection Curves for the Bending of a Clamped, Square,
Symmetric Cross-Ply (0/ 90/ 90/ 0) Plate under Uniform
Transverse Loading - Effect of Inplane Mesh Refinement ....................

Load-Deflection Curves for the Bending of a Clamped, Square,
Symmetric Cross-Ply (0/ 90/ 90/ 0) Plate under Uniform
Transverse Loading - Effect of Through—Thickness Mesh Refinement.

Load-Deflection Curves for the Bending of a Clamped, Square,
Symmetric Cross-Ply ( 0/ 90/ 90/ 0) Plate under Uniform
Transverse Loading - Comparison Among Different Theories and
Experimental Results .............................................................................

Quadrilateral Element Coordinate Description ......................................

xvi

154

155

155

156

156

174

Chapter 1

INTRODUCTION AND BACKGROUND

1.1 Introduction

AS analytical solutions are usually restricted to problems with simple geometry,
loading and boundary conditions, the finite element method is therefore useful when
dealing with complex problems. Using conventional three-dimensional (3-D)
displacement-based finite elements is very computationally expensive. Yet in the analysis
of composite laminates where local effects are important (e.g. embedded delaminations,
ply—dropoffs, free edges, cracks or other regions of 3-D stress fields), one must use a
theory based on 3-D kinematics. Most two-dimensional (2-D) plate/shell models are
therefore inadequate. However, with the more recent advent of such 2-D (quasi-3-D)
theories like the layerwise (e.g., Reddy (1987), Lu and Liu (1992), Liu (1995), Li and Liu
(1996)) and zig-zag theories (e.g., DiSciuva (1985, 1987), Cho and Parmerter (1993)),
these local effects are modeled satisfactorily. We can implement these more precise
theories to develop a robust finite element model that is more computationally efficient
than the conventional displacement-based continuum finite element model and yet
describe the local kinematics adequately.

The zig-zag theory has several important features that are ideally suited for our

purpose. Firstly, it possesses only a fixed number of degrees of freedom, regardless of the

 

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(‘03

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mm

number of layers in the laminate. Secondly, it is accurate and satisfies conditions such as
(i) the continuity of transverse Shear stress through the thickness of the laminate and (ii)
the homogeneous shear traction boundary conditions. Although these features imply an
almost ideal combination of accuracy and efficiency, there is one significant drawback to
this theory. The a prior assumptions of the displacement field require the transverse
deflection degree-of-freedom to be C 1 continuous. This restricts the choice of element
interpolation functions that can be used and is therefore very inconvenient for the
development of general purpose finite element models based on these theories.

The focus of this research will be on the development of a robust C0 finite element
model based primarily on the high-order zig-zag theory. It incorporates ideas from the
layerwise theories (Reddy, 1987; Barbero and Reddy, 1990; Li and Liu, 1996), the sub-
laminate analysis concept of Flanagan (1994), in conjunction with an interdependent
interpolation scheme of Tessler and Hughes (1983, 1985). The plate element will be
developed in a consistent approach that features exact satisfaction of boundary and
interface conditions for displacements and shear tractions. Averill and Yip (1996a)
presented a generalized form of the high-order zig—zag theory for beams in which the
continuity of the transverse deflection was relaxed by introducing a new variable along
with an appropriate constraint condition. Based on this generalized form of the theory, a
two-node beam element was developed that contained only C0 variables. The constraint
condition was then enforced via a combination of analytical and numerical means by
utilizing an interdependent interpolation scheme (Tessler and Dong, 1981) and the penalty
method, resulting in a simple, efficient and robust finite element beam model.

The new proposed plate finite element model will be based on modified forms of the

high-order zig-zag theories. These elements contain only engineering-type degrees of
freedom, displacements and rotations, and have the nodal topology of an eight-node brick
element, though they are based on plate kinematics. This, therefore, allows our proposed
finite element great versatility to model laminates of complex cross-sections. Depending
on the technique used in modeling the laminate, this finite element formulation will enable
us to achieve a solution similar to that of either the high-order zig-zag or layerwise theory.
If all the layers in the laminate are modeled as an equivalent mathematical layer (or
block), the current model becomes equivalent to the high-order zig-zag theories
(DiScuiva, 1987; and Averill and Yip, 1996b). If the number of mathematical layers is
equal to the number of physical layers in the laminate, it will yield the conventional
discrete layerwise theory (Reddy, 1987 and Li and Liu, 1996).

This type of plate element permits vertical stacking through the thickness of a laminate
with all the proper continuity conditions except for transverse normal traction. In addition
to being able to emulate the high-order zig-zag theory and layerwise theory, this unique
form of the proposed plate element can also be utilized to emulate a concept very similar
to the “local and global laminate” model (sub-laminate analysis) proposed by Pagano and
Soni (1983) to model a laminate consisting of many layers. They came up with a global
and local model whereby they divided the laminate into two parts; namely, (i) the local
region - region that is of interest and (ii) the global region - the remaining region. The field
equations in their model are based upon an assumed thickness distribution of stress
components within each layer of the local region and an assumed thickness distribution of
displacement components in the global region (effective properties are used). Although an

apparent loss of accuracy occurs in the calculation of transverse Shear stress components

Nil
Na:
8581
[O b

Star

in the global model, increasing the number of sub—laminates will obviously enhance both
the accuracy of transverse stress predictions as well as the local predictions.

The type of sub-structuring technique described above is especially important when
solid modeling of individual layers may not be feasible yet it is important to isolate a
region of critical behavior. This efficient local-global modeling of composite structures
has been employed widely (see for example, Lee and Liu, 1991; Kong and Cheung, 1995;

Yu et al., 1995; Li and Liu, 1996a).

1.2 Background: Laminated Plate/Shell Theories

Several recent survey papers summarize the approaches of plate theories (Bert, 1984;
Noor and Burton, 1989, 1990; Kapania and Raciti, 1989; Reddy, 1990). They consider the
problem of reducing the three-dimensional equations of the theory of elasticity to two-
dimensional plate/shell equations. The two-dimensional theories are obtained from the
three-dimensional elasticity theory by making a prior assumptions concerning the
variation of displacements and/or stresses through the thickness coordinate of the
laminate, yielding an equivalent single-layer plate theory. These theories based on the
method of expansion with respect to the thickness coordinate can be found in the work of
Mindlin (1961). The classical laminate plate theory (CLPT) is an extension of the classical
plate theory (CPT) to laminated plates. In both theories, the inplane displacements are
assumed to vary linearly through the thickness and the transverse displacement is assumed
to be constant through the thickness (i.e., transverse normal strain is zero) - Reissner and
Stavsky (1961), and Dong et al. (1962). With the anisotropic and non-homogeneous
nature of composite laminates, important effects of transverse shear and normal

deformation cannot be neglected, demanding improvements in the classical laminate plate

at:
10;.

her

theory (CPLT), which is based on the well-known Kirchhoff assumptions. As a result,
modeling approaches which take into account these effects have been the topic of research
in the last two decades.

A refinement to the classical laminate theory is provided by the first-order Shear
deformation plate theory (FSDT). The first-order shear deformation theory is commonly
known as the “Mindlin plate theory”, (Mindlin (1961) and Reissner (1945)). The first-
order shear deformation theory yields a constant value of transverse shear strain through
the thickness of the plate, and thus requires shear correction factors. Inclusion of first-
order transverse shear effects is often adequate as shown by Yang et al. (1966), provided
the laminate does not contain adjacent layers with drastically varying stiffness properties
(see Averill, 1994; Sun and Whitney, 1973).

Second and high-order shear deformation plate theories (HSDT) involve high-order
expansions of the displacement field (Reddy, 1984; and Barbero and Reddy, 1990).
Barbero and Reddy (1990) suggested a generalized strain-consistent third-order theory
and demonstrated that the displacement fields of many other “equivalent single layer”
third-order theories can be derived from it. The displacement field of HSDT
accommodates the vanishing of transverse shear strains (and hence shear tractions) on the
top and bottom of a laminate. This type of theory, however, generally results in
incompatible transverse shearing stresses between adjacent layers.

In summary, all the above theories (CLPT, FSDT, HSDT) differ only in their choices
of assumed displacement fields. Termed as “equivalent single layer” approach, all those
theories have a common drawback in that the thickness variation of displacements, and

. . . . 1 . .
thus strains, rs assumed to be continuous and smooth (1.6. C continuous). This

characteristic will lead to a discontinuous interlaminar stress field because of different
constitutive properties in adjacent plies. It therefore precludes the satisfaction of
continuity of transverse stresses at interfaces between adjacent layers of different
materials, and does not accurately reflect the kinematics in laminates. This may become
Significant for laminates that contain drastically different constitutive properties in
adjacent layers.

To overcome the deficiencies of “equivalent single layer” theories, discrete-layer (or
layerwise) theories (DLT) have been developed in which a unique displacement field is
assumed within each layer (see for example, Reddy, 1987; Toledano and Murakami, 1987;
Lu and Liu, 1992). These theories explicitly account for the layerwise construction of
laminates and allow the possibility of achieving continuous transverse shear stresses at all
layer interfaces.

The layerwise theory by Reddy (1987) treats each layer within the multi-layered
laminates as separate individual layers or combines several layers into sub-laminates. He
assumes that the displacement field is expanded within each layer using Lagrangian shape
functions used in conventional elasticity displacement finite elements. It is therefore a
quasi-3-D model based on plate kinematics as opposed to truly 3-D kinematics. Each layer
of the theory can be treated either as a mathematical or a physical layer. Several physical
layers can be combined into a sub-laminate and treated as a mathematical layer wherever
necessary (for computational savings). Because of the layerwise construction of this
theory, it can therefore model local effects like delaminations, ply terminations (i.e. ply-
dropoffs), ply splits, etc. The most significant aspect of this layerwise theory is that,

compared to the conventional 3-D displacement finite element model, it has a data

Vii

DOI
mu
[0 5
kill:
[nor
COns

C0m

structure that saves computational time while giving exactly the same results for
comparable meshes (see Robbins and Reddy, 1992). However, this theory contains a
number of degrees-of-freedom that is proportional to the number of layers in the laminate.
Consequently, for analysis of thick-section laminates involving hundreds of layers, the
computational effort required will still be enormous. The number of degrees of freedom
will grow rapidly with the number of layers and computer storage may force limitations
on the total number of layers analyzed.

The sub-laminate concept within the layerwise theory by Reddy (1987) that can be
employed to model several layers as one equivalent single layer is an attractive feature to
save computational costs and efforts. However, one drawback of this concept when several
layers are combined is that the continuity of interlaminar shear stress conditions will be
violated.

Liu and his co-workers (1995, 1996) extended the work of Reddy (1987) by adding
nodal rotations in addition to nodal displacements as independent variables. The nodal
rotations, however, are not required to be continuous across the composite interface so as
to satisfy interlaminar shear stress continuity, reflecting the correct nature of the laminate
kinematics. The theory gives excellent results for both stresses and displacements and,
more importantly, allows accurate predictions of transverse shear stresses from the
constitutive equations instead of having them recovered from the equilibrium equations as
commonly done.

The discrete layerwise theory (DLT) developed using plate kinematics requires only 2-
D finite elements. The advantage is that the element aspect ratio for these layerwise

models is restricted only in the two dimensions (in-plane coordinates), as opposed to all

three dimensions in the conventional elasticity finite elements. It is required for 3-D
displacement finite elements to have an element aspect ratio of one or close to one for
good converged results. To achieve this ratio, the thinness of each layer in the laminate
will force the other two dimensions of an element to be small in the case of 3-D elements.
Consequently, a very large number of 3-D elements will be needed to ensure a proper
mesh

There is another new class of laminate theory termed the zigzag theories which
incorporates the layerwise construction (different constitutive properties in each layer) of
laminates into the in-plane displacement kinematics. This was done by assuming
piecewise continuous functions through-the-thickness for in-plane displacement
components. These theories only involve a small number of degrees of freedom
(depending on the order of approximation of the displacement fields) unlike the layerwise
theories discussed above where the number of degrees of freedom is a function of the
number of layers in the laminate. This reduction in the number of degrees of freedom
circumvents the overwhelming difficulties and complexities associated with stress analysis
of multi-layered composite laminates.

The original idea can be found in the earlier works of DiSciuva (1985) where he
assumed a global linear variation superposed on a piecewise linear variation through the
thickness of the in-plane displacement. This theory is termed a first-order zig-zag theory
(FZZT). Although the theory enforces interlaminar shear stress continuity at each
individual interface analytically, the transverse shear stresses are constant through the
thickness because of the low order assumed for the displacement field. In addition, it does

not satisfy the traction conditions at the top and bottom surfaces. This drawback was

overcome by extending the theory to include higher-order terms for the in-plane
displacement field (see for example, Murakami, 1986; DiScuiva, 1987, 1992; Lu and Liu,
1992; Cho and Parmerter, 1993; Liu, 1995; Averill and Yip, 1996a). Refinement to the
first-order zig-zag theory was done by superposing a global cubic function instead of a
linear variation through the thickness onto the zig—zag function for the in-plane
displacement component. This improved the prediction of structural response in thick, and
especially, asymmetrical laminated structures (the form of DiSciuva’s theory (1993) is
strictly valid only for symmetric laminates). These are termed high-order zig-zag theories
(HZZT). Several other investigators have followed up with variations and slight
improvements on the above zig-zag theories (Xavier, Lee and Chew, 1993; Ling-Hui,

1994; and Murakami, 1986).

1.3 Background: Laminated Plate/Shell Finite Element Models

Modeling of thick-section laminates using conventional continuum-type finite
elements normally involves extremely large numbers of these elements to achieve accurate
predictions, and is therefore highly computationally expensive. Structural finite element
models derived using first-order (FSDT) as well as high-order (HSDT) shear deformation
theories (see Reddy, 1987) do not face this problem, but they are often inaccurate because
of two factors. First, they often do not adequately represent the effects of transverse shear
stresses and transverse normal stresses. Second, the commonly assumed linear variation of
inplane displacement components through-the-thickness of the laminate is only valid in an
average sense and is inadequate for accurately predicting the response of these structures,
especially in critical damage analysis. Higher order effects are therefore needed for

improved structural analysis. Bogdanovich (1991) presented a detailed discussion of the

10

various shortcomings of these computational models used in the analysis of thick
laminated composite plates. A detailed discussion of various theories and computational
models for composite laminates can also be found in Reddy and Robbins (1994).

Discrete layerwise theory (Reddy, 1987; Barbero and Reddy, 1990; Robbins et al.,
1991; Lu and Liu, 1992; Robbins and Reddy, 1993; Liu, 1995; Li and Liu, 1996) and
layerwise constant shear theories (see Srinivas, 1973; Epstein and Glockner, 1977; Epstein
and Huttelmaier, 1983; Barbero et al., 1990) were introduced to overcome the limitations
of shear deformation theories (FSDT and HSDT). These theories are based on a
distribution of displacements that is continuous through-the-thickness, while derivatives
with respect to the thickness coordinate are not necessarily continuous. Discontinuous
derivatives will result in discontinuous transverse Strains at layer interfaces. This in turn
will allow the satisfaction of interlaminar transverse stress continuity. These theories are
very accurate, but the “three—dimensional” elements derived from such theories generally
have a large number of degrees-of-freedom associated with them. This is because the
degrees-of-freedom are directly related to the number of physical layers in the laminate. In
fact, the elements of the discrete layerwise theories of Reddy (1987, 1994) can be viewed
as a ‘super-element’ of an assembly of continuum elements.

These discrete layerwise theories can be used in conjunction with other less refined
theories for more efficient local-global modeling of composite structures (see Lee and Liu,
1991; Kong and Cheung, 1995; Yu et al., 1995). Other notable recent “three-dimensional”
elements for laminate analysis include the layerwise constant shear element of Barbero
(1991) and Zinno and Barbero (1994) which have the form of an eight-node brick element

with constant transverse shear stresses and deflection through-the-thickness. This

ll

topology allows the flexibility of mesh refinement through the thickness, has the
versatility of modeling complex Shape structures, e.g. ply drop-off, lap—joints, tapered
beams, etc., and also maintains compatibility with conventional three-dimensional
continuum elements. Other models include those of Babuska et al. (1992) and Actis and
Szabo (1993) which use the hierarchical approach and Bogdanovich (1991) which uses

piecewise-polynomial deficient Spline functions to analyze composite laminates.

1.4 The Present Study

This dissertation will introduce a new technical theory and associated finite element
model for analysis of thick laminated composite and sandwich panels. This theory
essentially combines the discrete layerwise and zig-zag theories mentioned above. The
theory employs the sublaminate concept, in which each computational layer (or
sublaminate) contains several, even many, physical layers. Within each sublaminate, an
accurate approximation of the displacement field is employed that accounts for discrete-
layer effects without increasing the number of degrees-of-freedom as the number of layers
is increased. This is accomplished by satisfying analytically the continuity of transverse
shear stresses at layer interfaces as well as Shear traction conditions at both top and bottom
surfaces of each sublaminate. Because the resulting through-thickness variation of “in-
plane” displacements takes the form of a_ piecewise nearly linear function, the theory is
still essentially a zig-zag theory.

The operative degrees-of-freedom in the theory are located at both the top and bottom
surfaces of each sublaminate to facilitate the satisfaction of continuity conditions between
sublaminates. Averill and Yip (1996b) have employed this scheme for a layerwise beam

with success. This fundamental idea of recasting the proposed displacement field into

12

surface quantities rather than traditional mid-plane quantities without altering the
mechanics of the plate problem can be traced to Flanagan (1994). Flanagan (1994) sub-
divided a general laminate beam into sub-laminates to determine the strain energy release
rates in composites.

The accuracy and efficiency of the present theory is thus adaptable, depending on the
number of sublaminates chosen to model a given laminate. For most global structural
analyses, only one sublaminate is needed through the thickness of the entire laminate to
attain the desired accuracy of overall structural response measures. More sublaminates
may be used to increase the accuracy of overall structural response predictions or to
capture local effects such as interlaminar stresses. Depending upon the ply stacking
sequence and the type of global and/or local response measures that are sought, the
optimal number of sublaminate approximations required for accurate analysis of thick
multilayered composite laminates or sandwich panels will often be greater than one but far
less than the number of the layers in the laminate. The adaptable nature of the current

theory can thus be used to great computational advantage.

1.5 Interpolation Theory: The C-Concepts

We will also introduce the basic concepts needed to understand why some finite
elements fail while others succeed when modeling structural mechanics problems. The
concepts of continuity and completeness as understood in finite element practice are first
introduced. The continuity criterion ensures the compatibility of the displacement fields
across element edges. This can be reasonably achieved by ensuring the continuous
representation of the displacement fields across element edges if the element is CO. C1

elements like those derived from the Kirchhoff-Love theories of plates and Shells, where

 

 

511

hr

the
cor

fon

elem
sing
Shea
state
demc
consi
Pilma
(01151}
C 0112‘ in
that [h
CflSUre
la}: Fa
The

elemfm.

hat the]

13

strains are based on second derivatives of displacement fields, will require continuity of
first derivatives of displacement fields in addition to the displacement fields themselves.
The completeness criterion restricts the form of the assumed displacement field to ensure
the element can produce the strain—free rigid body motion as well as constant strain state
condition. These two criteria are seen to be insufficient to provide a complete and coherent
formal basis for the displacement type finite element method.

The poor behavior of conventionally formulated displacement-type C0 finite
elements, generally called ‘locking’, is often attributed to the high rank and non-
Singularity of the penalty-linked stiffness matrix. Shear locking is due to the inability of
Shear deformable elements to accurately model the curvatures within an element under a
state of zero transverse shearing strain (Averill and Reddy (1992)). Prathap (1994)
demonstrated that the correct rank and non-singularity required emerges directly from the
consistency of the discretized strain field approximations and are not necessarily the
primary cause of the poor behavior of the elements. He introduced the concepts of
consistency and correctness norms and collectively called them the C-Concepts, namely,
continuity, completeness, consistency and correctness. The consistency criterion requires
that the interpolation functions chosen to initiate the discretization process must also
ensure that any special constraints that are anticipated must be allowed for in a consistent
way. Failure to do so causes the model to be ‘locked’ out of the ‘correct’ solution.

The C—Concepts of Prathap (1994) will be utilized in the development of our new finite
elements based on the sub-laminate zig-zag theory. Numerical results will demonstrate

that the new elements developed using this new strategy are very accurate and robust.

14

1.6 6’ versus C‘ Finite Elements

The development of elements based on well-known classical theories like the Euler-
Bemoulli beam theory, the Kirchhoff-Love plate theory and equivalent shell theories will
result in C1 elements. These Cl theories will entail more restriction on the use of
approximation functions in the trial space besides having limitations in the range of
applicability. Cl continuity will require the continuity of slopes in addition to the
continuity of fields across element boundaries. They therefore must rely on cubic or higher
polynomials for the element shape or basis functions. AS such it is more attractive to have
elements that require C0 continuity and therefore require only simple basis functions to
satisfy the continuity of fields across element boundaries. It is with this in mind that our
new elements are specially formulated using a penalty-type formulation to require simple

0 . . .
C continuous basrs functions.

1.7 Interdependent Interpolation

Most conventional displacement type C0 finite elements are prone to “lock” in the thin
regime. This arises from the field inconsistency when equal order interpolations are used
for field variables which appear in different orders of its derivatives in the strain field that
has to be constrained. Thus, if one ensures that the strain field is consistently represented
by a proper a priori choice of unequal order interpolations for the contributing field
variables, there would not be ‘locking’. Tessler and Doug (1981) and Tessler and Hughes
(1983, 1985) derived a family of Timoshenko beam and Mindlin-Reissner plate elements
using this approach.

However, unequal order of interpolations typically gives rise to elements with nodes

having different degrees-of-freedom at each node. In the case of a Timoshenko beam

15

element with three nodes (two end nodes and a mid-node), it will have the deflection
degree-of-freedom, w defined at all three nodes and the rotation degree-of-freedom, 0
defined only at the two end nodes. A simple quadrilateral Mindlin plate element will
require eight nodes, four corners and four mid-side nodes. The deflection degree-of-
freedom, w will again appear in all eight nodes and the rotation degrees-of-freedom, 0x
and By only at the four corner nodes. Tessler and co-workers (1981, 1983, 1985)
overcame this problem by condensing out the undesirable middle or mid-side nodes by
explicitly enforcing the linear Kirchhoff mode in the case of the beam or through the use
of four differential edge constraints, i.e. the vanishing of the linear transverse shear strain,
in the case of a plate. This will result in an explicit deflection/rotation dependence.

In the present formulation, interdependent interpolation will be used both to alleviate
locking and to partially satisfy explicit constraints that are introduced to relax continuity

of the variables.

1.8 Organization of the Dissertation

The dissertation will be organized in such a way that each of the above concepts will
be introduced gradually. In Chapter 2, the basic concepts of the new technical theory and
the development of the finite element model will be introduced for the one-dimensional
beam problem. The beam problem is the simplest case that could be used to demonstrate
the underlying principles involved in the new technical theory.

Chapter 3 will introduce the interdependent interpolation concepts to alleviate shear
locking in Mindlin plate elements. In addition, the field and edge-consistency concepts
will be introduced to identify the inconsistent Strain fields that is causing elements to be

‘locked’ out of the solutions for constrained problems e. g. the vanishing of the transverse

16

normal and shear strains in thin plates.

Chapter 4 extends the one-dimensional laminate beam theory of Chapter 2 to a two-
dimensional laminate theory for plates. In Chapter 5, the independent interpolation
scheme and the field and edge-consistency concepts discussed in Chapter 3 will be
adapted and used to develop a robust plate finite element model using this new technical
theory.

Von Karman non-linearities are added to the element model in Chapter 6. Actual
experimental test results were used to compare the solutions predicted by the model. The
conclusions are in Chapter 7. The chapter summarizes the major differences between the
present theory and the existing discrete layerwise and zig-zag theories as well as their
derived finite element models. It also includes a section that discusses possible area of

research that can be pursued in the future.

 

d:

C01
all
lift

HE“

Chapter 2

LAYERWISE BEAM ELEMENT

As a preliminary study, we will first consider finite element approximations of the
Simplest structure, i.e. the beam element, using the new laminate theory with zig-zag
approximations. This is important as it will clearly illustrate the principle that is used in
deriving the new laminate theory. This chapter will also briefly cover the interdependent

interpolation concept that is used in beams.

2.1 Introduction

A new laminated beam theory is developed that can be considered a hybrid theory,
combining the general layerwise theories with independent layerwise kinematic
approximations and the efficient zig-zag theories in which the layerwise degrees of
freedom are eliminated by enforcing transverse shear stress continuity conditions. In the
new theory, the through-the-thickness approximation of the inplane displacement
components takes the form of a layerwise theory in which each layer is really a
sublaminate containing several, even many, physical layers. Within each sublaminate, a
zig-zag through-the-thickness approximation of the inplane displacement components is
taken in which the layerwise degrees of freedom are eliminated by enforcing continuity of
transverse stresses. Shear traction conditions at the top and bottom of each sublaminate are

also satisfied. The theory includes the effects of transverse normal strain, and is cast in a

17

18

form that is exceptionally well-suited for solution by the finite element method. An
accurate and convenient four-node planar element with beam-type kinematics is

developed and its utility is demonstrated.

2.2 Theory Formulation
In the present theory, the layerwise construction of laminated composite beams is
modeled as M sublaminates, with each sublaminate containing N m layers, where m is the

sublaminate number (see Figure l).

 

 

Z,z
,6 Pym/ """" 27)),

2 $333fi>§3§3§ 1"”
z, ————— Subaunm 1 ----- 4 21g:

 

 

 

Figure 1. Schematic of sublaminate and layer divisions in the laminate theory

The total number of layers in the laminate is then:

M .
Ntotal = 2 Nm (1)
m-

The thickness and material stiffness properties of each layer are arbitrary, and it is
assumed that adjacent layers are perfectly bonded together. In the following, 2 is a local
(sublaminate) thickness coordinate with its origin at the bottom of the sublaminate, while
Z is the global (laminate) thickness coordinate. The coordinate x is measured along the

length of the beam.

 

SL

su'

“he

Sub

l.

h

19

If the width of the beam is small, then 0),, txy, tz U are negligible, and a state of plane

stress can be assumed. The plane stress constitutive relations for the kth layer of the beam

take the form:
- - k k k - . k
0x() ”C(1)C(13)0 l8x()
OZ = C(],;) Cg? O 8z (2)
_sz_ o 0 C§§’_ .Yxa

 

 

 

 

 

 

The infinitesimal strain—displacement relations are:

(k) (’0
8m _ 3'11: 8m _ 9:9
x - x z — z (3)
(k) (k)
(k) a z aux

In each sublaminate, an independent displacement field is assumed in which the
through-the-thickness variation of inplane displacements is described by a cubic
polynomial in the local thickness coordinate with a piecewise linear (or zig-zag) function
superimposed upon it. The transverse deflection is assumed to vary linearly with the
transverse coordinate. The displacement components of the nth layer within the mth

sublaminate, where 1 S n S N m and 1 _<_ m S M , can be written as:

“(m ")(x _k20 zku ("1)”)+ ”Z (Z- 2 KO")
i=1 (4)

ugm’")(x, z) = z)w(m)(x)(l Th Z)+w(m)(x)(h—z— )

where the subscripts b and t refer to the bottom and top surfaces, respectively, of the mth
sublaminate, and hm is the total thickness of the mth sublaminate.

It is possible to eliminate the degrees of freedom 6m) in Eq. (4) by enforcing the

 

”an:

$11sz

“here

20

condition of transverse shear stress continuity at each layer interface. The condition of
shear stress continuity at the jth interface is:

(mi) ("1.1+ 1)

 

 

 

 

 

 

1,, = I... (5)
Making use of Eqs. (2-4) in Eq. (5), it is found that:
dw .
g, - a,(a, +311”), b,i22
(6)
. dwt dwb ,. ,.
+C‘(dx —E )+d,u3
where:
(i) t—l
C55 ..
i=[(i+1)_111+ 20]]
C55 j=l
(I) i-l
. C55 .
bi =[ (i+l)_112Zi+ 2 b1]
C55 j=
(I) i—l (7)
e.=[C55—1-zi+ a]
‘ (i+l) h 1
C55 ”' J:
(i) r—l
. C55 2
i = [ (1+1)—II3z,-+ 2d,]
C55 j=

Additional simplification of Eq. (4) can be achieved through satisfaction of the
transverse shear traction boundary conditions at the top and bottom surfaces of each

sublaminate. For sublaminate m, these conditions are:

 

 

("1.1) (m)

T = T

xz 2:0 b 8
mm.) _ (m) ( )
xz z= h - Tt

"I

where 12'"), rim) are the applied Shear tractions (or interlaminar Shear stresses, as the

 

fu

sul

21

case may be) at the bottom and top surfaces, respectively, of the mth sublaminate.

The displacement field is cast in its final form by introducing the variables:

(In)

 

“hm = “5cm, l) = ’70
2:0 9
(m) (mst) ( )
u, = ux
z=h,,,

 

Eqs. (8, 9) can be solved either analytically or numerically so that the displacement field
for the mth sublaminate can be expressed in terms of the operative degrees of freedom (all

functions of It only):

(m) (m) e(m) ("0 (m) Wm) 9(m)
t ’ t

(m)
ub ,Wb , b ,Tb ,ut , T

, , (10)
where again the subscripts b and t refer to the bottom and top surfaces, respectively, of the

sublaminate, and

(M) (m)
(M) __ dwb (In) _ dw,

9b -— ’t_Fx_

dx (11)

The displacement field now takes the form:

 

the

the

9&0

COU

pies

Cap}

22

uim’") = u(m)[l +p(lm)z 2+pf,_m)z3 + ”2 (z— 2,1)a "0]

i=1

n-l
(m) (m) 2 (m) 3 ( )
+ u, [—plz Z—pz Z -2 (Z‘Zi)aim ]

i=1

+0(m)[—z+p(3m)z+ 2+ pftm)z3 + "2 (z— z )bim)]

i=1

+9§m)[p<5m m) z2+ p(m)z3 + 2(z— z)c(-m )] (12)

i=1

n—l
2 3
+ tgm)[p£,m )z + pgm)z + pamz + 2 (z - Zi)dl(-m)]

i=1

.‘A

n-l
2
+r§m)[p 'g)z + p111); + 2(2— z)e(.m )]

i=1

("1. n) (In) 2 (m) 2
z hm t hm

In Eq. (12), pk (k = l to 11) and al., bi, ci, di, ei are functions of the layer shear stiffnesses
and thicknesses, and can thus be calculated a priori. Their functional forms are given in
Appendix A.

In the sublaminate displacement field described above, the functions of z that multiply
the degrees of freedom of (10) can be viewed as shape functions that describe the through-
the-thickness variation of these measures within the sublaminate. Thus, a layerwise
laminate theory can be developed in the form introduced by Reddy (1987), where now
each “layer” in the layerwise theory may contain several, even many, physical layers. Of
course, this concept was always possible to envision with the layerwise theories, but the
present formulation provides an extremely accurate and efficient approximation within

each sublaminate. The sublaminate approximations are connected by imposing continuity

dlSl

23

of displacements and transverse shear stresses at the interface between each sublaminate

region. The theory thus has the following properties:

(i)

(ii)

(iii)

(M

(V)

continuity of transverse shear stresses through the entire thickness of the
laminate is satisfied;

shear traction boundary conditions on the top and bottom surfaces of the
laminate are satisfied exactly;

a piecewise (layerwise) continuous through-the-thickness variation of the
inplane displacements is allowed, yet the number of degrees of freedom in each
sublaminate is independent of the number of layers in that sublaminate;

a linear through-the-thickness variation of the transverse deflection in each
sublaminate explicitly accounts for transverse normal strains and stresses;

only engineering-type degrees of freedom are used -- displacements and
rotations, plus shear traction terms that are always known on the surfaces and

unknown between sublaminates.

The accuracy and efficiency of the present theory is thus adaptable, depending on the

number of sublaminates chosen to model a given laminate. If only one sublaminate is used

through the thickness of the entire laminate, then the theory falls into the class of zig-zag

theories (see DiSciuva, 1985, 1986, 1987, 1993; Cho and Parmerter, 1993; Xavier, Lee

and Chew, 1993; Ling-Hui, 1994; Murakami, 1986; Averill, 1994; Averill and Yip,

1996a). If the number of sublaminates is equal to the number of layers in the laminate,

then the theory can be categorized with the more general layerwise theory of Reddy

(1987). In this case, a cubic layerwise through-the-thickness variation of the inplane

displacement components would yield extremely high accuracy (as in Lu and Liu, 1992).

 

fre

file.

“her

24

The optimal number of sublaminate approximations required for accurate analysis of thick
multilayered composite laminates or sandwich panels will often be greater than one but far
less than the number of layers in the laminate. The adaptable nature of the current theory

can thus be used to great computational advantage.

2.3 Finite Element Model

Because first derivatives of wb and w, appear in the displacement field of Eq. (12),
second derivatives of these variables are present in the strain energy functional, requiring
them to be C 1 continuous. In the current finite element model, this continuity requirement
is weakened by treating 0b and 0, as independent degrees of freedom and subsequently

imposing the constraints

dwb
8b = 91"}; = 0
(l3)
dw,
gt = et-E; = 0

via a combination of an interdependent interpolation scheme and the penalty method
(Lynn and Arya, 1974; Zienkiewicz, Owen and Lee, 1974; Reddy, 1980). All degrees of
freedom in the displacement field are then required to be only C0 continuous.

The principle of minimum total potential energy is employed to develop the finite

element model. For a constrained system, we have:

anp = so + 5v + Sg’qgfijdx]
nag 1.3)...) = 0

where U is the internal strain energy, V is the potential energy of external forces, and 71,, y,

(14)

25

are penalty parameters that enforce constraints in Eq. (13).

Substituting Eqs. (2, 3, 12, 13) into Eq. ( 14), the governing equations, boundary
conditions, and displacement-based finite element model can be developed. While the
finite element geometry could take the form of a two-node beam element with eight
degrees of freedom per node, it is advantageous to use the topology of a four-node, planar
element, as shown in Figure 2 (bottom). This topology allows the laminate thickness to be
conveniently subdivided and modeled by multiple finite elements (representing
sublaminates). It is thus possible to increase the accuracy of the finite element model, as
needed, to capture through-the-thickness gradients and transverse (interlaminar) stresses.
It is also possible to simulate delaminations using the redundant node concept.

The element formulation takes advantage of an interdependent interpolation concept
introduced by Tessler and Doug (1981) and recently used by other element developers
(Averill, 1994; Averill and Yip, 1996a; Friedman and Kosmatka, 1993). Except for the
element interpolation scheme discussed below, all aspects of the finite element
formulation follow the well-known standard procedures (see, for example, Reddy, 1993),
and the details are omitted.

In the constrained element formulation, the transverse deflection degrees of freedom
wb, w, are initially approximated using quadratic Lagrange interpolation functions, while
all other degrees of freedom are expanded using linear Lagrange interpolation functions.
Such an approximation scheme results in a six-node element in which each of the two
midside nodes contain a single degree of freedom associated with the transverse deflection
(see Figure 2, top). These midside nodes are eliminated by considering a modified form of

the constraint conditions in Eq. (13):

 

PM
the
the
C0n.
aim
the 5

Wk

26

d dwb) _
2319')? ‘ 0
d dw, _
210*? l " 0

By substituting the element approximations into Eq. (15), the midside degrees of

(15)

freedom wb3, w,3 may be determined in terms of the other nodal degrees of freedom,

resulting in a new interdependent element interpolation scheme:

2 2
wbs 2 wbij+ Z ijlT/j

i=1 i=1
2 2 (16)
(42,5 2 wtij+ 2 9,ij
i=1 i=1

where P1, P2 are the linear Lagrange interpolation functions,

1711 = (lg-<1 -&2) N2 = Ego—£2) (17)

are is the length of element e, and 7; is the natural axial coordinate in the element
(-1 5 § 5 1) . If the constraints in Eq. (13) are viewed as being composed of a constant
part and a linear (in x) part, then the above interdependent interpolation scheme satisfies
the linear part of the constraints identically. This makes the penalty function technique in
the present formulation more robust, because the penalty parameter enforces only the
constant part of the constraints. This approach effectively increases the order of the
element without introducing any additional nodes or degrees of freedom and eliminates

the shear locking problem so an exact order of integration may be used. A consistent force

vector is also obtained.

 

2.4 I)

SUbje

27

“11’ WI] “:2, Wt2
W
0:1, 1:rl '3 922’ 7:2
O
C
“bl, wbl W123 “122’ sz
9“. Tb] 9w sz
utl’ wtl “,2, "Q2
911’ Ill 9’2, 1’2

 

E::::i
p- _ __..L.:]

“bl, Wbi “b2, sz

 

91w Tb] 9w sz

Figure 2. Nodal topology of unconstrained element (top) and constrained
element (bottom)
2.4 Numerical Studies
Numerical results are presented for bending of a simply-supported laminated beam
subjected to a sinusoidally varying transverse load of magnitude one (see Figure 3).
Comparisons are made between predictions of models based on first-order shear
deformation theory (FSDT) (see Yang, Norris and Stavsky, 1966), the present layerwise

zig-zag theory (LZZT), and an exact elasticity solution of Pagano (1969).

 

Figure 3. Schematic of loading and support conditions

The material properties used in the analyses are as listed in Table 1, where E0 is a

28

reference value that is chosen here to be 1.0x106. Laminates composed of combinations
of these materials have been used previously (Toledano and Murakami, 1987), and provide
a good test of the models because they are distinctly different in their stiffness
characteristics. Material 1 is considered compliant in tension/compression and compliant
in shear. Material 2 is stiff in tension/compression and stiff in shear. Material 3 is stiff in

tension/compression and compliant in shear.

Table 1. Material properties used in the numerical analyses

 

 

 

 

 

E/Ea G/Ea
Material 1 1.0 0.2
Material 2 32.5 8.2
Material 3 25.0 0.5

 

 

 

 

 

The laminate configuration used in the present numerical study is described in Table 2.
It is an unsymmetrical five-layer laminate with seemingly random layer thicknesses and
material properties that vary drastically from layer to layer. This example provides a very
rigorous test of a laminate theory’s ability to model complex laminates subjected to
sharply varying loads. While results are presented here for only one laminate, it should be
noted that the current model has been tested on many problems with a wide variety of
lamination sequences and geometries, including sandwich beams. In each case,
comparisons between predictions of the present model and the exact elasticity solution

were at least as good as the comparisons presented here.

 

p11

CIT

29

Table 2. Description of laminate sequence

 

Laminate Layer Number Relative Thickness Material Number
(Volume Fraction)

 

 

0.100
0.250
0.150
0.200
0.300

5 layer random

MAWN—
DJ—‘UJN—

 

 

 

 

 

 

For a thick laminate with span-to-thickness ratio of four, the normalized maximum
deflection and the normalized maximum inplane normal stress predicted using a single
sublaminate are plotted in Figure 4 versus the number of elements used in the mesh. The
results are normalized by the exact solution of Pagano (1969) which has been modified for
beam analysis by specifying the appropriate constitutive equations for beams, Eq. (2),
instead of those for cylindrical bending. When only four elements are used, the error in
both cases is less than four percent. However, it appears that a nearly converged solution is
attained using approximately ten elements. Thus, all subsequent finite element results are
obtained using a uniform mesh of ten four-node elements (with full integration) for LZZT
and ten two-node elements (with interdependent interpolation of Tessler and Doug, 1981)
for FSDT. Note in Figure 4 that there is approximately five percent error in the predicted
stress when ten elements are used. This error exists because the current model does not
predict the exact result for this laminate when the aspect ratio, Uh, equals four. Thus, the
error is primarily due to assumptions in the theory as opposed to finite element
approximation errors. The accuracy of these predictions increases rapidly as the aspect
ratio of the beam increases.

In Figure 5, the predicted normalized center deflection versus the span-to—thickness

(hi

ant

Spa

“564

the.

30

ratio of the beam is shown. The deflection predictions of FSDT degrade Sharply for aspect
ratios less than about 50, largely because the material properties of adjacent layers vary
drastically. It should be pointed out that a shear correction factor of 5/6 (valid only for
isotropic beams) was used to obtain the FSDT results, which accounts for some of the
error. Predictions obtained using LZZT with one element (one sublaminate) through the
thickness are accurate for all aspect ratios greater than about two, and no shear correction
factor is needed. Due to the interpolation schemes used, there is no locking in either model
as the beam thickness (and thus the thickness of each element) decreases.

In Figures 6 and 7, the through-thickness distributions of inplane displacement and
inplane normal stress, respectively, as predicted by FSDT, LZZT, and Elasticity, are
plotted for a thick laminate having a span-to-thickness ratio of four. It can be seen that
LZZT, with only one element through the thickness, does an excellent job of predicting
the through-thickness distributions of both inplane displacements and stresses, even for
this very thick laminate. FSDT is not able to capture these variations. These two plots
highlight the importance of including the zig-zag through-thickness variation of inplane
displacements (and hence, inplane strains) for laminates in which the material properties
of adjacent layers vary considerably.

In Figures 8 and 9, the utility of being able to use multiple LZZT elements through the
thickness of a laminate is demonstrated for the prediction of transverse Shearing stresses
and the through—thickness distribution of the transverse deflection in a thick laminate with
span-to-thickness ratio of four. When five LZZT elements (one per physical layer) are
used through the thickness, the predictions of transverse shearing stress (calculated using

the constitutive equations, Eq. (2)) are indistinguishable from the exact solution, and the

 

fo

31

variation of transverse deflection is very good. Because increasing the number of
sublaminates in the model also increases the finite element discretization through-the-
thickness of the laminate, a greater number of degrees of freedom is required for such
refined analyses. It has been shown, however, that excellent results can usually be obtained
using only one element through the thickness of a laminate. Further, because inplane
normal stresses are predicted very accurately, transverse stresses can be obtained by the
now standard approach of integrating the two-dimensional equations of equilibrium

through the thickness (as in Averill and Yip, 1996a).

2.5 Summary

The new theory and finite element model described show excellent promise for
accurate, efficient, and convenient modeling of laminated and sandwich beams. The level
of accuracy in the prediction of through-the-thickness variations of displacements, strains,
and stresses can be varied by choosing the number of sublaminates used in the model. In
most cases, only one sublaminate, or one finite element through-the-thickness, is needed
to achieve the desired accuracy of both global deflections and local inplane stresses, even

for very thick laminates.

32

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.05 1 (0”)nu ............ A ..................... A:
r -A -------- 1
I ’ (0")cxact \. ‘
0.95 : i
0.9 1 w... 1
1 ”exact 1
0.85 f
0.8 r f
0.75 3 1
‘4 L 4A 4“ A l_ L k # L #4 A g 41
2 4 6 8 10
Number of elements in the mesh
Figure 4. Predicted normalized maximum deflection and normalized
maximum inplane normal stress versus number of elements in
the mesh
1 _ F l 4; 5 $'—é.>fl " .
0.9 h 1
0.8 1
w I
0.7 f
wexact » .
0.6 f 1
0.5 f T
L I
0.4 f j
1 J
10. 100. 1000. 10000.

L/h

Figure 5. Normalized center deflection versus span-to-thickness ratio of a

five-layer simply-supported beam subjected to a sinusoidal load

33

 

 

0.5-'f:1f"i MT
1 ' ‘ Elasticity

0'4 """""" FSDT l

0.3 . """ LZZT: 1 sublaminate

0.2

0.1

 

 

 

 

 

11
4s

Fifi

 

r

-0.1
-0.2
-0.3 -
-0.4
.05 --

 

 

r

 

 

 

 

 

 

1.051111014111051 1.5
ux10

Figure 6. Axial displacement versus normalized thickness coordinate at the
end of a simply-supported beam subjected to a sinusoidal load

 

P
U!
.1

 

 

Elasticity
0.4 ..... . ..... FSDT
0.3L —"‘" LZZT'Isublaminate

r

 

 

 

 

 

0.2
0.1

a-I h
II
a

 

FIN

-0.1
-0.2
-0.3
-0.4
-0.5 h.

V

 

Y

r

 

 

 

 

 

.10.

 

Figure 7. Axial stress versus normalized thickness coordinate at the midspan
of a simply-supported beam subjected to a sinusoidal load

0.5
0.4
0.3
0.2

0.1 '

a-m

-0.1
-0.2

.03-

-0.4
-0.5

 

 

 

 

 

' {x \ Elasticity “
l- \ ' \ \ ........... FSDT
; ,\ ---'- Lzznrsublamiuale
fl ‘ e ———— LZZTZ'3sublaminates
I — """ — LZZT: 5 sublaminates

 

 

 

 

Y

 

T

 

3‘1 D
II
&

 

 

 

 

 

Figure 8. Transverse shear stress versus normalized thickness coordinate at

0.5
0.4

0.3:

0.2

0.1 -

#IN

-0.l -

-0.2
-0.3
-0.4

-0.5 - .
6.1

Figure 9.

x=U10 of a simply-supported beam subjected to a sinusoidal load

 

V

V

*T a? fii ' v v v v v V7

 

 

 

 

Elasticity

 

 

u" —-—-- Lzzrz-rsublamiuaie
— - _ — LZZT: 3 sublaminates
_ “““ — LZZT: 5 sublaminates

 

 

 

 

 

 

A A A l A L A A l A

 

6.3 _6 6.4 6.5 6.6
wxlO
'II'ansverse deflection versus normalized thickness coordinate at
midspan of a simply-supported beam subjected to a sinusoidal

load

 

dis

orc

C0115

Chapter 3

MINDLIN PLATE ELEMENT

In the following discussion, we shall study the small deformation response of plates
using Mindlin plate theory, a popular theory in the application of finite element methods
for the bending of plate structures. This is a prologue to the more complex laminate theory
to be discussed later. This chapter will illustrate the field and edge consistency concepts
that are crucial in the development of robust quadrilateral plate elements without

introducing the complex nature of the displacement fields of laminate theories.

3.1 Introduction

The use of Mindlin plate bending theory for the development of plate finite elements is
popular, as only a C0 continuity requirement is imposed on the field variables (transverse
displacement, w and rotations, 6x and By). This allows the development of simple, low-
order elements containing three or four nodes with only three bending degrees of freedom
per node. Unfortunately, early studies showed that these simple low-order elements are
‘flexurally challenged’ (i.e., they ‘lock’) and exhibit violent stress oscillations in the thin
regime. Prathap (1994), in his review paper, provided explanations to the locking
behaviors exhibited by many such low order quadrilateral elements. Fried (1974) was the
first to provide the necessary insights as to why the displacement approach failed in such a

constrained problem, namely the vanishing of the transverse shear strains in the thin limit,

35

 

T:

di;
no
To
C01
CO!
Par

fOu

36

and he suggested the use of reduced/selective integration as a possible cure. Hughes, et al.
(1977) employed this reduced/selective integration technique on the shear strain energy to
produce the first ‘simple and efficient’ plate bending element that was free of locking.
However, this plate element was found to have two zero energy modes that caused the
element performance to deteriorate if the element was distorted and used in the general
quadrilateral form.

Recent studies by Tessler and Dong (1981), Tessler and Hughes (1983), Hughes and
Tezduyar (1981) and Zienkiewicz and Xu (1993) tried to alleviate locking by attempting
to satisfy the inconsistent kinematically derived strain field using an unequal order
interpolation of field variables (e.g. quadratic for w, linear for 6,r and By ). The use of
unequal order interpolation requires that elements have different degrees-of-freedom at
different nodes. In the present case, the quadrilateral plate element would have eight
nodes, with w defined at all eight nodes, 6,, and By defined only at the four corner nodes.
To remove the four mid-side nodes, Tessler and Hughes (1983) imposed four edge
constraints by requiring the tangential shear strains at the elemental boundaries to be
constant along each of the four edges. This is, in fact, equivalent to satisfying the linear
part of the tangential Kirchhoff shear strain constraint at each edge of the element. These
four constraints are used to condense out the four mid-side nodes, resulting in an
interdependency among the w field and the 0x and 9y fields, as opposed to the
conventional models where the three fields have independent interpolation schemes.
Zienkiewicz and Xu (1993) showed that their plate elements derived using mixed
formulations do not lock. However, the plate element of Tessler and Hughes (1983) using

the displacement approach needs to be supplemented by a residual energy balancing

37

technique using relaxation parameters to prevent locking in severe cases (Tessler, 1986).
These relaxation parameters are obtained by tuning the element response for specific test
cases. This energy balancing technique is now widely used. However, it is artificial in that
it introduces an error that compensates for another error without removing the original
error.

Recently, in addition to the more familiar completeness and continuity requirements,
Prathap and co-workers (1983, 1988, 1992, 1994) have performed extensive research on
the importance of the field and edge-consistency requirements and variational correctness
in the formulation to produce error-free, robust displacement-type finite elements. Prathap
(1994) collectively called them the C -concepts, namely, continuity, completeness,
consistency and (variational) correctness. He and his co-workers have developed robust
finite elements based on the C-concepts, including four-node (Prathap and Somashekar,
1988) and nine-node (Naganarayana et al., 1992) quadrilateral plate bending elements.

The assumed strain approach in which independent consistent shear strain fields, not
derived directly from the interpolations for the displacements (Hughes and Tezduyar, 1981
and MacNeal, 1982), is one way to satisfy the field consistency requirement. The
variational basis of the assumed strain methods was discussed by Simo and Hughes
(1986). Substitute shear strains have also been used by Dvorkin and Bathe (1984); and
Prathap and Somashekar (1988) to develop four-node elements as well as Huang and
Hinton (1984); Jang and Pinsky (1987) in their development of eight- and nine-node
Mindlin plate elements. Besides ensuring field consistency in their elements, Hughes and
Tezduyar (1981) were the first to use tangential shear strain interpolations on each of the

element edges to ensure edge-consistency in a general quadrilateral so that elements

 

Tbs

“he:

and:

38

assembled in an arbitrary grid would not lock. Other recent robust plate bending elements
include those by Crisfield (1983) (based on the classical plate theory), Bathe and Dvorkin
(1986), and Hinton and Huang (1986), wherein mixed interpolations for the shear strains
and tensorial transformation were used to achieve field and edge consistent elements.

In the current paper, the main focus is to improve the performance of the four-node
element MIN4 of Tessler and Hughes (1985) by identifying and eliminating the
inconsistent terms in the shear strain fields using an approach similar to that of Prathap
and Somashekar (1988). This will remove the need for ‘tunable’ relaxation parameters to
prevent locking in the thin regime. After performing such modifications, the consistent
shear strain fields of the new MIN4-CC element are, surprisingly, shown to be identical to
those developed by Prathap and Somashekar (1988) in the element QUAD4-CC. The
primary differences between the elements then appear to be the representation of
consistent load vectors and the post-processing of transverse deflection, which is quadratic
in the present element. The behavior of these and other elements are thoroughly illustrated

by numerical examples.

3.2 Formulation
The Mindlin theory for a linearly elastic isotropic plate is briefly summarized below.
The displacement components are:
ux(x,y,2) = -29,(x,y)

uy(x. y, z) = —29y(x. y) (18)

“Ax, y. z') = W(x, y)
where w(x, y) and rotations, 6x(x, y) and 9y(x, y) are defined at the plate’s midsurface

and the sign convention is as shown in Figure 10. All quantities are referred to a fixed

Sll

will

fall.

39

system of rectangular, Cartesian coordinates. A general point in this system is denoted by
either (x, y,z) or (x., x2, x3) , whichever is more convenient. Throughout, Latin and Greek
indices range from 1 to 3 and 1 to 2, respectively, unless otherwise stated.

From Eq. (18), the strain components will be:

e = —z9 E = -Ze 8.. =
xx x,x yy y.y «A. (19)
sz = (wry—6y) 7x2 = (wax-ex) ny = ‘z(0x,y+0y,x)

where a coma in the indices denotes partial differentiation.
Assuming cu is negligible, and with the generalized plane stress assumptions, the

stresses are given by:

E

 

 

on = _ 2[Z(9x,x+”9y,y)]
l-u
— E 9 9
on " _ 2[Z(U x,x+ ya” (20)
1—1)
on = —G[z(0x,y+0y,x)]
on = G(w,x-0x), 0y: = G(w,y-9y)

where E and G are the elastic and shear modulus, respectively, and t) is the Poisson’s
ratio.

Moment resultants of the stresses 6 and (5xy are defined as:

xx’ ny

Mxx = Izcxxdz = -D(9x’x+uey’y)
My), = JzO'yde = -D(09x’x+9y,y) (21)

l-u
M”, = Izoxydz = —DT(9x,y+9y,x)

where M n and M yy are bending moments, M is the twisting moment, and D is the

xy

bending rigidity defined by

.J’

 

40

3
12(1 — uz)
where h is the thickness of the plate.
Finally, the transverse shear force resultants, Q x and Qy are defined as:
Q, = [0,de = kZGh(w,x - ex)
(23)

Q), = joyzdz = kZGh(w,y-—9y)

The factor k2 in Eq. (23) is a shear correction factor that attempts to take into account the
actual non—uniform distribution of shear stress in the thickness direction. A value of

k2 = 1:2/ 12 is used for all computations here.

 

 

 

 

 

 

 

 

 

Figure 10. Sign convention for rotations and stress resultants

41

The equations of motion, the natural boundary conditions and the displacement—based

finite element are obtained using the principle of virtual work:
SW = BWW+ SW“, = O (24)

where Win, and W are the internal and external work done, respectively. The virtual

ext

strain energy of the plate in indicial notation is given by:
SWW = IoijfieijdV i, j = 1, 2, 3 (25)

where the integration is carried out over the volume, V, and summation of repeated indices

is implied. The virtual work due to external forces is:

SW“, = — J(q5w+ma89a)d£2+ j (MBanB)59adI'— j Qanaawdr
n r, r, (26)
(M3 = 1,2

where q, the distributed transverse loading in the z-direction, and distributed moments,

ma are given by:

q = Ipfzdz‘i'lczzlflf/ZZ
_ fwd +'.'(o I H: I
ml '— IZP x Z 2 xZ h/2 XZ -h/2) (27)

m2 = szfde + g(oyz|h/2 + OyzLh/z)

p is the density of the plate, and If = fig. is the body force per unit mass. M 0,3 and Qa

are the prescribed boundary moments and shear forces, respectively, 9 and F are the area

and boundary domains of the plate, respectively. Using the decomposition rule,
(M gunmen can be written as:

(1145034989,I = Mnn59n+Mnsaes (28)

where 56,I = Seana, 595 = 59—59,}; and 59 = seaga.

 

dl:

lh:

42

Now, let the moduli Dams be defined by:

01111: D, 01122 = ”D, 02211 = “D, D2222 = D

1—1) (29)
01212 = 01221 = D2112 = 02121 = TD

and all other terms are zero. Then using Eqs. (18-23) in Eq. (25), the virtual work equation

can be written as:

j 050,759,, 559.1, Bdo — j kGh(w,a — cameada
Q n

+ JkGh(w,a—9a)8w,adn = L1(59)+L2(5W) (30)
n

a,B,y,5 = 1,2

where the boundary and load conditions for the plate are written as two linear functionals:

L,(6Q) = —Jma50a dQ- j (Mnn69n+Mnsaes) dI‘
Q

’2 (31)

L2(5w) = — Iqow dQ— answ d1" (n, s: no sum)
(2 r,

F refers to the complete boundary. If F1 is the portion of the boundary where
displacements are specified and F2 is the portion where surface tractions are specified,
then I‘lul"2 = I‘ and I‘lnl‘2 = Q.

The equilibrium equations are obtained as:

MBovB" Q0: = ”ma

(32)
—Qa,a = q

 

de

“01

MOT.

43

and the following boundary conditions can be identified:

 

 

Simply
Clamped 3939:3314 supgloztwed-Zl Free Symmetric
symmetric
W = 0 W = 0 w = O Q = O Q = 0
93:0 MM=0 95:0 Mns=0 Mm=0
n: Mun: nn=0 Mun: 9n=0

 

Note that simply supported-1 (SS-1) is the appropriate simply-supported condition for
Mindlin-type plate elements to deal with general boundary types such as curved-
boundaries. Simply-supported-2 (SS-2) is based on thin-plate theory, and in cases where
there is no danger of over-constraining such as in simply-supported, thin rectangular

plates, SS-2 can be used to further eliminate degrees-of-freedom.

3.3 Finite Element Model

The finite element model utilizes the interdependent interpolation concept of Tessler
and Hughes (1983, 1985). The transverse deflection, w is initially approximated using
quadratic serendipity interpolation functions, N 1., while all other degrees-of-freedom are
approximated using linear Lagrange interpolation functions, Pi. This, however, will result
in mid-side degrees-of-freedom associated with w (a consequence of the higher order
interpolation functions used). The finite element therefore is not of a convenient form as
depicted by Figure 11a. The mid-side nodes are condensed out to achieve a uniform four-
node quadrilateral element. This is done by explicitly enforcing the linear part of the

Kirchhoff constraints,
YSZ’S = (W’s—95)”- = O (33)

along each side. A brief summary is outlined below (refer to Appendix B for details).

 

 

 

 

 

0 9
c k
(a) unconstrained element (b) constrained element
0 {w, 9x, 9),}
O { W}

Figure 11. Topology of the finite element model

Independent bi-linear in—plane displacements and rotations and a Serendipity (eight-

node) transverse deflection are initially assumed, i.e.,

8
w = 2N1”;
i=1
4 4
e,r = 213,9“. ey = 210,9”. (34)
i=1 i=1
4 4

where Pi(§, n) and N i(§, n) are respectively bi-linear and Serendipity shape functions,

and 9 By, and W, denote nodal degrees-of-freedom (refer to Figure 12).

0X
ll

xi’

9:

l

mu

 

 

Vat
a:

 

 

 

 

 

 

 

Figure 12. Quadrilateral element coordinate description

 

8V

“'1

3.4

DUI

45

The four mid-side w degrees-of-freedom are condensed out by the use of four differential

edge constraints:

(w._.,.-e,,,.)|§’n=il = 0 (se 10,111) (35)

where 1,, is the edge length and subscript k represents the corresponding edge as depicted
in Figure 12. Each of these constraints is explicitly solved for wk + 4 and back-substituted
into Eq. (34) (refer to Appendix B for details). This produces a four-node, quadratic,
coupled transverse deflection:
4
w = 2 [Piwi + Naiflgi + Nnieni] (36)
i: 1
where N§i and Nut” are given by:

1 I
N§1='N5’ N§2=-NE,1’ N§3=-ZN7’ Ng4=—N§3

4
l 1 (37)
N11] =ZN8, Nfl2=ZN6’ Nn3=—Nn2’ Nn4=_an
and
1 1
N5=§(1—§2)(1—T]), N6: i(1_112)(1+§)
l 1 (38)
N7=§(l_§2)(l+n), N8: i(1_nz)(1_§)
91;: and 9111' in 59- (36) are rotations corresponding to the natural (local) coordinate

system (see Figure 8.1). This approach effectively increases the order of the element

without introducing any additional nodes or degrees-of-freedom.

3.4 Consistent Shear Strain Fields
The interdependent interpolation scheme alleviates the shear locking problem but does

not eliminate it totally. This stems from the fact that constrained strain fields, i.e. the

46

transverse shear strains, are still not field consistent. In a general quadrilateral, it is
difficult to see the consistent form of the shear strains clearly because of the non-uniform
mapping from a natural (local) coordinate system into a Cartesian (global) system. In
order to see this clearly, the transverse shear strains need to be converted to the natural
coordinate system. Prathap and Somashekar (1988) termed the transverse shear strains in
the natural coordinate system as covariant shear strains. Examining the covariant shear
strain component, Ygz’ using Eqs. (34, 37, 38):
752 = (ng- 9g)
= Ppgwi + (Néi’é - 13199:: + Nni'éeni
= “—BIQFGHI +6T1249113 +9114} (39)

(l-Tl)
4

0+“)

+ {—egl—Bgz—Wl +W2}+ 4 {—9§3—9§4+w3—w4}

 

 

The coefficients of the quadratic term (1 —n2) are clearly inconsistent since they
contain only 6711' rotations alone and will therefore lead to spurious constraints and shear
locking when thin plates are modeled. Therefore, eliminating the inconsistent terms and
simplifying, a consistent shear strain, 752 is obtained. The consistent shear strain is

mathematically represented as:
7g: = Ppgwi + (Navg — P096 (40)
This shear strain is equivalent to interpolating the w degree-of-freedom using:
W = Piwi+N§i6§i (41)

instead of Eq. (36) with the interpolation strategy for the rest of the degrees-of-freedom
remaining unchanged.

In a similar manner, the consistent shear strain, '7,” can be derived from the

 

dc
SE
5q

511

C01

for

9011.
Intel
Sim

Bode

47

inconsistent shear strain, 7112 and is given by:

7112 = Pi’nwi+ (Nni’n-Pi)eni (42)
or with the w degree-of-freedom interpolated using:
w = Piwi + Nniflni (43)

This consistent field approach using the new assumed shear strain fields is similar to the
selective/reduced integration concept except that it only removes the relevant inconsistent
terms and therefore will not produce spurious zero energy modes. Prathap (1994)
demonstrated through the use of the Hu-Washizu theorem that the kinematically derived
strain field when replaced with a consistent constrained strain field is equivalent to a least-
squares smoothing operation. Simo and Hughes (1986) also showed that the assumed

strain method is variationally correct

3.5 Edge Consistency

Even with the choice of field-consistent transverse shear strain fields, edge-
consistency requirements must still be ensured to prevent locking in an element, especially
for general quadrilateral cases. It is essential that the tangential shear strain, Y” on an
edge be consistently matched or else there is a spurious constraint generated on the edge.
This can be achieved if the transformation from the natural coordinate to the Cartesian
coordinate systems are done in such a way that the edge constraints are preserved
consistently. Normal jacobian transformations of two adjoining elements at their own
integration points will redefine this consistency causing a mismatch of tangential shear
strain at the common edge. Prathap and Somashekar (1988) suggested the use of element

nodal coordinate transformations for the interpolation of the covariant—based shear strain

 

St:

N(

C0]

48

fields to preserve the consistency definitions of tangential shear strain, 7“ along a
common edge between two elements.

The nodal covariant degrees-of-freedom are transformed to the Cartesian counterparts

{7&2} = all 012 {VIZ} {VIZ} = b“ b12{Y§Z} (44)
ynz “21022 sz 7y: b21b22 Y112

where [a] = [b]_I . Following Eq. (19), the Cartesian coordinate based shear strain, 7x2

using:

can now be expanded as:

Yx (g’n) = W’ —9
Z x x (45)
= (wig... + w,.,n.,> - 9,

Introducing the covariant rotations, 9: and on into Eq. (45) and using the transformation
definition of Eq. (44), the Cartesian based shear strain, 7“ can be expressed in terms of

the two covariant based shear strains, 7:: and 7112

‘sz(§’ Tl) = (W,§ — 6§)§9x + (Win — en)n9x

= 7&3, + 7,211., (46)

= 75,sz ‘" Ynzbrz

and now Eq. (46) can be modified using the definitions of the consistent transverse shear

strain fields, '75: and 7112 from Eqs. (41 , 42) to give a consistent shear strain, E:

7,4611) = 7§zb11+7mb12
= (Pi,§wi + (Nip: -' Pi)9§i)b11+(P"nWi+(N

l

(47)
mm " Pi)9ni)b12

Nodal transformation must be canied out for the nodal covariant rotations, 6:1. and 9111' to
convert them back to the Cartesian coordinate definitions of the rotations, 6n- and Gyi.

Thus, the shear strain, '7“ in terms of On. and fly,- can be written as:

49

4
7.45.11) = 2‘, {9...1(a,.),(N§,.§— Fab“ +(a21),(N,,,.,, —P,-)b121

i=1 (48)
+ eyi[(al2)i(N§i’§ " Pi)b11+(a22)i(Nni’n _ Pi)bl2]

where by. are evaluated at the integration points and a l. j are computed at the nodal points.

Similarly, the shear strain definition for 7y: will be:

4
7,413.11) = 2 {e,.-[(a,,),(N§,-.§ - Fab21 + (a21),<N,,,-.,, - 10912221

i: 1 (49)
+ 9yi[(a12)i(N§i’§ " Pi)b21 + (022)i(N11i’11 — Pi)b22]

+ Wilpvgbzr + Pi,nb22]}
The consistent definitions of the Cartesian shear strain fields, in and 7n derived in this
manner will eliminate shear locking problems due to spurious constraints and as such an
exact order of integration may be used to evaluate the shear strain energy.

Interestingly, these two shear strain fields, 7x2 and '7” are identical to those of the
four-node element QUAD4-CC of Prathap and Somashekar (1988). Nonetheless, this new
element is different in two ways. One difference between the two element types is that
deflection obtained using this new interpolation is quadratic instead of bi-linear in
QUAD4-CC. A different force vector is also obtained by using this approach. This force
differs from the four-node element of Prathap and Somashekar (1988) by virtue of the
interdependent interpolation scheme used for the deflection. As such, the force vector for a
distributed face or edge load will involve the rotation terms because of the

interdependency of the rotations and transverse deflection (see Eq. 36).

50

3.6 Numerical Studies

The numerical experiments presented here are standard tests used by several authors to
verify the performance of new elements (see, e.g., MacNeal and Harder, 1985; Prathap
and Somashekar, 1988). The present element derived in this paper will be referred to as
the MIN4-CC (the CC is to denote that it is field- as well as edge-consistent). The results
of this element will be compared with the results using the QUAD4 element of the MSC/
NASTRAN (MacNeal, 1985), MIN4 element of Tessler and Hughes (1985), QUAD4-CC
element (field and edge consistent) of Prathap and Somashekar (1988) and the AQR8
element (assumed-stress quadrilateral) of Aminpour (1990).

Spectral analysis of the elemental stiffness matrix revealed three zero eigenvalues
associated with the three rigid body modes. No spurious zero energy modes are present.
This is to be expected since the stiffness matrix is obtained using a fully-integrated
scheme.

The Kirchhofir patch test is a test of the accuracy of the element to reproduce the pure
flexural mode. The patch is shown in Figure 13. Elements are of arbitrary shape patched
together to form a rectangular exterior boundary. As such, boundary conditions
corresponding to constant bending curvatures are easy to apply. The applied displacement
boundary conditions and the theoretical solution are also shown in Figure 13. The MIN4-
CC element passes the bending patch test and is able to reproduce the constant bending

moments and the surface stresses exactly for this arbitrary shaped patch.

51

 

 

 

 

 

 

 

y
4 3 1 0.04 0.02
a 2 0.18 0.03
2 3 0.16 0.08
1 x 4 0.08 0.08
_,
Boundary conditions:
w = —10~3(Jc2 + xy + y2)
0x = -10‘3(x/2 + y)
0), = —10-3(x +y/2)
Theoretical solution:
Bending moments per unit length: Mn = M”, = 1.1 1 1x104, M”. = 3.333xlO‘8
Surface stresses: OH = oyy = 10.667, oxy = 120.200

Geometric and material properties: a = 0.12, b = 0.24, thickness = 0.001, E = 1x106 , u = 0.25

Figure 13. Kirchhoff patch test for plates

Straight Cantilever Beam. The third test suggested by MacNeal and Harder (1985) is
solved for the three cases shown in Figure 14. The irregular element shapes are used to test

the combination of such shapes with linearly varying strains. The normalized tip

displacements in the direction of the loads are given in Table 3.

 

 

 

 

 

 

 

 

/ l l 1 T l J Geometric and Material Properties:
/J length = 6.0; width = 0.2; depth = 0.1
(a) Regular 7
E = 1.0X10 :1.) = 0.30
7 \ /45°>\\ /J<45°\ j Loading: unit forces at free end
/ . .
A (b) Trapezordal Theorctlcall
out-of-plane: 0.4321
twist: 0.03406
/ A45°
é / / / J / 1
l

 

(c) Parallelogram

Figure 14. Straight cantilever beam

 

pa

tw

116.1

153

Plar.

Table 3. Straight cantilever beam

52

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Tip load direction MIN4-CC MIN4 QUAD4-CC QUAD4 AQR8

(a) rectangular shaped elements
Out-of-plane shear 0.980 0.980 0.980 0.986 0.981
Twist 0.850 1.203 0.850 0.886 1.011

(b) trapezoidal shaped elements
Out-of-plane shear 0.963 0.893 0.963 0.968 0.965
Twist 0.805 1.248 0.805 0.896 1.029

(c) parallelogram shaped elements

Out-of-plane shear 0.978 0.975 0.978 0.977 0.980
Twist 0.778 1.117 0.778 0.890 1.159

 

 

All elements performed well with the regular mesh. The irregular-shaped elements
which contain considerable distortions cause performance deterioration of all the elements
except for AQR8 as can be seen from the results in Table 3. For all elements, both the
parallelogram and trapezoidal meshes caused unsymmetrical results at the free ends with a
twist load. This, however, is to be expected because of the unsymmetrical mesh. Values
reported above are therefore the mean values.

Curved Cantilever Beam. The curved cantilever beam as shown in Figure 15 is the
next test. This will also be a test of the effect of slight irregularity. Out-of-plane shear load
is applied at the free end to produce the out-of-plane deformations. The normalized out-of-

plane deflection for the curved cantilever beam is reported in Table 4.

53

Geometric and Material Properties:
inner radius = 4.12;
outer radius = 4.32;

are = 90° :
thickness = 0.1;
E = 1.0x107;
u = 0.25;

 

Loading: out-of-plane unit forces at tip

Theoretical tip deflection: 0.5022

 

F ired

Figure 15. Curved cantilever beam

Table 4. Curved cantilever beam

 

Tip load direction MIN4-CC MIN4 QUAD4-CC QUAD4 AQR8
Out-of-plane shear 0.944 1.023 0.944 0.951 0.956

 

 

 

 

 

 

 

 

 

 

Long Cantilever Test. This test was taken from Prathap and Somashekar (1988). Tip
out-of—plane shear loads are used to produce a constant variation of shear force and a
linear variation of bending moment. Different configurations are used to check for

distortion sensitivity. These are as shown in Figure 16.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

// / 40 Geometric and Material Properties:
% é \ length = 100;
0 50 ,00 /b 60 100 Width = 1.0;
depth = 0.1;
(a) (b)
10 E = 1.0x106;
/ /
/l) 90 [00 A0 100 Loading: unit forces at free end
(c) (4)
Theoretical:
“4000: (Mxx)| = 75 ; (Mu)2 = 25

Figure 16. Long cantilever beam

54

Normalized results obtained for the different configurations are identical to the four-
node element of Prathap and Somashekar (1988). The regular mesh configuration
produced results that agree with field-consistent 2-node linear shear flexible beam
elements. Linear bending moment can also be reproduced at the centroids of each of the
elements. However, in the distorted configurations (b, c and d), the elements can only
produce constant bending moments. Prathap and Somashekar (1988) reported that such
results indicate that the assemblage of two distorted elements acts only with the efficiency
of a single linear beam element. The effect of distortion will therefore lower the efficiency
of the two-element patch. Note, however, that even in extreme distortion i.e. collapsing the

quadrilateral elements into triangular elements (configuration d), locking is not present.

Table 5. Long cantilever beam test

 

 

 

 

 

 

 

 

 

Configuration w (tip) MIX M xx
.____=____=1 (element 1) (element 2)

(a) 0.938 1.000 1.000

(b) 0.750 0.667 2.000

(C) 0.750 0.667 2.000

(d) 0.750 0.667 2.000

 

Rectangular Plate. A rectangular plate will be used to test the convergence
characteristics of the new element under different boundary supports, aspect ratio and
loading conditions (see Figure 17). The test results are detailed in Tables 6 and 7. Uniform
convergence characteristics are clearly seen for all elements in all cases. In the case of a
clamped plate with aspect ratio of five, models that use MIN4 converge more slowly than

the other models.

 

 

 

 

 

 

 

55

Geometric and Material Properties:

a = 2.0; b= 2.0 or 10.0;

thickness = 0.0001;

E = 1.7472x107 ;

u = 0.3 ;

Boundaries = simply supported or clamped

 

 

 

 

7 sym Mesh = N x N (on 1/4 of plate);
fl—l | J Loadingzuniform load, q = 10_4 or
I b j center load, P = 4.0)(10—4
Theoretical solutions:
Boundary supports Aspect ratio Displacement at center of plate
b/a Uniform pressure Center load
Simple 1.0 4.062 1 1.60
Simple 5.0 12.97 16.96
Clamped 1.0 1.26 5.60
Clamped 5.0 2.56 7.23
Figure 17. Geometry, material properties, boundary conditions and

theoretical solutions for the rectangular plate tests

Table 6. Results for rectangular plate, simple supports: uniform load

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Normalized transverse defection at center

(a) aspect ratio = 1.0
Mesh MIN4-CC MIN4 QUAD4-CC QUAD4
2 x 2 1.079 1.035 0.978 0.981
4x4 1.019 1.009 0.995 1.004
6 x 6 1.008 1.004 0.998 1.003
8 x 8 1.005 1.002 0.999 1.002

(b) aspect ratio = 5.0
Mesh MIN4-CC MINL QUAD4-CC QUAD4
2 x 2 0.800 1.098L 0.971 1.052
4 x 4 0.992 1.001 0.978 0.991
6 x 6 0.996 1.001 0.991 0.997
8 x 8 0.998 1.001 0.995 0.998

 

 

 

3 d
For
def
Bur
cen-
Elfin
€ch
Very

56

Table 7. Results for rectangular plate, clamped supports: center load

 

Normalized transverse defection at center

 

 

(a) aspect ratio = 1.0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Mesh MIN4-CC MIN4 QUAD4-CC QUAD4
2 x 2 0.865 0.990 0.865 0.934
4 x 4 0.965 1.000 0.965 1.010
6 x 6 0.985 1.001 0.985 1.012
8 x 8 0.992 1.000 0.992 1.010
(b) aspect ratio = 5.0
Mesh MIN4-CC MIN4 QUAD4-CC QUAD4
2 x 2 0.318 0.634 0.318 0.519
4 x 4 0.825 0.699 0.825 0.863
6 x 6 0.911 0.795 0.911 0.940
8 x 8 0.946 0.860 0.946 0.972

 

 

 

 

 

 

 

To assess the robustness of the element, a square, simply-supported plate subjected to
a doubly-sinusoidal load was analyzed for span-to—thickness ratios, a/h, from 10 to 105.
For all cases, an 8x8 mesh was used in a quarter-model of the plate. The normalized center
deflection and maximum normal stress results (with respect to the elasticity solution of
Burton and Noor, 1994) are plotted in Figures 18 to 21. Stresses are evaluated at the
centroid of each element, which is the optimal stress recovery point for the MIN4-CC
element (see Barlow, 1976). There is no sign of numerical ill-conditioning even with
extreme thinness. Predictions of center deflection using MIN4 deteriorate rapidly in the
very thin regime for a plate aspect ratio of one. These results clearly show that the element

is very robust for both moderately thick and thin plates.

57

1.005 , r ' fi'fifi ' f ' r v--1l v v 4 ffifi..| , , V . 7" 1
E f:— ................ ‘ .................... A .................... A l
l,

0.995 j

 

 

 

 

0.99} {
iv : :
0.985} «1
0.98: 5

E + MIN4-CC 3

“-975 .' ~~~A MIN4 1

: . — - -- — QUAD4-CC 3

0.97 r j

 

 

 

 

 

 

 

”1000. ‘ 10000 7100000 .
a/h

Figure 18. Normalized center deflection versus span-to-thickness ratio of a
simply-supported square plate under sinusoidal loading (a/b = l)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1 V. f; .'._. '7’; ' '_" ‘ '_"
0.95 «
W 0.9 .
0.85- .
MIN4-CC i
t 1
0.8 _ MIN4 ,
. QUAD4-CC .
10. 100. 1000. 10000. 100000.
a / h

Figure 19. Normalized center deflection versus span-to-thickness ratio of a
simply-supported rectangular plate under sinusoidal loading (db
= 5)

58

 

 

 

 

 

 

 

 

 

 

] - - - 1 f
f ’ 4
0'99 A .................... A .................... ‘ ..................... A4
- ........... - —————————— - ........... -
0.98
6xx
0.” -
+ MIN4-CC
I —-I-— QUAD4-CC
A
I
0.95; ‘AU‘UAL - - - ...--..1 . ..--..1.
10. 100. 101/11,); 10000. 100000.
a

Figure 20. Normalized maximum normal stress versus span-to-thickness
ratio of a simply-supported square plate under sinusoidal

 

 

 

 

 

 

 

 

 

 

loading (a/b=1)
1 ~ w - + . . . - - 1
0.975 _ ____f____
0.95 -
0.925 .
an 0.9. 1
0375’ + MIN4-CC
111....
10. T “”100. ‘ ‘ 11000 0‘7710000.‘ ‘ 100000

Figure 21. Normalized maximum normal stress versus span-to-thickness
ratio of a simply-supported rectangular plate under sinusoidal
loading (a/b = 5)

 

611

.11

59

Circular Plate. A circular plate with clamped edges is commonly used to analyze an
element’s behavior in a moderately thick plate regime. Of particular interest is a solution
due to a center point load. According to Reissner’s theory, a logarithmic singularity exists
under the load. Figure 22 shows the coarse mesh of a symmetric quadrant used to analyze

the problem.

y 41
‘ Geometric and Material Properties:
radius, a = 5.0;

1 thickness, h = 0.1 or 2.0;
Boundaries = clamped

gym MCSh = 12,
E = 1.0x106;
u = 0.3
Loadingzuniform pressure, q = 1 or
center load, P = 4
sym Theoretical:
1 Uniform
_ i 2_ 2 2 _4_ 2 2_ 2
w- 64D[(a r) +l-‘Uh (a r)]
Center load

 

 

V).

 

 

 

8D 1 r]
2 "-
kGha a

 

P 2 2 2 a
W — m[(a -r )—2r In;-

Figure 22. Clamped circular plate (quadrant)

Figures 23-26 show the non-dimensionalized deflection along an edge of the clamped
circular plate. Results of center deflection show that the MIN4-CC element is capable of
reproducing accurate results for a clamped circular plate under uniform or center loads.
Note that the deflection curves obtained using the MIN4—CC and MIN4 elements are
quadratic in nature whereas only linear variations are predicted by the QUAD4-CC
element. However, this is only a result of the post-processing of deflection in the case of

MIN4-CC, and this could be done for the other elements as well.

 

-0.2

-0.4 1

£1

-0.6 P

 

    

Reissner Theory

 

 

 

 

 

 

-0.8 : + MIN4-CC
[ A - MIN4
. —--- — QUAD4-CC
-1 - . _
0 1 2 3 4 5

Figure 23. Normalized deflection (W = wl61tD/(Pa4)) along line of
symmetry (y=0) of a thin (a/h=50) clamped circular plate under
center load

 

 

 

 

 

 

 

 

 

0'7 T
I
-0.5'
l
-1- .
w I
-151- ‘
. ——- ReissnerTheory 1
: + MIN4-CC ,
-2 ' WA MIN4 1
p ' _..._ QUAD4-CC .
0 1 2 3 4 5

Figure 24. Normalized deflection (W =wl61tD/(Pa4)) along line of
symmetry (y=0) of a thick (a/h=2.5) clamped circular plate
under center load

 

 

 

   

 

 

 

 

 

 

0 7 V 4
—0.2 f -
-0.4 r -

_ l .
W 1 .
-0.6 - j
I 1
_0 8 ’_ Reissner Theory .‘
' » MIN4-CC ‘
» “A” MIN4 4
.1 L - - -- - QUAD4-CC .‘
0 1 2 3 4 5

Figure 25. Normalized deflection (W=w64D/(qa2)) along line of
symmetry (y=0) of a thin (a/h=50) clamped circular plate
under uniform load

 

 

 

 

 

 

 

 

0 . r .
l 1
'005 P "
, .
, l
W .1 . .
l ,- ' .A f
P /-/ .' T
r 7, ' _. " — Reissner Theory 1
-1.5- ,. —+— MIN4-CC :
Z ,-’-’ A" A MIN4 j
.‘ ............. ' —.--— QUAD4-CC 4
0 1 2 3 4 5

x

Figure 26. Normalized deflection (W = w640/(qa2)) along line of
symmetry (y=0) of a thick (a/h=2.5) clamped circular plate
under uniform load

 

VT

10 [l
001
33.;

the 1

CODd

reSul

Const

62

Morely’s Acute Skew Plate. The last test is to assess the behavior of the new element
under skew distortions (see Figure 27). Of particular interest is the acute 30° skew plate
simply-supported on all edges. An exact solution for the uniformly loaded condition has
been provided by Morley (1963) which indicated a singularity at the vicinity of the obtuse

vertices, with moments of opposite sign.

Geometric and Material Properties:

3:100;
thickness = 0.1;
YA 55-] or ss-Z angle, B = 30° or 60°

 

Boundaries: simply-supported (SS-1 or SS-2)
ss-l 33.; Mesh = Nx N
/ /a E = 1.0x10"; 0 = 0.3
S - r 55-2 é Loading: uniform load, q = 1

J] Theoretical:

 

 

 

S

a
qa‘ ‘3

w = 0.408—D—X10 (B = 30°)

4 —3
w = 2.505%X10 (B = 60°)

Figure 27. Morley’s acute skew plate

All solutions except for those obtained using MIN4 converged very slowly because of
the singularity and showed a much stiffer solution for the acute 30° angle case compared
to the solution of Morley (1963). Results obtained with MIN4-CC are comparable to the
nine-node element of Naganarayana et al. (1992). The choice of using a combination of
SS-l and 88-2 boundary conditions or purely SS-2 boundary conditions has an effect on
the results for the 30° angle case. Solutions using the combination of SS-l and SS-2
boundary conditions yield a stiffer result with a coarse mesh but both choices of boundary
conditions converged to about the same solution when the mesh was refined. The stiffer
result using the combination type boundary conditions could be traced to an over-

constraining of thin plate elements caused by SS-2. This phenomenon was also seen by

 

 

 

 

3.7

Cdg,
dish
appr

63

Rossow (1977) who reported that displacement based finite element models
underestimated the deflection at the center of the plate by almost 20 percent even with a
very refined mesh. Since the acute skewed plate is a numerically difficult problem, the

accuracy of the results can be considered to be fairly good.

Table 8. Normalized center deflection: uniform load

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(a) B = 30°
(i) Combination of 88-1 & SS-2 b.c. (ii) Purely SS-l b.c.
Mesh MIN4-CC MIN4 QUAD4-CC MIN4-CC MIN4 QUAD4-CC
2 x 2 0.811 1.637 0.487 1.037 2.030 0.622
4 x 4 0.756 1.118 0.699 0.956 1.187 0.877
8 x 8 0.774 1.021 0.762 0.856 1.052 0.841
16 x 16 0.818 0.999 0.815 0.841 1.018 0.838
32 x 32 0.863 0.995 0.862 0.876 1.008 0.875
(b) B = 60°
(i) Combination of 88-1 & SS-2 b.c. (ii) Purely SS-l b.c.
Mesh MIN4-CC MIN4 QUAD4-CC MIN4-CC MIN4 QUAD4-CC
2x2 1.185 1.362 0.711 1.359 1.619 0.815
4 x 4 1.014 1.074 0.925 1.041 1.11 1 0.948
8 x 8 0.987 1.028 0.965 0.992 1.038 0.970
16 x 16 0.990 1.009 0.985 0.991 1.016 0.985
32 x 32 0.986 1.003 0.985 0.986 1.007 0.985
3.7 Summary

 

The MIN4-CC element contains consistent shear strain fields as well as consistent
edge tangential shear strains, obviating locking even when the element is severely
distorted. Even though an interdependent interpolation scheme was used for the element
approximation of deflection, the present consistent strain fields were shown to be

equivalent to those developed by Prathap and Somashekar (1988). Thus, the main

 

difi

016

than

CC

Ihec

the c

64

differences between these two elements are in the consistent load vectors and in the
predicted transverse displacement distributions. Numerous numerical tests demonstrated
that the MIN4-CC element is both robust and accurate.

The assumed strain field approach that was successfully utilized to derive the MIN4-
CC element will next be used to develop a robust LZZB element using our new laminate
theory. It will ensure the satisfaction of both field and edge consistency requirements in

the element and prevent element locking in the thin regime.

Chapter 4

LAYERWISE PLATE ELEMENT - THICK PANELS

The field and edge consistency concepts have been introduced in the previous chapter.
It is now easier to introduce the new laminate theory for the bending of plate structures. In
this chapter we will solely concentrate on the formulation of the new zig-zag sub-laminate
theory for plates and the associated finite element model with direct application to thick
sandwich panels. The extension of the finite element model to thin plate bending

structures will be discussed in the next chapter which deals with the locking phenomenon.

4.1 Introduction

The layerwise plate element derived using the new laminate theory could take the form
of a four-node plate element. However, similar to the case of the layerwise beam element
discussed in Chapter 2, it is more advantageous to cast the element in the form of an eight-
node brick. This topology allows the laminate thickness to be conveniently subdivided and
modeled by multiple finite elements (representing sublaminates). It is thus possible to
increase the accuracy of the finite element model, as needed, to capture through-the-
thickness gradients and transverse (interlaminar) stresses. It is also possible to simulate
delaminations using the redundant node concept.

Both the theory and the finite element model have been developed in two forms. The

first form, called here LZZ3, is based on a high-order zig-zag theory and will be discussed

65

 

ht

di

[01
Var

EEK

66

in detail in this chapter. The finite element model LZZ3 contains seven degrees-of-
freedom per node: u, v, w, 0x, 0y, 1,, 1y . The first five degrees-of-freedom are the usual
ones used in plate/shell finite elements. The remaining two degrees-of-freedom, Ix, 13y ,
are the transverse shear tractions (or interlaminar shear stresses, as the case may be). If
only one sublaminate (element) is used through the thickness of a laminate, then these
transverse shear degrees-of-freedom can be eliminated at the element level, leaving only
the traditional five engineering degrees-of-freedom. The second form of the model, called
here LZZl, is based on a first-order zig-zag theory (see Cho and Averill, 1996) and
contains the usual five engineering degrees-of-freedom only. This model will not be

discussed here.

4.2 Theory Formulation

In the present theory, the laminate is composed of N T perfectly bonded layers stacked
together in the thickness direction. The thickness and material stiffness properties may
vary arbitrarily from layer to layer. These layers can be modeled as M sublaminates, with
each sublaminate containing N m layers, where m is the sublaminate number (see Figure

28). Mathematically, this is represented as:

NT= sz (50)

 

de

101

111116

3.8311

“Shel:

67

 

 

 

 

 

 

 

 

 

z is a local (sublaminate) thickness coordinate
Z,z with its origin at the bottom of the sublaminate
T Z is the global (laminate) thickness coordinate
ZM .
; v ° 1
' ////?}ZW///////‘/
Z ‘ 4 z
2 m "2‘”
Sublaminate 2 :
Wm z2(2)
Z Zr (2)
I T z0(2)
----- Sn blaminate—I— - — — —

Figure 28. Schematic of sublaminate and layer divisions

In order to facilitate the development of the theory, all expressions in the following
derivation pertain to the mth sublaminate. The sublaminate number designation is omitted
for brevity. The constitutive relations for the kth layer of the mth sublaminate with respect
to the laminate coordinate axes are given by:

T

 

 

 

 

k k k k-
to ‘(k) C(11) C(12) C(13) 0 0 C(16) Fe ‘00
"x C(k) Can 0 0 Cut) x"
Oyy 22 23 26 8yy
(10 (k)
0.22 = C33 0 0 C36 822 (51)
Ty: Cff,’ cf]: 0 7y.
13
xz Sym C151? 0 7.2
.1”; C(k) .7101.
66

 

 

The 13 anisotropic material constants in Eq. (51) can be expressed in terms of 9

independent constants. The layer strain-displacement relations using small strain

assumptions are:

= u, 8 = 12,}, 822 = W,Z

8 x Y)’

XX

(52)
M = w,y+v,z 112 = w,x+u,z ny — v,x+u,y

where a coma in the indices denotes partial differentiation.

 

50!
81's

1:111

6115

dim

68

In each sublaminate, an independent displacement field is assumed in which the
through-the-thickness variation of in-plane displacements is described by a cubic
polynomial in the local thickness coordinate with a piecewise linear (or zig-zag) function
superposed upon it. The transverse deflection is assumed to vary quadratically with the
thickness coordinate. The assumed displacement fields are initially assumed in the
following form:

3 n—l
It. .
ui")(x,y,z) = z z uk+ z (z—z,)§i
k=0

i=1

3 n-l
n k. .
k=0

i=1

1...): 0(1 —11+w:(%)+%(ill -1—1

where the subscripts b and I refer to the bottom and top surfaces, respectively, of the
sublaminate. h is the total thickness of sublaminate m. All quantities are referred to a fixed
system of rectangular, Cartesian coordinates. A general point in this system is denoted by
either (x,y,z) or (x1, x2, x3), whichever is more convenient.

The use of the non-conforming bubble function assigned to field variable uz is to
ensure the removal of the Poisson’s ratio stiffening effect whenever the full three-
dimensional constitutive relations are used. Poisson’s ratio stiffening effect is the
phenomenon that is observed in continuum elements when only one layer of linear
elements is used through the depth to model a region of flexure. This stiffening effect
originates from the bending energy terms and emerges from the inability of the

approximation functions to interpolate the transverse normal strain, 8 linearly through

22’

the depth of the plate (see Prathap, 1985).

th

sh

61:

am

“’he

69

Since the sole purpose of the bubble function is to eliminate the Poisson’s ratio
stiffening effect, terms involving W3 in the computation of transverse shear strain
relations are ignored. This simplification will eventually lead to W3 being a nodeless
variable (degree-of-freedom) and thus C-1 continuous in the resulting finite element
model. It is then condensed out at the element level.

It is possible to eliminate the degrees-of-freedom E,- and 1"], in Eq. (53) by enforcing
the conditions of transverse shear stress continuity at each interface. The conditions of

shear stress continuity at the jth interface, where z = 2]. , are:

(1') _ TU“) 1:(J') _ TU“) (54)

sz - xz yz - yz

Making use of Eq. (53) along with the infinitesimal strain-displacement and linear
elastic constitutive relations, Eq. (54) can be solved to give closed-form expressions for E,

and 01' in terms of the remaining degrees of freedom.

. . Bwb .. A 8w, aw,
‘11 = “11(V1'l'a—y )+azi"2+a3i"3+a4t(§; ‘a—y )

8w, Bwb)

.. aw, .. ..
+b“(ul+0—y )+b21“2+b3i“3+b4i(a_x _a—x

- = . +_ + . + . + ._ __
t 611 V1 8y Cztvz C3tl’3 C41 3y ay

(55)

. aw, .. . 8w, aw,
+dh-(ul-1-5; )+d2iu2+d3iU3+d4i('a—x- —§; )

where

 

Thes

Each

trans.

11min

70

(k) k—l k-l
apk = ak[fp + apq]+bkL cm]
q=l =1
(k) k—l k—l
bpk - bk[fp + 2 dpq]+ak[q2 bP‘l]
q=l =1
( ) k-l k-l
cpk = Ck fp + Z anJ-l-dkL cpq]
q= =1 (56)
(k) k-l k-l
dpk = d,[fp + Z dpq)+c,£z hm]
q=l :1
(k) 2
f(lk) _ 1 f3 — 32,,
(k) (k) zk
f2 - 22p f4 - Z

 

and

,. l k 1 k k 1 k

a. = —<C‘..* Til—Cf." ’635’1—1
Ak+1

~ 1 (k+1) (k) (k+1) (k)

bk = —(C55 C45‘C45 C55)
Ak+1

1 (621*"CS§’-Cf.§*"c§1’> <50
Ak+1

. 1 (1+1)

 

k k 1 k
(6.. C§.’- Cfo’ ’Cfts’1—1

Q,
R
H

Ak+1

(k+l) (k+1) (k+1)2
Ak+1=C44 C55 ‘(C45 )

These conditions therefore reduce the number of degrees of freedom by 2(Nm — 1) for
each sublaminate, leaving us with ten degrees of freedom per sublaminate.

Additional simplification of Eq. (53) can be achieved through satisfaction of the
transverse shear traction boundary conditions at the top and bottom surfaces of the

laminate. These conditions are:

 

“’41

C01}

71

 

 

 

 

To) _ T T1N.) 1:
xz z = 7-0 xb xz z = h xt

(58)
1(1) _ T (N...) _ T
yz z=zo yb yz z=h y!

where 111,, In, tyb, 1),, are the applied shear tractions (or interlaminar shear stresses, as

the case may be) at the bottom and top surfaces of the sublaminate.

The displacement field is cast in its final form by introducing the surface variables:

 

 

 

l A A
0:“51) =“o ”0:“;0 -
zo Z0
(59)
(N..) _ (N..)

 

Eqs. (58, 59) can be solved either analytically or numerically (see Appendix C) so that the
displacement field for each sublaminate can be expressed in terms of the operative degrees
of freedom (all functions of the inplane coordinates, x and y, only):

12 (60)

ab, vb, wb, 0x1), Byb, txb, tyb, u,, v,, wt, 0“, 0 xi, 12y,

yt’
where the subscripts b and t refer to the bottom and top surfaces of the sublaminate,

respectively, and

Bwb aw,

9x0 = '37 9x! = 3; (61)
awb aw,

0y, = 3'; 9,. = '87

The replacement of the partial derivatives of the deflection with the “new” rotations
was done to allow the assumed displacement degrees-of-freedom to remain C0
continuous. The constraints in Eq. (61) will have to be enforced explicitly by a penalty
formulation during the development of the finite element model.

The displacement field in Eq. (53) for the mth sublaminate can now be represented in

 

0V.

72

terms of through-thickness shape functions as:

( ) 4(k) k
ux" = ua (x, y)<b;")(z)

u‘"’ = 0‘*’(1,y)~r;""’(.) (62)

y “ a=1,2

u = w.(x.y)M.(z)+ng3<z)

2
where summation on repeated indices is implied. Index or is used to represent the top and
bottom surfaces with 1: bottom and 2: top; and

xk .1..2.3,,4,.5 A617
“(1): {u( 1, u( )’ u( ), u( ), u( )’ u( 1, “1 )}a = (u,v,w,9x,9y,1x,1y)a (63)

Here (bgm), ‘10:") and M a are shape functions of the thickness coordinate (2) only and are
known functions obtained either analytically or numerically by the process mentioned
earlier. These shape functions are listed in detail in Appendix C. Note that (1)8")and
W53") are always equal to zero and as such the inplane displacement fields, u x and u y are

independent of Wu and 1123.

4.3 Theory and Finite Element Model
The equations of motion, the natural boundary conditions and the displacement-based
finite element model of the mth sublaminate are obtained using the principle of virtual

work:

6w""’ = 8w‘.'"’+8w""’ = 0 (64)

mt ext
where the subscripts int and ext refer to internal and external work done, respectively.

The virtual strain energy of the single sublaminate is given by:

owl“) = Ioijfieijdv i, j = 1,2,3 (65)

mt

over the volume Vm of the mth sublaminate.

73

In order to reduce the continuity requirements on wb, w, , the rotational degrees-of-

freedom 9x1» 0 Byb, 0),, in Eq. (63) are introduced, thus eliminating all second-order

xt’

derivatives of the transverse deflection degrees-of-freedom in the internal work functional
and making all degrees of freedom C0 continuous. The rotational degrees-of-freedom are
coupled to the transverse deflection degrees-of-freedom through an interdependent
interpolation scheme (Tessler and Hughes, 1983 and 1985) and the constraints in Eq. (61)
are enforced by a penalty method.

Making use of Eqs. (62, 63, 65), the resultants of the sublaminate stresses

on, 0’”, on, T y, 1,2,1 y, can be obtained by defining:

N", z,
(Mxx):1k) = 2 [I 6xx(x9 y, z)¢£1k)(Z)dZ]

i=121-1

NM

(Myy):1k) = 2 [if oyy(x 2’12)?ka )(Z)d2]
i=1 Z,_ I
N

m z, (66)
(MN)? = 2(1 rxy(x, y,z)<1>(k)(z)dz

1:1 zt-l

NM
(MU)? = 2 (:1 rxy(x, y,z)‘1’(k )(z):z)
i=1 z,_

01 = 1,2; k =1,2,4,...,

74

N, z,
(Q90, = [I]: rxz(x, y, z)Ma(z)dz]

i= 1 1

N" z,-
(Qy)a = 2 £1 Tyz(x, y,z)M a(z)dz]

i= 1 —1

N, z,-
(Qz)a = [ I Ozz(x, y, z)Ma, Z(z)dz]

i: 1 7-1-1

Nun 2‘ (67)

(Qz) = 2(1 “a” y, Z)M3 z(z)dz]
121:

(R,)ff’= [I 1,,(x y.z)<b£,’,f (2)42

[I Tyz(x, y,z)‘P£1, :)Z(z)dz]

N.
(k) _
(Ry)a " Z
i'1 1—1

1,2; k =1,2,4,..

(I

Now introducing the virtual work due to external forces:

5W§f,’— = —1 00,611,. dV— 11,811,111“
v, r, i, j = 1,2,3 (68)

where p is the density, 13 = bif‘gi, g = rig, and n = n.2,. are the body force per unit mass,
the surface traction and unit normal respectively. 1“ refers to the complete boundary of the
mth sublaminate and F2 the boundary where surface tractions are specified. Due to the
sublaminate geometry, the boundary domain, 1" can be separated into two subregions such
that 1" = A u R, where A refers to the surface area of the top and bottom surfaces (or
interfaces) and R refers to the remaining surface area that defines the circumference of the

sublaminate. Using Eqs. (62, 68), the virtual work due to external forces is:

75

on
i

3.
I

- _ ”gpbifiui dz dA "1‘15“1 (M ‘10511, dz ds
A A R

-1[qo5wa + (m1 + .1130.ng 2180),] (M
A

(69)
_ ((1141,, + M,,)§f>.,aagf> ds — j [(Qn)a8wa + 9,803] ds
:2 32
01,0: 1,2; k = 1,2,4, ...,7
or refers to either top or bottom surface of the plate

where q, the distributed transverse loading in the z-direction, and the distributed
“generalized” moments, 21 and ma are given by:
901 = Ipsza dz+[ozzMa]g
q = Ipsz3 dz
(m1)? = jpbxoff’dznt 42"”):

xza

01 = 1,2;k = 1,2,4, ...,7 (70)

h

0

k k k
(m2)( ’ = J‘pbwa,’ dz+[tyz\rf,’]

a
M 113’ M 23, Q, and Q" are the prescribed “generalized” boundary moments and shear
forces and they are:

(k)

k k
(M,,)f,’ = (My)? (114.02" = (M,,),,
(k) _ (k) (k) _ (k)
(M21), - (Myx). (M22)... - (Myy)a (71)
(Q11)... = JrnzMadz Q" = IrnzM3dz

or =1,2;k =1,2,4,...,7
s1 refers to the portion of the top and bottom boundary contour s of the sublaminate where
displacements are specified and s2 is the portion where surface tractions are specified.

Making use of Eqs. (64, 65, 69), the virtual work equation can be written as:

 

To
the

gen

“he

76

 

 

 

0 - 8(Mxx)0k) MM”)? 8(Mxy)g()+a(Myx)0f)
- I - 8x + By + 3y 8):
-(R,)f,f"—(R,)ff’184f,f" (72)

(9,), 8(0),, . . A
— ax + a; -(Qz)a]8wa+(Qz)6w3]dA+L(ug‘))+L(wa)

a =1,2;k = l,2,4,...,7

 

where L028”), L(wa) are the boundary terms that result from integration by parts and
12:: ) is defined by Eq. (53). The boundary and load conditions for the mth sublaminate can
be written as two linear functionals:

. k A k k A k
L(ug‘)) = — 1(1):, +m2);)5u;) dA— 1(MIB+M25); 013611;) ds
A

32

L(w.) = - 114.5%. +0503] 44 - 11(2),». + (“2,6031 ds (73)
A 52

01 =1,2;k = l,2,4,...,7

and the governing equations in indicial notation are given by:

swa: (-Qp,0+Q3).. = q.

..(k). 2 (k) _ 2 (k)
5%. - 20:1(M10’1‘R0h " ‘20=1(’"B)a (74)
8W3: ’(Q3)a = ac

01,117 =1,2; k =1,2,4,...,7
To understand the boundary conditions associated with this higher-order laminate theory,
the degrees-of-freedom in the displacement field of Eq. (62) can be recast into a more

general form:

u = Hanna) ”(0,1,(0

x (I: :10 :0 :0 I: 1.2.3; <1 = 1’2 (75)
“y = anawna'l'fisa‘ysa

where the new set of degrees-of-freedom will be represented as:

77

( ( (
S S S
-1 _1_2_3 a=1,2 (76)
al.): (al. 1 u}. l. u}. l). = (14.9.1.1.

and subscripts s and n refer to the tangential and normal directions, respectively, on the

boundary surface. The shape functions, (DU) (1)“) ‘l’ig and WE}; are readily obtained

311’ nor

from the original shape functions (see Appendix D) using transformation equations for

Cartesian vectors, e.g.

u luxux+l u

n fly )7 (77)
u, = —lnyux+lnxuy
where In = cos(n, x) = dy/ds and l"y = cos(n, y) = —dx/ds are direction cosines

of the outward normal of the boundary.
Substituting Eq.(75) into the boundary and load terms associated with the iig‘)

degrees-of-freedom of Eq. (73) will yield:

L(ag<>) = —j[(ms)g>8rtw+(mn)g>5rtw] dA
A

(78)
+ 11(M.)§P517.o + (It/[01005171111] ds
52
where the new generalized moments are defined as:
’ l I l 11
(“1’51” = 1919910 b.1111 dz + [1.913. + 0.111 .
(l) (79)

1
(M 5)}? = IKOxxnx + txyny)<1>f3;+ (“trxynJr + oyyny)‘1’
1: 1,2,3; 01 =1,2; [3: s,n

1301] 612

 

 

de

935

685

COI

[rat

78

Now, the following sets of boundary conditions can be identified:

 

 

Simply Simply .
Clamped supportwl 3:553:35“ Free Symmetric“
um = 0 uml = 0 um = 0 (May) = 0 (1145):!” = 0
u... = 0 (May) = 0 (M39) = 0 (M33) = 0 u... = 0
wa=0 wa=0 wa=0 Qna=0 Qna=0
9.. = 0 (M053) = 0 9.. = 0 (Mag?) = o (Mpg?) = 0
0.. = 0 (Mpg?) = 0 (M,)f,2> = 0 (Mpg?) = 0 9.. = 0
I... = 0 (M053) = o t... = 0 (M03) = 0 (M03?) = o
I... = 0 (M553) = 0 (M333) = 0 (44353) = 0 I... = 0

 

*Symmetric conditions are only applicable to laminates with orthotropic materials that are stacked
with their principal material directions at either 0° or 90° to the coordinate axes (cross-ply lami-
nates are a special case of these specially orthotropic laminates).

The essential boundary conditions are similar to the plate/shell theory formulation with
the exception of the transverse “shear traction” (or interlaminar shear stresses, as the case
may be) degrees-of-freedom. A closer look at the boundary conditions will, however,
seem to yield an ambiguity in the boundary conditions associated with the “shear traction”
degrees-of-freedom, 1: m and Tna- These degrees-of—freedom appear explicitly in both the
essential and natural boundary conditions. In actual fact, there is no ambiguity. The
essential boundary conditions associated with the “shear traction” degrees-of-freedom for
the boundary types must always be complied with as dictated by the variational approach.
In cases where the shear traction conditions are known, (e. g., zero shear traction is
commonly encountered), it is therefore desirable (but not necessary) to specify the “shear

traction” degrees-of-freedom to correspond to this shear traction condition. However, in

 

ant
de:

“11

79

the event when the applied shear traction load is non-zero, it is imperative that the shear
traction must be included as an applied load using either Eqs. (70, 71) or Eq. (79). Also,
note that simply supported-1 (SS-l) is the appropriate simply-supported condition for the
plate elements with both general laminate and curved boundary types, and are very similar
to those of the Mindlin plate element. For cross-ply laminates or laminates with
orthotropic materials that are stacked with their principal material directions at either 0°
or 90° to the coordinate axes, SS-2 can be used to further eliminate degrees-of-freedom
since 15 and In are directly related to the shear strains through the kinematic assumptions.

While the finite element geometry could take the form of a four-node plate element
with fourteen degrees-of-freedom per node, it is advantageous to use the topology of an
eight-node brick element as shown in Figure 29a. This topology allows the substructuring
of the laminate into sublaminates for mesh refinement in the thickness direction. This will
increase accuracy and also permit the simulation of delamination using the redundant
node concept. Furthermore, using the eight-node brick element topology allows this new
3-D ‘structural’ element to be coupled with conventional 3-D continuum elements,
although compatibility would not be strictly satisfied in this case.

The finite element model utilizes the interdependent interpolation concept of Tessler
and Dong (1981), Tessler and Hughes (1983, 1985) and Tessler (1990). The transverse
deflection, wb, w, is initially approximated using quadratic interpolation functions, N j,
while all other degrees-of-freedom are approximated using linear Lagrange interpolation
functions, P j. This unequal interpolation scheme is a consequence of trying to satisfy the
Kirchhoff constraints in a consistent manner. This, however, will result in mid-side

degrees-of-freedom associated with wb, w, (a consequence of the higher order

 

4.4

final
31X 1

511101

80

interpolation functions used). The finite element therefore is not of a convenient form as
depicted by Figure 29b. The mid-side nodes are condensed out to achieve a uniform eight-
node brick element. This is done by explicitly enforcing the linear part of the constraints
of Eq. (53) along each side in a similar fashion as Tessler and Hughes (1983) (see
Appendix D for details). This approach effectively increases the order of the element
without introducing any additional nodes or degrees of freedom and alleviates the shear

locking problem so an exact order of numerical integration may be used.

a 8-node brick element
§% ( )

(b ) unconstrained element

   
  

(“1’ Vt) w,, ext’ eyt’ In: Ty!)

(“1r vh’ W1» 9x0, eyb’ "xh’ Iyb)

(c) constrained LZZJ element

 

mid-side nodes
condensed out

12::

Figure 29. Topology of the LZZ3 finite element model

 

 

7 (“1’ vl’ ”'1’ 0”, 0”, I“, tyt)7

 

2 (“b' Vb’ Wb’ 910’ eyb’ 1:rtb’tyb)2

4.4 Numerical Studies

Several numerical experiments dealing with anisotropic plates are presented. Spectral
analysis of the LZZ3’s stiffness matrix revealed six zero eigenvalues associated with the
six rigid body modes. No spurious zero energy modes are present. This is to be expected
since the stiffness matrix is integrated using a 3x3 Gauss quadrature rule “in the plane”.

Numerical results are presented for bending of three simply-supported laminated

 

 

ele
C0
501

61';

€161

Ihre

81

square panels subjected to a double-sinusoidally varying transverse load (see Figure 30).
The laminates are very thick, having a span-to-thickness ratio of four. Numerical examples
such as these are an excellent test of a model’s ability to capture local effects. Physically,
these examples are similar in many ways to local models of the region directly beneath an

impact load.

P—————————.———___..._.__’

  

double-sinusoidal
"""""""""""" transverse loading

Note: Load is distributed over entire top surface

 

Figure 30. Schematic of loading and boundary conditions for example problems

Numerical results are obtained using a quarter-model with discretization of 8x8
elements in the plane of the panel and one or more elements through the thickness.
Comparisons are made between predictions of the current model and an exact elasticity
solution (Burton and Noor, 1994). The stress distributions across the thickness are
evaluated at the center of the elements, since the center can be treated as the optimal
location for stress recovery (Barlow, 1976).

A Random Five-layer Panel. The first example is a random five—layer laminate. The
lamination scheme is given in Table 9 and the material properties used are listed in Table
10.

Three levels of thickness discretization were used. The first model employed a single
element through-the-thickness (a sublaminate containing all five layers), the second used

three elements through-the-thickness (one for the bottom layer (i.e. layer one), one for the

82

next to bottom layer and the third, for the remaining three layers) and the last model used

five elements (one for each physical layer) through-the-thickness of the entire panel.

Table 9. Lamination scheme for random five-layer panel

 

 

 

 

 

 

 

Layer No. Material Thickness
1 l 0. 10
2 2 0.25
3 3 0. 15
4 1 0.20
5 3 0.30

 

 

 

 

 

Table 10. Material properties for random five-layer and sandwich panel

 

Matl. 1 Matl. 2 Matl. 3 Core
E” 1.0e6 33.0e6 25.0e6 5.0e4

 

 

 

522 25.0e6 21 .0e6 1.0e6 1.5e5

 

E 1.0e6 21.0e6 1.0e6 5.0e4
33

 

 

 

0 0.01 0.25 0.25 0.01
12

0 0.25 0.25 0.25 0.15
23

0 0.25 0.25 0.25 0.15
13

 

G 0.5e6 8.0e6 0.5e6 2.17e4
12

 

G 0.5e6 4.0e6 0.2e6 4.20e4
23

 

G 0.2e6 8.0e6 0.5e6 2.17e4
13

 

 

 

 

 

 

 

A Sandwich Panel. The second example is a sandwich panel with five composite face
sheets on the top and bottom of a core material. The material properties are already listed
in Table 10. The lamination scheme is given in Table 11. Note that the five-layer face sheet

laminate is by itself a rather challenging laminate to analyze due to the variations of

83

material properties and layer thicknesses.

Three levels of thickness discretization were used. The first model employed a single
element through-the-thickness of the entire sandwich panel, the second used two elements
through-the-thickness - subdividing the laminate equally, and the other used three
elements through-the-thickness - one for the bottom face sheets (a sublaminate containing

five layers), one for the core, and one for the top face sheets.

Table 11. Lamination scheme for sandwich panel

 

 

 

 

 

 

 

 

 

 

 

 

 

Layer No. Material Thickness
l 1 0.010
2 2 0.025
3 3 0.015
4 1 0.020
5 3 0.030
6 Core 0.800
7 3 0.030
8 1 0.020
9 3 0.015
10 2 0.025
1 1 1 0.010

 

 

 

 

 

A Panel from TACOM ’s Composite Armored Vehicle. The third example utilized a
lamination scheme that is similar to that present in the US Army TACOM’s Composite
Armored Vehicle (CAV). The laminate can be divided into three sections, as described
below:

(1) Inner Shell. Four plies of S-2 Glass/Phenolic Fabric with stacking sequence

{ [902/02] }with thickness of 0.01 for the four plies stack. Thirty-seven plies of S-
2 Glass/855340 Epoxy Tows with stacking sequence {[0/ 90],

[45/—45/0/9O]4 , [45/0/45], [90/0/—45/45]4 } with thickness of 0.021.

 

001
lay
wit
m0

The

Tat

emph

84

(2) Armor Core. One layer of EPDM rubber (thickness = 0.0625) and one layer of
ceramic tile with inserts (thickness = 0.7).

(3) Outer Shell. Twelve plies of S-2 Glass/855340 Epoxy Fabric. Stacking sequence
is [0/90/45/-45/0/90]s with thickness of 0.01.

The laminate consists of 55 layers and is nearly two inches thick. To allow
comparisons between predictions of the current model and the exact elasticity solution, all
layers are oriented at either 0. or 900 from the reference axis by replacing all 45° plies
with 90° plies and the —45° plies with 0° plies. This is not a restriction of the current
model, however, which is capable of modeling completely general lamination schemes.

The material properties are listed in Table 12.

Table 12. Material properties for TACOM’s composite armored vehicle (CAV) panel

 

 

 

 

 

 

 

 

 

 

 

 

 

S-2 Glass/ S-2 Glass/ S-2 Glass/ EPDM
Phenolic 8553-40 8553-40 Ceramic
Fabric Tow Fabric R“b°°’
_ 1—__ = L i:
E” 3.0e6 6.2e6 3.0e6 3.0e3 5.0e6
£22 3.0e6 1.0e6 3.0e6 3.0e3 5.0e6
£33 1.2e6 1.0e6 1.1e6 3.0e3 1.25e5
1) 0.13 0.29 0.13 0.45 0.15
12
u 0.18 0.37 0.18 0.45 0.15
23
u 0.18 0.29 0.18 0.45 0.15
13
612 1.0e6 0.3e6 1.0e6 1.0e3 2.5e6
G 4.6e5 0.3e6 3.9e5 1.0e3 8.5e3
23
013 4.6e5 0.3e6 3.9e5 1.0e3 8.5e3

 

 

 

 

 

 

 

 

Once again, three levels of thickness discretization were used. The first model

employed a single element through the thickness of the entire panel, the second used four

85

elements through-the-thickness with each sublaminate representing the different sections -
one for the inner shell, two for the armor core, and one for the outer shell. The third model
used three elements t11rough-the~thickness with the two sublaminates in the armor core
combined and the rest similar to the second model.

The convergence characteristics of the different LZZ3 models were studied with
different mesh density. The center deflection at the mid-surface was used for comparison
even though the variation of the transverse deflection through-the-thickness for a thick
laminate can be substantial. This is readily seen in Figures 32, 38 and 46. Three
discretizations were used for all cases. The cases with one and three elements through-the-
thickness (sublaminates) are as described above. The two elements case is obtained simply
by subdividing the panel thickness into two equal parts. The convergence characteristics
for all the cases are depicted in Table 13.

Table 13 showed that for all cases even with a coarse mesh of 4x4, the results are
converged with the exception of the one-sublaminate model in the CAV panel case. For the
CAV panel, results clearly demonstrated that thickness refinement through the use of more
sublaminates is needed for laminates with such complicated layups. A note of caution is
that more thorough comparison between the different (sublaminate) models must, in
general, take into account through-the-thickness variation of the displacements and

stresses for thick panels.

86

Table 13. Results for square plate, simple supports: sinusoidal load

 

 

Normalized transverse deflection at center

 

 

(a) random five-layer panel

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Mesh l sublaminate 2 sublaminates 3 sublaminates
2 x 2 1.070 1.072 1.073
4x4 1.008 1.008 1.011
6 x 6 0.997 0.997 0.999
8 x 8 0.978 0.993 0.998
(b) sandwich panel
Mesh 1 sublaminate 2 sublaminates 3 sublaminates
2x2 1.131 1.120 1.133
4 x 4 1.054 1.033 1.056
6x6 1.040 1.014 1.042
8 x 8 1.035 1.007 1.037
(c) CAV panel
Mesh 1 sublaminate 2 sublaminates 3 sublaminates
2 x 2 0.863 1.052 1.057
4 x 4 0.822 0.999 1.003
6 x 6 0.815 0.989 0.993
8 x 8 0.812 0.986 0.989

 

 

 

 

 

 

Results of the analyses for all three examples are shown in Figures 31 to 51 for the 8x8
mesh discretization, where predictions by the LZZ3 models of the through-thickness

variations of inplane displacement, ux, transverse deflection, u inplane stress 0'”,

Z,

inplane shear stress ’1: transverse shear stress In and transverse normal strain, ezz are

xy’
compared to the variations predicted by three-dimensional elasticity. Even for these
extremely thick laminates, it can be seen that good predictions of ux, uZ , on and 1,0, are
obtained using only one element (sublaminate) through-the-thickness, with slight

improvements in these predictions when the thickness is discretized using three or more

elements. This is to be expected since the deflection, uz , and the inplane stresses, CH and

87

’1' can be captured adequately by zig-zag theories which are equivalent to our LZZ3

xy ’
model when one sublaminate is used.

In Figures 34, 41 and 49, the transverse shear stress is calculated using the constitutive
relations with no shear correction factors. In each example, more than one element
through-the-thickness is needed to capture the variation of sz adequately. The deviation
from the exact solution is more drastic in the results of the sandwich and CAV panels
(Figures 41 and 49) where these panels have material properties that are drastically
different in adjacent plies. The predictions of sz could be improved for the case when
one element is used through-the-thickness by recovering the stress using the three-
dimensional equations of equilibrium.

From Figures 36, 44 and 51, plots of the normal transverse strain ezz are shown.
Results of an is not expected to be good for models using small number of elements
through-the-thickness because of the simple approximation of the assumed displacement
field, uz. However, one can clearly see that the normal transverse strain, ezz will approach
the elasticity solution as more elements are used in the through-the-thickness direction.
This is especially seen in Figure 36 for the random five-layer panel when one sublaminate

is used to represent each layer. In general, however, accurate normal stress, 0' has to be

22 ’
recovered using the three-dimensional equation of equilibrium if the number of elements

used in the through-thickness direction is kept low.

4.5 Summary
The current LZZB model shows excellent promise for efficient and accurate analysis of
thick laminated composites and sandwich panels. This zig-zag approach has the following

desirable pr0perties: (i) interlaminar transverse shear stress continuity conditions are

88

satisfied; (ii) transverse shear tractions at the top and bottom surfaces of the plate are
satisfied exactly; (iii) transverse normal tractions at the top and bottom will tend to the
prescribed load value with an increase in the number of sub-layers; (iv) a small and fixed
number of degrees-of-freedom per sublaminate are needed to accurately describe the
kinematic behavior of complex sublaminate regions; (v) traditional engineering degrees-
of-freedom (displacements and rotations) are used; and (vi) the eight-node brick topology
permits the use of adaptive techniques for through-thickness discretization.

The casting of the technical theory in terms of surface quantities also allow the model
to represent the applied loads better. In most plate theories, surface tractions are applied at
the reference surface which usually corresponds to the mid-surface. This is acceptable for
thick plates but not for thin plates. For a thick plate as observed by the numerical
experiments, the through-the-thickness distribution of in-plane displacements, stresses as
well as the transverse stresses are non-symmetric with respect to the mid-surface.

Results showed that the current LZZ3 model is much more efficient and potentially
more accurate than 3D continuum-based models for analysis of laminated composites and
sandwich panels. The LZZ3 model is also computationally competitive with traditional

low-order plate elements.

89

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Elasticity
........... LZZ3: 1 sublamuzate
————— LZZ3: 3 sublaminates 4
a» — — — — LZZ3: 5 sublaminates
0.1 " // 7: = 4 +
2 ~.
7. ° "-
-o.1 L \"
-o.2 - \ J
-0.3 i
'004 \
.05LAAA- ,Anm ‘4 A A; ‘ .. . 44444
-o.4 -o.2 o 0.2 0.4 0-6
-6
uxXIO

Figure 31. Axial displacement versus normalized thickness coordinate of a

0.5
0.4

0.3-

0.2
0.1

3‘“!

-0.1 -
-0.2 -

-0.3
-0.4

.05_:;.

simply-supported square plate (random S-layer) subjected to
sinusoidal load

 

Y

 

 

 

 

 

Elasticity
LZZ3: 1 sublaminate

 

 

-‘--- LZZ3:3subIaminates
—--- LZZ3:5subIaminates

 

 

 

 

 

 

A +4 A A A A l A A A L

 

3.2 3.3 A ‘ 3.4 ‘ A 3.5!

-6
uzx10

Figure 32. Transverse deflection versus normalized thickness coordinate of a

simply-supported square plate (random 5-layer) subjected to
sinusoidal load

 

 

 

 

 

 

 

 

 

 

 

 

0.5 i T i I i ' l 7 f I I T
Elasticity
0.4 . ---------- LZZ3: 1 sublaminate i
----- LZZ3: 3 sublaminates
“-3 ' ——-— LZZ3:5sublamiuates
0.2
0.1 ' .a. = 4
h 4
5 o
-0.1 -
-o.2 -
-0.3 ~ «
-0.4
-o.5 -

 

 

 

 

 

Figure 33. Inplane normal stress versus normalized thickness coordinate of a
simply-supported square plate (random 5-1ayer) subjected to

 

 

 

 

 

 

  
  

 

 

 

 

 

 

 

 

 

 

sinusoidal load
05 _ vvvvvvvvvvvvvvvvvvvvvvvvvvvvvv .
0.4 . a .~ *
_ = 4
0.3 . h w
0.2
0.1 '
5 0 Elasticity
“ """"" LZZ3: 1 sublammate'
-0.1 D ,' -' ————— LZZ3: 3 sublammate' s ‘
,,,,,, :73"? --—- LZZ3:5sublammate° s
.002 ' , ' 1
.03 _ .........................
-0.4 ............. \
-o.s - ................................. ' «
' 1 e2 - 1 “0.8 .006 '004 '002 0
1x2

Figure 34. Transverse shear stress versus normalized thickness coordinate of
a simply-supported square plate (random S-Iayer) subjected to
sinusoidal load

a-m

0.5

0.4 -

0.3
0.2
0.1

-o.1 ~
-o.2 -
-o.3 ~

-0.4

-o.s~

91

 

r

vvvvvvvvv

 

 

 

T

 

Elasticity

LZZ3: I sublaminate
— -- - - L223: 3 sublaminates
- - — - LZZ3: 5sublaminates

 

 

 

 

 

 

 

 

 

 

AAAAAAAAAAAA

AAAAAAAAAAAAAAAAAAAAAA A

 

 

Figure 35. Inplane shear stress versus normalized thickness coordinate of a
simply-supported square plate (random 5-layer) subjected to
sinusoidal load

 

i
J

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.5 - i .\ W
0.4 - a = 4 .- .‘.
h .' \
0.3 . '\
0.2 "
0.1 - .
Z 0 l \.
7. ,x
_0.l . I .\
-o.2 - ‘\
Elasticity
.0'3 P \ """""" LZZ3: I sublaminate .
_0 4 -~-‘- LZZ3:3sublamiuates
' .- \ — — —- - LZZ3: 5 sublaminates
-0.5 - A - . . - A . l . - . . - . . A
0 0.2 0.4 0.6 0.8
-6
23x10

Figure 36. 'Il'ansverse normal strain versus normalized thickness coordinate
of a simply-supported square plate (random S-Iayer) subjected to
sinusoidal load

92

 

 

 

 

 

   
  
   
   
  

 

 

 

 

 

 

 

 

 

 

 

0.5»' ' ' fi' - ~ - - . . - , - a f
04 xx
0'3“ Elasticity
02. ...... ”23:13“! 0 I
' —-—-— 1.23:2:ch . l
01. ——-— LZZ3:3sublamiuates _/ +
. ./
z
i.- 0
-o.1»
a
-0.2 7'_4
-0.3r
-0.4
«0.5L 4 . A
-l

uxx10-6

Figure 37. Axial displacement versus normalized thickness coordinate of a
simply-supported square plate (sandwich) subjected to sinusoidal
load

 

 

 

 

0.4
0.3 . Elasticity

"""""" LZZ3: I sublammate'
0.2 ' ----- LZZ3: 2 sublaminates
01L --—- LZZ3:3sublamiuates

 

 

 

 

a-IN

V

41.1
-o.2
-o.3
-o.4
.05

1

I'

 

 

 

. A A. 1 _A. A A L

 

—6
uzx10

Figure 38. 'h'ansverse deflection versus normalized thickness coordinate of a
simply-supported square plate (sandwich) subjected to sinusoidal
load

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.5 VVVVVVVVVVVVVV 1
”K
0.4
0.3 P
0.2 " g =
h 4
0.1 '
E 0
h
-0.1 L Elasticity i
_01 _ """""" LZZ3: 1 sublaminate
-'- - - LZZ3:2sublamiuates
'03,, “-—" LZZ3:3SflbWS
43.4 M'fi
fl
-0.5-.".-:-.-...4---7 ................ 1
-30 -20 -10 0 10 20 30
0

Figure 39. Inplane normal stress versus normalized thickness coordinate of a
simply-supported square plate (sandwich) subjected to sinusoidal

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

load
0.5 a
0.49
0.465
5. 0.45
h
— Elasticity
0.43 ........... ”23.13“”. I —-1
- . — ~ - LZZ3: 2 sublaminates
- — — - LZZ3: 3 sublaminates
0.4-30 4+0A3‘310‘AA‘20AA‘A304
0'

Figure 40. Inplane normal stress versus normalized thickness coordinate of a
simply-supported square plate (sandwich) subjected to sinusoidal
load - top face sheet

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0-5 ’ '..'__'_'_'_L—'—'—'--'-'-3--'——'-'—'-—'_—'_—'—_L—'._—I_l.L..Z_'._'..i._‘..'_;_'-'_' ‘ ' '- «
'EI:___ .r'
004 T1.
I
0.3 . i J
0.2 ' 1 fl = 4
h
001 . Ill .4
II'
2 : ‘
it 0 I
. W.
.0 2 _ .’ """"" LZZ3: I sublammate’ i
' 1|; ----- LZZ3:2sublaminates
.03 . 2 - - — - LZZ3: 3 sublammates
.004 ”_‘_ -—- l\"-..
-o.5 a - . - - -‘.“.‘:Z‘if‘.“f.‘::f‘f'—f—T—"f:‘i‘fj:“Tf. f- . . . - .
-0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0
1'
xz

Figure 41. 'Ii‘ansverse shear stress versus normalized thickness coordinate of
a simply-supported square plate (sandwich) subjected to

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

sinusoidalload
0.5i"ififivv""";'VYV'YYTYTV'f‘
\
0.4
03L
0.2 * g: 4 i
0.1'
5 0
" L
'04 Elasticity
.0 2 b """"" LZZ3: 1 sublammate' &
’ -‘—-- LZZ3:2sublaminates
4.3L —--- LZZ3:3sublaminates 1
-0.4
\ MA
-0.5 A n A . ‘. l A L - . . . A,
-10 -5 0 5 10
Txy

Figure 42. Inplane shear stress versus normalized thickness coordinate of a
simply-supported square plate (sandwich) subjected to sinusoidal
load

95

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.49
0.465
E 0045
h
Elasticity
0,43 """""" LZZ3: 1 sublaminate ——<
----- LZZ3:2sublaminates
—— -- LZZ3: 3sublalninates
0.4 AAAAAA - A A 4 A - . - . . ...... - n A -
-10 -5 0 5 10
I”

Figure 43. Inplane shear stress versus normalized thickness coordinate of a
simply-supported square plate (sandwich) subjected to sinusoidal

 

 

   
 

 

 

 

 

 

 

 

 

 

 

load - top face sheet
0.5~'-.:—s' ' i I
0.4 i ‘C"-~ \
03- “"'~--.\
0.2- ‘
0.1.
E. 0
h
-0.1- i
—— Elasticity
-0.2 - """""" LZZ3: 1 sublaminate
- - - - - LZZ3: 2 sublaminates
-0.3* ———- LZZ3:3sublamiuates .
-0.4 .
-o.5--LA.-.----.3.--.3
0 5 10 15
.5
czleo

Figure 44. 'Ii-ansverse normal strain versus normalized thickness coordinate
of a simply-supported square plate (sandwich) subjected to
sinusoidal load

96

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 5 _ ............ f- - - ................ ‘
Elasticity
"""""" .LZZ3:1subhunbuue
--- - - LZZ3:3sublaminates
— — - - LZZ3: 4sublaminates
J
5
h
4&5 ' . -- - - . -454 I .-ss-l,- ............... kg‘
-7.5 -5 -2.5 0 2.5 5 7.5
—6
u x10

Figure 45. Axial displacement versus normalized thickness coordinate of a
simply-supported square plate (CAV) subjected to sinusoidal load

 

 

 

 

 

 

 

 

 

 

 

 

 

0.5: '7?" 7' 'Tfi' ' 'fijfi' -
0.4r '
0.3- 4
a
0.2- 5 -4
0.1P
z
- 0-
h
.001. \ .
:' | 1
”'2' 5' l 9 Elasticity
.03_ ' ,' 1 """""" LZZ3:Isublanunate
;’ I , _._-_ LZZ3:3sublaminatcs
-0.4- g l,’ ,- — —- — — LZZ3: 4 sublaminates
-o.s--':---.A---.--------/-{- -- ---------A-
42.5 45 47.5 50 52.5 55 57.5

—6
uzx10

Figure 46. 'Ii'ansverse deflection versus normalized thickness coordinate of a
simply-supported square plate (CAV) subjected to sinusoidal load

97

 

0.5 ' f ' ' 7 Y
004 F .3: 'l
0.3 ~ Elasticity

a _ 4 """"" LZZ3: 1 sublanuuate'
0.2 ' f, " - - - - - LZZ3: 3 sublaminates *
0 l I} - — — - LZZ3: 4 sublaminates

4t1
4L2-
4L3»
4L4»
4L5--. ll--

 

 

 

 

 

 

 

 

3-IN

 

 

 

 

 

 

 

 

 

 

-20.

 

Figure 47. Inplane normal stress versus normalized thickness coordinate of a
simply-supported square plate (CAV) subjected to sinusoidal load

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 f . - 1 ~ - .
-0J. fl = .
h 4
-o.2 -
5.
h
-0.3 *
Elasticity
I ~~~~~~~~~~ LZZ3: 1 sublaminate
-' —---- LZZ3:3sublaminates
4’" ' I” J — - — — L223.- 4sublaminates
.20 A ‘ A .10 A A ‘ A 0 L A A A 10 ‘ A A #720 4*
0

Figure 48. Inplane normal stress versus normalized thickness coordinate of a
simply-supported square plate (CAV) subjected to sinusoidal load
- outer shell

 

 

0.4 -

v - 1 v v v v V r v v r V fiv f v V r 1 r v—v— ff? ......
0.5; i
.
e
.
.
/
v
A

I
’ q

. .

.

.

 

0.3 " —— Elasticity
2 4 ----------- LZZ3:1 sublammate'
0.2 , .

h --—-— LZZ3:3sublanunates
01_ ---- LZZ3:4sublaminates

 

 

 

 

 

FIN

-o.1 -
-o.2
.03
-o.4 -
-o.s -

 

 

 

 

 

 

Figure 49. 'h'ansverse shear stress versus normalized thickness coordinate of
a simply-supported square plate (CAV) subjected to sinusoidal

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

load
05_ . . . r as - - . . . . . . . . . . . . . .
0.4) T'- "
0.3r a
_=4
0.2~ h
0.1. J
5 o- .
'oel’
Elasticity
-0.2 i """ ' """ LZZ3: I sublaminate
_03 b ----- LZZ3: 3 sublaminates
— — — - LZZ3: 4sublaminates
-o.4- J
-2 0 2 4 6
1:,

Figure 50. Inplane shear stress versus normalized thickness coordinate of a
simply-supported square plate (CAV) subjected to sinusoidal load

99

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.5 - 7 f? f a T
0.4 -
0.3 P a at
0.2 - 7s = 4
0.1 .
E 0 / .
II
-0.1 - .
_03 . Elasticity
: """""" LZZ3: 1 sublaminate
.03 i ----- LZZ3: 3sublanu'nates 1
-- —- LZZ3:4subla.minates
-0.4 *- .
-o.5 » 3 - - - - . - - - - . - a - A . l s
0 20 40 60
—6
5:1le0

Figure 51. Transverse normal strain versus normalized thickness coordinate
of a simply-supported square plate (CAV) subjected to sinusoidal
load

Chapter 5

LAYERWISE PLATE ELEMENT - THIN PANELS

The layerwise plate element developed in Chapter 4, like many conventional
displacement-based C0 -plate finite elements, is plagued by the shear locking phenomenon
when utilized in the thin regime. This chapter will primarily deal with the locking
problem. The assumed strain field approach described in Chapter 3 for a Mindlin C0 -plate
element will be adapted for use with the layerwise C0 -plate element to eradicate locking

in the thin regime.

5.1 Introduction

In the previous chapter, a new technical theory and associated finite element model
were developed for the analysis of thick laminated composite and sandwich panels. This
theory, like most refined theories, is primarily intended for thick plate applications where
the effects of the transverse shear and normal stresses cannot be ignored (i.e., classical
laminate theory is no longer accurate). Although the refined model developed from the
theory may be primarily intended for thick plate situations, it is critical that this model
exhibit robustness even for plates in the thin regime. In addition, the eight-node topology
of the new element permits refinement in the thickness direction. This through-the-
thickness refinement using several elements can also lead to very large span-to-thickness

element ratios regardless of thick or thin plate geometries. This chapter discusses the

100

101

techniques used to make the finite element model of the new sublaminate zig-zag theory
robust for both thick and thin plate applications.

This chapter will focus on the critical problems and techniques to resolve shear
locking, Poisson’s ratio stiffening and stress oscillations that usually plague three-
dimensional displacement-based finite elements developed using general higher-order
displacement fields. The causes of the shear locking phenomenon in thick plate elements
have always been either the presence of an inconsistent transverse shear strain field (see
- Somashekar et al., 1987) or a lack of edge-consistency of the tangential transverse shear
strains (Prathap and Somashekar, 1988; Mohan et al., 1994). The use of reduced and
selective integration techniques (Zienkiewicz et al., 1971; Pawsey and Clough, 1971;
Hughes et al., 1977, Averill and Reddy, 1992), hybrid and mixed methods (Lee and Pian,
1978; N car and Andersen, 1977; and Spilker and Munir, 1980) and modified shear strain
methods (Hughes and Tezduyar, 1981; Hinton and Huang, 1986; and Crisfield, 1984) are
but some of the many ways used to alleviate or remove the shear locking phenomenon
encountered by plate finite elements in the thin regime. A detailed discussion of the
various techniques was reviewed by Prathap (1994). Using the field and edge-consistency
concepts proposed by Prathap (1994), Yip et al. (1996) developed a robust and accurate
four-node C0 Mindlin-type plate element for the homogeneous and isotropic case. The
fundamental concepts used to develop this four-node plate element will be utilized to
derive a field- and edge-consistent eight-node brick element from the “inconsistent”

element described in Chapter 4.

102

5.2 Formulation
In the previous chapter, the assumed displacement fields in a sublaminate were
represented in terms of through-thickness shape functions as follows:
. k k
u,(x. y. z) = uf.’(x.y><bf,’<z)
k = 1, 7

a = 1,2
uz(x.y,z) = wa(x.y)Ma(z)+W3M3(z)

A k k
uy(x, y, z) = (4:1)(x, y)‘l’fll )(z) (80)

where summation on repeated indices is implied. Index on is used to represent the top and
bottom surfaces with 1: bottom and 2: top; and the degrees-of-freedom are represented as:

4k) AU) ,.(2) ..(3) ..(4) ..(5) ,.(6) «(7)
it“: {u ,u ,u ,u ,u ,u ,u }a = (u,v,w,0x,0y,‘cx,1:y)a (81)

(1);“, (P3) and M a are shape functions of the thickness coordinate (z) only and are known

functions obtained either analytically or numerically (see Appendix C for details).

5.3 Interdependent Interpolation

The rotational degrees-of-freedom, 0 m and 0 are coupled to the transverse

ya

deflection degrees-of-freedom, wa through an interdependent interpolation scheme
(Tessler and Hughes, 1983 and 1985) given by:
w,ll = Piwiai—N .0 i = 1,...,4, a = 1,2 (82)

x1 xiu

+ N y ,0 y “1
Here Pi are the bilinear Lagrangian interpolation functions (details in Appendix B).

This interpolation strategy for Wu can be expressed in terms of the natural (local)
coordinate system instead of the global (Cartesian) coordinate system (see Figure 52):
Wt: = Piwia+N§i9§ia+Nni0 i = 1, ...,,4 a = 1,2 (83)

nia

Here N fit and Nni depend on quadratic Serendipity functions, N i + 4 and are given by:

103

1 1 (84)
N11] -ZN8, an 4N6’ Nn3——Nn2, Nn4z—Nn]
and
1 1
N5=§(1—§2)(1—n), N6: i(1_n2)(1+§)
1 1 (85)
N7=§(1—§2)(l+‘n), N8=§(l"n2)(1—§)
By and 9m- in Eq. (83) are rotations corresponding to the natural (local) coordinate

lk
'2'
depending on the corresponding edge k. .5 refers to the tangential direction of the line edge

system. Note also that .5 = ijsg where j: = is the line jacobian and g = éor g ,

(for convention, refer to Figure 52).

ex
ii

 

 

 

Figure 52. Quadrilateral element coordinate description

5.4 Field Consistent Shear Strain Fields

The above interdependent interpolation scheme alleviates the shear locking problem
but does not eliminate it totally. The element developed using this scheme still locks in the
very thin regime. The source of this shear locking phenomenon was identified to be

inconsistency of the transverse shear strain fields with respect to the inplane coordinates, x

104

and y. There are also inconsistencies in the thickness coordinate, 2 but the integrated effect
of the transverse interpolation inconsistencies is insignificant compared to the in-plane
interpolation inconsistencies (Robbins et al., 1991). This stems from the fact that when the
span-to-thickness ratio is large, terms associated with the thickness coordinate, z, will be
small. As such, it is only necessary to make the transverse shear strain fields field
consistent with respect to the inplane coordinates to prevent the shear locking
phenomenon.

In a general quadrilateral, it is difficult to see the consistent form of the shear strains
clearly because of the non-uniform mapping from a natural (local) coordinate system into
a Cartesian (global) system. In order to see this clearly, the transverse shear strains need to
be converted to the natural coordinate system. Prathap and Somashekar (1988) termed the
transverse shear strains in the natural coordinate system as covariant shear strains.

The transformation of any fields from the covariant to Cartesian based coordinate

systems can be given by the following:

Yfiz sz
{ }= m{ } (as)
7n: sz

where [J] is the Jacobian matrix and is defined as:

a a x, ,

[J]=|:11 12 = §Yj (87)
“21 “22 x’n %

The Jacobian inverse is:

- b b 1 a —a
[J] 1 = 11 12 = '7' 22 12 (88)
b21 I722 ‘021 “11

where I] I is the determinant of the Jacobian matrix.

105

Now consider the Cartesian coordinate based transverse shear strain, 7x2:

Ms, 11) = um+ u... (89)
= it)“,z + (uz,§ ,x + uz,n1‘|,x)
Because of the presence of terms from the inverse Jacobian, it is much easier to introduce

some new in-plane displacement fields, it: and "n so that the in-plane displacement field,

u x can be rewritten as:
ux = “@1711 +unb12 (90)

or in terms of its new set of degrees-of-freedom in the covariant frame as:

a, = fiéfldJégb” +ag’;¢§,’;b12 1: 1,2,3; a = 1,2 (91)

where the new set of degrees-of-freedom will be represented as:

171): (17 9,173,173”) =(u,9,13)
é: é t in §§§a (”1,2 (92)

Here (Pg; and (1):; are still shape functions of the thickness coordinate (2) only and are

given in detail in the Appendix D.
Making use of the form for the in-plane displacement, u x , in Eq. (90), the Cartesian
based shear strain, 7“ , of Eq. (89) can be simplified as:

'sz(§:n) = (ufi’z+uz’§)bll + (un’z+uz’n)b12 (93)
= 7§zb11+7nzb12

where the definitions §,x = bll and nu = b12 were used and we have now expressed

the Cartesian based shear strain, szi in terms of a set of covariant based shear strains,

We.» 71"}. Ignoring the bubble function 1323 in the transverse shear strains, the covariant

shear strain, when expanded using Eq. (91) becomes:

106

7&2 = u§a¢£2 z 'i' G§a¢£2 Z "I" T§a¢£312 Z + Wa,§bl(1

_ (1) (2) 3) a = 1,2 (94)
7n: = afield)11th + enaq’na. z + 1110,44,“, z + Wa’nMa
or written in a concise form:
7&2 = EEQCDéQ’ Z + Wa,§04(x (95)
7m = Hg’&¢#&’z+wa,nMa 1: 1,2,3; 01 = 1,2

where the definition in Eq. (92) was used.

Likewise, for the shear strain field, 'sz , there is again a need to introduce some new in-
plane displacement fields, fig and 1’2n so that the in-plane displacement field, uy can be
rewritten as:

or in terms of its new set of degrees-of-freedom in the covariant frame as:

u, = afiwéfibn +ag';\p§,';b,, 1: 1,2,3; a = 1,2 (97)

where shape functions, ‘i’é’g, and Wild are listed in the Appendix D. In a similar manner
as above, the shear strain 'sz can be expressed in terms of a set of covariant based shear

strains, {7:2, ym} . Mathematically, this is as shown below:

lsz(§" Tl) = “y’z + ul’)’
= uy’z + (uzsggsy + uz’nn 9y) (98)
= (12:32 + uz,§)b21 +0711”. "l" “Z,n)b22
= igzbzl + 7nzb22
where
7§2 = aégqlgoi, z + Wa’éMa (99)

y“, = ag’&‘P§"&,z+wa,nMa 1: 1,2,3; on = 1,2

107

In the finite element formulation, bilinear Lagrangian interpolation functions,
Pi(§, 11) are used to interpolate the degrees-of-freedom, 1751"), or equivalently

Ego), and £25113, of Eq. (92) with the exception of Wu:

.... " l
_ P1471820 (100)

i=1,...,4;k =1,2,4,...,7;l=1,2,3;a =1,

4")- «k. -l_. -l. -l
“a — Piuia)’ “£31 — Piaget, 14";

The interdependent interpolation strategy is used for the degrees-of-freedom, W01 (see Eq.
(83)).
Using Eqs. (83, 94, 99) with the small strain assumptions, the covariant shear strain

component, 7%, identified with 71:; is:

u§,z 4" avg
Pi(u§ia¢£g Z + egia‘p 31), Z + Tgia¢é2 Z) + Ma(P‘-’ ng-a + Nat-@951“ + Nni’ gent-a)

Ygz

(1 - 2
——8n—2{_9nla— en2a— 9113a _ 9'q4a}IMOl

+§[(1—n){<1>£2z(ug2a - a...) + (95,... - Gamma. + M.)
+ calms... - 1,1,1} +11 + 1116424113.. - at...)
(2) (3) (101)
”91:30: — 054a)(<1>§a’ z + Ma) + (1):“, 2(Tg3a — 1:40,) }]
+ (iglhcbg; 2(“g1a + “@200 + (11224030, + 03:20,)
+ «>82, it... + it...) + Mam... - w...»
+ (1.1—“2mg gum + 11540,“ cg; 4953,, + 9:400
+ «>233. .112... + it...) + Maw... — w...»
On examining Eq. (101), the coefficients associated with the quadratic term (1 — 112)
will clearly lead to an inconsistent shear strain field, 7§z since they contain only 6111‘

rotational degrees-of-freedom. This will cause spurious constraints and shear locking in

the thin regime where 7&1 tends to zero. The inconsistent terms came from the

108

interdependent interpolation scheme for degreescof-freedom wa. Leaving out the

inconsistent terms, a consistent ' can then be written in a concise form,
£2

- - (I)
1&2 = P .4219ng + (’5':th N§i’§9§ia)Ma

i= 1,...,4; l=1,2,3; a=1,2

(102)

The inconsistent terms came from the interdependent interpolation scheme for degrees-of-
freedom wa . The consistent transverse shear strain, E2 of Eq. (102) can be derived using

the following interpolation for Wu:
Wu = Piwia'l'Ngl-egia l: 1, ...,4, on = 1,2 (103)

instead of Eq. (83) with the interpolation strategy for the rest of the degrees-of-freedom
remaining unchanged.
In a similar manner, the consistent shear strain, 3112 can be derived from the

inconsistent shear strain, 7712 , identified with 'sz , and is given by:

- _ - (1)
I'll - Piufili)a¢lla-Z+ (Pi’nwia+Nni’nenia)Ma (104)

with the wa degrees-of-freedom, now interpolated using:

ya = Piwia+Nni0nia i: 1, ...,4, a = 1,2 (105)
The other two consistent transverse shear strains, igz and :Ynz obtained from the
inconsistent covariant based shear strains, 7&2 and y," identified with sz of Eq. (99) will

be of a similar form to Eqs. (102, 104) and they are:

A _ (l)
_ Ptugiqufia, Z + (Ppgwia 'l' Ngi,§e§ia)Ma

~<
W
N

l

_ — 1 (I) 106
i112 - Piuilib‘llnmz + (Pi’llwia + Nfli’flenia)Ma ( )

i= 1,...,4; l=1,2,3; 01:1,2

Strictly speaking this set of new shear strain fields, {3:2, inz’ 35,2, :Ynz} is still not field

109

consistent. They are only field consistent with respect to the in-plane coordinates.
However, to make them totally field consistent (which may not be necessary at all as
reported by Robbins et al., 1991) is simple. Any term associated with the thickness
coordinate, 2 that is of an order higher than one in the set of shape functions,
{(1)22 z, <14]: 2, W122“, ‘10:; 2} must be removed. But this would severely compromise
the accuracy of the model in applications to thick plates. As will be demonstrated through
numerical results, such a strict enforcement of field consistency is not necessary for the
present model.

This consistent field approach using the new assumed shear strain fields is very similar
to the selective/reduced integration concept except that it only removes the inconsistent
terms. Straightforward selective/reduced integration will not work as it removes a lot more

terms than necessary in the strain fields and introduces spurious zero energy modes in the

process.

5.5 Consistent Penalty Constraints
In the new laminate theory and finite element formulation, four constraints were
introduced to produce a C0 element. The constraints correspond to the top and bottom

rotations and are given as follows:

Cm = (Wu—ex)“ Cw = (my—9y)Cl a = 1,2 (107)

These constraints are enforced by the penalty method in the finite element formulation.
Note that the form of Eq. (107) is very similar to the form of the shear strain fields
obtained from Mindlin plate theory in Chapter 3. These penalty constraints therefore need

to be checked for consistency just like the above transverse shear strain fields. Using the

 

110

relation from Eq. (86), the constraints in Eq. (107) can be recast into their covariant

counterparts as:

can: = “11Cxa+012Cya

18
C (0)

not = “21cm "' “22Cya

Following the above procedures, the covariant based penalty components, Cga , will

be examined:

C501 = (W’§ — 9901

(1 - 2)
= +{-9nla+en2a'en3a+en4a} (109)

(1- +11)

1
4’ 4n){-9§1a _ 9€201 - ”’10: + W201} + (_T—{_e§3a — 9&4“ + w3a_ w4a}

 

Again, the coefficients of the quadratic term (1 —n2) are inconsistent as they are
functions of the rotational degrees-of-freedom 0,"- alone and therefore must be eliminated
to prevent any spurious constraints. Eliminating the inconsistent terms and simplifying,
the consistent penalty component, Cga can be obtained by interpolating the wa degrees-
of-freedom using exactly the same form as in Eq. (102) above instead of Eq. (83) with the
interpolation strategy for the rotational degrees-of-freedom remaining unchanged. Cg“

can then be written in a concise form,
6.3,, = Ppgwia + (N§,,§—P,.)9§ia (110)

In a similar manner, the penalty constraint component, Cm can be derived from the

mconmstent component, cm! as:

Cnz = mem+ (NW, —P,.)en,.a (111)

with the wa degrees-of—freedom interpolated using Eq. (105) above. These derived fields

are totally field consistent since the thickness coordinate. 2, does not explicitly appear in

111

any of the equations.

5.6 Consistent 'li'ansverse Normal Strain Field

A problem that is normally encountered with finite elements that utilize the three-
dimensional constitutive relations is the rapid deterioration of the computed transverse
normal stress as the span-to-thickness ratio increases. This problem is especially severe in
modeling of isotropic plates under flexure. Predicted bending deflections are typically
stiffer by about 10%, depending on the value of Poisson’s ratio, when one element is used
through-the-thickness. This stiffening effect, however, is not apparent if (1) reduced
(plane-stress) constitutive relations are used or (ii) when two or more elements are used
through-the-thickness or (iii) for laminated plates. Prathap (1994) termed the effect as
Poisson’s ratio stiffening. This phenomenon was attributed to an interpolation
inconsistency that prevents the finite element model from simulating a state of zero
transverse normal stress in the presence of general non-zero bending strains (see Robins et
al., 1991). Addition of the bubble function using a nodeless variable W3 in the
displacement field, uz , as seen in Eq. (80) eliminates the stiffening effect. This, however,
does not ensure the field—consistency in the definition of the transverse normal strain, ezz.

Besides the normal Kirchhoff constraints, i.e. the vanishing of the transverse shear

strains, there is another conStraint related to the vanishing of the transverse normal stress,
82s = "(C313xx "' C328yy + C367xyVC33 (112)
In the thin limit, the transverse normal strain must be consistent with the combined
Poisson effects from the dominating in-plane strains. Notice that 8zz exhibits a higher
order in-plane interpolation (quadratic) and a lower order transverse interpolation (linear)

than 8 8),), and ny' In general, these interpolation inconsistencies will prevent Eq.

xx’

112

(112) from being satisfied exactly and a deleterious numerical constraint will be imposed.
Therefore, these inconsistencies in the transverse normal strain field must be eliminated.
Again, using the same argument as in the transverse shear strains, only the in-plane
interpolation inconsistencies need to be removed as the contribution to the strain energy
by terms involving the thickness coordinate z is minimal.

The in—plane normal strains, an and e the in-plane shear strain, 'ny as well as the

Y)”

transverse normal strain, Eu are expanded in terms of the covariant based degrees-of-
freedom in a similar fashion as in Eq. (101). Omitting all the tedious algebraic processes,

the consistent transverse normal strain, ézz , is found to be:

ézz = {(1-§)[W1+ W4 + (6111 —6714)]a

+ (l + §)[w2 + 1123 + (0,]2 - 9713)]11

(113)
+(1+11)[w3+w4+(0&4—0g3)]a}Ma,z/8+W3M3,z or = 1,2
or in a concise form:
ézz(§, Tl) = (Piwia + 11750:“, + 1171".0§im)Mm,z + W3M3,z (114)
i = 1,...,4,0l =1,2
where the interpolation functions are:
P] = (2—é—n1/8; fig. =(1-n1/8; 1V... = (1—§)/8;
F2 = (2+E3-n1/8; figs = -(1-n)/8; 7% = (1+§)/3; (115)
P3 = (2+§+n)/8; ITI§3 = -(1+n)/8; Kim = -(1+§)/8;
134 = (2-§+n)/8; 175,4 = (1 +n)/8; 171,4 = —(1-§)/8;

This is different from what is normally obtained using a l-point or reduced quadrature rule

to evaluate the original transverse normal strain, 82::

113

8ZZ = uZ’Z

(Piwia+N§19§ia+Nnienia)Ma’z+W3M39z (116)
i=1,...,4, or =1,2

Again, the “consistent” strain field, in is only in-plane interpolation consistent and not

transverse interpolation consistent (linear for E cubic for the other three strains, 8 e

Z: ’ xx, yy

and ny ). It can be made totally field consistent even with respect to the thickness
coordinate, 2 by removing all terms associated with quadratic 2 terms or higher in the

contributing a y and ny strain fields when evaluating the strain energy contribution

xx’ 8y
of the transverse normal strain, ézz. For the same reasons given previously for the
transverse shear energy terms, this last step is not necessary as the contribution of the

transverse normal strain field to the overall strain energy is very minimal in the thin

regime.

5.7 Edge Consistency

Even with the choice of field-consistent transverse shear strain fields, edge-
consistency requirements must still be ensured to prevent locking in an element, especially
for general quadrilateral cases. The consistent matching of the tangential shear strain, 752 ,
on any common inter-element edge is crucial. Any deviation will give rise to spurious
constraints on the edges when the elements are non-rectangular. Normal jacobian
transformations of two adjoining elements at their own integration points will redefine this
consistency causing a mismatch of tangential shear strain at the common edge. Prathap
and Somashekar (1988) showed that using element nodal coordinate transformations for
the interpolation of the covariant-based shear strain fields and, in this case, the covariant-

based penalty constraint fields as well, will preserve the consistency definitions of

114

tangential shear strain, 752 , as well as the penalty constraint fields, C along a common

sa’

edge between two elements.
So for the inconsistent Cartesian based transverse shear strain, 7x2, the set of
inconsistent transverse shear strain fields, {7&2’7112} of Eq. (93) will now each be

replaced with their own consistent definitions from Eqs. (103, 104).

7xz(§’ Tl) = Y§2b11+7mb12

.. (1)
= (Piué21¢ga, z + (Pi’éwia + Néi’gegia)Ma)b” (117)

- (l)
+ (PiugliLCDna’ z + (Pi’nwia + Nni’nenia)Ma)b12

i = 1,...,4; l=1,2,3; a =1,2
Following the procedure of Prathap and Somashekar (1988) (see also Yip et al. (1996)),
nodal transformation must be carried out to convert the set of nodal covariant degrees-of-
freedom, {limp rim-a} (see Eq. (92)) back to their Cartesian coordinate definitions, 17115,)
of Eq. (81). Using Eqs. (86, 117), the Cartesian based shear strain, 7x2, after the nodal
transformation will become:

4
_ 1 1
sz(§’ T1) = Z {“ia[(all)ipi¢(§02, zbll + (“2011095102, zb12]

i=1

1 1
+vioj(an)ip,.<r>‘§c:,1;ll +(a22).P.<1>( ’ 1912]

l 1 1111.2

3 3
+Txia[(all)ipi¢(§ozzbll +(021)1P‘q’( ) 1712]

l ' Tia. Z

(3) (3) (118)
+ Tyiaual2)ipi(b§(1,zbll + (a22)ipi¢na,zb12]
(2) (2)
+ exia[(all)i(Pi<p§a, z + Néi’fiMa)bll + (“21)i(Pi¢na, z + Nni’nMa)b12]
(2) (2)
+ 0,,a[(a12),(P.-<I>ga- Z + N§,.§Ma)b1, + (a22)1.(1>,.<1>,mM + Nanawlz]

Where by. are evaluated at the integration points and al.j are computed at the nodal points.

Similarly,

115

4
7W0; 11) = 2 {uia[(all)ipiq’éla),zb21 + (a21)iPi‘P$|l(:,zb22]

i=1

1 1
+Via[(012).-P1‘P(§oz,zb21+(022)'P"¥( ) [’22]

l 1 1111.17.

(3) (3)
+ "xia[(a11).-P iwéa. zb21 4‘ (“21)1P1‘Pna. zb22]

(119)

(3) (3
+ TyimI(“12)ilpiwtm. zb21 + (“22);P1‘Pnoin22]

2 2
+ 6...I<a1.>.—<P.-‘Pé.3,. + NssMobs + (.,,),(mgg, . + N....,M.>b221

(2) (2)
+ Oyia[(al2)i(Pi‘P§a, Z + Ngjnga)b21 + (a22)i(Pl‘PT|a, Z ‘l" Nni,nMa)b22]

+ Wialpi,§b21+ Pi’anZIMa} = 1, 2
This is repeated for the Cartesian coordinate based penalty constraints, C m which are
written as:

Cxa(§’ Tl) = (W’x—ex)a
(120)
= ((W’§ 9x+w9nn9x)—ex)a a = 1, 2
Introducing the covariant rotations, 9g and 9n into Eq. (120) and using the
transformation definition of Eq. (86), the Cartesian based penalty constraints, Cm can

now be expressed simply in terms of the two covariant based penalty constraints, Cs,“ and

Cna

Cxa(§’n) (w9g-e§)a§9x+(wvn—en)an9x
C§a§9x+ Cnan’x (121)

= Cgabll + Cnablz

The above equation is then replaced with the consistent definitions of the penalty

 

constraint components, Cg“ and CW from Eqs. (110, 111) to give a field consistent C m :

an“; ll)= Egab11+ Enabn
=(Pi'§wia + (Ngi’§_Pi)0§ia)bll + (Pi’nwia+ (Nni’n -Pi)0nia)b12 (122)
i = 1,...,4; a =1,2

116

Once again, nodal transformation must be carried out for the nodal covariant rotations,
0b- and 9111' to convert them back to the Cartesian coordinate definitions of the rotations,
0 xi and 0”- . Thus, the penalty constraint component, Cm expressed in terms of 0“. and

0),,- can be written explicitly as:

4
Exa(§in) = 2 {exia[(all)i(N§i’§_Pi)bll + (021)1(Nni’n ‘Pi)b12]
" =‘ (123)
+ eyia[(012)i(N§i’§ —Pi)bll + (“22).‘(Nni’n ‘ Pi)b12]
+wialpi’fibll +Pi,nb12]} a = 1,2

The penalty constraint definitions for Cm are of a similar form to the above equation and

are given by:
4
Eyrfigt 11) = z {9x)a[(al 1 )i(N§1'1§ - Pi)b21+(021)i(NTli’Tl — Pi)b22]
i: l (124)
+ 9,,a[(a,2),(N§,,§ — P,.)b21 + (a22)i(Nm-,n — P0522]
+ wia[Pi,§b21+Pi,nb22]} 01 = 1, 2

Likewise, the consistent normal strain, ézz after transforming the covariant rotations into
the Cartesian rotations will be given by:

4
£2,205.11) = 2 {9,.,1(a1,),IT/§,+ (anti/“,1

i= 1
+ 9y1a[(012),-1V§,- + (022)1Nni]
wiaPi }M

(125)

(1’2 + W3M3’z

The consistent definitions of the Cartesian shear strain fields, 7x, and 7),, , normal
strain field, ézz as well as the penalty constraints, Em and CW derived in this manner
will eliminate locking problems due to spurious constraints and as such an exact order of

integration may be used to evaluate the shear strain energy.

117

5.8 Numerical Studies

First, a spectral analysis of the new, field consistent layerwise finite element model
was performed to check for spurious deformation modes. An unconstrained element yields
six zero eigenvalues corresponding to the six independent rigid body modes (three
translations and three rotations).

The new finite element was subjected to patch tests proposed by MacNeal and Harder
(1985) to determine its suitability for arbitrarily distorted element shapes. Though the new
finite element has an eight-node brick topology (it could also take the form of a four-node
plate element except that it will lose the ability for stacking in the thickness direction; see
Chapter 4 for discussion), it is based on plate kinematics and as such, it is more
appropriate to use the patch tests designed for plate elements.

Membrane and Kirchhofi Patch test. This is an accuracy test of the element to
reproduce the pure stretching and flexural modes. The patch is shown in Figure 53.
Elements are of arbitrary shape patched together to form a rectangular exterior boundary.
As such, boundary conditions corresponding to constant bending curvatures are easy to
apply. The applied displacement boundary conditions and the theoretical solution are also
shown in Figure 53.

The patch is modeled with a single layer of elements through the thickness. The
present element passed both patch tests. It is able to reproduce both the in-plane strains
and stresses exactly for the membrane patch test as well as the constant bending moments
and the surface stresses exactly for the Kirchhoff patch tests for this arbitrarily shaped
patch.

Circular Plate. A circular plate will be used to check the performance of the LZZB

118

model for a non-uniform mesh. This problem was discussed in Section 3.6. A circular
plate with clamped edges for two load and span-to-thickness ratios will be analyzed.
Figure 54 shows the mesh of a symmetric quadrant used to analyze the problem. Two
through-thickness discretizations are used: one and two sublaminates. Their results will be
compared with the exact solutions based on Reissner-Mindlin theory (see Hughes et al.,

1977).

YT b

 

 

 

 

 

 

 

 

I Node x y
4 3 1 0.04 0.02
a 2 0.18 0.03
2 3 0.16 0.08
’ x 4 0.08 0.08
—>
(a) Membrane Plate Patch Test
Boundary conditions:
u = 10_3(x+ y/2)
v = 10'3(y+x/2)
Theoretical solution:
—3
e”: cyy=1xy =10 , on[ = o”, = 1333. ‘ny = 400

(b) Bending Plate Patch Test
Boundary conditions:

w = -10-3(x2+xy+y2)

0

X

—1o’3(x/2 + y)

e, —1o‘3(x + y/2)
Theoretical solution:
Bending moments per unit length: Mn = M” = 1.111x10'7, Mxy = 3.333x10'8

Surface stresses: on = a” = 20.667, Oxy = 10.200

Geometric and material properties: a = 0.12, b = 0.24, thickness = 0.001, E = 1x10‘5 , u = 0.25

Figure 53. Patch test for plates

119

y A Geometric and Material Properties:
1 radius, a = 5.0; thickness, II = 0.1 or 2.0;
Boundaries = clamped

Mesh =12;
E = 1.0x106;
W". 1) = 0.3

Loading:uniforrn pressure, q = 1 or
center load, P = 4
1: Theoretical:
; Uniform

 

 

 

 

 

 

 

sym 2 4
= _q—[ 2_ 2 _ 2 2_ 2]
l w 640“ r)+l_uh(a r)
a Centerload
w = -—P—[(a2-r2)-2rzlne— 80 ln-C]
161“) r kGha2 a

513

where D = _—
12(1—v2)

Figure 54. Clamped circular plate (quadrant)

Figures 55 and 56 show the non-dimensionalized deflection (normalized to the thin
plate solution) along an edge of the clamped circular plate for the LZZ3 models under the
two different load conditions as well as span-to-thickness ratios. For the thin clamped
plate under center or uniform loads, both models are capable of reproducing accurate
results. For a very thick plate of span-to-thickness ratio 2.5, results correlated well with
the exact solution away from the predicted logarithmic singularity in the point load case
but not very well (error of 10%) in the uniform load case. The finite element solutions also
showed variation of the deflection in the through-thickness direction for thick plate cases.
Through-thickness deflection can differ by as much as 20% for the center point load case
and 4% for the uniform load case (see Figures 55 and 56 for the deflection of the top and
bottom surfaces of the plate). In view of the fact that we are trying to model a very thick
plate, results obtained by both models are deemed to be reasonable. Also these results
reinforce the results of the earlier patch tests in that the LZZ3 models can perform well for

non-uniform mesh.

120

 

 

Reissner Theory
1 sublaminate ‘
2 sublaminates j

 

 

 

 

 

 

 

Figure 55. Normalized deflection (W = w161tD/(Pa4)) along line of symmetry
(y=0) of a clamped circular plate under center load

 

 

 

Reissner Theory

 
 

 

 

 

 

 

 

+ 1 sublaminate .
“HA“ 2sublaminates ‘
. tap (loading) surface j
0 l 2 3 4 5
x

Figure 56. Normalized deflection (171 = w640/(qa2)) along line of symmetry
(y=0) of a clamped circular plate under uniform load

121

Numerical results are again presented for the bending of two simply-supported
laminated square panels (of Chapter 4) and an isotropic plate subjected to a double-
sinusoidally varying transverse load. The material property of the isotropic panel is the
same as in the patch tests. The laminated panels chosen for the tests are the sandwich and
TACOM’s composite armored vehicle (CAV) panels. The detailed lamination sequence,
geometric and material properties for both panels are listed in Section 4.4. This time, the
laminates are thin panels having a span-to-thickness ratio of ten and one thousand. The
latter case will be an excellent test of a model’s ability to accurately simulate bending in
very thin laminates.

Numerical results are obtained using a quarter-model with various discretizations of
the plane of the panel and either one or more elements through-the-thickness.
Comparisons are made between predictions of the current models, a finite element model
based on first-order shear deformation (Mindlin) theory and an exact elasticity solution
(Burton and Noor, 1994). The Mindlin-type element selected is a nine-node Lagrangian
element with selectively reduced integration and appropriate shear correction factors, (see
Whitney, 1973; Chatterjee and Kulkami, 1979).

The simply-supported, square plate under double-sinusoidally varying transverse load
will be used to test the convergence characteristics of LZZ3 models for different mesh
densities and the two span-to-thickness ratios. The center deflection at the mid-surface
was used as a comparison measure. Three thickness discretizations were used for all cases.
For the laminated panel, the cases with one and three elements (sublaminates) through-
the-thickness are as described in Chapter 4. The two elements case is obtained simply by

subdividing the panel into two equal parts. The results of all the cases are shown in Table

122

14, where predicted deflections are normalized by the exact solution.

Table 14. Results for square plate, simple supports: sinusoidal load

 

 

Normalized transverse deflection at center

 

Normalized transverse deflection at center

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(a) isotropic panel
(i) span-to—thickness ratio = 10 (ii) span-to-thickness ratio = 1000

Mesh 1 sublaminate 2 sublaminates 3 sublaminates l sublaminate 2 sublaminates 3 sublaminates
2 x 2 1.071 1.073 1.071 1.061 1.067 1.067
4x4 1.022 1.021 1.018 1.015 1.016 1.017
6x6 1.013 1.011 1.008 1.007 1.007 1.007
8x8 1.010 1.008 1.005 1.004 1.004 1.004

(b) sandwich panel

(i) span-to-thickness ratio = 10 (ii) span-to-thickness ratio = 1000

Mesh 1 sublaminate 2 sublaminates 3 sublaminates 1 sublaminate 2 sublaminates 3 sublaminates
2 x 2 1.095 1.094 1.095 1.058 1.065 1.059

4 x 4 1.026 1.024 1.026 1.015 1.015 1.012
6x6 1.014 1.011 1.014 1.006 1.007 1.007
8x8 1.009 1.006 1.009 1.004 1.004 1.004
(c) CAV panel

(1) span-to-thickness ratio = 10 (ii) span-to-thickness ratio = 1000

Mesh 1 sublaminate 2 sublaminates 3 sublaminates 1 sublaminate 2 sublaminates 3 sublaminates
2 x 2 1.052 1.072 1.079 1.060 1.060 1.060

4 x 4 0.993 1.012 1.020 1.014 1.014 1.014

6 x 6 0.975 0.999 1.009 1.006 1.006 1.006

8 x 8 0.971 0.997 1.005 1.002 1.003 1.003

 

 

In all cases, converged results are obtained even for a coarse mesh of 4x4. Note that for the
span-to-thickness ratio of ten which is moderately thick, more thorough comparison
between the different (sublaminate) models must, in general, take into account the
through-the-thickness variation of the deflection (see for example, Figure 61).

Plots of the normalized center deflection for span-to-thickness ratios from four to 104
are shown in Figures 57 to 59 for two of the LZZ3 models as well as the nine-node
Lagrangian FSDT element. The first LZZ3 model selected employed a single element

through-the-thickness of the entire panel, and the other used three elements through-the-

123

thickness. The results clearly demonstrated the superiority of the LZZ3 models as
compared to finite elements based on first—order shear deformation theories. This is
especially obvious in the moderately thick TACOM’s CAV panel which has a very
complicated layup scheme. The LZZ3 models exhibit no shear locking or Poisson’s ratio
stiffening phenomenon when applied to plates in the thin regime. However, when the
element span-to-thickness ratio exceeds 103 , there is a need to increase the computational
precision (e.g. double precision to quad. precision) due to numerical ill-conditioning of
the stiffness terms. This is especially true in modeling thin plates with finite elements
possessing full three-dimensional capability. There is, therefore a limit imposed on the
span-to—thickness ratio beyond which the resulting system of equations will be
numerically unsolvable unless some form of “relaxation” parameters are used. This
numerical ill-conditioning is in no way related to the shear locking phenomenon attributed
to field- and edge-inconsistencies but is strictly related to the machine finite word length
and it is termed “machine locking” by Briassoulis (1993). The study also showed that the
deflection results are relatively insensitive to the choice of value of penalty parameters
used to enforce the constraints of Eq. (41). A low value, however, will yield additional
zero eigenvalues during the eigenvalue test.

The through-the-thickness variations of displacements and stresses for the two
laminated panels as predicted by the two LZZ3 models as well as the nine-node
Lagrangian FSDT model are shown in Figures 60 to 88. Plots show comparison among the
three finite element models and the three-dimensional elasticity solution for through-
thickness variations of inplane displacement, u x , transverse deflection, uz , inplane stress
6

inplane shear stress I transverse shear stress In and transverse normal strain,

xx’ xy’

124

cu. For both the symmetric (sandwich panel) and unsymmetrical (CAV) layups with
span-to-thickness ratio of a thousand, the FSDT as well as the LZZ3 models yield accurate
results for the inplane displacement, u x (Figures 75 and 82) but not for the transverse
shear stress I xz (Figures 79 and 86) with the exception of the three elements through-the-
thickness LZZ3 model. In general, accurate inplane displacements will lead to accurate
inplane stresses (see Figures 77-78, 80, 84-85, 87). However, for moderately thick
laminates with span-to-thickness ratio of ten, the FSDT model is inadequate for both the
sandwich and CAV panels.

In the case of IJrz , predictions using the FSDT model with Whitney’s shear correction
factors are very poor, as expected. As for the LZZ3 models, more than one element
through-the-thickness is needed to capture the variation of IJrz in each of these rather
complicated examples. The transverse shear stress shown in Figures 64, 72, 79 and 86, is
computed using the three-dimensional constitutive relations with no shear correction
factors. Predictions of the normal transverse strain, cu improve as more elements are used
in the through-the-thickness direction. Nevertheless, as mentioned earlier, it is more

common to recover the normal stress, 0'ZZ using the three-dimensional equations of

equilibrium.

5.9 Summary

The present LZZ3 model using the assumed strain field approach has been shown to be
robust and accurate for application to thick and thin panels. It passes the membrane and
Kirchhoff patch tests and has no spurious energy modes. This new finite element model is
therefore a viable alternative to the elements derived using discrete layerwise or zig-zag

theories as well as the conventional continuum elements.

125

 

 

1.035 E

 

 

 

. —e— FSDT
i A LZZ3:Isublamiuate
1-03 r — E1- - LZZ3: 3 sublaminates
1.025
u 1.02

Z

 

”z’exact 1.015 :-

 

 

 

 

 

1.01:
I
lows: ——————— g_ —r H h “"5" ~ ~ _ _ a:
b 4
1) A A1
1 - - -..---s - A 2A..-.\/ . - -----y - - .....y
1 10. 100. 1000. 10000.

a/h

Figure 57. Normalized center deflection versus span-to-thickness ratio of a
simply-supported square plate (isotropic) subjected to sinusoidal

 

 

 

 

 

load
1.] if T fivfi T
+ FSDT
: WA LZZ3:Isublaminate
1.08 . - El - LZZ3:3sublamiuates
1.06 L
”Z

 

uz’exact l 04 L

1.02

 

 

 

 

 

‘ fl- - q‘ n " nfig,“ vn w-s f" "fii
1 L . . - .4-..1 . g2 AAAHY - . -...-.<.{\ r . A_....y
1 10. 100. 1000. 10000.

a/h

Figure 58. Normalized center deflection versus span-to-tbickness ratio of a
simply-supported square plate (sandwich) subjected to sinusoidal
load

 

1.15:r “H":
° . A LZZ3:Isublatninate 1
C --El~- LZZ3:3sublaminates I
1.05 r .
u » 1
z 1 _ r .............. _ n “”46"
u El’ 1
z’exact 1
0.95 :1 g
0.9 E r
0.85 : .
. 1
’ l A /J\A AL A A A AAJ A A AAA! A A A A_L ‘
l 10. 100. 1000. 10000.
a/h

126

 

 

 

 

 

 

 

 

 

 

Figure 59. Normalized center deflection versus span-to-thickness ratio of a
simply-supported square plate (CAV) subjected to sinusoidal load

 

v v v v 1 Y v v 1 1 v v v *v 1 v Y r

 

 

 

 

 

 

 

 

 

 

 

 

0.4
0.3 ~
0.2 -
0.1 -
5 o
h L
-o.1 E, ,. i0
-0 2 _ """""" FSDT
’ ————— LZZ3: I sublaminate
_03 1 — - — — LZZ3: 3 sublaminates
41.4
41.5 - - -----------------
-6 .4

 

Figure 60. Axial displacement versus normalized thickness coordinate of a
simply-supported square plate (sandwich) subjected to sinusoidal

load

127

 

 

 

 

 

 

1L5 r ' ' " ' ' ' ' ' ' ' ' 1' si\"
' : 1
_— E’aSfiCi‘y \' ' / L
0'4 ---------- FSDT 1 , I
0.3 . — ' — ' - LZZ3: I sublaminate 1 /,/’ '
- — - - LZZ3: 3 sublaminates ' //’

 

 

 

 

 

 

 

A 214 A A 216

I 212 ‘
uzx10_6

A 210 ‘

Figure 61. 'D'ansverse deflection versus normalized thickness coordinate of a
simply-supported square plate (sandwich) subjected to sinusoidal

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

load
115) ..- **I* * « IIIII f """ .Lj;;;.;' 4
0.4
0.3 r 1
0.2L 2 =
h 10
0.1 r
5 0
-0.1 * Elasticity 1
........... FSDT
412 ----- .Lzzssrsabuuuauuc ‘
.0.3L —’—— LZZ3:3sublaminates
.0'4 A5
415- i‘ff'TILL- - .- ;- . -- --» ------------------ -
-75 -50 -25 0 25 50 75 100
0x}?

Figure 62. Inplane normal stress versus normalized thickness coordinate of a
simply-supported square plate (sandwich) subjected to sinusoidal
load

128

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.5 vvvvvvvvv 4 - - - ~ 4 - - 44 . ...................
0.49
.
7. - 10
0.465 / i
h """""" FSDT
''''' LZZ3:1sublaminate
0.43 .4 — - — - LZZ3: 3 sublaminates
0.4 AAAAAAAAAAAAAAAAAAA 4-
-75 -50 -25 0 25 50 75 100
0’”

Figure 63. Inplane normal stress versus normalized thickness coordinate of a
simply-supported square plate (sandwich) subjected to sinusoidal
load - top face sheet

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.3 .
0.2 - ‘ g = 10 1
1 E y
0.1 ’ 1
.1. o '
-0.1 T Elasticity
"""""" FSDT
'0-2’ '. -'-‘- LZZ3:Isublaminate ;
_03 _ 1‘ — - - - LZZ3: 3 sublaminates j
-o-4 ..................... % ...........
-05 .;..:..:.;.;I.:-.:'-:'. ...... :‘.-..-..: AAAAAAAAAAAAAAAAAA
-1 5 -1 25 -1 -0.75 -0.5 -0 25 0
TIZ

Figure 64. Transverse shear stress versus normalized thickness coordinate of
a simply-supported square plate (sandwich) subjected to
sinusoidal load

129

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.5-{4' *
0.4 \
0.3»
0.2- 2:10
0.1?
E. 0
'0'1‘ Elasticity
. ---------- FSDT
.0'2 — ' - ‘ — LZZ3: I sublaminate
.03. -——— LZZ3:3sublaminates 4
-0.4 \
057‘1
-40 -20 0 20 40
Ix]

Figure 65. Inplane shear stress versus normalized thickness coordinate of a
simply-supported square plate (sandwich) subjected to sinusoidal

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

load
0.5 - - . fl 444444 - : - - - -
0.49 ’
a
7! — 10
0.465
E, 0.45
h
: Elasticity ___q
0.43 ........... FSDT
: ———— LZZ3: 1 sublaminate
— — - - LZZ3: 3 sublaminates
0.4 A n . A_ A A n - A A A : A A A A a A A A A n A
.40 -20 0 20 40
In

Figure 66. Inplane shear stress versus normalized thickness coordinate of a
simply-supported square plate (sandwich) subjected to sinusoidal
load - top face sheet

130

 

 

 

 

 

 

 

 

 

 

 

 

05’ Eta
0.4 '1, “‘\ i
\. l
"'3' ‘. —- Elasticity E ‘
0.2 ~ ‘. —‘—‘— LZZ3:Isublattttnatc 1
‘ — - - - LZZ3: 3 sublaminates 1
0.1 - 1). :
z ‘ '
1'. 0 1. :
-0.1 r ‘- g 1
\ 1
-0.2 ~ 1. i 7... = 10
_03 . ‘- i
‘ 1
-0.4 '1. 4
4,5 31
--0----5----m---A15--

.5
Esz10

Figure 67. Transverse normal strain versus normalized thickness coordinate
of a simply-supported square plate (sandwich) subjected to
sinusoidal load

 

 

 

 

 

 

 

 

 

 

 

 

0.5—TV """ ;,'\'-_. """" r f f T f - r Y T
0.4 ’
0.3 . Elasticity
....... FSDT
0.2 . ''''' LZZ3: 1 sublaminate
‘\ —— —- LZZ3:3sublamiuates
0e] 7 '. \
5 0 -
h
-0.1 >

 

 

 

 

 

Figure 68. Axial displacement versus normalized thickness coordinate of a
simply-supported square plate (CAV) subjected to sinusoidal load

 

 

 

 

 

 

 

 

 

 

 

0 .5 _.‘ - . - . . - . - . - ......... ‘ . . e
0 4 ‘1‘ i '1
o ' " l
. I
. l
0'3 ‘-\ Elasticity /
0.2 , 1 ---- LZZ3: I sublaminate ,’
. - — - - LZZ3: 3 sublaminates //
0.1 ' '. /
z ‘ ’l
— 0 " 1 / /
h t \
-0.1 - .' 1‘
' 1
-0.2 L 1’ -6 g a
1 uz = 1523x10 (FSDT) , _ = 10
.03 . , ’1 h
I
-0.4 i .1. [I]
-0.5--’---. ------------ I / ----- -.A
930 940 950 960 970
111x10-6

Figure 69. lransverse deflection versus normalized thickness coordinate of a
simply-supported square plate (CAV) subjected to sinusoidal load

 

 

 

 

005 I. '''''''''''''''''
0.4 r
03 t Elasticity
........... FSDT
0.2 ' """" LZZ3: I sublaminate

 

 

 

— — - — L223.- 3 sublaminates /
/ l.
I

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

‘75

Figure 70. Inplane normal stress versus normalized thickness coordinate of a
simply-supported square plate (CAV) subjected to sinusoidal load

 

 

132

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

o ...............................
v [I
..... l : ,
/
-0.1
-0.2
Z a _
r. 7. - 1°
-0.3
/ 7. ..
I . ' --" _‘ Elasticity _
-0.4 1 . j _ _ _: ........... FSDT
j, - ’ ''''' LZZ3: I sublaminate
' 1:11;} —— -— LZZ3: 3sublamiuates
‘0 5 AAAAAAA 1/ A A A 1 A" AAAAAAAAAAAAAAAAA L
-75 -50 -25 0 25 50 75
Oxx

Figure 71. Inplane normal stress versus normalized thickness coordinate of a
simply-supported square plate (CAV) subjected to sinusoidal load
- outer shell

 

 

0.4 r

0.31 Elasficily
FSDT

0.2 r —" ' - LZZ3: I sublaminate
0 1 - - — - LZZ3: 3 sublaminates

 

 

   

 

 

 

 

 

R'IN

-0.l -
-0.2 r
-0.3 ~
-0.4 .
-0.5 . -

 

 

 

Figure 72. 'h'ansverse shear stress versus normalized thickness coordinate of
a simply-supported square plate (CAV) subjected to sinusoidal
load

34"

0.5
0.4
0.3
0.2

0.1 ~

-0.1 -

-0.2

-0.3-

-0.4
-0.5

133

 

V

 

1

V

V

le

fi‘lfi

 

 

 

 

 

Elasticity
FSDT T
" ' — ‘ - LZZ3: I sublaminate
- — - — LZZ3: 3 sublaminates ‘

 

 

 

 

 

 

 

.10.

Figure 73. Inplane shear stress versus normalized thickness coordinate of a
simply-supported square plate (CAV) subjected to sinusoidal load

0.5

 

v v 77 vvvvvvvvvvvvvvvvvvvvvvvvvvv

 

 

 

 

 

 

 

 

 

 

 

 

 

. \\

0.4 - ‘. ‘

\ \

. \
0.3 ’ \. \

\ fl =
0.2 r -\ \ h 10 w
0.1 ~ \ \ .

\ \
5 o» ‘r /
h
-o.1 r
-0.2 - Elasticity 4
_03_ ““ LZZ3:1sublaminate
- — — - LZZ3: 3 sublaminates
-0.4 -
-0.5 * .............................. ‘
20 30 40 50 60
—6
euxlo

 

Figure 74. Transverse normal strain versus normalized thickness coordinate
of a simply-supported square plate (CAV) subjected to sinusoidal
load

I34

 

 

 

 

 

 

 

 

 

 

 

 

 

0.5 Kr'fi'flf'hn "f'r'fi"""""-
0.4
0.3’ 1
0.2' g: 3
h 10
0.1r .
5 0
h
'0-1F Elasticity
........... FSDT
'0'2' ----- LZZJ:Isublaminate
_03 _ —- —- LZZ3:3sublaminatcs .
'0e4 \
-0.5*.........L-.Ln.s ...A.ssan.s.....:
-0.075 -0.05 -0.025 0 0.025 0.05 0.075
a
I

Figure 75. Axial displacement versus normalized thickness coordinate of a
simply-supported square plate (sandwich) subjected to sinusoidal

 

 

 

 

 

 

 

 

 

 

 

 

 

load

05*? 1 f f f f T f : 'u‘
I | I
0.4 : i 1:
0.3»: g i

I I
0.2“ 2:103 i l-
h - l
0.1» : E
2 ° T: :
.01. 9 :-
' Elasticity . :
-: ........... FSDT . h
'0'2 g - - - - - LZZ3: 1 sublaminate : {
-0.3-E —--- LZZ3:3sublaminatcs i :.
’ . l
-0.4 : i 4:
-o.s-5-.---A.E--A.L-A..-!.-'-

51.9 51.95 52 52.05
u
Z

Figure 76. 'lransverse deflection versus normalized thickness coordinate of a
simply-supported square plate (sandwich) subjected to sinusoidal
load

135

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.5V ' , ' ' ' r ' V V L V f - r f ' ' f '7"
0.4 l
0.3? ‘
.- J
0.2 7', = 103
0.1b *
5 0
'0-1' Elasticity
........ .. FSDT ‘
'02 —"‘- LZZJ:IsubIamiuate
.03. -—-— LZZ3:3subIaminates .
-0.4 / I
-o.sh"_.:.rr.nn....-s...4
-l -0.5 0 0.5 l
0'

Figure 77. Inplane normal stress versus normalized thickness coordinate of a
simply-supported square plate (sandwich) subjected to sinusoidal

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

load
0.5..-7..--c--,v-,vaa
0.49
0.465
3:103 [
5045 h
h .
Elasticity
0.43— ___________ FSDT
"—‘— lZZ3:13ublalmuate'
-~ -- L223: 3sublamiuates
0'4-1“‘k-o.s%flioku‘o.sé 1

xx

Figure 78. Inplane normal stress versus normalized thickness coordinate of a
simply-supported square plate (sandwich) subjected to sinusoidal
load - top face sheet

 

 

 

 

 

 

 

 

05- """ 1:; 1.? 1:.;.;;;;;_;:_<::_v . ~ . v . -
M“
0.4 .‘
l
0.3 . ,'
0.2 - (_l = 103 .
h E
001 h J
z o ‘
h
.001 p Emmi”
........... FSDT
'0‘2 ' u ----- LZZ3:1sublaminate 3
-0.3 - l r — -- LZZ3: 3 sublaminates *

 

 

 

 

 

 

 

 

 

 

 

-l75 -150 425 -100 -75 -50 -25 0

Figure 79. 'D'ansverse shear stress versus normalized thickness coordinate of
a simply-supported square plate (sandwich) subjected to
sinusoidal load

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.5 * k i ' ' - ' , - . 9
0.4 1
0.3 ’
0.2L 2 = 3
h 10
0.1 ~ .
5 0
h
-0.1 ’ Elasticity
"""""" FSDT
'0-2' “"'— LZZ3:Isublammat' e 4
.03_ ——-- LZZ3:3sublaminates
-0.4 \
-o.s-.-Ahr....r an-..c.zai-
-0.4 -0.2 0 0.2 0.4
I”

Figure 80. Inplane shear stress versus normalized thickness coordinate of a
simply-supported square plate (sandwich) subjected to sinusoidal
load

 

 

 

 

 

 

 

 

 

 

 

 

 

0.5_-\-I.--.V.V-H.flfl H--_-s-fvws.
0.4 '33
0.3L 'x.\ .
0.2r ‘-\_ 2: 3 J
\\ h 10
091b ‘\'
E 0 '\ \
h \
-0.1 \.\
Elasticity x.
-0.2- —'--- LZZ3:1sublantinatc ‘-\.
.03. --—- LZZ3:3sublamtnates \,\.\
~0.4 “\.\3
-0.5-.-. his-.-u.c-c- U--.Jns+_u..-‘:
-0.015 -0.01 -0.005 0 0.005 0.01 0.015
8

22

Figure 81. 'Il‘ansverse normal strain versus normalized thickness coordinate
of a simply-supported square plate (sandwich) subjected to
sinusoidal load

 

 

sssssss v v v .,....,v v .
oesr\ 1
d

0.4 r

_ Elasticity
03 ........... FSDT
0.2 “‘—‘— LZZ3:Isublaminatc
— - - - LZZ3: 3 sublaminates

 

 

 

 

 

0.1

Y

 

 

FIN
é
/

r

.0.1
-0.2 -
-0.3 -
-0.4

-0.5~L.-.--.A-.s--s-.+---.-.n-.*
-0.2 -0.1 0.1 0.2 0.3

10

#lfi

T

 

 

 

 

E:

Figure 82. Axial displacement versus normalized thickness coordinate of a
simply-supported square plate (CAV) subjected to sinusoidal load

 

 

 

 

 

 

 

 

 

 

 

 

05 -' ' * E ' a- fl . ~l.

1 l

0.4. , l‘

0.3* 3 l :

a _ l I

0.2. ,..—10 I E.

0.1- : i

Z a .

T. 0’ l i‘

. ' u.

-0.1 I :

.o.2- —— Elasticity . :.

----- FSDT I I

.0.3- ----- LZZ3:Isublaminate :.

— - — — LZZ3: 3 sublaminates ' I

-0.4- I :

051.1.1-.a-11u.11...119..-:4
170.1 170.2 170.3 170.4 170.5

uz

Figure 83. Transverse deflection versus normalized thickness coordinate of a
simply-supported square plate (CAV) subjected to sinusoidal load

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.5_ ............ .,ss../.fi,.d
0.4’ -
. Elasticity
0'3 """""" FSDT
0.2. —“"‘ LZZ3:1sublaminatc
0 1 — — -- LZZ3: 3sublaminates
5 o. / .
h
w [E 1
.02L 1 7
e f
I j7 g = 103
'0o3' / 1 h
-0.4- / ,’
/
0511-111..-//.-L-LT.-. ......
-300000 -200000 -100000 0 100000 200000
0’
xx

Figure 84. Inplane normal stress versus normalized thickness coordinate of a
simply-supported square plate (CAV) subjected to sinusoidal load

139

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

o .................. - ........ c
-0.1 I;
a 3 / 1
ii = 1“ / 1
-0.2
/
E. I
-0.3 [I
l
/ l Elasticity
_0.4 f ........... FSDT _
7 ""‘ LZZ3:1sublaminat¢
[ - - - - LZZ3: 3 sublaminates
I
_05 ..... WJ/l .......
-300000 -200000 -l00000 0 100000 200000
0

Figure 85. Inplane normal stress versus normalized thickness coordinate of a
simply-supported square plate (CAV) subjected to sinusoidal load
- outer shell

 

P
u-
1

 

0.4 '
0.3 .
0.2 '
0.1 -

 

 

     

3'IN

 

 

Elasticity
........... FSDT

*‘—“ LZZ3:1sublaminatc 4
- - — — LZZ3: 3 sublaminates

-0.l* /'
-0.2r ‘x.
-0.3+ °"\.\,
-0.4~

 

 

 

0‘.
‘.
‘.
‘-
‘.
‘.
~

' - .
I
l A A A A I A A A L A A L A a j A A A A l A %L 4

 

 

 

-250 -200 -150 -100 -50 0

Figure 86. Transverse shear stress versus normalized thickness coordinate of
a simply-supported square plate (CAV) subjected to sinusoidal
load

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.5; ........ C ............
0.4-
0.3- g _ 103
0,2. h
0.l~
.2- 0b \ 4
h
-0.1 .
-0.2* .
Elasticity
'04“ ------ FSDT
.04_ —-—-- LZZ3:1sublamiaate
' —--— LZZ3:3sublantiuatcs
-0.5.1\
450000 400000 -50000 0 50000 100000
1:
xy

Figure 87. Inplane shear stress versus normalized thickness coordinate of a
simply-supported square plate (CAV) subjected to sinusoidal load

 

vvvvvvvvvvvvvvvvvvvvvv

0.5
0.4
0.3
0.2 ~
0.1 -

 

 

 

   

Elasticity
"""" LZZ3: 1 sublaminate
- - - - LZZ3: 3 sublaminates

 

 

 

 

 

a-IN

-0.1 .
-0.2 -

'004
-0 5 .
. AAAAAAAAAAAA

 

 

 

 

 

Figure 88. Transverse normal strain versus normalized thickness coordinate
of a simply-supported square plate (CAV) subjected to sinusoidal
load

Chapter 6

NON -LINEAR MODEL

In this chapter, some aspects of the formulation and solution of geometrically non-
linear problems will be treated. The study will concentrate on small strain/moderate
rotation problems, typically encountered in slender, flexible structures. The solution will
be based on a Total Lagrangian formulation, and the procedure adopted to trace out the

complete load-deflection path is the Newton-Raphson method.

6.1 Introduction

Most of the early work on geometric non-linearity related primarily to the linear
buckling problem (see Crisfield, 1994 for his review on geometric non-linear problems).
For genuine geometric non-linearity, ‘incremental’ procedures (or forward Euler) were
originally adopted by Turner et al. (1960) using the ‘geometric stiffness matrix’ in
conjunction with an updating of coordinates.

For non-linear analysis, a number of ‘alternative’ strain measures are used. The
common strains are the rotated engineering, Green’s, Almansi’s and rotated log strains.
These strain measures are then used in the virtual work equation to develop the finite
element model resulting in either the ‘Total Lagrangian’, ‘Updated Lagrangian’ or ‘Co-
rotational’ formulations. These formulations are identical from a continuum mechanics

point of view, the only difference being the choice of reference configuration. However, in

141

142

discretized form, applied to a structural element based on some approximating theory, the
performances of the various formulations are usually quite different (see Mattiasson et al.,
1986).

Composite structures are designed such that their in-plane stiffnesses are almost
always much greater than the transverse stiffnesses. It is therefore pertinent to consider
only the von Karman non-linear strains (a special case of the Green’s strains) which will
account for moderately large deflections and small strains. The inclusion of the von
Karman non-linear strains will allow geometric non-linear analysis of laminated
composite plates using the Total Lagrangian approach.

The non-linear algebraic equations of the non-linear model are solved by iterative
methods, which will be discussed in Section 7.3. In iterative methods, the expressions
involving the dependent variables and their derivatives are computed using their values
from the previous iteration, so that the integrals can be evaluated (by numerical integration
methods).

As in linear analysis, problems occur with shear locking and, again as with linear
analysis, these can be improved by various methods such as the use of (selective) reduced
integration or the assumed strain field approach as discussed in Chapter 3. We will once
again adopt the approach of the assumed strain field approach for the layerwise plate

element to prevent shear locking.

6.2 Theory Formulation
The displacement field given in Eq. (62) with respect to the initial configuration is

reproduced here for convenience:

143

a“) = 17(k)(X, Y)<I>ff’(2)

x a

(n) __ ..(k) (k) k = 1, ...,7

uy _ ua (X, Y)‘Pa (Z) a = 1,2 (126)
14:") = wa(X, Y)Ma(Z)+w3M3(Z)

with the shape functions (by), ‘l’g‘l, M a and M 3 defined in Appendix C.

In general, the upper-case X, Y, Z and the lower-case x, y, 2 will relate to the initial and
current coordinates respectively. In the present work, this distinction is not really
necessary since the Total Lagrangian procedure always refers measures to the initial
undeformed configuration. The stress measure that is conjugate to the Green’s strain
tensor, E is the second Piola-Kirchhoff stress tensor, .5 and they are given as:

1.5 = %(F‘T-E—l); 12,, = %[%§%_8"] (127)

and

_ poaXi 8X1. 128
'7 — pdxksk’dxl ( )

 

p T
S = B905") 4.5-(5"); S

Using the second Piola-Kirchhoff stress tensor, .5 and the Green’s strain tensor, E as a
suitable pair of energy conjugate stress and strain measures, the virtual work equation is
given by:

j SUM,j dVo = j tifiui dFo+ j pbifiui dVo (129)
V F V

0 0 0

V 0 and F0 are the volume and surface area, respectively, in an arbitrary known reference
configuration. The vectors, Q and g are the body forces per unit volume and the surface
tractions, respectively in the initial undeformed state.

If the in-plane strain terms, (e.g., (aux/8.302 and (Buy/8X)2 ), are small compared

144

to one, then they can be considered negligible, and the Green’s strain tensor, E of Eq.

(127) degenerates to the von Karman strains. They can be expressed as:

E Bux 1 BuZ 2
xx - 327 +502?) w
au u
- y z
2Eyz 5—2- +37 2E),z

Variation of Eq. (130) leads to:

dfiux
8Eix:=:;§
dfiu
SEW = 37 y
afiuz

SEZZ = -—Z'
dSu
ZSEYZ = 7 y
afiux

25Exz = a—Z
dfiux

26Exy = 5'?

2 duz
) Ezz=a—Z

Bu

_ x
icy-37+

au

___y 1(a_“z
'ay+2ay
Bu auz

I

first

(130)
Bu

__y+

8X

2E (auz auz

$53?

)

duz dfiuz
+ (a7 a7
(BuZ dfiuz

)
)

+5???

(131)

Bfiuy

duz dfiuz
a")? (

asaz duz
+_.__
ax aY (

+3737

) l

The constitutive relations relating the Lagrangian-Green’s strain and second Piola-

Kirchhoff stress for the case of small strains and large rotations are of a similar form to Eq.

(51). The proof is outlined in Appendix E.

145

 

 

 

 

F -
PS w.) Cfi’j’c‘é’c‘g’ 0 0 Cfi’g’ -E '(k)

x" C“) C“) 0 0 C“) "’
Syy 22 23 26 E”

(k) (k)

53, = C33 0 0 C36 Ezz (132)
SYZ C2? C3? 0 2Eyz
S 2E

XZ Sym C(Slg) O 12
SM, 2E ,
_ ._ (k) _ n.

 

 

where the index k is to denote the layer number.
If the non-linear (von Karman) strains, Eq. (130), with the constitutive relations, Eq.
(132), are to be included in the virtual work equation, Eq. (129), the procedure discussed

in Section 4.3 does not change, except that the resulting algebraic equations will be non-

linear. The Principle of Virtual Work yields:

 

 

0- Mail" My)? ems)? await"
’I" 3X+8Y+BY+6X
- (Raff) - (R,)ff’16a;"’

(9,), ale),
' ax + 31’

 

- (Qz)a]5wa + (flaws (133)
a BWB a BWB a aw,
‘[a—x[(”x)asa—x ] * 517W mm; ] +a—x[(Nyx>as'a—y ]
3w
+ a—aY-[(ny)aB-a7p]]wa]d.4 + may)
(1,3 = 1,2;k =1,2,4,...,7;l =1,...,7
where Lag”) are the boundary terms. The stress resultants are as defined in Eqs. (66,

67) except that the stresses are replaced by the second Piola-Kirchhoff stresses and the

reference configuration is the initial undeformed configuration. The additional stress

resultants that are not defined by Eqs. (66, 67) are:

146

N i
(Ix/x)”, = 2 053m, Y, Z)Ma(Z)MB(Z)dZ]

i=1 0

N i
(Ny)aB = 2 OS”(X, Y, Z)Ma(Z)MB(Z)dZ] (134)

i=10

N i
(”Mala = 2 styor, Y, Z)Ma(Z)MB(Z)dZ] = (Nyx)aB
i=1 0

The boundary and loading conditions for the plate are:

mag“) = — jun, + m2);"’5ag" dA — j (Ml B + M23);k)n38ilflk) as
A

32

L(wa) = — I[qa5wa + 315%] dA — j [(Qn)a8wa + Qnaw3] as
A 52 (135)

—j [(Nn)anB,n + (an)anB,s]5wa dS
s
0t,|32=1,2;k = l,2,4,...,7
where the definitions of Eqs. (70, 71) were used after again replacing the stresses with the
second Piola-Kirchhoff stresses. The subscripts n and 5 refer to the outward normal and

tangential directions respectively and they are with respect to the initial undeformed

configuration. Note that the following expressions were used to obtain the final form of

Eq. (135):
N" = 11513, + 2nylnxlny + Nylfiy
NS = (Ny' Nx)lnxlny + ny(13X +13y)
.3. = l 5.3—-1 9— (136)
ax nx an "Y 83
a _ a a
3): — lny 524-1,” .a—S

where [M and I"), are the direction cosines of the outward normal of the boundary in the

initial configuration.

147

The finite element model for the layerwise plate theory is obtained by substituting the

element approximations, Eq. (137), into the virtual work equation, Eq. (133).

u = Piui v = Pivi
w = PlWi+in9xi+Nyi9yi
6x = Pie”. 0y = Piflyi (137)
1,, = Pit“. ry = Pity.
l = 1, .. ,4

Pi are the bi-linear Lagrangian functions and N N yi are defined in Appendix B.

xi’
The finite element model can then be expressed in a matrix form,

[Ke]{de} - {Fe} = 0 (138)
where [K e] , {d‘} and {F e } are the elemental stiffness matrix, nodal degrees-of-
freedom vector and generalized force vector, respectively (see Appendix F for details).

Similar to the linear layerwise finite element model, the inconsistent transverse shear
strain, normal strain and penalty constraint fields must be replaced by their consistent
counterparts, Eqs. (118—119, 123-125) in the non-linear model to eliminate the shear

locking and Poisson’s ratio stiffening effects.

6.3 Newton-Raphson Iteration Method

The non-linear algebraic equations of the non-linear model are solved by iterative
methods. The element stiffness matrix in Eq. (138) is non-linear and unsymmetric when
the non-linear von Karman strains are included. Therefore, the assembled non-linear
equations will be non-linear and unsymmetric. The assembled non-linear equations must
be solved, after imposing the boundary conditions, by an iterative method, which seeks an

approximate solution to the algebraic equations by linearization. The two most common

148

methods are the direct iteration and the Newton-Raphson iteration methods. The latter
method is probably the most widely used because of the symmetry of the tangent stiffness
matrix obtained.

The Newton-Raphson iterative scheme is described in many books (see for example,
Ochoa and Reddy (1992); Crisfield (1994)). The method is based on a Taylor series
expansion of the algebraic equations, Eq. (139), about the known solution, {d}i at the ith
iteration. The global finite element equations can be written as:

[K({d})]{d}-{F} = 0 (139)
For an approximate solution {d}i :

{R} = [K]{d}-{F}¢0 (140)
where {R} is called the residual and is generally not equal to zero unless convergence

has been attained. Expanding {R} about {d}i , we obtain

 

{R(d”‘)} = {R(di)}+[3§§i]'<{d}i+l—{d}")
(141)
32{R} l'+1 l' 2
+2’—,[-a——{d},] ({d} —{d}) +

Since our goal is to minimize the residual {R(d‘)} , we can write Eq. (141) as:

0~ {R(d") )i}+[KT ]{Ad"}+0({Ad"})2 (142)
where [K T] is the tangent stiffness matrix (or geometric stiffness matrix),
.- _ .- _ 3{R} ‘

[KT] - [KTM )1 - [am] (143)

(the stiffness coefficients of [K T] are listed in Appendix F). If the finite element model is
derived using a variational principle, it can be shown that the tangent stiffness matrix,

[K T] , will always be symmetric even if the direct stiffness matrix, [K] , is symmetric for

149

non-linear problems. Neglecting quadratic terms and higher, the truncated Taylor series

can be rewritten as:

{Adi} = -[K"T]"{R(di)}
. _1 . . (144)
= -[K1(d')] (([K(d‘)]{d'} - {F}))
and the total solution at the (i + 1 )th iteration is given by:
{di+ 1} = {d‘} +{Ad’} (145)

The iterative solution of Eqs. (144, 145) is continued until a predetermined criterion
and tolerance, 8 (normally chosen in the range, 10.5 S 8 S 10—1 ) is satisfied. Two typical

error critieria are:

 

 

. , , . _ ({Adi}T{Adi})l/2
Displacement cntenon. IIdII2 _ ({di}7{di})1/2 < (146)
- - - . _ ({R’}T{R’})“2
Force (orResrdual)cr1ter10n. IIRII2 — ({F}T{F})1 /2 S

There are many other convergence criteria such as the energy norms, but the above two
criteria often are the most useful.

The method described above yields the solution for a given load, but it is often
desirable to know the load-deflection path. Also, these methods may be inefficient, even
non-convergent, for highly non-linear problems. It is therefore desirable to use
incremental strategies to solve most non-linear structural problems. Eq. (139) can be

rewritten as:
[K({d})] = MFO} (147)
where {F 0} is a reference load and A. is a scalar parameter. The methodology does not

change except that the load is applied in increments. For the kth increment, we can rewrite

Eq. (144) as:

150

{Adi} = -[KT(dl)1;‘<<[K(dl)1{dl}-x,.{F,})) (148)
where

{d)‘;+'} = {61%;} +{Adi}

(149)
A, = 1,,_,+A).,

and {d3} is the solution at the end of the last increment. In general, it is good to have
three to six iterations per increment. This is controlled by the load increment AM.

Note that at the beginning of each iteration the tangent stiffness matrix and residual
vector must be updated using the latest available solution, {di} or {dfi} . Rather than
updating the tangent stiffness during every iteration, it is common to make the

approximation:
[Kidk = [thc (150)

and as such update [K T] only at the beginning of each increment. Since the tangent
stiffness matrix is kept constant and only the residual vector is updated during the
iterations within each increment, there will be computational savings. Such an approach is

called the (incremental) modified Newton-Raphson method.

6.4 Numerical Results

Geometric non-linear responses of composite plates can be very different from
isotropic, metallic plates. The responses of composite plates are highly dependent on the
lamination scheme and boundary conditions and can be very significant even at small
loads and deflections. Two laminates were selected to assess the accuracy of our model.

The first is a single-layer orthotropic plate and the second is a cross-ply (0/90/90/0)

_LI-rlm

 

151

laminate. The cases selected are attractive because test results exist for the non-linear
behavior of square plates with all four edges either simply supported or clamped under
uniform transverse loading (see Zaghloul and Kennedy, 1975; cited in Ochoa and Reddy
(1992)). The geometries and material properties for the 0° ply that were used in the
experiments are as follows:

(i) Orthotropic plate:

a=b=12,h=0.138

E, = 3x106, E2 = E3 = 1.28x106
6 (151)

(ii) Cross-ply laminate (0/90/90/0):

a=b=12,h=0.096

E, = E3 = 1.8282x106, E2 = 1.8315x106
6 (152)

v,2 = v23 = v,3 = 0.2395

A quarter plate model will be used to analyze the two cases. Different meshes were
used to determine the convergence characteristics of the new layerwise zig-zag element,
LZ23 for the two cases. In Figures 89 and 92, the load-deflection curves of different
inplane discretizations are shown. A mesh of 4x4x1 for both cases yielded converged
results in both cases. Through-thickness refinement using four sublaminates (each
representing a physical layer) reinforced the fact that a mesh of 4x4x1 is adequate to
model both cases. The load—deflection curves for the through-thickness refinement for
both cases are shown in Figures 90 and 93.

In Figures 91 and 94, the present theory was compared with solutions obtained by the

152

classical laminated plate (CLPT) and the first-order shear deformation (FSDT) theories in
addition to the test results obtained by Zaghloul and Kennedy (1975). A mesh of 8x8xl
was used for the comparison. For the simply-supported orthotropic plate, LZZT as well as
the shear deformation element yield results that are in excellent agreement with the
experimental data (see Figures 89 to 91). The classical laminate bending element under-
predicts deflection.

The load-deflection curves obtained from the quarter-plate model for the clamped,
cross-ply laminate are shown in Figures 92 to 94. The finite element solutions of the LZZ3
model under-predict the deflection (almost 15% at the maximum load of 2) when
compared with the experimental data. This trend is also seen in the shear deformation
element (see Figure 94). In fact, the load-deflection curves by the two different theories
are almost identical. Since mesh refinements of the LZZ3 model showed that the results
are converged. This therefore lead us to the same conclusion as Ochoa and Reddy (1992).
They attributed the poor prediction of their shear deformation element to be due to
inaccuracies in the representation of the material stiffnesses as well as the actual boundary

conditions of the experiment.

6.5 Summary

The prediction of the load-deflection path by the non-linear LZZ3 model does not
differ much from the shear deformation element of Ochoa and Reddy (1992). This is to be
expected since the span-to-thickness ratios of the two experiments are large. In Chapter 5,
we have demonstrated that the first-order shear deformation element perform adequately
for large span-to-thickness ratios except in cases of laminates with drastically different

materials in adjacent layers or large number of plies. Therefore, in order to evaluate the

153

true performance of the non-linear LZZ3 model, one must also compare the in-plane
displacements as well as the (Cauchy) stresses. However, because of lack of experimental

data we are unable to do so.

154

 

05,-.--fs---,----,-ss
D

 

 

 

 

 

 

 

— Experimental f‘:
I <
. —e~— 4x4x1 A .-/'
0.4 ' A 8x8x1 ,..!
. — a— I6x16x1
, linear ,3, ,
0.3 r A "/ 1
”max ,. non-linear 1
1'5; ,... v
0'2 ' ,;. ‘ /€‘/’: +
0.1 ’ 4., I
; /
1 ' I
0 - T
0 0.5 1 1.5 2
load, 2»

Figure 89. Load-deflection curves for the bending of a simply-supported, square,
orthotropic plate under uniform transverse loading - effect of inplane
mesh refinement

 

 

 

 

 

 

 

 

0.5 ' ' i ' ' v v v v v 1 f v v V r ‘ J
— .A.
Expenme' ntal
0 4 ’ —9— 4x4x1 ' 4:: / 4.
O b ' . . '/
. "A " 4x4x4 _-/ ,
1 —EJ- — 8x8x1 . ._./.-,‘ J
C — «a— 8x8x4 1W” "/ i
0.3 L -/ “
b ' é '
w 4
max .r‘" non-linear _ .,— .
h »/ t‘ ”’5 i v- ‘
0.2 _ / 1/
L a, .A i
1 a
0.1 - .,, .
l '4 4
/ J
i
‘ J
o A A g A A A A A A a A A A A r A A A A 1
0 0.5 1 1.5 2
load, 1.

Figure 90. Load-deflection curves for the bending of a simply-supported, square,
orthotropic plate under uniform transverse loading - effect of
through-thickness mesh refinement

155

 

0.5

 

 

 

 

 

 

 

F 9‘
b ‘ .
: —- Expenmental f. (- " .
b
0.4 - ‘9_ CL" I, _
[:1
—-El— LZZ3 , ff
. linear
0.3 - a
. ,...”
w ""
r" . .
W“ e. a non-linear - A.
0 .4”.- ,.M"" a 4
0.2 - .,.- ,... ,
1 .' .02 . I , m" ‘ 3 g Q .
L ._ - e v ,
L '43; V 1
I u r ~/¢ <
001 . 1,;/% 1
0 . A A A A A A A A A A AA A A AA 1 A A A . s
0 0.5 l 1.5 2
load, 2»

Figure 91. Load-deflection curves for the bending of a simply-supported, square,
orthotropic plate under uniform transverse loading - comparison
among different theories and experimental results

 

 

 

 

 

 

 

 

0.5 f a V V V Y ' ‘ f V T if ' ' V V ' " r T
h . 1
. _— Experimental
0 4 ’ —9— 4x4x1 A,
. - '71
. . --¢.-- ‘ 8x8x1 A .,v’ ,
’ —- E1- — 16x16x1 /
. . ._ I»; ’ .
0,3 - linear ,-” .
. , . I
w I
max 1 ‘3? 1
L /
0-2 t 4" non-linear
t A .3. ‘
i .4.» .— 5;- " .c. .—
) ’3 ,A i‘ i“
001 P ’. 5
r ’3- "
) /’~’
1
0 . A A A A J A A A A L A A A A L A A A A l
0 0.5 1 1.5 2

load, 1

Figure 92. Load-deflection curves for the bending of a clamped, square,

symmetric cross-ply (0/ 90/ 90/ 0) plate under uniform transverse
loading - effect of inplane mesh refinement

156

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.5 a f a e r a fi .
I —— Experimental I
. —e— 4x4x1 fl .
0'4 f A 4x4x4 _ 4’0 J
’ —EJ- — 8x8x1 .,Iwfi’ -
I — «tr — 8x8x4 , f ”’
0.3 ‘ linear ,2”
”max . 7; ' i I, 2
0.2 . non-linear J
A. - -19.: - 2 .3. ‘ ’ J
0.1 - , , .. ~
J /: J 1
0 ’ A A g 1 1 A A A A n A A A A n J
0.5 1 1.5 2
load, 1
Figure 93. Load-deflection curves for the bending of a clamped, square,
symmetric cross-ply (0/90/90/0) plate under uniform transverse
loading - effect of through-thickness mesh refinement
0.5 fl . - . f , . ,
J —-—-— Expenme’ ntal " :
0.4L CL” ” .293
. A FSDT ééf -, A
’ _’B" LZZ3 .’_:.)‘-' “J J
. ., .
0.3 ’ linear e __./Q J
w : ’. 7'9 J
max C ’. / :‘fir non-linear A. ,J
0_2 . [.91 ¢ 6 ' .
r I e, .
: :‘2' ’4 ...... ' 'A“ _xéqu—“A
J /, flex-W'A’gé" }
0'1 T /fl, “Aer-J" " 1
; i". 1
0 0.5 1 1.5 2
load, 1

Figure 94. Load-deflection curves for the bending of a clamped, square,

symmetric cross-ply ( 0/ 90/ 90/ 0) plate under uniform transverse

loading - comparison among different theories and experimental
results

Chapter 7

CONCLUSIONS

Two main objectives were accomplished —- the development of a new technical theory
and a new, robust C0 plate finite element for sandwich and composite panels. The new
technical theory has some common features of other approaches of laminate theories, in
particular the discrete layerwise and the zig-zag theories. However, in the formulation
presented here, the use of surface quantities to represent the degrees-of-freedom as well as
the satisfaction of the non-homogeneous transverse shear traction at the top and bottom of
the sublaminate is a significant departure from the other two types of theories. A review of

the most relevant aspects of those differences are summarized in the next section.

7.1 Review

Our technical theory as well as the discrete layerwise and zig-zag theories are assumed
displacement approaches. Discrete layerwise theories do not generally satisfy the
interlaminar continuity of transverse shear stresses explicitly (see Li and Liu (1996) for an
exception to this) while both the new theory and the zig-zag theories do satisfy these
conditions exactly. However, the discrete layerwise theories will achieve this in the limit
with increasing number of sub-divisions.

The concept of stacking elements through the thickness is readily achieved by

elements derived from discrete layerwise theories as the only condition that must be met is

157

158

the continuity of displacements. This is because there is no specific requirement for the
continuity of interlaminar transverse shear stresses. The zig-zag theories which have their
reference planes at the mid-planes cannot be stacked. The elements using zig-zag theories
must meet both the displacement continuity as well as the continuity of transverse stresses.
This cannot be done since the zig-zag theories satisfy only homogeneous shear traction
(traction-free) conditions at the top and bottom surfaces. Any stacking will only result in
‘zero’ or discontinuous shear traction between two sublaminates. In the present theory, the
non-homogeneous shear traction conditions are satisfied exactly. This is achieved through
the shear traction degrees-of-freedom that can be left unknown at the interface where two
sublaminates are stacked on top of one another.

The new theory allows the development of an eight-node brick finite element model
because it uses surface degrees-of-freedom. Essentially, the theory is still based on plate
kinematics and uses an approach that is in many ways opposite to what is normally done,
that is, to degenerate a three-dimensional body into a two-dimensional surface. Casting of
the technical theory in terms of surface quantities, however, has the advantage of allowing
the model to represent exactly how the surface tractions are applied. In the case of the zig-
zag theories, surface tractions are applied at the reference surface which corresponds to
the mid-surface. This is acceptable for thin plates but not for thick plates. As observed in
our numerical experiments (see also Bogdanovich (1991)), in a thick plate with symmetric
laminate scheme, the through-thickness distribution of in—plane displacements, stresses
and transverse stresses are non-symmetric with respect to the mid-surface. This is due to
the presence of transverse normal stress in the plate.

The assumption that transverse displacement is independent of the through-the-

159

thickness coordinate must be used with caution. The maximum deviation of transverse
displacement from the constant value is in the range of i12 — 18% for laminated as well
as sandwich panels of up to 55 layers and having span-to-thickness ratio of 4 as shown by
our numerical computations. This deviation narrowed to less than 2% for span-to-
thickness ratio of 10. Nevertheless, this assumption generally is acceptable in many
practical problems if the span-to-thickness ratio is large and the material properties are not
drastically different as in sandwich panels. However, neglecting the transverse normal
stress through the assumption of inextensible normal strain will result in the inability to
predict failure modes initiated by this stress component. The choice of a quadratic
displacement field in the thickness direction for our theory overcomes this problem and is
adequate as revealed by comparisons between the exact solutions (Pagano, 1969; and
Burton and Noor, 1994) and our finite element computations.

The layerwise theories result in elements with a large number of degrees-of-freedom.
The finite element model derived using the new theory only has a small and fixed number
of degrees-of-freedom while retaining the desired accuracy just as in the zig-zag theories.
This is very attractive and efficient for large computational models. Furthermore, our
theory will yield traditional engineering (displacement and rotation) degrees-of-freedom.
These engineering degrees-of-freedom will allow easier imposition of boundary and
loading conditions.

In summary, it has been demonstrated through numerous numerical tests that the finite
element model derived using this new theory is a viable alternative to the elements derived
using discrete layerwise or zig-zag theories as well as the conventional continuum

elements. It passes the patch test and is accurate and robust.

160

7.2 Areas of Future Research

The approach to using surface quantities to ‘regenerate’ plate finite elements into
three-dimensional elements is new and attractive. Some areas of current research include
extending the theory to the generalized shell theory as well as to heat transfer theories.
There is therefore a lot of potential for future research to expand this concept.

As for the derivation of finite element models, there is a need to consider things such
as material non-linearity and thermal effects, plasticity (in metal forming applications) and
damage (for delamination problems). The development of a six-node wedge element using
this theory is also needed. The reason is that for some geometric boundary types, the
wedge element types sometimes are preferred over the eight-node brick element type.

An important area of future research is to eliminate the restriction of the existing finite
element formulations to flat plates. This is because the existing finite element has the
restriction that the thickness coordinate must be straight. The curved shell element under
development will alleviate this problem but yet still does not allow modeling of tapered
structures or plates with variable thickness. Therefore, the formulation of a more general
topology for the plate element will create greater flexibility in the model.

Accurate transverse normal stress is important for analyzing crack problems or
problems that have failure modes associated with this stress component. The model
accurately predicts displacements and stresses other than the transverse normal stress.
Another area of possible research is therefore to improve the transverse normal stress
prediction in the model.

The new finite element model has potential for use in a lot of applications. The

element can be used for explicit and implicit dynamic analysis, structural optimization as

161

well as metal forming processes. In addition, this element will be able to perform damage
analysis with the availability of new adaptive techniques like the superposition method
(Fish, 1992; Kim et al., 1991) to improve the quality of finite element calculations in

regions of unacceptable errors.

BIBLIOGRAPHY

BIBLIOGRAPHY

Actis R.L. and Szabo B.A., 1993, “Hierarchical Models for Bi-directional
Composites”, Finite Elements in Analysis and Design, Vol. 13, No. 2/3, pp. 149-168.

Aminpour M.A., 1990, “A Four-Node Assumed-Stress Hybrid Shell Element with
Rotational Degrees of Freedom”, NASA Contractor Report 4279.

Averill R.C. and Reddy J.N., 1992, “An Assessment of Four-Noded Plate Finite
Elements Based on a Generalized Third-Order Theory”, International Journal for
Numerical Methods in Engineering, Vol. 33, pp. 1553-1572.

Averill R.C., 1994, “Static and Dynamic Response of Moderately Thick Laminated
Beams with Damage”, Composites Engineering, Vol. 4, pp. 381-395.

Averill R.C. and Yip Y.C., 1996a, “Development of Simple, Robust Finite Elements
Based on Refined Theories for Thick Laminated Beams”, to appear in Computers and
Structures.

Averill R.C. and Yip Y.C., 1996b, “An Efficient Thick Beam Theory and Finite
Element Model with Zig-Zag Sublaminate Approximations”, to appear in AIAA Journal.

Babuska I., Szabo B.A. and Chang R.L., 1992, “Hierarchial Models for Laminated
Composites”, International Journal for Numerical Methods in Engineering, Vol. 33, No.
3. PP. 503-535.

Barlow J., 1976, “Optimal Stress Locations in Finite Element Models”, International
Journal for Numerical Methods in Engineering, Vol. 10, pp. 243-251.

Barbero EJ. and Reddy J.N., 1990, “An Accurate Determination of Stresses in Thick
Laminates Using a Generalized Plate Theory”, International Journal for Numerical
Methods in Engineering, Vol. 29, pp. 1-14.

Barbero E.J., Reddy J .N. and Teply IL, 1990, “General TWO-Dimensional Theory of
Laminated Cylindrical Shells”, AIAA Journal, Vol 28, No. 3, pp. 544-553.

162

163

Barbero E.J., 1991, “3-D Finite Element Modeling of Laminated Composites by
Layerwise Constant Shear Elements”, in Enhancing Analysis Techniques For Composite
Materials, NDE-Vol. 10, Winter ASME Conference 1991, Eds. Schwer L., Reddy IN. and
Mal A., pp. 229-237.

Bathe KI. and Dvorkin EN, 1986, “A Four-Node Plate Bending Element Based on
Mindlin/Reissner Plate Theory and a Mixed Interpolation”, International Journal for
Numerical Methods in Engineering, Vol. 21, pp. 367-383.

Belytschiko T. and Hsieh E.J., 1973, “Non-Linear Transient Finite Element Analysis
with Convected Co-ordinates”, International Journal for Numerical Methods in
Engineering, Vol. 7, pp. 255-271.

Belytschiko T. and Glaum L.W., 1979, “Applications of Higher Order Corotational
Stretch Theories to Nonlinear Finite Element Analysis”, Computers and Structures, Vol.
10. pp. 175-182.

Bogdanovich A.E., 1991, “Spline Function Aided Analysis of Inhomogeneous
Materials and Structures”, in Local Mechanics Concepts for Composite Material Systems,
IUTAM Symposium, Blacksburg, VA., Eds. Reddy J.N., Reifsnider K.L., Springer Verlag,
pp. 355-382.

Briassoulis D., 1993, “The Performance of a Reformulated Four-Node Plate Bending
Element in Moderately Thick to Very Thin Plate Applications”, Computers and
Structures, Vol. 47, No. 1, pp. 125-141.

Burton W.S. and Noor A.K., 1994, “Three-Dimensional Solutions for
Thermomechanical Stresses in Sandwich Panels and Shells”, Journal of Engineering
Mechanics, ASCE Vol. 20, 10, pp. 2044-2071.

Cho M. and Parmerter R.R., 1993, “Efficient Higher Order Composite Plate Theory
for General Lamination Configurations”, AIAA Journal, Vol. 31, pp. 1299-1306.

Cho Y.B. and Averill R.C., 1996, “An Improved Theory and Finite Element Model for
Laminated Beams Using First Order Zig-Zag Sublaminate Approximations”, submitted
for publication.

Chattetjee SN. and Kulkarni S.V., 1979, “Shear Correction Factors for Laminated
Plates”, AIAA Journal, Vol. 17, No. 5, pp. 498-499.

Crisfield M.A., 1983, “A Four-Noded Thin-Plate Bending Element Using Shear
Constraints - A Modified Version of Lyons’ Element”, Computer Methods in Applied
Mechanics and Engineering, Vol. 38, pp. 93-120.

164

Crisfield M.A., 1990, “A Consistent Co-Rotational Formulations for Non-Linear,
Three-Dimensional, Beam-Elements”, Computer Methods in Applied Mechanics and
Engineering, Vol. 81, pp. 131-150.

Crisfield M.A., 1994, Non-Linear Finite Element Analysis of Solids and Structures,
Vol. 1: Essentials, John Wiley and Sons.

DiSciuva M., 1985, “Development of an Anisotropic, Multilayered, Shear-Deformable
Rectangular Plate Element”, Computers and Structures, Vol. 21, pp. 789—796.

DiSciuva M., 1987, “An Improved Shear-Deformation Theory for Moderately Thick
Multilayered Anisotropic Shells and Plates”, Journal of Applied Mechanics, Vol. 54, pp.
589-596.

DiSciuva M., 1986, “Bending, Vibration and Buckling of Simply Supported Thick
Multilayered Orthotropic Plates: An Evaluation of a New Displacement Model”, Journal
of Sound and Vibration, Vol. 105, pp. 425-442.

DiSciuva M., 1993, “A General Quadrilateral Multilayered Plate Element with
Continuous Interlaminar Stresses”, Computers and Structures, Vol. 47, pp. 91-105.

Dvorkin E.N. and Bathe K.J., 1984, “A Continuum Mechanics Based Four-Node Shell
Element for General Non-Linear Analysis”, Engineering Computing, Vol. 1, pp. 77-88.

Epstein M. and Glockner PG, 1977, “Nonlinear Analysis of Multilayered Shells”,
International Journal for Numerical Methods in Engineering, Vol. 13, pp. 1081-1089.

Epstein M. and Huttelmaier HR, 1983, “A Finite Element Formulation for
Multilayered and Thick Plates”, Computers and Structures, Vol. 16, pp. 645-650.

Fish J, 1992, “The s-Version of the Finite Element Method”, Computers and
Structures, Vol. 43, NO. 3, pp. 539-547.

Flanagan G., 1994, “A General Sublaminate Analysis Method for Determining Strain
Energy Release Rates in Composites”, AIAA-94-1356-CP, presented at the AIAA 1994
Conference, pp. 381-389.

Fried 1., 1974a, “Finite Element Analysis of Incompressible Materials by Residual
Energy Balancing”, International Journal of Solids and Structures, Vol. 10, pp. 993-1002.

Fried 1., 1974b, “Residual Energy Balancing Technique in the Generation of Plate
Bending Finite Elements”, Computers and Structures, Vol. 23, pp. 409-431.

Friedman Z. and Kosmatka J .B., 1993, “An Improved Two-Node Timoshenko Beam
Finite Element”, Computers and Structures, Vol. 47, pp. 473-481.

165

Hinton E. and Huang H.C., 1986, “A Family of Quadrilateral Mindlin Plate Elements
with Substitute Shear Strain Fields”, Computers and Structures, Vol. 23, pp. 409-431.

Huang H.C. and Hinton E., 1984, “A Nine-Node Lagrangian Mindlin Plate Element”,
Engineering Computing, Vol. 1, pp. 369-379.

Hughes T.J.R., Taylor R.L. and Kanoknukulchal W., 1977, “A Simple and Efficient
Finite Element for Plate Bending”, International Journal for Numerical Methods in
Engineering, Vol. 11, pp. 1529-1543.

Hughes T.J.R. and Tezduyar TE, 1981, “Finite Elements Based Upon Mindlin Plate
Theory, with Particular Reference to the Four-Node Bi-linear Isoparametric Element”,
Journal of Applied Mechanics, ASME, Vol. 48, pp. 587-596.

Jang J. and Pinsky P.M., 1987, “An Assumed Co-variant Strain Based 9-Noded Shell
Element”, International Journal for Numerical Methods in Engineering, Vol. 24, pp.
2389-241 1.

Jiang J. and Olson M.D., 1994, “Large Elastic—Plastic Deformations of Slender
Beams: Co-Rotational Theory vs. von-Kérman Theory”, Computational Mechanics, Vol.
159 pp- 117‘128.

Kim Y.H. and Lee S.W., 1988, “A Solid Element Formulation for Large Deflection
Analysis of Composite Shell Structures”, Computers and Structures, Vol. 30, No. 1/2, pp.
269-274.

Kim Y.H., Levit I. and Stanley G., 1991, “A Finite Element Adaptive Mesh
Refinement Technique that Avoids Multipoint Constraints and Transition Zones”, CED-
Vol. 4, Iterative Equation Solvers for Structural Mechanics Problems, ASME, pp. 27-35

Kong J. and Cheung Y.K., 1995, “Three-Dimensional Finite Element Analysis of
Thick Laminated Plates”, Computers and Structures, Vol. 57, No. 6, pp. 1051-1062.

Ling-Hui H., 1994, “A Linear Theory of Laminated Shells Accounting for Continuity
of Displacements and Transverse Shear Stresses at Layer Interfaces”, International

Journal of Solids and Structures, Vol. 31, pp. 613-627.

Lee CY. and Liu D., 1992, “Layer Reduction Technique for Composite Laminate
Analysis”, Computers and Structures, Vol. 44, No. 6, pp. 1305-1315.

Lee SW. and Pian T.H.H., 1978, “Improvement of Plate and Shell Finite Elements by
Mixed Formulations”, AIAA Journal, Vol. 16, pp. 29-34.

Li X. and Liu D., 1996, “A Laminate Theory Based on Global-Local Superposition”,
to appear in Communication of Numerical Methods in Engineering.

166

Li X. and Liu D., 1996, “A Generalized ZigZag Theory for Laminate Plate Analysis”,
to appear in AIAA Journal.

Liu D., 1995, “2in Theory for Composite Laminates”, AIAA Journal, Vol. 33, No.
69 pp. 1163‘1165.

Lo K.H., Christensen R.M. and Wu E.M., 1977, “A High-Order Theory of Plate
Deformation - Part 2: Laminated Plates”, Journal of Applied Mechanics, Vol. 44, pp. 669-
676.

Lu X. and Liu D., 1992, “An Interlaminar Shear Stress Continuity Theory for Both
Thin and Thick Composite Laminates”, Journal of Applied Mechanics, Vol. 59, pp. 502-
509.

Lynn RP. and Arya SK, 1974, “Finite Elements Formulated by the Weighted Discrete
Least Squares Method”, International Journal for Numerical Methods in Engineering,
Vol. 8. pp. 71-90.

MacNeal R.H., 1982, “Derivation of Element Stiffness Matrices by Assumed Strain
Distributions”, Nuclear Engineering and Design, Vol. 70, pp. 3-12.

MacNeal RH. and Harder R.L., 1985, “A Proposed Standard Set of Problems to Test
Finite Element Accuracy”, Finite Elements in Analysis and Design, Vol. 1, pp. 3-20.

Mattiasson K., Bengtsson A. and Samuelsson A., 1986, “On the Accuracy and
Efficiency of Numerical Algorithms for Geometrically Nonlinear Structural Analysis”, in
Finite Element Methods for Nonlinear Problems, Eds. Bergan P.G., Bathe K]. and
Wunderlich W., Springer-Verlag Berlin, Heidelberg

Mohan P.R., Naganarayana B.P. and Prathap G., 1994,“Consistent and Variationally
Correct Finite Elements for Higher-Order Laminated Plate Theory”, Composite
Structures, Vol. 29, pp. 445-456.

Morley L.S.D., 1963, Skew Plates and Structures, Pergamon, Oxford.

Murakami, H., 1986, “Laminated Composite Plate Theory with Improved In-Plane
Responses”, Journal of Applied Mechanics, Vol. 53, pp. 661-666.

Naganarayana B.P., Prathap G., Dattaguru B. and Ramamurty TS, 1992, “A Field-
Consistent and Variationally Correct Representation of Transverse Shear Strains in the
Nine-Noded Plate Element”, Computer Methods in Applied Mechanics and Engineering,
Vol. 97, pp. 355-374.

Noor AK. and Andersen C.M., 1977, “Mixed Isoparametric Finite Element Models of
Laminated Composite Shells”, Computer Methods in Applied Mechanics and
Engineering, Vol. 11, pp. 255-280.

167

Ochoa 0.0. and Reddy J .N., 1992, Finite Element Analysis of Composite Laminates,
Kluwer Academic Publishers, Netherlands.

Pagano N.J., 1969, “Exact Solutions for Composite Laminates in Cylindrical
Bending”, Journal of Composite Materials, Vol. 3, pp. 398-411.

Pawsey SE. and Clough R.W., 1971,“Improved Numerical Integration of Thick Shell
Finite Elements”, Intemational Journal for Numerical Methods in Engineering, Vol. 3,
pp. 545-586.

Peng X. and Crisfield M.A., 1992, “A Consistent Co-Rotational Formulation for Shells
Using the Constant Stress/Constant Moment Triangle”, International Journal of
Engineering Science, Vol. 35, pp 1829-1847.

Piskunov V.G., Verijenko V.E., Adali S. and Summers E.B., 1993, “A Higher-Order
Theory for the Analysis of Laminated Plates and Shells with Shear and Normal
Deformation”, International Journal of Engineering Science, Vol. 31, N o. 6, pp. 967-988.

Piskunov V.G., Verijenko V.E., Adali S. and Tabakov RY., 1994, “Transverse Shear
and Normal Deformation Higher-Order Theory for the Solution of Dynamic Problems of
Laminated Plates and Shells”, International Journal of Solids and Structures, Vol. 31, No.
24, pp. 3345-3374.

Prathap G. and Viswanath S., 1983, “An Optimally Integrated 4-Noded Quadrilateral
Plate Bending Element”, International Journal for Numerical Methods in Engineering,
Vol. 19, PP. 831-840.

Prathap G., 1985, “The Poor Bending Response of the Four-Node Plane Stress
Quadrilateral”, International Joumal for Numerical Methods in Engineering, Vol. 21, pp.
825-835.

Prathap G. and Somashekar BR. 1988, “Field- and Edge-Consistency Synthesis of a
4-Noded Quadrilateral Plate Bending Element”, International Journal for Numerical
Methods in Engineering, Vol 26, pp. 1693-1708.

Prathap G., 1994, “The Displacement-Type Finite Element Approach - From Art to
Science”, Progress in Aerospace Science, Vol. 10, pp. 295-405.

Ramm E. and Matzenmiller A., 1986, “Large Deformation Shell Analysis Based on
Degenerate Concept”, in Finite Element Methods for Plate and Shell Structures, Volume I .'
Element Technology, T.J.R. Hughes and E. Hinton, eds., Pineridge Press Int., Swansea,
UK, pp. 365-393.

Reddy J .N., 1980, “A Penalty Plate—Bending Element for the Analysis of Laminated
Anisotropic Composite Plates”, International Journal for Numerical Methods in
Engineering, Vol. 15, pp. 1187-1206.

168

Reddy J .N., 1984, “A Simple Higher-Order Theory for Laminated Composite Plates”,
Journal of Applied. Mechanics, Vol. 51, pp. 745-752.

Reddy J.N., 1987, “A Generalization of Two-Dimensional Theories of Laminated
Composite Plates”, Communications in Applied Numerical Methods, Vol. 3, pp. 173-180.

Reddy J.N., 1993, An Introduction to the Finite Element Method, Second Edition,
McGraw-Hill, Inc., New York.

Reddy J.N. and Robbins DH, 1994, “Theories and Computational Models for
Composite Laminates”, Applied Mechanics Review, Vol. 47, No. 6, Part 1, pp. 147-169.

Robbins D.H., Reddy J.N. and Krishna-Murty A.V., 1991, “On the Modeling of
Delamination in Thick Composites”, in Enhancing Analysis Techniques For Composite
Materials, NDE-Vol. 10, Winter ASME Conference 1991, Eds. Schwer L., Reddy J .N . and
Mal A., pp. 133-149.

Robbins DH. and Reddy J .N., 1993, “Modeling of Thick Composites Using a
Layerwise Laminate Theory”, International Journal for Numerical Methods in
Engineering, Vol. 36, pp. 665-677.

Rossow M., 1977, “Efficient Co Finite Element Solution of Simply-Supported Plates
of Polygonal Shape”, Journal of Applied Mechanics, ASME, Vol. 44, pp. 347-349.

Simo J.C. and Hughes T.J.R., 1986, “On the Variational Foundations of Assumed
Strain Methods”, Journal of Applied Mechanics, Vol. 53, pp. 51-54.

Somashekar B.R., Prathap G. and Babu CR, 1987, “A Field-Consistent, Four-Noded,
Laminated, Anisotropic Plate/Shell Element”, Computers and Structures, Vol. 25, No. 3,
pp. 345-353.

Spilker R.L. and Munir N.I., 1980, “The Hybrid-Stress Model for Thin Plates”,
International Journal for Numerical Methods in Engineering, Vol. 15, pp. 1239-1260.

Srinivas S., 1973, “Refined Analysis of Composite Laminates”, Journal of Sound and
Vibration, Vol. 30, pp. 495-507.

Sun CT. and Whitney J .M., 1973, “Theories for the Dynamic Response of Laminated
Plates”, AIAA Journal, Vol. 11, pp. 178-183.

Tessler A. and Dong S.B., 1981, “On a Hierarchy of Conforming Timoshenko Beam
Elements”, Computers and Structures, Vol. 14, pp. 335-344.

169

Tessler A. and Hughes T.J.R., 1983, “An Improved Treatment of Transverse Shear in
the Mindlin-Type Four-Node Quadrilateral Element”, Computer Methods in Applied
Mechanics and Engineering, Vol. 39, pp. 311-335.

Tessler A. and Hughes T.J.R., 1985, “A Three-Node Mindlin Plate Element with
Improved Transverse Shear”, Computer Methods in Applied Mechanics and Engineering,
Vol. 50. pp. 71-101.

Tessler A., 1986, “Shear-Deformable Bending Elements with Penalty Relaxation,” in
Finite Element Methods for Plate and Shell Structures, Volume 1: Element Technology,
T.J.R. Hughes and E. Hinton, eds., Pineridge Press Int., Swansea, UK, pp. 266-290.

Tessler A., 1991, “A Higher-Order Plate Theory with Ideal Finite Element Suitability”,
Computer Methods in Applied Mechanics and Engineering, Vol. 85, pp. 183-205.

Toledano A. and Murakami H., 1987, “A Composite Plate Theory for Arbitrary
Laminate Configurations”, Journal of Applied Mechanics, Vol. 54, pp. 181-189.

Turner M.J., Dill E.H., Martin H.C., Melosh R.J., “Large Deflection of Structures
Subject to Heating and External Load”, Journal of Aeronautical Science, Vol. 27, pp. 97-
106.

Verijenko V.E., 1993, “Nonlinear Analysis of Laminated Composite Plates and Shells
Including the Effects of Shear and Normal Deformation”, Composite Structures, Vol. 25,
pp. 173-185.

Wempner G, 1969, “Finite Elements, Finite Rotations and Small Strains of Flexible
Shells”, International Journal of Solids and Structures, Vol. 5, pp. 117-153.

Whitney J.M., 1973, “Shear Correction Factors for Orthotropic Laminates Under
Static Load”, Journal of Applied Mechanics, ASME, Vol. 40, No.1, pp. 302-304.

Xavier P.B., Lee K.H., and Chew CH, 1993, “An Improved Zig-Zag Model for the
Bending of Laminated Composite Shells”, Composite Structures, Vol. 26, pp. 123-138.

Yang P.C., Norris CH, and Stavsky, Y., 1966, “Elastic Wave Propagation in
Heterogeneous Plates”, International Journal of Solids and Structures, Vol. 2, pp. 665-
684.

Yip Y.C., Averill R.C. and Tessler A., 1996, “A Simple, Efficient and Robust C0
Mindlin-Type, Four-Node Plate Element”, submitted for publication.

170

Yu G., Tzeng G.Y., Chaturvedi S., Adeli H. and Zhang S.Q., 1995, “A Finite Element
Approach to Global-Local Modeling in Composite Laminate Analysis”, Computers and
Structures, Vol. 57, No. 6, pp. 1035-1044.

Zaghloul S.A. and Kennedy J.B., 1975, “Nonlinear Behavior of Symmetrically
Laminated Plates”, Journal of Applied Mechanics, Vol. 42, pp. 234-236.

Zienkiewicz O.C., Taylor R.C. and Too J .M., 1971, “Reduced Integration Techniques
in General Analysis of Plates and Shells”, International Journal for Numerical Methods in
Engineering, Vol. 3, pp. 275-290.

Zienkiewicz O.C., Owen D.R.J., and Lee K.N., 1974, “Least Square Finite Element
for Elasto-Static Problems. Use of ‘Reduced’ Integration”, International Journal for
Numerical Methods in Engineering, Vol. 8, pp. 341-358.

Zienkiewicz CC. and Xu 2., 1993, “Linked Interpolation for Reissner-Mindlin Plate
Elements: Part I - A Simple Quadrilateral”, International Journal for Numerical Methods
in Engineering, Vol. 36, pp. 3043-3056.

Zinno R. and Barbero E.J., 1994, “A Three-Dimensional Layerwise Constant Shear
Element for General Anisotropic Shell-Type Structures”, International Journal for
Numerical Methods in Engineering, Vol. 37, pp. 2445-2470.

Zinno R. and Barbero E.J., 1995, “Total Lagrangian Formulation for Laminated
Composite Plates Analysed by Three-Dimensional Finite Elements with Two-
Dimensional Kinematic Constraints”, Computers and Structures, Vol. 57, No. 3, pp. 455-
466.

 

APPENDICES

APPENDIX A

 

APPENDIX A

 

 

 

 

 

_ 1 13 - -9
Pi " '5' P2 = g p3 ' 5
- —a ”
p4=C+b'Z-‘ p5=_§ p6=—p4
1 ~ - “ (A-l)
p7 = -(_l) = _‘1 p - —9—+hg
C55 p3 E 9 C(sls) C
23 21 -2
P10 = ‘= P11 = +17:
c (N...) c
C55
21,-1343,
cl. = hard-5+6, bi = aa,+&,a—&, (A.2)
d 8 (1,2;—a, e _ 2a + 21,21
1 _ a,- (1) ‘ _ J (N...)
C55 C55
Nm-l N,,—l
a = a[1+ Z 21,] 13 = d[2hm+ Z 1),]
‘=' “ (A3)

171

 

3"!

at

172

N,,,—1
—hm+ch,3n+ 2 (hm—2,)(&,a—e,)
i=1
—1
—ch,3n — N2 (hm —z,)(&,a—e,)
i=1
Nm—l

hi, - 13h; - 2, (hm — z,)(&.13 — 13,-) (A4)
1': 1

1M _Ni(hm_

(l)
C i=1

(1%th + N2 (hm fish-{(i—

C55" i=1

E)

El

2)

L
¥__/

 

APPENDIX B

 

173

APPENDIX B

lntmlmndcnflntemlation

This appendix is relevant to both Chapters 2 and 3 which deal with the use of
interdependent interpolation functions for the MIN4-CC and LZZ3 finite element models.
The details of the derivation of these interdependent interpolation functions will be shown
here.

The interdependent interpolation scheme enforces the linear part of a constraint of the

form:

9) = O (B-l)

(W’s - .
along each element side. In the case of the LZZ3 model, it may be for either the top or
bottom surface. The constraint of Eq. (B.1) can be associated to the vanishing of the
transverse shear strain in Chapter 3 for the Mindlin plate theory. However, in the laminate
theory of Chapter 4, the constraint is not equal to the vanishing of the transverse shear
strain. It does not have any physical meaning. Its sole purpose is to introduce a new
variable, 0,, or 9), so that the new laminate theory becomes a C0 instead of a C l theory.
The discussion that follows is valid for both the four-node Mindlin as well as the top

and bottom surfaces of the eight-node LZZ3 finite element models. The subscript on that

usually differentiates the two surfaces in the case of the LZZ3 model will be omitted here

for clarity:

 

I-
'I

 

 

174

co
.rn

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3\\‘ /£r— “4*
V I3 5 n
ll b34 7n
" 9 b23 ‘5 9
12 a ——>> n
8
bu ’4
Y
“41 I “12

 

 

_ . #x ————>—> 0).
“a" ‘i'J‘r bi; = ””1

Figure B.l Quadrilateral element coordinate description

The xy coordinate and the 5,1] coordinate systems are refer to as the “global” and
“local” coordinate systems, respectively. Independent bi-linear rotations and a Serendipity

(eight-node) transverse deflection are initially assumed, i.e.,

8
W = 2Nl(xry)wi
i=1

4 4
mo.» = Zia-(awe..- e,(x,y> = Zia-(awe,- (13.2)

i=1 i=1

4 4
x = 2Pi(§,n)xi y = 2P1(§r ")Y,

i=1 i=1

where Pi(§, n) , and N ,.(fi, 11) are respectively bi-linear and Serendipity shape functions,

and 0 By, and w,» denote nodal degrees-of-freedom. Because of mapping of the global

xi’

coordinates x and y into the local §, 1] coordinates as depicted in Eq. (B.2), then

Pi(§’ Tl)=Pi(xaPa(§, 11): yaPa(§’ 11))
and

N,(§.n)=1iI.-(xaPa(§,n),yaPa(§,n))

 

175

The bilinear functions are given by:

P.- = ,1,“ + §.§)(1 mm) (13.3)

and the biquadratic serendipity functions by:

N, = (1,10+§,lé)(lHum—(1-§2)(l+n,~11)-(1—112)(1+§.~§)]§3n.-2
(3.4)

1 1
+§(1—§2)(1 +nm)(1—§3)n?+§(l—n2)(l +§,«§)(1—n?)§3
where (5.11) <—: {—(l, 1)] .
For side 1-2, let s denote the coordinate running along an element edge as shown in

Figure B]. The constraint along this side (11 = —1) is:

9)

’s— s |n=—1

(3.5)

(w

Because of the anisoparametn'c interpolation, different degrees-of—freedom will appear at
the corners and mid-side nodes (see for example Figure 29). To achieve a uniform four-
node configuration, the four mid-edge w degrees-of-freedom are then condensed out by

the use of four differential edge constraints:

(mm—93’s”g Mil = 0 (s6 [0, 1,1) (3.6)

where [k is the edge length. The differential constraint equation is written for each
element edge. Each of these constraints is then explicitly solved for wk + 4 and back-
substituted into Eq. (B.2).

So for the edge 1-2, the differential constraint will be given by:

(W’ss—esrs)ln =—1 = 0 (B7)

Computing the following edge derivatives:

176

§>
I

— Wl‘n = —l
= N‘wiln = _l

gut-nu»,+§§<§+1)w2+(1-§2)w5

,5={(.,..)..G..).-2.W.}i

l
= (wl + W2 — 2w5)—.§

3

(B8)

E)

 

n=-l

A
W’SS

 

n=—1

where ds = jsl lag and js is the edge jacobian.

n :
Now, the tangential edge rotation, also known as covariant rotation when normalized by its

length (refer to Figure BI) is:
9 = 0 n (B9)

and will be approximated using bilinear functions similar to the Cartesian rotations, 0,C

and 0),. The derivatives of the tangential edge rotation along side 1-2 will be:

e, = s|n=_1
= Piesi|n=_l
1 3.10
= §[(1-§)951+(1+§)9521 ( )
A l 1
63’s = §[052_esl]7

S n = -1
Substituting Eqs. (B8) and (BIG) into Eq. (B.7), the mid-side w5 degree-of-freedom is

found to be:

1 1 .

where j12 = js 1 is used to denote the line jacobian corresponding to edge 1-2. The
n = -

line jacobian can be obtained in the following manner:

177

 

 

 

 

i = x|11=-l = Pixi|n=—l 3) = y|n=_l : Piyi|n=—l
i=lIx-x] y=l[y—y]
’s 2 2 1 ’s 2 2 l
. _ J» 2 . 2
J,|n=_l — (x.,) +(y.,) (3.12)
x —x 2 — 2
=12 ., 1» y)
2 2
=11/2

In general, the subscripts is one of the variables E, or n . Curve s generally corresponds to
one of the edges of an element of reference on which parameter s is defined. The other
mid-side wk + 4 degrees-of—freedom can be obtained by cyclic changes of indices

1 -—> 2 —-> 3 —> 4, etc. from Eq. (B.l 1) and can be easily verified to be:

1 1 .
W6 = §(W2+W3)‘Z(9s3“952)123

1 l .

l l .
“’8 = §(W4+W1)‘g(951‘954)141

wk + 4 in terms of covariant rotations is readily obtained from Eqs. (B.1 1, B.l3):

1 1
l 1
1 1 (3.14)
W7 = §(w3 + W4) — 2(6g3 — 9&1)
1 l
by making the appropriate substitution of the corresponding edge transformation. Eqs.

(B.ll and BB) or Eq. (B.l4) can also be cast in terms of Cartesian rotations by using

simple transformation equation for Cartesian tensors:

178

9

10+l0,

n nx x ny y (3.15)
85 = —ln).04+lm,0y
where lmr = cos(n, x) = dy/ds and I"), = cos(n, y) = —dx/ds are direction cosines

of the outward normal of the boundary. The following expression for the wk 4, 4 degrees-

of-freedom are then obtained:

l1(2cz-xl)<e,249,1)+(y2-y1)(9,.2—e,.,)1

1
w5 = §(W1+W2)"8

W6 = $092 + W3) " '18'[(x3 ‘x2)(9x3 — 9x2) + (Y3 “Y2)(9y3 — 9,.2)]
(13.16)

W7 = %(W3+W4)—%[(X4‘x3)(9x4‘9x3)+(Y4‘Y3)(9y4‘0y3)1
w8 = 5m, + w,>—§1<x,—x4)(9,,— e,...)+(y1—y.)<e,.l — 9,.)1

Substitution of the mid-side wk 4, 4 degrees-of—freedom using Eq. (B.l6) back into Eq.
(3.1) for the displacement field, w will produces a four-node, quadratic, coupled

transverse deflection:

Xl Xl

4
w = 2 [P,w,+N .e .+Ny,.ey,.] (3.17)

i=1

where N xi and N )4. depend on N i + 4 and edge projections on x and y axes and are given

by:

1
x1 = —§["12N5‘a41N8] N

l
yl = —§[b12N5 - b41N8]

1 l
Nx2 = “‘8'1‘123N6—012N5] Ny2 = _§[b23N6—b12N5]

1 l (B.18)
Nx3 = ‘§[a34N7‘023N6] Ny3 = ‘glb34N7‘b23N61

1 l
Nx4 = —§[a41N8—a34N7] Ny4 = —§[b41N8-b34N7]

where

179

1-2 _.
211%)(111)

1

— -(1-112)(1+§)

2
1

" -(1—§2)(1+n)

2

1

_ ._ 2 ..
2(111)11 é)

(B.l9)

APPENDIX C

180

APPENDIX C

Shaneliunctinns

The shape functions in the displacement field, Eq. (62) are obtained after substituting
Eqs. (51-53, 55-57) into Eqs. (58, 59) and solving them analytically so that the remaining
degrees-of-freedom (£22, £23, 92, and 93 ) in the in-plane displacement field for the mth
sublaminate can be expressed in terms of the new operative degrees-of-freedom of Eq.
(63), i.e. ( ub, up vb, v,, be, 9n, Byb, OW, Txb, In, tyb, Ty, ).

Let

X = {172, 173, 92, and 1‘23}T
and
U = {ub, up vb, v,, 9x1» 0,”, Gyb, Byprxb, Ix,,1yb,’ty,}T.

So Eqs. (58, 59) can be written as:

—1
= 'ikUk (01)

where indices i,j= l, ...,4 and k = 1, ..., 12
and

181

Nm—l
2 1’21 R12 -
i:
NM-l
zsz‘J' 2 “21 R14 =
i=1
N,,,—1
ZZN,+ 2 d2i R22 -
i=1
Nm—l
€21 R24 ‘
i=1
N,,,—1
2
1N3 2 (Zia—Z005.- R32 =
i=1
Nm-l
2 (Zn-2962.- R34 =
i=1
NM-l
2 (sz—Zilbzi R42 '
i=1
N—l
2
ZN"... z (ZNM-Z,)a2i R44 =

i=1

NM—l

2 b3.

i=1
NM—l

32va + 2 a3,-

i=1

N,,,—1 (C2)
3
sz+ 2 (zNM—z,)d3i
i=1
NM-l
2 (ZN 2,)631.
i=1
NM—l
Z (ZN "Zi)b3t
i=1
N-l

3
2 <z~,.—z.-a..->
i= 1

182

Nm—l
T15 = 2 b4i

i=1
M.,—1

1+ 2 a4,-

i=1

"I
:3
II

NM-l
-(1) «(1) -(1)
1,, = Q45 — 2 (hue... —a,,-Q4s)

i=1

Nm—l
“(1) AU)

*(1)
Tl,” = -Q55 - 2 (al.-Q55 -b1,-Q45)
i=1
Nm—l

T25: 1+ 2 (141

i=1

Nm-l
T27 = 2 C41

i=1

Nm-l
-(l) «(1) -(1)
T29 = Q44 — 2 (due... -c,.-C4s)

i=1
Nm-l
-(l) -(l) .(1)
T2,“ = Q45 - Z (61.-Q55 -d1,-Q45)

i=1

T31 = ‘1
Nm—l
T35 = 21v.“ 2 (Zn/{20%
i=1
NM—l

737 = 2 (W.,—10041

i=1

NM—l
“(1) «(1)
T39 = _ZNMQ44 - Z (ZNm—ZixduQM -

i=1

T16 = _T15
T18 = ‘T17
A(Nm)
T1, 10 = "Q45
A(Nm)
T1,12 = QSS
T26 = ‘725
T = —T
28 27 (C3)
A(N,,.)
T210 = Q44
AN...)
T112 = ‘Q45
T32 = 1
Nm-l

T36 = ‘ Z (sz—zgd“

i=1

T38 = ‘T37

“(1)
€11Q45)

183

N,,-1
(1) (I) “(1)
T3,“ = ZNMQ45 — 2 (ZN... --Z )(611st -d11Q45)

i=1

—1
T45 = Ni (ZNM- 1’41 T46 = ‘T45
i= 1
~,,_1 ~,.-1
T47 = ZNm+ 2 (ZNM‘Z1W41 T48 = ' 2 (ZNA-Zilam'

i=lN1=1 (C.3)

..(1) am (1)
T49 = ZN" Q45 - N": (ZN... -Z)(b11Q44-a11-Q45)
i=1
1 1
..(1) (1) ( )
T411 = _ZN». Q55 - N2 (ZN... -Z1)(011Q55- b11'Q45)

i=1

(1) CJJJ)
all other T11 =0 and Q45 = A”
k

The displacement field of Eq. (62) after introducing the interpolation functions (Eq.

(B.1)) can be written as:

u = aJJ’P .(x. y)<I>f,"’(z)
u = 1190111,, y)\PJ"’(z) (C4)
(w1ai’1(x.y) + amt/.111 y) + 9,..N..-(x. y))M.,(z)

k
II

where indices 01 = 1, 2; i = 1, ...,4; k = 1, 2, ..., 7 and id = 4(a— l) +i .Indices
on identify whether the degree-of-freedom is at the top or bottom, k indicates the degree-

of-freedom and ia, the local element node number. u(.k ) w- 0

ia’ 111’ xia and eyia are the

nodal degrees-of—freedom. Note that wia = 1253’: 1212513 where 813 is the Kronecker

delta. Similarly, we can write 9x101 = 1753-: 12MB“ and Gym

= 4(5) 1.21105]:5

“id:

The shape functions, (by), ‘1’? and M a are defined as follows:

184

oJ," = 1+rJ,"
of,” = o

o?” = —z+I“J,"’
all other of) = Ff,“

1112) = 1.132)

A

w

v
I

- 0
‘1’?) = —z+A(15)

all other ‘1’? = A3)

M1 = (1-1;) ((2.5)
ZN"
Z
M _ _
2 ZN,"
M3 _ 4(1 “5.)
ZN... ZN...
where
N,,,—1

(k) _ 2 3 .

Fa - rum:Z +’21sz + Z (Z—zi)(rl,kad2i+r2,kad3i+r3,kac2i+r4,kac3i)’
i=1
NM—l

2 3 b b .

a ’3,kaZ +’4,ka '*' 2 (Z—zi)(rl,ka 2i+r2,ka 31+r3,kaa2i+r4,kaa3i)’
i=1

and koc = 2(k—1)+a;witha =1,2;k = l,2,...,7.

>
A
R-
V
II

APPENDIX D

185

APPENDIX D

Won:
The shape functions, <D£2(z), <D§"&(z), ‘P£Q(z) and ‘I'J‘J&(z) are related to the shape
functions, ¢&k)(z) and ‘I’ékl(z) of Eq. (75) through the transformation matrix of Eq.

(81) in the following way:

$111) = (¢1xl)+¢<(z2)b21/b11)
4’1?) = (¢h4)+¢&5)b21/b11)
(D1?!) = (¢&6)+<I>&7)b21/b“)
(1)112 = (¢&J)+<I>&2)b22/b12)
(1)13 = ((1)114) + ¢&5)b22/b12)
“’19.? = (4’8?) + ¢&”b22/b12)

‘l’ég = (W&2)+‘P&”b“/b21)
‘Péfl = (‘P&5)+‘P&4)b“/b2])
will = (‘P&7)+‘P&6)b“/b21)
W112 = (W842) + Whinbiz/bzz)
W11? = (W&SJ+‘P&4)b12/b22)
“11130? = (TQM‘I’h‘SJbiz/bzz)

multiplied with the factors bl2 and b2] respectively.

(D.1)

Note that bl2 and 1221 may both be equal to zero, but (D1181 and ‘ngl are always pre-

APPENDIX E

186

APPENDIX E

_l_‘I',,l‘AillLl [AA 11!. 1-1 ‘nAt 111‘ as fmll ° . are

Botatians

It is important to find out how the second Piola-Kirchhoff stress and strain tensors are
related to the Cauchy stress and strain tensors for the case of small strains/large rotations.
Mattiasson et al. (1986) showed that there is a definite relation between the quantities. The
proof is reproduced here for completeness.

The relation between the reference position vector, X , the current position vector, x,

and the displacement vector, y is given by:
5=X+u; xi=Xi+ui (13.1)

The Lagrangian-Green strain, E can be expressed as:

l l

where the comma denotes differentiation with respect to the initial configuration, and 8,].
is the linear Lagrangian strain tensor.

Consider a small part of a continuum in a fixed Cartesian coordinate system,
X ., i = l, 2, 3 , with base vectors 2,. Applying a finite rotation to a vector, d2; through an

orthogonal rotation tensor, 13 , yields:

dX‘ = ads; IdXI = ldfl = d5 (13.3)

$ All!
Here X i , i = 1, 2, 3 define another Cartesian coordinate system with base vectors, 1,- ,

which initially coinciding with X,- , but rotate with the body.

187

*
Now, giving the body an infinitesimal deformation so that the vector (1X turns into

d5:
d5 = 011121 = dx:Z: (E.4)
The following relation is obtained:
dx + d1; = dx" + d1.“ (13.5)

where d u is the relative displacement vector between two neighboring particles.

Let Idxl = ds. The difference ds2 — d52 in terms of the Lagrangian strain tensor, E , is:
2 2 i i 1
is —dS = 2 124.1121,de = 2 EijdXidXJ (E.6)

Here, it must be emphasized that the displacement gradient dui/de is of finite
magnitude, while duI/dx; « 1 by virtue of the infinitesimal deformation assumption.
This implies that the quadratic part of the strain tensor E; can be neglected. One can

therefore write E-- = 81.. Noting that dXi = dx:,we find Eij = E1

l] j tj , and finally

This result shows that for a case of small strains, but large rotations, the Lagrangian strain
tensor components are equal to the linear Lagrangian strain components in a system co-
rotating with the particle.

Using the polar decomposition theorem, the deformation gradient tensor, If can be

multiplicatively decomposed into a rotation tensor, 8 and a stretch tensor, Q as:

ax.
E = 131] where Fij = 337’- (E8)
I

For infinitesimal strain, the stretch tensor can be expressed as:

188

U z 1 + e4 (13.9)

where e is a small number and 41:13 = 1:1. The deformation gradient can thus be

approximated by:
F = R (E.10)
Noting that ratio of densities, p/ p z 1 in the case of infinitesimal strain, the second

Piola-Kirchhoff stress tensor can then be written as:

p T
S = 3005") ~9-(E‘J)

(E.11)
z B . g . ET
The component forms of the Cauchy stress tensor, 9' are:
9 = OijZiZj = 013-21; (E12)

A* o o I 0 I
where [1 18 a base vector in the co-rotating CarteSian coordinate system.
*
The matrices of tensor components 011' and 6U are thus related by the normal

transformation laws:

7' t
[0] = [R] [0 ][R1 (E-13)
where [R] is the matrix of the components of R in the fixed system. Combining Eqs.

(EU and E.13), we get:

[S1 = 1R1<1R171€11R1>1R17 = [6"] (E14)
The above equation shows an all important result that for small strains, the second Piola-
Kirchhoff stress components are equal to the Cauchy stress components in a system co-
rotating with the particle.

This implies that a constitutive relation, formulated for the case of infinitesimal strain

189

and rotation, and relating Cauchy stress and engineering strain or rates of these quantities,

can be used unaltered to relate Lagrangian strain and second Piola-Kirchhoff stress. For

instance, constitutive relations given in the form:

(E. 15)

11
n
.26-

f(§) or C:

Q
II

can be written as:

S = f(E) or S (5.16)

11
'0
151

APPENDIX F

190

APPENDIX F

1 ’ v ’ i

The finite element model of Eq. (138) when written in the expanded form is:

4 .

 

 

 

 

 

 

11011 1K‘211K1311K141 11051 1K‘611K'711K181 1.111 {f‘}
1K2'11K221 1K2311K241 1K2511K2611K271 [W] (.121 {f2}
111/311111321 1x33] 1x34] 1x351 1106111671 1108] M3} {f3}
1161111691 1K431 1x44] 1x45] 1x46] 1x47] [W]. W} ,=. {f4} , (1.,)
1K“) 1x5211K531 1x541 1K5511K5611K5711K581 {451 {f5}
[KM] [Km] 1K6311K641 1K6511K661 1K6711K681 {d6} {f6}
1x711 1x72] 1x73] 1x74] 1W] 1x76] 1W] [W] {cm {f7}
_1K8'11K8211K8311K8411K8511K8611K8711K881 {d8} J {f8} 1
where
1dll=1u1l§ {d2}={V‘-};
1231:1111; {d4}={9,1}; .
l= 1,2,...8 (F.2)
11:51:19,}; 1.1614311;
{d7}={T,1}; {d8}={W3};

A special note is that {W3} will be condensed out at the element level since it is a

nodeless variable. Expressing Eq. (F. 1) in indicial notation, we obtain for the ith iteration:

01,13 = l,2,...,8

(Kfl’dff = (F‘ff , k = 1,2, ...8 (if a, 13:8) (R3)
” 1 (de=8)

Here summation of k, B is implied. The tangent stiffness, [K T] of Eq. (140) can then be

obtained from Eq. (F.3) by:

191

a a (17 i 0111311=1.2,...,8
T1514“: [(87.1% Kim 1 . F4
( 1k) (JmadP+ 3d? 61",] j,k,m ={1.2,...8(ifa,[3,y¢8) (.)
l (lfa,B,‘Y=8)

The stiffness components of [K] and [K T] are obtained from the left hand side of
the virtual work equation of Eq. (126) while using the constitutive relations, Eq. (129), to
relate the second Piola-Kirchhoff stresses and the Green’s strain for large rotations, small

strains.

1 5,615,. dVO = j €415,512}. dVo
V V

0 0

(F5)
i,j,k =1,2,...,6

Here, for convenience, contracted notation has been used instead of the normal tensorial
notation,
S] = Sxx; 52 = Syy; S3 = Szz; S4 = Syz; S5 = 52:2; S6 = Sxy; (F6)
E1 = E”; E2 = E”; E3 = E22; E4 = 2E”; E5 = 2E“; E6 = 2Exy
and Cik are the elastic coefficients defined in Eq. (129). The direct stiffness, [K] is
therefore obtained by adding the individual stiffness contributions of each “strain” product
from Eq. (ES). This is represented mathematically as:
13 6 6 B
(1 (1
Kij = 2 Z Cmnlkij lmn (F7)
m = 1n = 1
where [kgfihn are defined in Appendix G and Cm" are the elastic coefficients. The

tangent stiffness, [K T] , is obtained using a similar approach and are listed in Appendix

H.

APPENDIX G

192

APPENDIX G

i ' n ' rm n st in
To derive the direct stiffness matrix, it is easier to look at the individual strain products

of Eq. (F.5). Using Eqs. (123, 127-129, 134) and simplifying, we obtain, for example:

an (I)
1 13,85, dVo = 81.“, (j p, ,p 1., @9811!) av)”,
V. V.
6“” 1P ¢<k>P 5 N 5 N 5 M
+ “it: 15 1x a1 13x 13+ xj,X 14+ yj,X 15] 13(“2.x)
V .

0

+ Ma[Pi, 1151.3 + ”1.111514 + Nyi, X8k5](uz, x)

x P. <1) 0
( "x 6 (GI)
1 (I)
+ iMfllpja X813 '1' ij’ X5l4 + NYj, X515](uz, X))] deujB
i,j=1,2,...,4
k,l=1,2,...,7
01,13 = 1,2
to = 4(a—1)+i = l,2,...,8
jB = 4(B—1)+j=1,2,...,8
where 511‘ is the Kronecker delta. In general, Eq. (G. 1) can be expressed in the form:
k,l=1,2,...,8
m,n =1,2,...,6
kl 1 id _ 1,2,...8(ifk¢8)
JEnaEm dVo = 8dla[kia,jfl]mndjfi _ 1 (if k = 8) (6'2)
V

0

'13- 1,2,...8(ifl¢8)
J ' 1 (ifl=8)

where nodal degrees-of-freedom, di-B are as defined in Eqs. (F2, F3) and [kg 143],” is

the stiffness contribution from the strain products, E m, En. Comparing Eqs. (GI) and

193

(6.2), we obtain:

kl
1km. 113111 = 1P1, XPj, xq’th‘I’él) dV
V

1
+ J [2P1 X¢J(1k)[Pj' ,5,3 + ij,x514 + N”. X515]Mfl(uz’ X) (6.3)

'1’ [P1, x5k3 "' in, 2151.4 + Nyi, x5k51Ma(“z, X)

l
x (P13 xd’él) + 5MB[PJE x813 + ij, X514 + Nyj’ X515](uz, 20)] dV

where the first and second integrands represent the linear and non-linear contributions to
the overall stiffness matrix. The other stiffness components can be derived in a similar

manner and they are as follows:

kl
”101113112 = [P1, XPj, rq’lkaél) dV
v

0

1
+ Hip,- xcpgcqu’ r513 +ij,r514+Nyj, 1'5151M13(“z, r)
v, '

(G.4)
‘1' [P1321513 + Na, x5114 ‘1' Nyi, X5k5]Ma(uz, x)
1
x (Pi. 4311141),» §M5[Pj’ ,8,3 + 1v,r 1., 1514 + N”, ,5,5](uz, 0)] dV
kl — — —
1km, 113113 = [P1, xd’thMB, 2119,53 + ”11514 + ””5151 (W
V0
‘1' 1 [M 01M 13, 2“” i, x5k3 + ”111,115“ + Nyi, x5k5] (G.5)
v

0

x [131.63 + 71745,, + Nyj515](uz, X)] dV

Notice that the consistent normal strain field, 8 Eq. (125), has been utilized and the

22’

following interpolation functions have been introduced:

194

N”. = [(all)iN§i+(a21)iNni]

_ __ _ (no sum on i) (G.6)
N”. = [(a12)l.N§i + (a22)iNm.]

and definitions of fig? Nni are given in Eq. (115).

k8
”1011113 = 1P1,x¢¢(zk)M3,z (IV
V" ((3.7)
+ I[MaM3,Z[Pi, x513 + ”1.121514 'J' Nyi, x5115] (“2. x” dV
v

kl
1km, 1.5116 = 1 Pi, xogcnpj, yogi) +Pj, xwgn dV
V

0

1
+ l[§MB(Pi. X911“ '1' Malpi, x5k3 + in, x5k4 + Nyi, X5k5](uz, x»
v

0

(G8)
x (1P1, x513 "' ij, x514 '*' Nyj, x515](“z, y)
+ [P1, 1'513 ‘1' Nu, 1'514 + Nyj, r515](“z,x))

‘1‘ Ma[Pi, x5k3 'J’ in, x514 + Nyi, x5ksllpj, 1%” "' P1, x‘i’é’Wz. x)] dV

kl
[kia, 113121 = 1P1, YPj,X\¥t(1k)¢él) ‘1‘,
v

0

l
+ I [51”, 3111mm, + ~.,-,.6..+ Nyj,x515]MB(“z,x)
V .

0

(G.9)

+ Malpi, r5k3 + in, r514 + Nyi, 1'56““; 1’)

1
x (Pi. xd’fsl) + §[Pj, x513 + ij’ x514 + Nyj, X5151“; 19)] dV

195

kl
”101113122 = JPi,YPj,Y\P1(1k)\Pél) dV
v

0

l
V, '

(G.10)
+ [P1, r513 + er,y514 + Nyi, y5kslMa(“z. r)
1
x (PI, 31,!) + 5143113., ,5,, + N, 1., ,8,4 + N”, Y815](uz, ,0] dV
kl — — —
1km, 113123 = [P1, Y‘Pg‘)MB, 21])1513 + ”11514 + ””5151 (W
V0
1’ 11541111111215.1513 + Na, r514 + Nyi, Y8k5] (6'1 1)
V0
x [131.13,3 + N285” + Nyjagnuz, Y)] dV
k8
”101.1123 = [P1, yq’hk)M3,z (IV
V" ((3.12)

+ IlMaM3,Z[Pi, r5113 + in, r5k4 '1' Nyi, Y5k5] (“a Y)] dV
v

0

kl
[1,4 1.4126 = (P1, figure, yogupj, xwgn dV
V

1
+ 1 [51111391, 1"?th + Ma[Pi, Y5k3 + in, Y5k4 + Nyi, Y8k5](uz, 1'»
v0 ((3.13)
X ([Pj, x513 '1' Nu, x514 ‘1' 1vyj,X515](uz. r)
'1' [P1, Y513 + ”1.1, r514 + Nyj, Y515](“z,x))

'1‘ Malpi, r513 "’ in, 1'514 ‘J' Nyi, Y6k5][Pj, y‘pél) + P}, x‘l’é”1(uz, 1'” “JV

kl - — —
[k,a, 1.413, = j P j, chgwa, 2110,61, + 11114.5,4 + Nyifiks] dV
V

0

1 _ _ " (3.14
'1'- J[§MBMQ’Z[Pl5k3+NXi5k4+Ny16k5] ( )

x [P1311613 + Nx1,x514 + Nyj, x515](“z, X)] dV

196

81
[k1,113131 = [Pfix‘bénMade
v

0

l
+ l[§MBM3.Z[Pj, X513 +ij,x514+Ny,-,x5151(uz,11)] (W
Va

kl — — -
[kia,jB]32 = [P1, Y‘PénMazlprsu + N115“ ‘J' NyiSkS] dV
v

0

1 _ _ _

81
[k1.1'13]32 = [1943501431 dV
V

0

1
+ J‘[§MBM3, Z[PJ'.Y813 + ij, 1’514 + Nyj, 13151042, y)] 611’
V0

kl — — —
1km, 113133 = [Mot 2MB, Z[P1‘5k3 + ”11514 + NyiakS]
V0

x [131.8,3 + ij8,4 + Nyjsgj dV

k8 — — —
”101.1133 = 1M3,2Ma, 21131-5113 +Nx15k4+Ny18k51 dV
v

0

81 - — —
[k1, 11,133 = 1 M3, 2MB, 291513 + 111145,4 + Nyjag] dV
V

0

”$131133 = [(M3,z)2 (IV
V

0

(G. 15)

(G. 16)

(G. 17)

(G.18)

(G. 19)

(G20)

(G.21)

197

kl - — —
[km ”3],, == A'111,Z[Pj,l’(bél)+ P j, x‘I'éJJHP1513 + N11511: + N 1.815] dV
V y

0

1 _ _ _
V

x([Pj,x513+ij,x514+Nyj,x515](“z.r)

81
[1,, 1.13136 = (M31110, yogi) +PJ.’ MU] (W
V

1 G23
+ [ [5M3 ZMB([PJ.’ Xian-AN“; X5,4+Ny 1., 115151041, ,) J l
v, ’
+ [P1, r513 +ij, y514 +Nyj, Y515](“z.x))] (W
The consistent form of the transverse shear strains, Eqs. (118, 119) must be used to

evaluate the stiffness components for the non-linear model. For convenience, the

transverse shear strains will be expressed in a concise form as:

— (k) «(1‘), - _ (k) ,.(k)
7.. = inaum, y. — Lyman, ((3.24)
where the interpolation functions, Liz, LS; are readily obtained from Eqs. (118, 119)

and are given as:

(l) (1) (1)
ina = [(011)1P1¢§a,zb11+(“21)1P1¢ b12]

“(1.2
(2) (1) (1)
ina = [(a12)1pi¢§a,zbll +(022)1Pi¢na.zb12]
(3)
ina = [Pvgbn +P1’nb12]
(4) _ (2) (2)
ina " [(011)1(Pi¢§a,z + Néi’fiMalbll + (021)1(Pi¢na,z + Nni’nMalbIZ] (6'25)
(5) (2) (2)
ina = [(a12)1'(Pi(b§a, z + NfiirgMalbll '1' (“22)1(Pi(bna,z + Nni'nMa)b12]
(6) (3) (3)
ina = [(011)11’1‘1’501, zbll + (“21)1Piq’na. 21,12]
7 3 3 .
L5,“), = [(012)1P1¢(§d,zb11 + (022)1P1‘Pg‘1izb12] (no sum on i)

198

(l) (1) (1)
Lyia = [(011)1P1‘Pga, zb21 +(021)1P "1' [’22]

I “(1.2

(2) (1) (1)

Lyia = [(012)1P1‘Pga.zb21+(022)1P1‘Pna,zb22]
(3)

Lyia = wia[Pi’§b2l +Pi’nb22]

4 2 2
Lg“; = [(011)1(P1‘PéoZz+N§11§Malb21+(0201031161,;+Nn1’nMa)b22] (G26)

5 2 2
L( ) = [(012)1(Piqléa),z+N§12§Ma)b21 +(022)1(Pi‘y( ) +Nni’nMalb22]

M m. z

6 3 3
L( ) = [(011)1P1‘1’éo1),1b21+(021)-P"¥( ) b22]

yia 1 1 na,z
L3; = [(012)1P1‘P22,1b21+(022)1P1‘1’$.3J,1b221 (no sum on 1)

1ka 1.5144 = I 4243),, dV (G.27)
Va

[kg 1.11145 = 14’:ngng (6.28)
Vo

[kt-1,111.. = [lei-‘11.]... (6.29)

[£341,155 = [42491, dV ((3.30)
V

kl
[kia,jB]61 = IP13 X¢Bl)[Pi, y‘Pg‘) + Pi’x‘llakll dV
V

1
+ HIP!) JADE) + 5541311)}, X513 + ij, X514 ‘1' Nyj, 115151041, 11))
V0

x Ma([Pi, x5113 "' ”111,115“ + Nyi, 115/15““; 1')
‘1' [P1, Y5k3 "' in, r5114 'J' Nyi, r5k5](“z, x»

1
+ iMfllpj’ X5” + N“? X5“ + Nyj. X5k5][Pi. Yq’hk) ‘1' P1, x‘Péf’Muz, x)] (W

(G.31)

199

kl
”(01.113162 = 1P1. ”10m. y¢ékJ+P1,x‘I’&"’l W
V

kl
(kid, jB]63

1
+ j [(10]; $104,511,315, ,8,, + ij, ,8,4 + NH, Y815](uz, ,))
V0

x Ma([Pi, x5113 + ”111,115“ + Nyi, x5ks](“z, y)

"’ [P13115113 + Na, 1’514 + Nyi, r5k5](“z, 21))

(G.32)

1
+ §M111P1 1513 + N 514 + N 5115191. Yog") + P1,X‘P&k)](uz, V] W

xj,Y yj,Y

= 1 MB, Z[P,., ,(pgc) + P, X‘I’&’<)}[P 113,, + Fig.8“ + Ivy-515] W

V

0

+ [[MGMB, 21131-513 + NUS” + Nyjfifi]
Vo

X (1P1, x5k3 + ”111,115“ + Nyi, x5k5](“z, y)
+ “’1, 1’513 ‘J‘ N111, y5k4 ‘1' Nyi, r5ksl(“z.x))] (IV

(6.33)

k8
[k1a,1]63 = 1M3,2[P1, Y<I>1J,") +P1.X‘P((xk)] dV
v

0

+ l[M3. ZMa([P1, 1151.3 ‘1’ Nx1,x514 + Nyi, 11815101,, y) (G34)
V0

+ [P1, r5113 + Nx1,y5k4 + Nyi, 75115105, x) )] dV

200

kl
[k,a, 1.5166 = 1 [P1, yoga + Pi, fighupj, yog) + P 1., MO] (1V
V

1
+ j [5143(1104 Y<i>g<> + P, X81510]
V

0

+ Ma([Pi, x5113 '1' ”111,115“ "’ Nyi, x5151“; r)

+ [P1, 1151.3 + in, r5k4 + Nyi, r515](“z,x)) (6.35)
X (1P1, x513 + ij, X514 ‘1’ Nyj, x515](“z, r)

+ [P131313 '1' ~11}, r5114 '1’ Nyj, 1'55““; 11))

+ [P13 y<1>é’) +Pj, X91110]

X Ma([Pi, x5113 + in, x5114 + Nyi, 21515105, 1')

+ [P1, Y8k3 + N111, r514 + Nyi, y5115](“z,x))] dV

all other [ka 1111.... = 0 (G36)

APPENDIX H

201

APPENDIXH
11 |S|°m Mlllv K, E 11]

The tangent stiffness components are obtained via Eq. (E4). They are as follows:

T kl kl
J kiijJH = [kimjbln
l
+ l[‘2'Pi. X¢t(1k)[Pj,X513 +ij,X814+Nyj, x5151M13(uz,x)
v

0

+ (MaMB[P1, X513 ‘1' ”xi, x514 '1' Nyi, x5115”

X [P1,x513 +ij,X514+Nyj,X615][(uz,X)2+ (“x,x)]] dV (H-l)
i,j =1,2,...,4
k,l = l,2,...,7
01, = 1,2
ion = 4(01—1)+i = l,2,...,8
113 = 4(B—1)+j = l,2,...,8
T kl kl
1 kia,jB]12 = [1908113112
1
"’ [[5]: X¢hk)[Pj, 1513 +ij, y514+Nyj, r5151M13(“z.y)
v, ’
+ (MaMBIP1,x51¢3 + in,X8k4+Nyi,X8k5]) (H2)
1
X (5(1)); 1513 ‘1' N11}, 1'514 'J’ Nyj, r515)(“z,x)(“z. r)
1
+ (P 1., ,5,3 + ij.’ X51, + Nyj’ X515)|:(uy, ,) + 5(114 Y)le dV
r kl kl
[ kia,jl3]l3 = [kia,jB]13
"‘ [[MaMBIP1,x51¢3 +Nx1,x5k4+Ny1, x515] (H-3)
v

0

x [P13 111513 + ij’ x514 + Nyj’ x515](“z, 2)] (1V

202

r kl kl
[ kia,j13]16 = [kidjblm

1
+ I [5(P1,x<l>g‘)+ Ma[Pi’ x513 + ”111,115“ + N111, 115/15““; x))
V0

xMfl([PJ-’X5,3 +ij,x514+Nyj,x515](“z, ,1) (HA)
+ “DJ. y613 + ij’ y6,4 + N”, y615](uz, x))
+ MaM31P1,x513 + N 5,,4 + M44515]

X [Pj,x513 +ij, 11814 + Nyj,x515][ux, 1’ + uy,X + (uz,X)(uz, 1’)“ dV

xi, 1:
r kl kl
[ km, 10121 = 1km, 111121
1
‘1' I [2P1- ,‘1’1JJIP1,x513 + ”111', x514 + Nyj, 1151515411“; x)
v, ’
+ M 61M (31” i, ran ‘J’ N xi, y5k4 + Nyi, r5k5] (H5)

1
x (5(Pj, X513 + ij, X514 + Nyj, x515)(“z, x)(uz, y)
1 2
+ (P1, 1’813 + Nun/514 '1' Nyj, y515)[(ux, X) + i012, X) ])] dV

T kl kl
[ kia,jB]22 = [kidjfllzz

1
V0 ’

(H.6)
'1' (MaMB[P1, r5113 ‘1' ”xi, 1151.4 ‘1' Nyi, r5115”
x [P13Y813 + ij, ,6“ + N”; y515][(uz, y)2 + (uy, Y)]] (W
T kl kl
[ kia, 113123 = [kid 113123
+ I[MaMB[Pi, Y5k3 + Nx1,r5k4 + Nyi, Y5k5] (H-7)
v

0

X [P1, 1'513 + N11}, 1514 + Nyj, 1515““; 2)] (IV

203

T kl kl
[ kim113126 = [hm-11121,

1
+ 1 [5031, 1"?th "’ Malpi, Y5k3 + ”111, 1’514 + Nyi, y5k5](uz, y))
V0

x MB([PJ-’ ,6,3 + ”11111514 + N”, X65101, y) (H8)
'1’ [P1, 7513 + ij, y514 “J” Nyj, r515](“z. 11))
+ MaMfllpi, 1'513 + Na, r514 + 1Vyi,1’5k5]

X 1P); 1'513 "’ ij, r514 + Nyj, 1251511141, r + “y, x + (“8 XX“; 1')“ CW

T kl kl
[ kia,jB]3l = [kidj13131

1 — _. _
+ 1 [5M ,1 4MBIP1513 + IVA-51.4 + N,.5.sl (H9)
V0
X [P1, x513 4' ij, 11514 + Nyj, x515](“z, x” dV

where NJ", N . are defined in Eq. (G.5).
yr

T 81 81
[ k1,}13131 = “1.113131
1 (H.10)
v, ’
T kl kl
[ kra, 18132 = “‘11:, 16132
1 _ _ _
+ 1 [5M 1, 21411191513 + ”..,-514 + ”,..-5.51 (11.11)
v, '
x [P12 1513 + ij’ r514 + Nyj, 1'55““; Y)] dV
T 81 81
[ k1,;13132 = ”1.113132
(H.12)

l
V, ’

204

T kl kl
[ kia,jB]36 = ”101.1356

1 _ _ _
+ J [55401. ZMB[Pi5k3 "’ Nx15k4 + Nyi6k5] (H. 13)

X ([Pj, x513 "' ij, x514 + Nyj, x515](“z, y)
+ [P]; y513 +ij, y514 +Nyj, y515](“z,x))] (W

T 81 81
[ kl,jB]36 = [k1,jbl36

l
+ I [5M3 ZMWPJ: x513 + ij. x514 + Nyj.x5151(uz, y) (11.14)
v, ’

+ [P111513 + ij’ YE),4 + Nyj’ ,615](uz, x) )] dV

T kl kl
[ kia,jB]6l = [kimjfllm

l
+ I [EMOIMBIPJL X513 + ij, X514 + Nyj, x515]
Va

X ([P 1, 115113 + ”111,115“ + Nyi, 7515]“; x)2

+ [P8115113 + ”n, x5114 + Nyi, x5k5](“z, x)(“z, y))
(H.15)

+ MaMfisz, x)2 + (ux’ 11)]

X ([Pi, x5113 + ”n, x5k4 "' Nyi, X8k5][Pj, Y5k3 + ij, y5k4 '*' Nyj, 115115]
+ ”’1, r5113 + ”111,115“ "‘ Nyi, YakSHPj, x513 + ij, X514 + Nyj, x5151)

l

+ 5M 3“” j, x5113 + N 5114 + N 51:51“D 1, chg‘) + Pi, #9)]("5 X)] “W

xj,X yj,X

205

T kl kl
[ km. 113162 = [kimjfiltsz

l
+ I [EMGMBIPJL Y513 + ij, Y514 + Nyj, y515]
V0

X (IP13 x5113 4' in, x5114 + Nyi, x5k5](“z, y)2

"' [P1, y5k3 "’ in, y5k4 + Nyi, 751151“; x)(“z, y))
(H.16)

1
+ MGMB[§(uz’ ,)2 + (11% g]
X (“’13 x5113 + ”111,115“ + Nyi, x5k5][Pj, 115113 + N
+ [P1, Y8k3 + in, y5k4 + N

1

x j, Y5k4 + ”N. Y5k5]

yi, Y8k5][Pj, x513 + ij, x514 + Nyj, x5151)

51:4 + N 5115]“); Y‘I’g‘) +Pi,X‘P((xk)](uz, fl] dV

xj,Y yj,Y

T kl kl
[ kia,jB]63 = [kia,j[3]63

+ [MaM (uz )
Jo B ’2 (H.17)

X (“D1, x5113 4' in, x5114 "‘ Nyi, X6k5][Pj. Y5k3 + ij. Y5k4 + Nyj. 1’5“]

+ [P1, y5k3 4' in, 115184 + Nyi, YakSHPj, x513 + ij, x514 + Nyj, 115151)] 4"

T kl kl
[ ioz,jB]66 = [kia,jB]66
l
+ j [5541391. ,cbgk) + Pi, xwgn
V

0

X ([Pj, x513 + ij, 111514 + Nyj, x515](“z, y)
+ [P]; 75113 + ij, 115114 "' Nyj, 115/15““; 11))

l
+ iMaMBHPj. x513 + ij. X614 + NH" X5’5](uz' Y)

+ [P]; Y5k3 + ij, 215114 "’ Nyj, Y5k5](“z, X))

X ([P 1, X8k3 4' in, x5114 "’ Nyi, x5k5](“z, y)

4' [P13115113 + ”11115114 + Nyi, Y5k5](“z, x))

+ MaMpKux, y) + (My, x) + (uz, x)(uz, y)]

X ([P 1, X513 + in, x5114 + Nyi, 115/151”}; Y5k3 + ij, y5k4 "’ N y j, Y5k5]

"' [P1, Y5k3 + ”111,115“ + Nyi, 115/15]”); x513 + ij, x514 + N 5151)] (W

(H.18)

yj. X

T ' ' kl
all Other [ kt}, 181nm = [kim jBlmn (H.19)

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