ubc_1990_a1 k66

SUPER FINITE ELEMENTS FOR NONLINEAR STATIC AND

DYNAMIC ANALYSIS OF STIFFENED P L A T E STRUCTURES

By

T A M U N O I Y A L A S T A N L E Y K O K O

B. Sc. (Hons), University of Ife, Nigeria, 1982

M . Eng., University of Nigeria, Nsukka, Nigeria, 1986

A THESIS SUBMITTED IN

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T H E UNIVERSITY OF BRITISH COLUMBIA

October 1990

T A M U N O I Y A L A S T A N L E Y K O K O , 1990

In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.

Department

The University of British Columbia Vancouver, Canada

Date

DE-6 (2/88)

Abstract

The analysis of stiffened plate structures subject to complex loads such as air-blast

pressure waves from external or internal explosions, water waves, collisions or simply

large static loads is still considered a difficult task. The associated response is highly

nonlinear and although it can be solved with currently available commercial finite

element programs, the modelling requires many elements with a huge amount of input

data and very expensive computer runs. Hence this type of analysis is impractical at

the preliminary design stage. The present work is aimed at improving this situation

by introducing a new philosophy. That is, a new formulation is developed which is

capable of representing the overall response of the complete structure with reasonable

accuracy but with a sacrifice in local detailed accuracy. The resulting modelling is

relatively simple thereby requiring much reduced data input and run times. It now

becomes feasible to carry out design oriented response analyses.

Based on the above philosophy, new plate and stiffener beam finite elements are

developed for the nonlinear static and dynamic analysis of stiffened plate structures.

The elements are specially designed to contain all the basic modes of deformation

response which occur in stiffened plates and are called super finite elements since

only one plate element per bay or one beam element per span is needed to achieve

engineering design level accuracy at minimum cost. Rectangular plate elements are

used so that orthogonally stiffened plates can be modelled.

The von Karman large deflection theory is used to model the nonlinear geometric

behaviour. Material nonlinearities are modelled by von Mises yield criterion and

associated flow rule using a bi-linear stress-strain law. The finite element equations

are derived using the virtual work principle and the matrix quantities are evaluated by

n

Gauss quadrature. Temporal integration is carried out using the Newmark-/? method

with Newton-Raphson iteration for the nonlinear equations at each time step.

A computer code has been written to implement the theory and this has been

applied to the static, vibration and transient analysis of unstiffened plates, beams and

plates stiffened in one or two orthogonal directions. Good approximations have been

obtained for both linear and nonlinear problems with only one element representations

for each plate bay or beam span with significant savings in computing time and costs.

The displacement and stress responses obtained from the present analysis compare

well with experimental, analytical or other numerical results.

111

Table of Contents

Abstract ii

List of Tables ix

List of Figures xiv

Acknowledgements xv

1 Introduction 1

2 Literature Review 4

2.1 Analytical Methods 4

2.2 Numerical Methods 6

2.2.1 Finite Difference and Finite Element Methods 6

2.2.2 Finite Strip Method 8

3 Description of the Super Finite Elements 9

3.1 Introduction 9

3.2 Super Finite Element Discretization 11

3.3 Displacement Functions 14

3.3.1 Plate elements 15

3.3.2 Beam elements 18

3.4 Compatibility, Convergence and Order of Accuracy 21

4 Theoretical Formulation and Analysis of Problem 23

4.1 Introduction 23

iv

4.2 Equations of Mot ion 23

4.3 Finite Element Formulation 25

4.3.1 Introduction 25

4.3.2 Shape Function Matrices 26

4.3.2.1 Plate elements 26

4.3.2.2 Beam elements 28

4.3.3 Strain Displacement Relations 29

4.3.4 Constitutive Relations 30

4.3.5 JMass and Damping Matrices 34

4.3.6 Stiffness Formulation 35

4.3.7 Load Vector 37

4.3.8 Torsion Beam Element 39

4.3.8.1 Stiffness matrix for beam torsional element 40

4.3.8.2 Mass matrix for beam torsional element 44

4.4 Numerical Integration 46

4.5 Temporal Integration 48

4.6 Computer Code 51

5 Static Analysis Results 53 5.1 Introduction 53

5.2 Unstiffened Plates 54

5.2.1 Square Plate I with Simply Supported Edges 54

5.2.2 Square Plate I with Clamped Edges 59

5.3 Beams 65

5.3.1 Rectangular Beam with Simple Supports 65

5.3.2 Rectangular Beam with Clamped Ends . 72

5.4 Stiffened Plates 73

5.4.1 Clamped 2-Bay Stiffened Plate I 73

v

5.4.2 Clamped DRES Stiffened Panel 79

5.4.3 Clamped DRES1B Stiffened Panel 95

5.4.4 Simply Supported 2x2-Bay Stiffened Plate I 106

6 Vibration Analysis Results 120

6.1 Introduction 120


6.2.1 Square Plates with Various Edge Conditions 121

6.2.2 3-Bay Continuous Plate 123

6.2.3 2 x 2-Bay Continuous Plate . 124

6.3 Beams 125

6.3.1 Rectangular Beams with Various Boundary Conditions . . . . 125


6.4.1 2-Bay Stiffened Plate II with Clamped Boundaries 127

6.4.2 3-Bay Stiffened Panel with Clamped Edges 130

6.4.3 DRES Stiffened Panel 133

6.4.4 2x 2-Bay Stiffened Plate 134

6.4.5 2x4-Bay Stiffened Plate with Clamped Edges 135

6.4.6 4x4-Bay Stiffened Plate 137

7 Transient Analysis Results 141

7.1 Introduction 141


7.2.1 Square Plate I with Simply Supported Edges 142

7.2.2 Square Plate I with Clamped Edges 144

7.2.3 Simply Supported Square Plate II Subject to Triangular Load 146

7.3 Beam Example 149

7.3.1 Rectangular Beam with Simple Supports 149

vi


7.4.1 Clamped 2-Bay Stiffened Plate II 151

7.4.1.1 Step Load 152

7.4.1.2 Blast Load 154

7.4.2 Simply Supported 2-Bay Stiffened Plate II 155

7.4.2.1 Step Load 156

7.4.2.2 Blast Load 158

7.4.3 HOB 315 Loading on Clamped DRES Stiffened Panel 160

7.4.4 Clamped DRES1B Stiffened Panel 162

7.4.5 Clamped 2x2-Stiffened Plate II 166

7.4.6 Clamped 4x4-Bay Stiffened Plate 169

8 Summary and Conclusions 183

Bibliography 187

A Shape Functions 194

B Strain-Displacement Matrices 198

C [fi] Matrices for Plate and Beam Elements 202

D Formulas for J, I z z , J0 and T 205

vn

List o f Tables

4.1 Strain rate parameters for steel and aluminum 33

4.2 Sampling points and weights for Gaussian integration 47

5.1 Linear elastic response of simply supported Square Plate I 55

5.2 Linear elastic response of clamped square plate I 59

5.3 Linear elastic response of simply supported rectangular beam . . . . . 68

5.4 Linear elastic response of clamped rectangular beam 72

5.5 Linear elastic response of 2-Bay Stiffened Plate I 77

5.6 Deflections and strain energy in linear elastic DRES stiffened panel . 87

5.7 Deflections and strain energy in linear elastic DRES1B panel 98

5.8 Nonlinear elastic response of DRES1B panel 103

5.9 Nonlinear elastic-plastic response of DRES1B panel 103

5.10 Nonlinear elastic response of 2x2-Bay Stiffened Plate I 115

5.11 Nonlinear elastic-plastic response of 2x2-Bay Stiffened Plate I . . . . 115

6.1 Eigenvalues of square plates with various edge conditions 122

6.2 Natural frequencies of 3-bay continuous plate 124

6.3 Natural Frequencies of 2x2-Bay Continuous Plate 126

6.4 Natural frequencies of rectangular beams 127

6.5 Comparison of net number of variable in 2-bay stiffened plate . . . . 129

6.6 Natural frequencies of 2-bay stiffened panel 130

6.7 Comparison of net number of variable in 3-bay stiffened plate . . . . 132

6.8 Natural frequencies of 3-bay stiffened panel 133

6.9 Natural frequencies of DRES stiffened panel 134

viii

6.10 Natural frequencies of 2 x 2-bay stiffened plates 135

6.11 Natural frequencies of 2 x 4-bay stiffened plate 137

6.12 Net number of variables in 4x4-bay stiffened plate 139

6.13 Natural frequencies in 4x4-bay stiffened plate 140

ix

List of Figures

3.1 Schematic representation of an orthogonally stiffened plate 10

3.2 Assemblage of plate and beam elements 12

3.3 The super finite elements 13

3.4 Shape of the Lagrange polynomials 17

3.5 Shape of the Hermitian polynomials 18

3.6 Shape of the function 19

3.7 Shapes of the sin 27r and sin47r functions 20

4.1 Bi-linear stress-strain relationship 31

4.2 Stress-strain relation of strain-rate sensitive material 34

4.3 Shapes of typical blast loads 38

4.4 Torsion beam element 41

4.5 Beam cross-section 48

5.1

5.9 ax at 0.2 N / m m 2 in nonlinear elastic clamped Square Plate I 66

5.10 ax at 0.8 N / m m 2 in nonlinear elastic clamped Square Plate I 67

5.11 Linear elastic-plastic response of simply supported rectangular beam . 69

5.12 Large deflection response of simply supported rectangular beam . . . 70

5.13 Stresses in large deflection analysis of simply supported rectangular

beam 71

5.14 Large deflection response of clamped rectangular beam 74

5.15 Stresses in large deflection analysis of clamped rectangular beam . . . 75

5.16 Configuration of 2-Bay Stiffened Plate I 76

5.17 In-plane displacement in 2-Bay Stiffened Plate I 78

5.18 Panel centre displacement in 2-Bay Stiffened Plate I 79

5.19 Stiffener centre displacement in 2-Bay Stiffened Plate I 80

5.20 Displacement shapes along CB of 2-Bay Stiffened Plate I 81

5.21 Configuration of DRES stiffened panel 82

5.22 Super element models for DRES stiffened panel . 83

5.23 ADINA discretization of DRES stiffened panel 84

5.24 Displacement of point D in DRES stiffened panel 85

5.25 Displacement of point C in DRES stiffened panel 86

5.26 Normal stress perpendicular to stiffener at 0.5 psi . 89

5.27 Normal stress parallel to stiffener at 0.5 psi 92

5.28 Configuration of DRES1B panel 96

5.29 Models for DRES1B panel 97

5.30 Linear elastic in-plane displacements in DRES1B panel . . '. 99

5.31 Linear elastic bending displacements in DRES1B panel 100

5.32 Linear elastic normal stresses in DRES1B panel 101

5.33 Nonlinear elastic in-plane displacements in DRES IB panel 102

5.34 Nonlinear elastic bending displacements in DRES1B panel 104

xi

5.35 Nonlinear displacements of points D and E in DRES1B panel 105

5.36 Large deflection elastic-plastic displacement profiles in DRES IB panel 107

5.37 Nonlinear elastic normal stresses in DRES1B panel 108

5.38 Normal stress ax in nonlinear elastic DRES1B panel 109

5.39 Normal stress ay in nonlinear elastic DRES1B panel 110

5.40 Details of 2x2-Bay Stiffened Plate I I l l

5.41 Bending displacements in linear elastic 2x2-Bay Stiffened Plate I . . 112

5.42 In-plane displacements in linear elastic 2x2-Bay Stiffened Plate I . . 113

5.43 Normal stress o~x in linear elastic 2x2-Bay Stiffened Plate I 114

5.44 Normal stress o~y in linear elastic 2 x 2-Bay Stiffened Plate I 116

5.45 Nonlinear elastic-plastic u displacement profile along y = 15 in in sim-

ply supported 2x2-Bay Stiffened Plate I 117

5.46 Nonlinear elastic-plastic v displacement profile along x 7.5 in in

simply supported 2x2-Bay Stiffened Plate I 118

5.47 Nonlinear elastic-plastic w displacement profile along x = 7.5 in in

simply supported 2x2-Bay Stiffened Plate I 119

6.1 Configuration of 3-bay continuous plate 123 ^

6.2 Configuration of 2 x 2-bay continuous plate 125

6.3 Configuration and discretizations for 2-Bay Stiffened Plate II 128

6.4 Configuration and discretizations for 3-bay stiffened plate 131

6.5 Configuration and discretization for 2x4-bay stiffened plate 136

6.6 Configuration of 4x4-bay stiffened plate 138

7.1 Dynamic relaxation response of linear elastic simply supported Square

Plate I 144

7.2 Transient response of simply supported Square Plate I 145

7.3 Transient elastic response of clamped Square Plate I 146

xii

7.4 Transient nonlinear elastic-plastic response of clamped Square Plate I 147

7.5 Details of Square Plate II 148

7.6 Central displacement history of simply supported Square Plate II . . 149

7.7 Transient linear elastic response of simply supported rectangular beam 151

7.8 Transient linear elastic-plastic response of simply supported rectangu-

lar beam 152

7.9 Loads applied to 2-Bay Stiffened Plate II 153

7.10 Linear elastic response of clamped 2-Bay Stiffened Plate II due to step

load 155

7.11 Nonlinear elastic response of clamped 2-Bay Stiffened Plate II due to

step load 156

7.12 Nonlinear elastic-plastic response of clamped 2-Bay Stiffened Plate II

due to step load 157

7.13 Panel centre displacement of clamped 2-Bay Stiffened Plate II due to

blast load 158

7.14 Stiffener mid-point displacement of clamped 2-Bay Stiffened Plate II

due to blast load 159

7.15 Linear elastic response of simply supported 2-Bay Stiffened Plate II

due to step load 160

7.16 Panel centre displacement in nonlinear elastic analysis of simply sup-

ported 2-Bay Stiffened Plate II due to step load 161

7.17 Stiffener mid-point displacement in nonlinear elastic analysis of simply

supported 2-Bay Stiffened Plate II due to step load 162

7.18 Panel centre displacement in nonlinear elastic-plastic analysis of simply

supported 2-Bay Stiffened Plate II due to step load 163

7.19 Stiffener mid-point displacement in nonlinear elastic-plastic analysis of

simply supported 2-Bay Stiffened Plate II due to step load . . . . . . . 164

xm

7.20 Panel centre displacement in simply supported 2-Bay Stiffened Plate

II due to blast load 165

7.21 Stiffener mid-point displacement in simply supported 2-Bay Stiffened

Plate II due to blast load 166

7.22 Blast load on D R E S Stiffened Panel 167

7.23 Panel centre displacement of blast loaded D R E S Stiffened Panel . . . 168

7.24 Stiffener mid-point displacement of blast loaded D R E S Stiffened Panel 169

7.25 Discretizations of D R E S I B panel 170

7.26 Displacements of points D and E in D R E S 1 B panel due to blast load 171

7.27 Displacements of points C and F in D R E S 1 B panel due to blast load 172

7.28 Displacement along y = 24in in D R E S 1 B panel due to blast load . . . 173

7.29 Displacement along G D in D R E S 1 B panel due to blast load 174

7.30 Displacement along B C in D R E S I B panel due to blast load 175

7.31 Configuration of 2 x 2 Stiffened Plate II 176

7.32 Linear elastic response of 2 x 2 Stiffened Plate II - Step Load 177

7.33 Nonlinear response of 2 x 2 Stiffened Plate II - Step Load 178

7.34 Response of points A , B and G in 4 x 4 Bay Stiffened Plate - Rectan-

gular Pulse 179

7.35 Displacement profiles along H G F in 4 x 4 Bay Stiffened Plate - Rect-

angular Pulse 180

7.36 Displacement profiles along D B A in 4 x 4 Bay Stiffened Plate - Rect-

angular Pulse 181

7.37 F ina l displacement profiles along beams D B A and E C B in 4 x 4 Bay

Stiffened Plate - Rectangular Pulse 182

xiv

Acknowledgements

The author wishes to express his gratitude to his supervisor Dr. M . D. Olson for

his guidiance, advice and valuable time during the preparation of this thesis. The

valuable suggestions offered by Dr. D. L. Anderson at various stages of the work are

appreciated. The author would also like to thank Dr. R. Houlston of the Defence

Research Establishment, Suffield, Alberta for providing the ADINA results for the

DRES IB panel. The author is also appreciative of the many useful discussions he

had with Dr. J . Jiang who also provided some of the comparison dynamic finite strip

results. The support and encouragement offered by P. Kumar, R. B. Schubak, Dr. J .

D. Dolan, Dr. A. Filiatrault and all the author's friends are also much appreciated.

Financial support of the Natural Sciences and Engineering Research Council of

Canada in the form of a Research Assistantship from the department of Civil Engi-

neering, University of British Columbia is gratefully acknowledged.

xv

Chapter 1

Introduction

Stiffened plates are structural components consisting of plates1 reinforced by a sys-

tem of orthogonal beams2 (or ribs) to enhance their load carrying capacities. These

structural components find wide application in various kinds of structures. For ex-

ample, in naval architecture stiffened plates are used in the construction of the hulls

of ships while in the aircraft industry they are used in constructing the fuselage of

aircraft. Stiffened plate construction is also found in bridges, buildings, railway cars,

large transportation carrier panels and storage tanks.

There are situations in which a stiffened plate structure might be subjected to

complex loads such as air blast pressure waves from external or internal explosions,

water waves, collisions or simply large static loads. These loads can induce large

deformations which stress the material well over the elastic limit to cause significant

plastic deformations in the structure. Hence in the design/analysis of the structures

geometric and material nonlinearities must be taken into account.

The static or dynamic large deflection elastic-plastic analysis of stiffened plates is

a difficult task. Solution of the problem by analytical techiques is practically impos-

sible due to the intractable task of integrating the governing differential equations of

motion. On the other hand the problem can be solved by numerical methods based 1 Plates are flat surface structures whose thicknesses are small compared to their other dimensions.

Of particular interest in this study are rectangular thin plates whose thicknesses are less than a tenth of the least other dimension.

2 A beam is a structure whose length is large compared to its other dimensions and carries load primarily by bending.

1

Chapter 1. Introduction 2

mainly on the finite element technique. Currently, there are available some com-

mercial finite element programs capable of performing nonlinear analysis of stiffened

plates. However, with these regular finite element programs, the modelling requires

many elements with a huge amount of input/output data and very expensive com-

puter runs. This type of analysis is impractical at the preliminary design stage.

The work presented in this thesis is aimed at improving the situation by introduc-

ing a new analysis/design philosophy. It is the objective of this thesis to develop a

simplified numerical formulation which is capable of representing the overall response

of the complete structure with reasonable accuracy but with a sacrifice in local de-

tailed accuracy. This will result in a relatively simple model which requires much

reduced input data and run times and makes it feasible to carry out design oriented

analyses.

On the basis of the above philosophy, new plate and stiffener beam finite ele-

ments are developed for the nonlinear static and dynamic analysis of stiffened plate

structures. Using polynomial as well as continuous analytical displacement functions,

the elements are specially designed to contain all the basic modes of deformation

response which occur in stiffened plate structures and are called super finite elements

since only one plate element per bay or one beam element per span is needed to

achieve engineering design level accuracy. Rectangular plate elements are used so

that orthogonally stiffened plates can be modelled.

Following a brief literature review, presented in Chapter 2, the new super elements are described in Chapter 3. The super element discretization and the displacement functions are discussed and the chapter ends with a discussion on the justification for

the choice of the displacement functions using a study of the compatibility, conver-

gence and order of accuracy analysis of the elements.

In Chapter 4 the finite element matrix quantities are derived for the super el-ements. The mathematical derivations conducted in the chapter include details of

Chapter 1. Introduction 3

the Gaussian integration, the Newton-Raphson iterative scheme and the implicit

Newmark-/3 temporal integration procedure. The chapter also highlights some de-tails of the computer code developed to implement the theory.

The numerical investigations carried out to verify the new formulation are pre-

sented in Chapters 5, 6 and 7. Chapter 5 deals with the static response, Chapter 6

the vibration analysis and Chapter 7 is devoted to transient applications. Unstiffened

plates, beams and plates stiffened in one or two mutually perpendicular directions

are considered. For the static and transient applications, various combinations of

geometric and material nonlinearities have been incorporated but only linear elastic

vibrations have been carried out in Chapter 6.

Finally, Chapter 8 gives the summary and conclusions derived from the present

study and ends with some suggestions for future research.

Chapter 2

Literature Review

2.1 Analytical Methods

The classical thin plate theory is well known for the linear elastic analysis of isotropic

unstiffened plates. For large deflection analysis of such plates the large deflection

theory by von Karman is usually employed. The applications of these methods to

analysis of plate structures is well documented in [1].

A comprehensive study of the historical development of analytical procedures for

the analysis of stiffened plates has been presented by Troitsky [2]. It is clear from

the review that the first known attempt at the analysis of stiffened plate structures is

due to Boobnov in 1902. In his work Boobnov applied stress analysis procedures to

two-way stiffened plates and treated the problem as a beam on an elastic foundation

and obtained design charts for stiffened plate structures.

The development of the orthotropic plate theory by Huber in 1914 made significant

contributions to the analysis of stiffened plates. Although the theory was developed

for naturally orthotropic plates1 it can also be applied to stiffened plates by treating

them as being equivalent to an orthotropic plate of constant thickness. In order for

the orthotropic theory to apply it is required that 1 Natural orthotropy is used to describe the situation in which a body possesses different elastic

properties in orthogonal directions. On the other hand the elements of the body may be arranged in proper geometric configurations such that it exihibits different properties in orthogonal directions. This is called structural or technical orthotropy. Orthogonally stiffened plates are examples of such structures.

4

Chapter 2. Literature Review 5

1. The material of the stiffened plate follow Hooke's law.

2. Plane sections remain plane before and after bending and the out-of-plane dis-

placements be small compared to the plate thickness.

3. The stiffeners be evenly and closely spaced.

4. The stiffeners be disposed symmetrically with respect to the mid-plane of the

plane.

The last assumption is difficult to meet in practice since for most stiffened plate

structures the stiffeners (or ribs) are placed asymmetrically with respect to the mid-

plane of the plate. As a consequence, the inherent assumption of a strain-free mid-

plane is grossly violated by such structures and hence the orthotropic plate theory

cannot be applied rigorously to these structures unless certain modifications are made.

A more rigorous extension of the orthotropic plate theory to eccentrically stiffened

plates has been conducted by Pfluger in 1947. Using the force-displacement relations

for a plate element with ribs on one side he obtained an eighth order differential

equation involving only a displacement component. This method has been employed

by several investigators and Clifton et al [3] have developed an exact theory, based

on the method, for plates with ribs of open or closed box sections.

An important feature associated with stiffened plates subjected to bending loads

is the shear lag phenomenon. This pheneomenon describes the situation where the

normal stress is maximum at the stiffener location and shows a lag with increasing

distance from the stiffener. The analysis of this behaviour has been given considerable

attention by some investigators and some of these have been discussed by Troitsky [2]

and Timoshenko and Goodier [4].

There are very few analytical solutions of stiffened plates exhibiting geometric non-

linear behaviour. Troitsky [2] has reported on the large deflection theories developed

by Vogel in 1961 and Abdel-Sayed in 1963. These theories are, essentially, extensions


of the large deflection theory for orthotropic plates, following the von Karman large

deflection theory for isotropic plates. Solutions exist to only a few problems.

Modern structures have to be designed into the plastic range to take advantage of

the extra load-carrying capacity afforded by the ductility of the material. However,

incorporating the theory of plasticity into a stiffened plate theory presents enormous

difficulties if an analytical solution is sought. The problem can be simplified by

assuming a rigid-plastic material behaviour. The yield line analysis is based on this

assumption and proceeds with assumed collapse mechanisms. With this method

solution to some plate structures have been obtained [5].

A major difficulty in the use of analytical methods is due to the onerous task

of integrating the governing differential equations of motion. The situation is worse

when geometric and/or material nonlinearities are involved. However, with numerical

techniques it has been possible to obtain fairly easily approximate solutions suitable

for engineering purposes. These numerical methods are discussed in the next section.

2.2 Numerical Methods

2.2.1 Finite Difference and Finite Element Methods

The finite difference and finite element methods are well known in the analysis of engi-

neering problems. In the finite difference method, the governing differential equations

of motion are replaced by a set of difference equations written for a finite number

of grid points into which the domain of the problem is divided. The resulting set

of algebraic equations are solved simultaneously for the finite number of unknown

parameters at the grid points and this represents an approximation to the exact so-

lution [6] . Webb and Dowling [7] have applied the method to the large deflection

elasto-plastic analysis of discretely stiffened plates.

In the finite element method, the governing equations are also replaced by a set


of algebraic equations which are obtained by discretizing the continuum into a finite

number of elements. The accuracy of the solution usually depends on the number

of elements and the order of the trial polynomial functions used to approximate

the displacement/stress variations within each element. The finite element method

is by far the most versatile of all numerical methods as it can handle problems with

complicated geometry, boundary conditions or loadings very easily. Most importantly,

with finite elements the ribs of a stiffened plate need not be symmetrically placed with

respect to the midplane of the plate or be densely and equally spaced since the method

is quite capable of simulating the response of plates with discrete stiffeners easily.

Several researchers have applied the finite element method to linear elastic static

and dynamic analysis of stiffened plates. Extension of the method into the large

deflection elastic-plastic range have also been investigated [8,9,10]. Several all purpose

finite element programs have been developed in recent times. Some of these programs

have capabilities for static and dynamic large deflection elastic-plastic analysis of

stiffened plates. For example the VAST (Vibration and Strength Analysis Program)

and ADINA (Automatic Dynamic Incremental Nonlinear Analysis) programs have

recently been used to analyze stiffened plates subjected to air blast loading [11].

Inspite of the existence of these finite element programs there are very few publi-

cations dealing with orthogonally stiffened plates subjected to large static or dynamic

loads capable of inducing geometric/material nonlinearities. The reason for this is

that a complete nonlinear finite element analysis requires the use of huge input data

and very expensive computer runs. It is for this reason special finite elements such

as finite strips have been developed to analyze certain classes of problems. These are

discussed in the following subsection.


2.2.2 Finite Strip Method

The finite strip method, developed by Cheung [12], is suitable for the analysis of

structures with regular boundaries. The structure is divided into a finite number of

strips and, unlike the finite element method in which polynomial functions are used

in all directions, the finite strip method uses continuously differentiable analytical

functions in one direction and polynomials in other directions. Furthermore, the con-

tinuous functions are stipulated to satisfy a priori the kinematic boundary conditions

at the ends of the strips.

Details of the application of the method to static and dynamic analysis of plate

structures are well documented in [13]. The extension of the method to nonlinear anal-

ysis of unstiffened plates has also been investigated by some researchers [14,15,16,17].

Only recently, the method has been extended to static and dynamic large deflection

elastic-plastic analysis of stiffened plates [18,19,20]. However, these applications are

restricted to one-way stiffened plates. Attempts have been made to model orthog-

onally stiffened plate using compound strips [21,22] but these have been limited to

linear elastic problems.

The major advantage of the finite strip method over the more versatile finite

element method is its simplicity. It requires smaller amount of input data and core

memory and uses much reduced run times compared to the finite element method.

In the present study, the advantages of the finite element and finite strip methods

are combined to develop a relatively simple formulation capable of representing the

response of stiffened plates subject to intense static or dynamic loads with reasonable

accuracy. The method uses special finite elements called super finite elements which

are macro elements having both polynomial and continuous analytical displacement

functions in all in-plane directions. The description of the new elements is presented

in Chapter 3.

Chapter 3

Description of the Super Finite Elements

3.1 Introduction

The class of structures which are of interest in this work include rectangular plates

reinforced by stiffener beams placed in one or two mutually perpendicular directions,

as typified by the structure in Figure 3.1. Each panel bay (eg. A B C D ) is modelled

by a rectangular plate element and the stiffeners (such as A B , BC) are treated as

beam elements running along the edges of the panels. The displacement fields for the

plate and beam elements have been carefully chosen to simulate all possible linear and

nonlinear deformation modes for the elements acting together as in stiffened plates

or separately as simple or continuous beams or plates with all possible boundary

conditions. The elements are termed super finite elements since only a single element

is required to model the basic response. The details of the elements and the associated

displacement fields are presented in this chapter.

In Section 3.2 the super element discretization is presented and the degrees of

freedom associated with each element are described. Section 3.3 gives details of all

the displacement fields for each element and Section 3.4 highlights the justification

for the choice of the displacement functions.

9

Chapter 3. Description of the Super Finite Elements 10

y. v

i A i : B :

J i D i : c :

1 z ,w Plan ^ p | a t e

c.

beams

J

Jl

X, u

elevation

Figure 3.1: Schematic representation of an orthogonally stiffened plate


3.2 Super Finite Element Discretization

Figure 3.1 shows details of an orthogonally stiffened plate. By the present formulation

each panel bay is represented by a super plate element and each stiffener span by a

super beam element. A panel bay and two adjacent beams are isolated in Figure 3.2

to illustrate the assemblage of the elements in the structure. Details of the nodal

variables associated with each element are presented in Figure 3.3.

The plate element has 9 actual nodes, numbered 1 to 9. The 'extra nodes' labelled

uio> u n , v i 5 a r e actually the amplitudes of the trigonometric functions used to

model the in-plane displacements (described in Section 3.3) and these are lumped at

the mid-side and central nodes labelled 5 to 9. Each of the four corner nodes has

six variables the two in-plane displacements, u, v; the out of plane displacement,

w; the two slopes, wx, wy and the twist, wxy, where wx = dw/dx, etc. The positive

directions of u, v and w are shown in Figure 3.2. Each of the mid-side nodes numbered

5 to 8 also has six variables u, v, w and the normal slope wx or wy, together with

two additional in-plane variables. For example, the mid-side node, numbered 5, has

the variables u, v, w, wy, uw and Ui3; node 6 has the variables u, v, w, wx, vn

and v14; and similarly for nodes 7 and 8. The central node, numbered 9, has seven

degrees of freedom u, v, w, u u , v12, ui5, and v15. Over all, the plate element has

55 degrees of freedom as shown in Figure 3.3(a).

The beam element (Figure 3.3(b)) has 3 actual nodes numbered 1 to 3. In analogy

to the plate element, the 'extra nodes' labelled u4 and u5 are amplitudes of the in-

plane displacement and these are taken as variables at the middle node. For bending

and axial action alone, a beam in the ^-direction has three variables u, w, and wx

at the end nodes and four variables u, w, u 4 and u5 at the middle node. However,

for problems in which there is significant presence of torsion in the stiffener beams,

additional variables are included to approximate the torsional rotation, 6 and lateral


Figure 3.2: Assemblage of plate and beam elements


4

wio

1*14 i * n

7

" 1 5 U12 n 9

v12 VX5

1 1*13

Degrees of Freedom:

Jy) wxy

X, u

At nodes 1,2,3,4 - u,v,w,wx,wy At nodes 5,7 - u,v,w,wy; plus (1*10,1413) and ( i 4 n , i 4 i 4 ) , respectively At nodes 6,8 - u,v,w,wx; plus (I ; II , I>14) and ( ^ 1 0 , ^ 1 3 ) , respectively At node 9 - u,v,w; plus i * 1 2 , v12, W i s , U 1 5

(a) plate element

1 1*5 i * 4 3 2

Degrees of Freedom: At nodes 1,2 - u,v,w,wX,9,9X At node 3 - u,v,w,9; plus 1*4,1*5

(b) beam element in x direction

Figure 3.3: The super finite elements


bending displacement, v. The additional variables are v, 9, 9X at the end nodes and

v and 6 at the middle node.

For beam problems, the nodal variables are located along the beam centroidal

axis and in this case torsional displacements have been ignored so that the beam

element has a total of 10 variables. However, in stiffened plate structures, ,all nodal

variables are assumed to be located at the centroidal plane of the plate. This way,

no additional degrees of freedom are introduced at a beam-plate connection [23]. In

structures with negligible torsional displacements, the stiffener beam element still has

10 variables but the number is increased to 18 if torsion is significant.

Inter-element continuity1 between two adjacent plate elements or a plate and a

beam element is ensured at the three nodes along the plate edge, while continuity

between two adjacent beam elements is provided at the beam end nodes.

3.3 Displacement Functions

The displacement fields for the super elements have been carefully chosen so that all

possible displacement modes in a stiffened plate structure can be modelled fairly ac-

curately with only one super element per bay or span. To achieve this goal continuous

analytical functions (usually, trigonometric and hyperbolic functions) are 'smeared'

with the usual finite element polynomial functions in a fashion similar to the finite

strip formulation, except that, in this case, no boundary conditions are satisfied a

priori and the analytic and polynomial functions all run in the two in-plane direc-

tions. The super elements thus combine the simplicity of the finite strip method and

the versatility of the regular finite element method. The displacement fields chosen

for each element are described in the following subsections.

1 The continuity is defined in Section 3.4.


3.3.1 Plate elements

Figure 3.3(a) shows a typical super plate element of length a and width b. The nodal

variables are as indicated in the figure and the element has 55 degrees of freedom.

The displacement fields associated with the element are given by

u 10

u = Nfm + sin 27r[L 1 (7 ? ) ,L 2 (77) , L3(T))\ <

u12

+ sm4ir[Ll(v),L2(-n),L3(y)} <

Ul3

Ul5

(3.1)

v = N-Vi + [LT_(0, L2((), L3()] sin 2wn I

VlO

v v12 /

} +

+ [L1(aL2(aL3(()]sm4Trrl{

Vl3

'14

Vl5

(3.2)

w = N ^ + U)[Hi(v),H2(V),Ha(r,),H4(V)]< Wy 5

Wy7

} +

+[H1((),Hi{C)tH3{aHA((Mv)

w$

w. xS

w6

Wx6

+ # 0 M K (3.3)


where i 1, 2, . . . , 9; j = 1, 2, . . . , 16; u, v are the two in-plane displacements; w is

the flexural bending (out-of-plane) displacement and

= x/a and 77 = y/b.

The quadratic Lagrange interpolation polynomials are given by:

HO = n 2 - H + 1

L2{() = 2 e - i (3.4)

These quadratic polynomials are shown in Figure 3.4. The cubic Hermitian polyno-

mials are given by:

H2(() = a{i - 2e + e)

HsU) = 3(2 - 2 3 (3.5)

and their shapes are as shown in Figure 3.5. (j> 1S the first symmetric vibration mode

of a clamped beam and is given by:

4>(0 [o;(sinh fi( sin/x) + (cosh// cos/z)]/ (3-6)

where p = 4.7300407448,

cos p cosh fl sinh p sin p

anc

(p = a(sinh0.5/x sin0.5/i) + (cosh0.5/i cos0.5/x)

The shape of the (f> function is shown in Figure 3.6.


1.2

"20.0 0.5 1.0

x/a

Figure 3.4: Shape of the Lagrange polynomials

iV", TV" are products of the Lagrange interpolation polynomials and JVj" are prod-

ucts of the Hermitian polynomials. These are given explicitly in Appendix A. u,, are the nodal variables in the x, y directions, ipj are the corner node lateral displace-

ments, slopes and twists; all attached to the midplane of the plate. The corner node

lateral displacement vector is given by

{%/J}t = [wl,wxl,wyl,wxy-L,w2,wX2,wy2,wxy2, (3.7)

w3, wx3,wy3,wxy3, w4, wx4,wy4, wxy4]

where wx = dw/dx, etc.

The shapes of the trigonometric functions sin 2 ^ and sin47r used to model

the in-plane displacements are shown in Figure 3.7.


x/a

Figure 3.5: Shape of the Hermitian polynomials

3.3.2 Beam elements

Figure 3.3(b) shows a super beam element of length a, with its degrees of freedom.

The membrane and flexural displacement fields referred to the centroidal axis of a

beam in the ^ -direction are given by

u3

> + u 4 sin 2TV( + u5 sin 4ir +

+e[H[{(),H'2(t),H'3(t),H'4(t))l

Wx2

+ e4>\)wz (3.8)


= [ t f i ( 0 , # 2 ( 0 , # 3 ( 0 . - ^ ( 0 M w2

wx2

(3.9)

where the primes denote differentiation with respect to x and e is the distance between

the centroidal axis of the beam and the mid-plane of the plate.

The effect of torsion and lateral bending in the stiffener beam element has been

included in some cases and the rotation, 6 and lateral displacement, v fields are

approximated, respectively, by


1.5

1.0 h

0.5

0.0

-0.5 h

-1.0

-1.5

sin(27rx/a)

sin(47rx/a)

0.0 0.5 x/a

Figure 3.7: Shapes of the sin27r and sin4.7r functions

1.0

* = [ffl(0,tf2(0,#s(0,#4(0]

01

9X1

9x2

( \ Vl

+ ^ 3

= [ii(0.ij(0.i(0] V2 { V3 )

(3.10)

(3.11)

where 6i,9xi, ...v3 are the beam nodal rotation, twist or lateral displacement vari-

ables as illustrated in Figure 3.3(b)


3.4 Compatibility, Convergence and Order of Ac-

curacy

Consider the w displacement field for the panel A B C D in Figure 3.1. The basic

response of the panel with clamped boundaries all round is represented by the last

term 0(^)^>(T/) in Equation (3.3). Then to allow for support movements and to ensure

compatibility of displacements and slopes between adjacent panels the 16 degree of

freedom plate bending element, developed by Bogner et al [ 24 ] , having the bi-cubic

polynomial functions (first term of Equation 3.3) is employed. The other two terms

in Equation 3.3 are included firstly to match the plate and beam displacements along

the plate edges and secondly to allow various boundary conditions to be modelled

along the edges. The w displacement field is C 1 continuous and hence will ensure

convergence of the solution of linear plate bending problems with the error in strain

energy being of order 0(l4) where Z is a characteristic element dimension.

Bi-quadratic shape functions are used for the in-plane displacements u and v in

order to obtain an order of accuracy in energy consistent with the out of plane dis-

placements, C continuity between adjacent elements also being ensured. The sine

terms in the in-plane displacement fields are included to capture nonlinear geomet-

ric effects sin 2 7 r being suitable for simply supported boundaries and sin 4 7 r for

clamped boundaries [18,19]. These functions are essential in providing a good ap-

proximation to the distribution of membrane stresses in large deflection elastic-plastic

analyses as demonstrated by some of the results of the analyses in Sections 5.2.2 and

7.2.3. The sine functions are multiplied by quadratic shape functions in the other

direction to maintain the order of accuracy, ensure compatibility between plate and

beam displacements and also to capture shear lag effects.

In analogy to the plate, the basic response of the beam element A B with clamped


boundaries is represented by (f>(). Then to allow for arbitrary end motion and to en-

sure compatibility between elements, the cubic polynomials are included. A consistent

order of accuracy (0(Z4)) in strain energy is also ensured. The choice of the in-plane

displacement field also ensures compatibility between beam and plate displacements,

continuity of displacements between adjacent beam elements and a consistent order

of accuracy.

The basic rotational response of the stiffener beam element A B with clamped ends

is also approximated by () to correspond to the wy displacement field along the edge

A B of the plate element A B C D . Then for arbitrary end rotation and for compatibity

between elements the cubic Hermitian polynomials are included in Equation 3.10.

This rotation field is also C 1 continuous and the error in the linear torsional strain

energy is of order 0(/ 4 ) . The lateral bending displacement field, v is chosen to be

quadratic for compatibility with the in-plane displacement field in the plate. Con-

sequently, it does not provide C 1 continuity which is normally required for beam

bending. However, this is probably of little consequence since the effect of lateral

bending is expected to be very small.

Chapter 4

Theoretical Formulation and Analysis of

Problem

4.1 Introduction

The theoretical formulation and method of analysis of the problem is presented in this

chapter. First, the governing equations of motion are derived in Section 4.2. Then in

Section 4.3 the finite element formulation is introduced. Here, the super elements de-

scribeed in Chapter 3 are used in conjuction with the strain-displacement and consti-

tutive relations to derive the finite element matrix quantities. The Newton-Raphson

iterative scheme used to obtain the tangent stiffness matrix is also presented. The nu-

merical integration scheme used to evaluate the matrix quantities is briefly discussed

in Section 4.4, while Section 4.5 highlights the temporal integration scheme. Finally,

Section 4.6 focuses on some important aspects of the computer implementation.

Because the analysis procedures presented in this chapter are widely available in

the literature, only brief summaries are presented to highlight the important aspects

that apply to this work.

4.2 Equations of Motion

The governing equations of motion are obtained via the principle of virtual work. If

a deformable body subject to a set of arbitrary loading and boundary conditions is

23

Chapter 4. Theoretical Formulation and Analysis of Problem 24

in equilibrium, the principle of virtual work stipulates that

Wint + Wext = 0 (4.1)

where Wint and Wext are kinematically admissible small variations in the internal

virtual work and external virtual work, respectively. When the applied loading is

temporal in nature d'Alembert's principle is employed to include the inertial forces.

Thus ignoring body forces the equations of motion can be expressed as

Jv[{d}Tp{d} + {d}TKd{d} + {e}T{*})dV - Js{d}T{q}dS = 0 (4.2)

where {c}, {e} are the stress and strain vectors, p the mass density, is a viscous

damping parameter, q is the applied surface traction, V and S are, respectively,

the volume and surface area and {d} is the displacement vector. The tilde ( * ) is

used to denote a virtual change in the given quantity with respect to a generalized

displacement and the superimposed dot [( ' ) = (d/dt)} denotes differentiation with

respect to time. For static loads, the velocity and acceleration terms {d} and {d} in

Equation (4.2) vanish.

In Equation (4.2) the internal virtual work Jv{e}T{cr}}dV has been stated in terms

of the true (Eulerian) stresses o~ij which are defined in the deformed configuration.

This is made possible by the assumption that the strains considered in this work are

small and hence the true stresses cr^ are approximately equal to the Kirchhoff stresses

Sij which are defined in the undeformed configuration [25].

Equation (4.2) can be specialized for plate or beam structures by appropriately

defining the stress, strain and displacement terms. For plate structures,

UV = f ( { d } ) = k x , e y , 7 * J

(4.3)


where


torsional stiffness and mass matrices for the stiffener beam elements are treated sepa-

rately in Section 4.3.8 since the effect of stiffener beam torsion has not been included

in all cases.

4.3.2 Shape Function Matrices

4.3.2.1 Plate elements

The displacement fields for the plate elements are given by Equations (3.1), (3.2) and

(3.3). These equations can be written collectively as

u

v

w

= [N]{Se} (4.4)

where {8e} is the element nodal displacement vector which for the plate element is

given by

(4.5)

{6e} T = l u i , V l , W l , W x l , W y l , W x y l , U 2 , V 2 , W 2 , W x 2 , W y 2 , W x y 2

U 3 , V 3 , W3, W x 3 , W y 3 , W x y 3 , UA, V 4 , W 4 , W x 4 , Wy4, Wxy4

u5,v5, w5,wy5, uw,u13,u6, v6, w6, wx6, vn,v14

u7, v7, wr, wy7, uu, u14,u8, v8, w8, wx8, vw,vl3

u9, v9, w9, u12,v12, u15, v15\

and [TY] is the shape function matrix which for the super plate element is given by

[N} =

7Y" 0 0 0 0 O i V j O 0 0 0 0

0 Nf 0 0 0 0 0 7V2V 0 0 0 0

0 0 Ng Ng Ng Ng 0 0 Ng Ng Ng Ng


0

0

0

0

N?

0

0

The shape functions, i V " , . . . in Equation (4.6) are defined exphcitly in A p -

pendix A .

In the regular finite element method, a shape function for any nodal variable has a

unit value at that node and zero value at all other nodes. In the present formulation

only the functions TV", . . . , Ng\ N",... ,N% and N, ... , N satisfy this requirement

and.hence, in a finite element analysis, the magnitudes of the nodal variables corre-

sponding to these shape functions will represent the magnitudes of the corresponding

displacements at the respective nodes. For example, in any analysis, the magni-

tudes of Wi, wxi, wyi,wxyi represent the actual out-of-plane displacement, slopes

and twist at node 1; and similarly for other corner nodes. Also, the magnitudes of

Ui, u2, . ,UQ\ VI, v2, ,v9 give the actual in-plane displacements at the respective

nodal positions.

However, the remaining functions N?0,... , JV^; Nf0, ..., 7V"5 and N7, ... , N5 do

not satisfy the requirement of having a unit value at one node and zero at all other

0 0 0 0 0 N% 0 0 0 0 0

0 0 0 0 0 Nv4 0 0 0 0

0 Nw i V 1 0

Nw J v 12

0 0 Nw i v 1 3

i v14 Mw

i V 1 5 Nw

0 0 0 i V i o J V 1 3 iV 6" 0 0 0 0 0

Nv5 0 0 0 0 0 NX 0 0 ( V i i J V 1 4

0 Nw i \ 1 7 Nw 0 0 0 0 i V 1 9 i V 2 0 0 0

0 0 0 -


nodes. This is due to the presence of the analytical functions in these displacement

functions. The variables corresponding to these shape functions merely represent

displacement amplitudes at the respective nodes. To obtain the actual displacements

at these nodes, in any analysis, the contributions from other shape functions have to

be included. For example, w5, wy5 do not automatically represent the out-of-plane

displacement and normal slope, respectively, of node 5. The displacement and normal

slope at this node will include contributions from these amplitudes as well as those

from other shape functions at that point.

4.3.2.2 B e a m elements

The in-plane and out-of-plane displacement fields for the beam element (Equations

(3.8) and (3.9)) can be combined as

u

w = [N){Se}

where the element displacement vector in this case is given by

(4.7)

{ e } T = l u 1 , W 1 , W x l , U 2 , W 2 , W x 2 , U 3 , W 3 , U 4 , U 5 \

and the shape function matrix for the super beam element is

(4.8)

[N] = N Ng1 N N iV 5 m N N? N N? N 10 0 Nl 0 Nh3 NbA 0 Nb5 0 0

(4.9)

The shape functions appearing in Equation (4.9) are presented in Appendix A. The

superscripts m and b denote membrane and bending, respectively.

Again, the function Nb does not rigourously satisfy the requirements of a shape

function and hence the variable corresponding to this function, w3, does not auto-

matically represent the displacement at node 3, as discussed for the plate element


case.

4.3.3 S t r a i n Disp lacement Re la t ions

In this study, it is assumed that the plates are thin and the beams slender so that

the effect of shear deformation in these structures is negligible. Large deflection

effects are taken into account by including first order nonlinearities in the strain

displacement relations (following von Karman theory). By this theory it is assumed

that the deflections are equal to or larger than the plate or beam thickness, but still

small relative to the other dimensions (a or 6) of the plate or beam. For the plate the

strain-displacement relations are

du d2w 1 dw 2 x = dx~~Z~dx^ + ^~dx~>

dv d2w 1 dw ^ , = * - * V + 2 ( * ) ( 4 1 0 )

^ du ' dv d2w dw dw ^xy x y dy dx dxdy dx dy

where ex, ey and exy are the strain components in the x y plane and ^ x y is the

engineering shear strain. For a beam spanning the x or y direction the relevant

relation is the first or second of Equations (4.10). Using these equations and taking a

small variation with respect to the generalized displacements, the virtual strain vector

{e} in Equation (4.2) can be related to the virtual displacements, {8e}, symbolically

as

{i_} = [[B] + [Co(6.)]]{6e}

where [B] is the linear strain-displacement matrix given by

(4.11)


\B\

-2- 0 -z&-dx dx2

jsi-9 2

0 a_ ay JL JL

. dy dx -2z dx&y

[N] (4.12)

[Co] is the nonlinear strain-displacement matrix given by

[Co]

dNT" dNV dx dx

dN? dNJ dy dy

, dN dNJ dNV dm I 1 1 J 1 1 \ w . \ dx dv dx dv I 1

(4.13)

dx dy dx dy ' 3

i,j = 1, 2, .. .25. The relevant expressions for beam elements are obtained by drop-

ping the appropriate terms in Equations (4.12) and (4.13). See details in Appendix B.

4.3.4 C o n s t i t u t i v e Re la t ions

The assumption of elastic material behaviour is no longer adequate in the design

of modern structures since emphasis is placed on savings in weight. The theory of

plasticity provides more realistic estimates of the load carrying capacities of struc-

tures. To keep pace with modern trends the material of the stiffened plate structure

is assumed to be elastic-plastic with a bilinear stress-strain relationship.

Figure 4.1 shows the elastic-plastic (hardening) model (for uniaxial conditions)

adopted in this study. E and ET are, respectively, the elastic and plastic modulii;

c>o, e0 the uniaxial yield stress and strain, respectively; and ee, ep, respectively, rep-

resent the elastic and plastic strains. For the situations considered the elastic and

plastic strains are assumed to be of the same order of magnitude. In plasticity theory

three important aspects of material behaviour, namely, the yield criterion, the flow

rule or normality condition and the hardening rule, must be specified to define the

constitutive relations. These are described briefly in the following paragraphs.

The yield criterion specifies the state of stress which causes yielding of the ma-

terial. For a multiaxial state of stress this is equivalent to the state in which an


Figure 4.1: Bi-linear stress-strain relationship

equivalent effective stress first exceeds the uniaxial yield stress of the material. The

yield function, F has the general form

F({*},K) = 0 (4.14)

where K is a positive parameter and in general K and F({a}) depend on the existing

level of plastic strain and the plastic strain history. The von Mises yield criterion is

adopted here since it predicts well the behaviour of metals [26,27,28]. This criterion

assumes that yielding occurs when the distortion or shear strain energy equals the

distortion energy at yield in simple tension and the yield condition is given by

\ " * ) a + (*2 - + ( '3 - * i ) 2 ] = al (4.15)

where, C j ,


after the onset of yielding. An associated flow rule is employed. This assumes that

the incremental strain vector at any point on the yield surface is normal to the yield

surface at that point, as opposed to a non-associated flow rule where the strain vector

takes any other direction. The associated flow rule results in an incremental plasticity

theory.

Most structural materials exhibit strain hardening behaviour, in that, the yield

surface changes as yielding progresses. Hardening rules describe how the yield sur-

face changes with yielding. Two commonly used hardening models are the kinematic

and isotropic hardening models. In the kinematic hardening model, the yield sur-

face maintains its original size but translates in stress space as yielding progresses.

This model accounts for the Bauschinger effect, which is a situation in which the

yield surface appears to translate in stress space under cyclic loading situations. In

the isotropic hardening model the yield surface retains its initial shape but expands

uniformly about the origin of the stress space. Although this model ignores the

Bauschinger effect it is widely used in engineering practice due to its simplicity and

is employed in this formulation.

Based on the plasticity theory described above the incremental stress-strain rela-

tions can be written as [29]

{da} = [DT]{de} (4.16)

where {da} and {de} are the incremental stress and strain vectors, respectively, and

[DT] is the elastic-plastic constitutive matrix given by

[DT] = [D] - [D]{A}{A}T[D}[E' + {A} r[I>]{A}]- 1 (4.17)

[D] is the elasticity matrix, {A} = dF/d{a}, F is the yield function, E" = ET/(1-^-)

is obtained from the bi-linear stress-strain curve of Figure 4.1 and E, Ej are the elastic

and plastic modulii, respectively.


The stress increment due to a given strain increment can now be obtained from

Equation (4.16) and the state of stress at the end of any iteration step is the sum of

the stress increment and the stress from the previous iteration. Yielding takes place

when the equivalent effective stress, cre exceeds the uniaxial yield stress,


a

e

Figure 4.2: Stress-strain relation of strain-rate sensitive material

Strain rate effects can be easily incorporated in elastic-plastic subroutines [20,27],

although these have not been included in the elastic-plastic algorithm in this work.

However, strain rate effects have been taken into account in some cases by upgrading

the static yield stress in accordance with Equation (4.18).

4.3.5 Mass and Damping Matrices

Having established the shape functions, strain-displacement and stress-strain rela-

tions, attention will now be focused on the derivation of the finite element matrix

quantities. In this section the mass and damping matrices will be derived.

Using Equation (4.11) and the substitution {d} = [N]{Se}, Equation (4.2) can

now be written for each element as


{Se}TJv [[N]Tp{N]{Se} + [Nf Kd[N]{Se} + [[B] + [C0}f{*}} dV = {Se}T ^[NfqdS

(4.19)

Since {8e}T is arbitrary, Equation (4.19) can be written as

K K U + [ce]{Se} + / [[B] + [C0]}T{


displacements. The equation can now be expanded in a Taylor series about a known

solution {S0} so that Equation (4.23) is expressed as

{*(*,)} + 8 W g { e } ({S}-{S0}) + . ..={?} (4.24)

Ignoring higher order terms and defining the incremental nodal displacement vector

as {AS} = {S} {S0}, Equations (4.24) can be recast as

[kT}{AS} = {p} - {$(S0)} (4.25)

where [kT] =


For a beam element in the x-direction the entries in [fl] are expressed as

dNb dNb

where i,j = 1, 2 , . . . , 10; r, s = 1, 2, . . . , 5 are selected to correspond to the appropriate

w variables, with those corresponding to the u variables being zero.

Details of the [fl] matrices are given in Appendix C.

4.3.7 Load Vector

In this thesis it is assumed that the structure is subject to uniformly distributed loads

only. That is, the spatial variation of the load q is not considered.

In the dynamic realm, q is a function of time. Blast type dynamic loads are

considered, and in general, the temporal variation of the load can be expressed as

q(t) = qm(l - i / r ) e x p ( - A 1 i / r ) 0


Figure 4.3: Shapes of typical blast loads


area, S. For beam elements q is in units of force per unit length and the integration

is carried over the beam length. In a stiffened plate the load is usually applied on the

plating so there is no loading on the stiffener beams.

Only vertical loads (loads in the out-of-plane direction) are considered in this work,

so the consistent load vector will not contain entries in the locations corresponding

to the in-plane variables. The intergral in Equation (4.32) is evaluated exactly and

the load vector for each element type is given below:

Plate Elements

M T = f|f LO, 0,36, 6, 6,1,0, 0 , 3 6 , - 6 , - 6 , - 1 , 0, 0, 36, - 6 , - 6 , 1 , 0, 0, 36, 6, - 6 , - 1 ,

0, 0, 72*, 12*, 0, 0, 0, 0, 72*, - 1 2 * , 0, 0,

0, 0, 72*, - 1 2 * , 0, 0, 0, 0, 72*, 12*, 0, 0,

0,0,144*2, 0,0,0, 0J (4.33)

where a, b are the length and width, respectively, of the plate element, q is the value

of the load level and the constant factor * is given by

y 1 l * = / )d( [a(cosh p + cos p 2) + (sinh fi sin u))

Jo


symmetric loads are considered, but has significant effect on the linear vibration

response of stiffened plates (see Chapter 6 for details of the response results). It has,

therefore, been considered necessary to include the effect of stiffener beam torsion for

dynamic problems.

In this section, the beam torsional stiffness and mass matrices are derived explicitly

from the strain energy and kinetic energy integrals. These matrices are calculated in

a separate subroutine in the computer program and the beam torsional stiffness and

mass matrices are conceptually added to the global stiffness and mass matrices in the

usual finite element fashion.

It should be noted that only linear torsional displacements have been considered

on the asumption that these displacements are small. Also, coupling of bending and

torsional deformations has not been considered in the formulation for simplicity. A

more rigorous formulation might consider the effect of finite bending on the twisting

of the beam and also an investigation of lateral buckling of the beam.

4.3.8.1 Stiffness matrix for beam torsional element

Consider a beam element in the a; direction. The degrees of freedom for torsion and

lateral bending are as shown in Figure 4.4. The variable v represents displacement in

the y direction (lateral displacement). Consider the lateral movement of the beam

(Figure 4.4). The beam centroidal displacement vc is given by

vc = vT + e6 (4.35)

where vr is the lateral displacement at the top of the beam and is equal to the v-

displacement along the edge of the plate. Along an edge of the plate v is quadratic

(see Equation (3.2)), hence is chosen to be quadratic to ensure compatibility of

displacements. Using Equations (3.10) and (3.11) the torsional rotation and lateral


displacement fields can be expressed as

0 < > =

VT

Hi H2 H3 HA 4>() 0 0 0

0 0 0 0 0 Lx L2 L3

Oi

0x1

02

0x2

03

Vl

V2

V3

(4.36)

V, , 0 ! , 0 x 1 V 3 , 0 J V*2, 0 2 . 0 x 2

Figure 4.4: Torsion beam element

The shape functions in Equation (4.36) have been described in Chapter 3. Note

also that for a beam in the x-direction, 9 = wy.

The strain energy in the beam due to torsion, lateral bending and warping (in-

duced by torsion) is given by


U = \CJ r 92xdx + \EIZZ [\vc,xx)2dx + \ET r 92xxdx Z Jo Z Jo Z Jo

= \GJ [ae2xdx + \EIZZ f (4xi + 2evTtXXetXX + e2e2xx)dx + ^ ET f e2xxdx Z Jo Z Jo Z Jo

(4.37)

where a is the beam length, G is the shear modulus, J the torsional constant, T

the warping constant and I Z Z is the moment of inertia about the z z axis. The

symbolism ( > x ) implies differentiation with respect to x. Formulas for the calculation

of J, T, I Z Z for various beam cross-sections are given in [31] and the relevent ones

are quoted in Appendix D. Equation (4.35) has been used to obtain the expanded

form of Equation (4.37).

By taking the first variation of the strain energy with respect to the generalized

displacements the torsional stiffness matrix is obtained as

[k] = GJ [ae2xdx + EIZZ r v2xxdx+EIzze I" vTiXX6iXxdx + E(e2IZZ+T) f 92xxdx Jo Jo Jo Jo (4.38)

Utilizing the shape functions in Equation (4.36) and carrying out the integrations in

Equation (4.38) the torsional stiffness matrix can be written as

[k] = [k,] + [k2] + [*3] + [fc4] (4.39)


where

[*i] = GJ

6 a 6 a 0 5 10 5 x10 0 a 10

2a2 15 a 10 a

2

30 pxa 6 a 6 a 0 5 10 5 10 0 a 10 a

2

30 a 10 2a

2

15 -pia

0 0 -uta Pi

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

16EIZ2

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 1 1 - 2

0 0 0 0 0 1 1 - 2

0 0 0 0 0 - 2 - 2 4

[*s] = EIzze

0 0 0 0 0 0 0 0

0 0 0 0 0 - 4 - 4 8

0 0 0 0 0 0 0 0

0 0 0 0 0 4 4 - 8

0 0 0 0 0 0 0 0

0 - 4 0 4 0 0 0 0

0 - 4 0 4 0 0 0 0

0 8 0 - 8 0 0 0 0

(4.40)

(4.41)

(4.42)


[*4 E(e2Izz + V)

12 6a -12 6a 0 0 0 0

6a 4a 2 6a 2a 2 0 0 0 0

-12 6a 12 6a 0 0 0 0

6a 2a 2 6a 4a 2 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

where px = 0.5231643586, p2 = 4.877697585, and u3 - 198.4629301.

4.3.8.2 M a s s m a t r i x for b e a m tors iona l e lement

The kinetic energy in the beam due to torsion and lateral bending is given by

T = \w*p (\jc62 + Av2c)dx (4.44) I Jo

where J c is the polar moment of inertia about the centriod, A is the area of the beam

cross section, CJ is the natural frequency of the beam and p the mass density. Substi-

tuting Equation (4.35) into Equation (4.44), and defining Jo as the polar moment of

inertia about the top of beam the kinetic energy can now be expressed as

T = \ L O 2 P [a[Jo02 + A(vT + 2evT8)]dx (4.45) 2 Jo

Hence, by variational techniques the torsional beam mass matrix is obtained as

pa /a ya [m] = pj0 / 02dx + pA vTdx + pAe / vT6dx (4.46) Jo Jo Jo

Utilizing the shape functions in Equation (4.36) and carrying out the integrations

in Equation (4.46) the torsional mass matrix can be written as

[m] = [ m i ] + [m2] + [m3] (4.47)


where,

[ m i ] = pJo

13a 35

11a2 210

9a 70

13a2 420 p4a 0 0 0

11a2 210

a 3 105

13a2 420

a 3 140 p5a

2 0 0 0

9a 70

13a2 420

13a 35

11a2 210 p4a 0 0 0

13a2 420

a 3 140

11a2 210

a 3 105 -p5a

2 0 0 0

p4a p5a2 p4a -p5a2 p6a 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

\m2 pAa

~3u~

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 4 - 1 2

0 0 0 0 0 - 1 4 2

0 0 0 0 0 2 2 16

[ m 3 ] = pAae

0 0 0 0 0 l l a2

60 a 60

a 3

0 0 0 0 0 a2

60 0 a 2 15

0 0 0 0 0 a 60 l la 60

1 3

0 0 0 0 0 0 a2

60 a 2 15

0 0 0 0 0 p7a p7a psa

11a2 60

a 2 60

a 60 0 p7a 0 0 0

a 60 0

11a 60

a 2 60 pTa 0 0 0

a 3

a 2 15

1 3

a 2 15 p8a 0 0 0

(4.48)

(4.49)

(4.50)

where p4 = 0.2615821802, p5 = 0.0562872124, p6 = 0.396478085, p7 = 0.0364333308

and p8 = 0.4502976989.


4.4 Numerical Integration

The well known Gaussian integration scheme is adopted in the evaluation of the

volume integrals in Equations (4.21), (4.22) and (4.26). Let / = f(,7]X) be the

integrand in each of the volume integrals, then by the integration scheme the volume

integrals can be presented as

j f fQ f(t,V,C)dtdVdC = llllllji^o^dfjdc ~ E E E ^ W ( 6 , % , a ) (4.5i)

k j i where , fj, are the transformed normal coordinates which vary from 1 to +1,

i, Vj> Ck are the sampling points in the x, y, z directions, respectively, W{, Wj,

are the wieghting factors in the respective directions.

The weights and sampling points are selected to minimize the error in the integral.

The scheme employed depends on the form of the integrand, / . In any one direction,

n sampling points can integrate a polynomial of order (2n 1) exactly [29]. However,

for the problem at hand this formula cannot be applied as the integrand, in general,

consists of products of polynomial and circular or hyperbolic functions. Based on

previous experience [18,19,32] and a numerical experiment conducted in this work,

5 integration points are employed in each in-plane direction. In the out-of-plane

direction a 2-point integration scheme is used for elastic analysis. However, for plastic

analysis it is necessary to upgrade the number of sampling points to 4, in order to

capture the plastic stress distribution across the plate or beam thickness [33]. The

location and weights of the sampling points are presented in Table 4.2. A more

comprehensive list can be found in [34].

For beams of I or T cross-sections an additional point is introduced for each

flange. In the evaluation of the volume integrals for a beam element, the integrand is

evaluated at the sampling point in the beam direction and this is multiplied by the


Table 4.2: Sampling points and weights for Gaussian integration

/ + 1 / (0 t f = w / ( 6 ) 1=1

n W 2 0.57735 02691 89626 1.00000 00000 00000 4 0.86113 63115 94053

0.33998 10435 84856 0.34785 48451 37454 0.65214 51548 62546

5 0.90167 98459 38664 0.53846 93101 05683 0.00000 00000 00000

0.23692 68850 56189 0.47862 86704 99366 0.56888 88888 88888

corresponding weighting coefficient and a weighted area. The weighted area is equal

to the product ^hwtwW{] where hw, tw are, respectively, the height and thickness of

the beam web and Wj is the weighting factor corresponding to the sampling point in

the thickness direction. For an extra point in a flange, the weighted area is simply

equal to the area of that flange.

If the beam cross-section is rectangular or has equal flanges, the centroidal axis

coincides with the normal coordinate, axis. But for T or I sections with unequal

flanges these axes do not coincide and it is necessary to transform the normal coor-

dinate variable about the beam centroidal axis.

Consider the beam shown in Figure 4.5. Let A A be the centroidal axis of the

beam section and B B the centroidal axis of the web alone. Also, let ( c , be the

running variables measured with respect to the axes A A and B B , respectively. Then,

Cc = C - d

But

d = ( C 2 - / 2 ) - y

= h - C l - h - \ { h - h - h )


Hence,

Cc = C + ^ ( 2 C l - & - / ! + / , )

with = ^Chw. For the extra point in a flange, the variable is measured from the

beam centroid to the middle of the flange.

B

h w 2

L

B

Figure 4.5: Beam cross-section

4.5 Temporal Integration

The element matrix quantities are assembled conceptually in the usual finite element

fashion to obtain the global equations of motion:

[M}{6} + [C}{6} + Jv[[B] + [C0]}T{a}dV = {P} (4.52)

where, [M], [C] are the global mass and damping matrices, respectively, and {P} is

the global load vector. Equations (4.52) are nonlinear ordinary differential equations

in time.


In this section attention is focused on the time integration of the equations of

motion (Equations (4.52)). For nonlinear problems, direct integration, or step-by-

step, methods are preferred to modal or other methods. In the direct integration

methods, the time derivatives appearing in the governing equations are replaced by a

finite difference approximation. Two main groups of direct integration methods are

available. These are the explicit and implicit schemes. Explicit schemes only require

satisfaction of the equations of motion at the previous time step in order to advance

the solution to the next time step. In contrast, implicit schemes require information

from both the previous and next time steps.

There are several direct integration methods available in the literature [27,35,36]

and, in general, the choice of the scheme to use depends on the nature of the problem

at hand. In this thesis, the implicit Newmark-/3 method [37] is employed. The

method is second order accurate and is unconditionally stable for linear problems, in

that the solution does not grow without bound even when a large time step is used.

Unconditional stability of the method also applies for nonlinear problems, although

there is no rigorous proof in this case [27]. As a consequence of this condition the

method allows for the use of large time steps. Hence, time steps could simply be

based on the lowest fundamental frequency (highest period) rather than on the highest

frequency (which requires more effort to evaluate accurately) as would be the case

if a conditionally stable explicit scheme is used. This is particularly useful in the

present work since one of the aims of the study is to develop a simple formulation

which requires reduced run times.

The Newmark-/? method is also self starting, in that it requires only the initial

data to commence the solution and is thus a single step method - a desirable feature.

The method has also been applied successfully to many nonlinear structural problems

[27].


One major disadvantage of an implicit integration scheme such as the Newmark-

(3 method is that it requires more storage space and computational effort in each

time step. However, the use of large time steps compensates for this deficiency. The

implementation of the integration scheme now follows.

The governing difference equations are given by

[M}{6}n+1 + [C}{8}n+1 + f({S}n+1) = {P}n+1 (4.53)

Wn+l = + A[(l - 7 ) { * } B + 7tf}n+l] (4-54)

Wn + i = Wn + (At){6}n + (At)2[( - (3){6}n + 0{8}n+1] (4.55)

where [M], [C] are the global mass and damping matrices and the subscripts refer to

the time step number, At is the time step interval, 7, f3 are parameters depending

on the integration scheme. For the Newmark scheme adopted here, 7 = 0.5 and

f3 = 0.25. The nonlinear term /({} n +i) represents the internal force vector at the (n + l)th time step. Using a Taylor's series expansion this term can be expressed as

/(Wn+i) = / ({*}) + [KT]{AS}N+1 (4.56)

where \KT\ is the global tangent stiffness matrix and

{ A S } n + 1 = {*}n+i - {S}n (4.57)

is the incremental displacement vector.

The acceleration vector is obtained from Equation (4.55) as

{8}n+l - f3(Aty

which on substitution into Equation (4.54) gives

{A8}-(At){6}n-^(l-2(3){6}T (4.58)

W + i = ^ ) { M > + i + ^ g 2 ^ } " + ^2/3 7 ) W " ( 4 - 5 9 )


Substituting Equations (4.55), (4.58) and (4.59) into Equation (4.53), the difference

equation can be written as

[K]{A6}n+1 = {P}n+l (4.60)

where, [K] is an effective stiffness matrix given by

[K} = 1 W] + ^ [C] + [KT] (4.61) J3(At)21 1 (3At

and {P} is the effective load vector given by

{p} = ,{Pu1-m) + [ M ] ( ^ { ^ + ^ { % ) +

+ [ C ] ( ^ p W n + At2^i{S}n) (4.62)

For linear problems, [KT] = [K] and f({S}n) = [K]{6}n and the incremental

displacement term {A6}n+i in Equation (4.60) has to be replaced by {c^} n + i and the

effective load vector in this case is given by

4.6 Computer Code

A computer code named NAPSSE (Nonlinear Analysis of Plate Structures by Su-

per Elements) has been written to implement the theory. The program is written

in F O R T R A N language. It is presently functioning on an I B M 3081K main frame

computer and a GA-386L microcomputer.

The program has the capabilities to perform static, vibration and transient analy-

sis of unstiffened plates, beams and plates stiffened in one or two orthogonal directions.


For static and transient analyses all possible combinations of material and geometric

nonlinearities can be specified. For these nonlinear analyses, the solution is obtained

by an iterative procedure and convergence is achieved if the maximum norm, defined

by \SAi/Ai\max, or the Euclidean norm, defined by i=? (A; ) 2 / D|=i (A t - ) 2 , is less

than an acceptable tolerance specified by the user, where, A ; is the solution of the

nodal displacement variable i, SAi the correction factor for that variable and n the

total number of nodal variables.

Since not all nodal variables represent the actual displacements at the correspond-

ing nodal points, the program furnishes additional output for the displacements at

specified locations within the plate or beam elements. These displacements have been

computed using the nodal solution vector in conjunction with the values of the re-

spective shape functions at the predetermined locations. The program can also give

information on stresses and strains at the Gauss points, if required.

Chapter 5

Static Analysis Results

5.1 Introduction

In this chapter the super element formulation discussed in Chapter 4 is used to inves-tigate the response of various plate structures subjected to applied static loads. The

governing equations of motion for this case reduce to

where {P} is the global load vector, and the equations are solved by Newton-Raphson

iteration as discussed in Chapter 4. Although problems with nonlinear geometric and/or material behaviour are of

primary interest, the linear elastic responses have also been investigated to provide a

fuller understanding of the response characteristics of the new elements. In this case,

the governing equations solved are

where [K] is the global stiffness matrix and no iteration is required here.

The applied loading is assumed to be uniformly distributed in all cases. For linear

elastic analysis the load is applied in one load step while for nonlinear analysis the

load is applied in several steps and accumulated up to the full load.

In keeping with the objectives of this work only one element is used to represent

a panel bay or beam span as the case may be. However, some exceptions to this

(5.1)

[K]{8} = {P} (5.2)

53

Chapter 5. Static Analysis Results 54

general rule have been made, in a few instances, to study the convergence properties

of the super elements or to employ discretizations similar to those used by other

investigators.

The problems analyzed are categorized according to structure type. First, unstiff-

ened plates are analyzed in Section 5.2 and then beams acting alone, with no plating,

are considered in Section 5.3. Finally, Section 5.4 focuses on the response of plates

stiffened in one or two mutually perpendicular directions.

5.2 Unstiffened Plates

5.2.1 Square Plate I with Simply Supported Edges

The dimensions and material properties of the square plate are as given below:

dimensions = 100 mm x 100 mm x 1 mm

elastic modulus, E = 205,000 N / m m 2

Poisson's ratio, v = 0.3

The plate is simply supported all round. A uniform pressure load is applied and the

structure is modelled by one super plate element. This model has 55 gross and 15

net degrees of freedom. The boundary conditions are applied as follows:

At the four corner nodes all variables except the twist variable are constrained.

At the four mid-side nodes, all variables except the normal slope (e.g. wy5 at

node 5) are set equal to zero.

At the middle node (node 9) all variables are left free.

(See Figure 3.3 for location of nodes and variables). Thus the structure is constrained

against in-plane motion.

Chapter 5. Static Analysis Results 55

First, a linear elastic analysis is carried out for an applied load of 0.1 N / m m 2 .

The panel centre deflection and strain energy are presented in Table 5.1 together with

those obtained by the finite strip [19] (one mode solution) and exact [1] methods. It

is observed that the super element analysis is in excellent agreement with the finite

strip solution. However, it over estimates the exact central deflection by 1.2% and

under estimates the exact strain energy by only 0.4%. This excellent agreement of the

one-super element solution with both the finite strip and exact analyses demonstrates

its viability.

Table 5.1: Linear elastic response of simply supported Square Plate I

Central Strain Deflection (mm) Energy (Nm)

Present 2.1887 0.4517 (% Error) (1.15) (-0.4).

Finite Strip 2.1890 0.4512 Exact 2.1639 0.4535

Contours of the crx stress at the bottom surface of the plate are plotted in Fig-

ure 5.1. Note that the stress values have been computed only at the Gauss points

and these have been extrapolated linearly (depth wise) to the bottom or top surface.

Only the stresses computed at the 25 in-plane sampling po

ubc_1990_a1 k66

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