finite element static,dynamic and stability analyses of arbitary stiffened plates - jan barik

FINITE ELEMENT STATIC, DYNAMIC ANDSTABILITY ANALYSES OF ARBITRARY

STIFFENED PLATES

A thesis submitted to Indian Institute of Technology, Kharagpur

for the award of the degree of

Do˝or of Philosophyin

Engineering

by

Manoranjan Barik

Department of Ocean Engineering and Naval ArchitectureIndian Institute of TechnologyKharagpur - 721 302, India

January, 1999

With dedication toMy Ailing Mother

With love toMy wife Trushna

And little ones Trushita and MonishaWho endured all the sufferings silently

And looked for this day patiently

* INDIAN INSTITUTE OF TECHNOLOGYKHARAGPUR 721302, INDIA

DEPARTMENT OF OCEAN ENGINEERINGAND NAVAL ARCHITECTURE

Professor Madhujit MukhopadhyayB.E., Ph.D., D.Sc.

TELEX : 06401-201 ITKG IN

GRAM : TECHNOLOGY KHARAGPUR

PHONE :(91) (03222) 55221-55223

(91) (03222) 77390-77392

9=;

(6 lines)

Extn. : Office 4468, Res. 7468

Direct: (91) (03222) 77902 (Res.)

FAX : (91) 3222-55303

(91) 3222-55239

E-mail : [email protected]

Certificate

This is to certify that the thesis entitled ‘FINITE ELEMENT STATIC,DYNAMIC AND STABILITY ANALYSES OF ARBITRARY STIFFENEDPLATES’ being submitted to the Indian Institute of Technology, Kharag-pur by Mr. Manoranjan Barik for the award of the degree of Doctor ofPhilosophy in Engineering is a record of bonafide research work carriedout by him under my supervision and guidance, and Mr. Barik fulfills therequirements of the regulations of the degree. The results embodied inthis thesis have not been submitted to any other University or Institutefor the award of any degree or diploma.

Madhujit Mukhopadhyay

Acknowledgements

I express my sincere gratitude toProfessor L. R. RahejaandProfes-sor S. C. Mishra, the Ex-Heads, Department of Ocean Engineering andNaval Architecture, Indian Institute of Technology, Kharagpur, who dur-ing their headships extended all the computational and other related fa-cilities of the Department to make my progress of work smooth.

I owe a lot toProfessor S. K. Satsangi, Professor and Head, De-partment of Ocean Engineering and Naval Architecture, whose out of theway provisions and help made my work to see the end at last, amidst allthe obstacles.

The authority ofRegional Engineering College, Rourkelagrantedme the study leave and theMinistry of Human Resources and Develop-ment, Government of Indiaprovided me the Fellowship to carry out theresearch work. Their provisions are highly acknowledged with thanks.

I extend my heartfelt thanks toDr. O. P. Sha, Assistant Professor,Department of Ocean Engineering and Naval Architecture, who virtuallymade me the owner of his Personal Computer.

Dr. A. H. Sheikh, Assistant Professor of the Department was alwayseager to extend his analytical ability without any hesitation. My sincerethanks are due to him.

With all the humbleness, I gratefully acknowledge the valuable sug-gestions received on various occasions fromProfessor S. Majumdar,Professor, Department of Civil Engineering, Indian Institute of Technol-ogy, Kharagpur.

Special thanks are due toMr. Parimol Kumar Roy without whosehelp the completion of the thesis would have been delayed considerably.

v

vi ACKNOWLEDGEMENTS

I acknowledge the help received in various forms from all the facul-ties and staff members of the Department whose excellent cooperationmade my stay here a homely, pleasant and enjoyable one.

Krishna andPrusheth, the LATEX lovers did marvelous jobs to han-dle in my own way, the commas and semicolons of the LATEX. I sincerelyacknowledge their invaluable help.

The works could not have seen such a happy ending without thelov-

ing cooperation ofAbhinna, who helped me in taking the final prints. Ifeel short of words to thank him.

I express my sincere thanks to myChurch Members at RourkelaandKharagpur and the IIT Christian Fellowship Members, particularlyJamesandPatrick who held me up through their fervent prayer supportthroughout my research work.

The sweet presence of my co-scholars,Satish, Asokendu, Sushanta,Murthy, Chaitali, Sangita ... made my stay at the Institute a memo-rable one, for together we suffered the moments of trauma, together wetriumphed over the success, together we shared the moments of joy andhappiness and the greatest of all was that we understood each other betterthan any body else.

Above all, I express my deep sense of gratitude tomy ProfessorandsupervisorProfessor Madhujit Mukhopadhyay, Professor, Departmentof Ocean Engineering and Naval Architecture, Indian Institute of Tech-nology, Kharagpur, whose constant encouragement, guidance and thetime I spent along with him was invaluable to me. There were momentswhen he pushed me forward, enough to stumble, so that I may rise upand stand upright on my own on firm ground. And often he dragged meforward just to enable me to reach my goal. I adore him for his many ex-cellent qualities and feel myself blessed to work under him, for workingwith him was never a burden, rather a pleasure, the moments of which Iwill be carrying along with me throughout my life’s journey.

Indian Institute of TechnologyKharagpur (Manoranjan Barik)

Contents

1 INTRODUCTION 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 The Objective and Scope of Present Investigation . . . .5

2 REVIEW OF LITERATURE 92.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Review on Bare Plates . . . . . . . . . . . . . . . . . . 9

2.2.1 Static Analysis of Bare Plates . . . . . . . . . .10

2.2.2 Free Vibration Analysis of Bare Plates . . . . . .17

2.2.3 Stability Analysis of Bare Plates . . . . . . . . .24

2.3 Various Methods of Analysis of Stiffened Plates . . . . .30

2.4 Review on Stiffened Plates . . . . . . . . . . . . . . . .32

2.4.1 Static Analysis of Stiffened Plates . . . . . . . .33

2.4.2 Free Vibration Analysis of Stiffened Plates . . .36

2.4.3 Stability Analysis of Stiffened Plates . . . . . .43

3 MATHEMATICAL FORMULATION 473.1 The Basic Problems . . . . . . . . . . . . . . . . . . . .47

3.2 Proposed Analysis . . . . . . . . . . . . . . . . . . . .48

3.2.1 The Basic Assumptions . . . . . . . . . . . . .49

3.2.2 The Transformation of the Coordinate . . . . . .49

3.3 Arbitrary Bare Plate Bending Formulation . . . . . . . .52

3.3.1 The Displacement Function . . . . . . . . . . .52

3.3.2 Elastic Stiffness Matrix Formulation . . . . . . .53

3.3.2.1 Stress-Strain Relationship . . . . . . .53

vii

viii CONTENTS

3.3.2.2 Strain-Displacement Relationship . . .54

3.3.2.3 Stiffness Matrix of the Bare Plate Bend-ing Element . . . . . . . . . . . . . . 57

3.3.3 Consistent Mass Matrix of the Bare Plate Element58

3.3.4 Geometric Stiffness Matrix of the Bare Plate El-ement . . . . . . . . . . . . . . . . . . . . . . . 59

3.3.5 Boundary Conditions for the Bare Plate . . . . .61

3.3.6 Stresses at the Nodes of the Bare Plate . . . . . .65

3.4 Arbitrary Stiffened Plate Element Formulation . . . . . .65

3.4.1 The Displacement Function . . . . . . . . . . .66

3.4.2 The Plate Element Formulation . . . . . . . . .67



3.4.2.3 Elastic Stiffness Matrix of the Plate El-ement of the Stiffened Plate . . . . . .70

3.4.2.4 Consistent Mass Matrix of the Plate El-ement of the Stiffened Plate . . . . . .71

3.4.2.5 Geometric Stiffness Matrix of the PlateElement of the Stiffened Plate . . . . .73

3.4.3 The Stiffener Element Formulation . . . . . . .81

3.4.3.1 Coordinate Transformation for the Stiff-ener . . . . . . . . . . . . . . . . . . 81



3.4.3.4 Elastic Stiffness Matrix of the StiffenerElement . . . . . . . . . . . . . . . . 86

3.4.3.5 Consistent Mass Matrix of the Stiff-ener Element . . . . . . . . . . . . . . 87

3.4.3.6 Geometric Stiffness Matrix of the Stiff-ener Element . . . . . . . . . . . . . . 88

3.4.4 Boundary Conditions for the Stiffened Plate . . .93

CONTENTS ix

3.4.5 Stresses in the Stiffener . . . . . . . . . . . . . .96

3.5 Consistent Load Vector . . . . . . . . . . . . . . . . . .97

3.6 Solution Procedures . . . . . . . . . . . . . . . . . . . .97

3.6.1 Static Analysis . . . . . . . . . . . . . . . . . .97

3.6.2 Free Vibration Analysis . . . . . . . . . . . . .98

3.6.3 Stability Analysis . . . . . . . . . . . . . . . . . 99

4 COMPUTER IMPLEMENTATION 1014.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . .101

4.2 Application Domain . . . . . . . . . . . . . . . . . . . .102

4.3 Description of the Programme . . . . . . . . . . . . . .104

4.3.1 Preprocessor . . . . . . . . . . . . . . . . . . .104

4.3.1.1 functioninput() . . . . . . . . . . . . 104

4.3.1.2 functionnodgen() . . . . . . . . . . . 107

4.3.1.3 functionstcod() . . . . . . . . . . . . 107

4.3.1.4 functionconnect() . . . . . . . . . . . 109

4.3.1.5 functionband() . . . . . . . . . . . . 109

4.3.1.6 functionxycod() . . . . . . . . . . . . 109

4.3.1.7 functionsfr1() . . . . . . . . . . . . . 109

4.3.1.8 functionrgdplt() . . . . . . . . . . . 110

4.3.1.9 functionstifin() . . . . . . . . . . . . 110

4.3.1.10 functionrgdstf() . . . . . . . . . . . . 110

4.3.2 Processor . . . . . . . . . . . . . . . . . . . . .111

4.3.2.1 functionform-stif-mass-geom() . . . 111

4.3.2.2 functionelm-stif-mass-geom() . . . . 111

4.3.2.3 functionelm-stf-mass-geom() . . . . 113

4.3.2.4 functionglobal-stif-mass-geom(). . . 113

4.3.2.5 functionglobal-stf-mass-geom(). . . 113

4.3.2.6 functionglobal() . . . . . . . . . . . . 113

4.3.2.7 functionelm-load() . . . . . . . . . . 113

4.3.2.8 functiongbl-load() . . . . . . . . . . 113

x CONTENTS

4.3.2.9 functionbnd-stif() . . . . . . . . . . . 114

4.3.2.10 functionsfr2() . . . . . . . . . . . . . 114

4.3.2.11 functionbmat() . . . . . . . . . . . . 114

4.3.2.12 functiondbmat() . . . . . . . . . . . 114

4.3.2.13 functionsolve() . . . . . . . . . . . . 114

4.3.2.14 functionr8usiv() . . . . . . . . . . . 114

4.3.3 Postprocessor . . . . . . . . . . . . . . . . . . .116

5 NUMERICAL EXAMPLES 119

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . .119

5.2 Arbitrary Bare Plates . . . . . . . . . . . . . . . . . . .120

5.2.1 Static Analysis of Arbitrary Bare Plates . . . . .120

5.2.1.1 Rectangular Plates Under Uniformly Dis-tributed Load . . . . . . . . . . . . . .121

5.2.1.2 Rectangular Plates Under ConcentratedLoad . . . . . . . . . . . . . . . . . .122

5.2.1.3 All Edges Clamped Rhombic Plates Un-der UDL . . . . . . . . . . . . . . . . 123

5.2.1.4 All Edges Simply Supported RhombicPlates Under UDL . . . . . . . . . . .124

5.2.1.5 Annular Sector Plate Under Concen-trated Loads . . . . . . . . . . . . . .127

5.2.1.6 Circular Plate Under Different Load-ings and Boundary Conditions . . . .128

5.2.2 Free Vibration Analysis of Arbitrary Bare Plate .130

5.2.2.1 Free Vibration of Rectangular and SquareBare Plates . . . . . . . . . . . . . . .131

5.2.2.2 Free Vibration of Bare Skew Plates . .132

5.2.2.3 Free Vibration of Trapezoidal Bare Plates135

5.2.2.4 Free Vibration of Triangular Bare Plates138

CONTENTS xi

5.2.2.5 Free Vibration of Bare Annular SectorPlates . . . . . . . . . . . . . . . . . .140

5.2.2.6 Free Vibration of Bare Elliptical andCircular Plates . . . . . . . . . . . . .142

5.2.3 Stability Analysis of Arbitrary Bare Plates . . . .145

5.2.3.1 Buckling of Uniaxially Compressed Sim-ply Supported bare Rectangular Plates145

5.2.3.2 Buckling of Uniaxially Compressed ClampedBare Rectangular Plates . . . . . . . .147

5.2.3.3 Buckling of Biaxially Compressed ClampedBare Rectangular Plates . . . . . . . .147

5.2.3.4 Buckling of Simply Supported Bare Rect-angular Plates Uniaxially Compressedby Triangular Load . . . . . . . . . .148

5.2.3.5 Buckling of Uniaxially Compressed BareSkew Plates . . . . . . . . . . . . . .148

5.2.3.6 Buckling of Uniformly Compressed BareCircular Plates . . . . . . . . . . . . .151

5.3 Arbitrary Stiffened Plates . . . . . . . . . . . . . . . . .151

5.3.1 Static Analysis of Arbitrary Stiffened Plates . . .151

5.3.1.1 Square Plate with a Central Stiffener .152

5.3.1.2 Cross Stiffened Rectangular Plate . . .155

5.3.1.3 Rectangular Multi-Stiffened Plate . . .162

5.3.1.4 Rectangular Slab with Two Edge Beams165

5.3.1.5 Stiffened Skew Bridge Deck . . . . .170

5.3.1.6 Stiffened Curved Bridge Deck . . . .173

5.3.1.7 Circular Plate with a Central Stiffener178

5.3.2 Free Vibration Analysis of Arbitrary Stiffened Plates180

5.3.2.1 Free Vibration of Concentrically Stiff-ened Clamped Square Plate . . . . . .180

xii CONTENTS

5.3.2.2 Free Vibration of Eccentrically Stiff-ened Clamped Square Plate . . . . . .182

5.3.2.3 Free Vibration of Cross Stiffened SquarePlate . . . . . . . . . . . . . . . . . .183

5.3.2.4 Free Vibration of Eccentrically Stiff-ened Rectangular Plate . . . . . . . .183

5.3.2.5 Free Vibration of Rectangular Multi-stiffened Plates . . . . . . . . . . . . .185

5.3.2.6 Free Vibration of Multi-stiffened SkewPlates . . . . . . . . . . . . . . . . . .185

5.3.2.7 Free Vibration of Trapezoidal StiffenedPlates . . . . . . . . . . . . . . . . . .191

5.3.2.8 Free Vibration of Concentrically Stiff-ened Annular Sector Plates . . . . . .192

5.3.2.9 Free Vibration of Eccentrically Stiff-ened Annular Sector Plate . . . . . . .194

5.3.2.10 Free Vibration of Circular Stiffened Plates197

5.3.2.11 Free Vibration of Elliptical StiffenedPlate . . . . . . . . . . . . . . . . . .197

5.3.3 Stability Analysis of Arbitrary Stiffened Plates .201

5.3.3.1 Buckling of Square Stiffened Plates . .201

5.3.3.2 Buckling of Simply Supported Rect-angular Stiffened Plates . . . . . . . .202

5.3.3.3 Buckling of Rectangular Stiffened Plateswith Different Boundary Conditions .202

5.3.3.4 Buckling of Skew Stiffened Plates withDifferent Boundary Conditions . . . .206

5.3.3.5 Buckling of Uniformly Compressed Di-ametrically Stiffened Circular Plates .206

6 CONCLUSIONS 209

CONTENTS xiii

6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . .2096.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . .2116.3 Further Scope of Research . . . . . . . . . . . . . . . .212

Preface

Plates are in extensive use as one of the important structural elementsin the modern day structures in civil, marine, aeronautical and mechan-ical engineering. These plates may assume arbitrary shapes dependingon their structural behaviour, the area of application and the type of ser-vices they are put to. Though they have wide applications without anyrib reinforcement, but various engineering structures demand economyin weight with enhancement of strength through stiffening of the platedstructures. When the arbitrarily shaped bare and the stiffened plates arein service, they are subjected to the static lateral load, the dynamic loadand the inplane load. To investigate the actual behaviour of the platesunder these loads, rigorous analysis is required to assess the strength andstability under various boundary conditions.

In the present era of super speed number crunching machines, numer-ical methods have found their way into the structural analysis because ofthe non-amenability of analytical solutions for complex structural prob-lems such as arising out of the arbitrary shape of the plates. Among thesenumerical tools, the finite element method has been proved to be the mostversatile and powerful one because of its generality and capability to han-dle structural and geometrical complexities with ease. Several numberof commercial softwares and in-house codes have been developed us-ing the finite element method for carrying out the structural analyses.But most of these packages have inadequate facility for efficient stiff-ener modelling, improper specification of boundary conditions in case ofa curved boundary and loss of generality in the mesh division processbecause of the stiffener position in the plate. Moreover, these codes are

xv

xvi PREFACE

not susceptible to easy modification in case of need. On the other hand,though there are plenty of elements developed so far in the finite elementdomain, many have been found to be inadequate and inefficient in someway or other for analyzing plates of arbitrary geometrical configurations.The present investigation is an attempt to accommodate the unstiffenedand the stiffened plate problems of arbitrary shapes by developing newefficient elements.

A plate bending element has been developed following the philoso-phy of isoparametric element to enable the analysis of arbitrarily shapedbare plates. The basic element considered for the development of this ele-ment is the simplest 12 degrees of freedom rectangular plate bending ele-ment popularly known asACM element. Bare plates of many geometricalconfigurations have been analyzed for static, dynamic and stability mak-ing use of this new element. For analyzing arbitrary stiffened plates, an 8degrees of freedom rectangular plane stress element has been combinedto the basicACM element. The stiffener modelling has been done con-sidering a curved general element which can be placed anywhere withinthe plate element which removes the restraint of positioning the stiffen-ers along the nodal lines. The static, free vibration and buckling analyseshave been performed on arbitrary plates with eccentric and concentricstiffeners using this stiffened plate bending element.

The thesis has been presented in six chapters. It also includes thebibliography section showing the important references concerned withthe present investigation.

Chapter 1 includes the general introduction and the scope of presentinvestigation.

The review of literature confining to the scope of the study has beenpresented in theChapter 2. The general methods of analysis of the stiff-ened plates have been briefly addressed in this chapter.

The Chapter 3 comprises the mathematical formulation of the twoelements. The elastic and the geometric stiffness matrices and the mass

PREFACE xvii

matrix for the plate element and the stiffener element have been formu-lated separately. The boundary conditions have been implemented byconsistently formulating the stiffness matrices of the boundary line andadding them to the global stiffness matrix.

TheChapter 4 briefly describes the computer programme implemen-tation of the theoretical formulation presented in Chapter 3. The differentfunctions and the associated variables which have been used in writingthe codes inC++ language have been presented in brief. A few numbersof flowchart of the computer codes have been illustrated.

Several numerical examples which include the static, the free vibra-tion and the stability analyses of bare and the stiffened plates of variousgeometries have been presented in theChapter 5 to validate the formu-lation of the proposed method. Attempt has been made to include a widespectrum of problems of diverse geometrical plate shapes. The resultshave been compared wherever possible and the discrepancies in themhave been discussed.

TheChapter 6 sums up and concludes the present investigation. Anaccount of possible scope of extension to the present study along with alist of publications has been appended to the concluding remarks.

At the end, some important publications and books referred duringthe present investigation have been listed in theBibliography section.

List of Symbols

Although all the principal symbols used in this thesis are defined in thetext as they occur, a list of them is presented below for easy reference. Onsome occasions, a single symbol is used for different meanings depend-ing on the context and thus its uniqueness is lost. The contextual expla-nation of the symbol at its appropriate place of use is hoped to eliminatethe confusion.

English

As cross sectional area of the stiffener[Bp] strain matrix for plate element of stiffened plate[Bs] strain matrix for stiffener element of stiffened plate[Bu] strain matrix for bare plate elementdx, dy element length in x and y-directiondv volume of the element[Du] rigidity matrix of bare plate element[Dp] rigidity matrix of stiffened plate element[Ds] rigidity matrix of stiffener elementE modulus of elasticity{f} acceleration field vector{FI} nodal inertia force parameter{fku} reaction component per unit length of bare plateFx, Fy, Fxy inplane forcesFS axial force in the stiffenerG modulus of rigidityIs second moment of area of the stiffener

xix

xx LIST OF SYMBOLS

|J | jacobianJs torsional constant of the stiffener|Jst| jacobian of the stiffener[K] global elastic stiffness matrix[KG] global geometric stiffness matrix[Ku]e elastic stiffness matrix of the bare plate bending element[KuG]e geometric stiffness matrix of the bare plate bending element[Ku] global elastic stiffness matrix of the bare plate[KuG] global geometric stiffness matrix of the bare plate[Kp]e elastic stiffness matrix of the stiffened plate element[KpG]e geometric stiffness matrix of the stiffened plate element[Ks]e elastic stiffness matrix of the stiffener element[KsG]e geometric stiffness matrix of the stiffener element[Kp]b stiffness matrix of the boundary of stiffened plate[Ku]b stiffness matrix of the boundary line of the bare plateku, kv, kw translational restraint coefficientkα, kβ rotational restraint coefficient[M ] global consistent mass matrix[Mu]e consistent mass matrix of the bare plate bending element[Mp]e consistent mass matrix of the stiffened plate element[Ms]e consistent mass matrix of the stiffener elementMs bending moment of the stiffenerMx,My,Mxybending moments of the plateNi(s, t) cubic serendipity shape functionsNu, Nv, Nw finite element shape functionsNθξ

, Nθη finite element shape functions{P} global load vector{P}e element load vectorq load intensitys-t axis system of the plate in the mapped domains1 length along the boundary

xxi

Ss first moment of area of the stiffenert thickness of the plateTs torsional moment of the stiffeneru, v inplane displacementsu´, v´, w´ displacements at midplane of the platew out of plane displacement{w} acceleration vector in z-directionxi, yi Cartesian nodal coordinatesx, y, z global axis systemx1-y1 local axes at the point of a curved boundaryx´-y´ local axes at any point of a curved stiffener

Greek

α angle between thex´ -y´ andx-y axes systemβ angle between thex1-y1 andx-y axes system{δ}u nodal displacement vector of bare plate{δ}p nodal displacement vector of stiffened plate{δ}u nodal acceleration vector of bare plateξ-η axis system of the element in the mapped domain{σ}u stress resultant vector of bare plate{σ}p stress resultant vector of stiffened plate{σ}s stress resultant vector of stiffenerσx, σy, τxy stresses at a point{ε}u generalized strain vector of bare plate{ε}uG geometric strain vector of bare plate{ε}p strain vector of stiffened plate element{ε}pE elastic plate strain vector{ε}pG geometric plate strain vector{ε}s strain vector of the stiffener{ε}sE elastic stiffener strain vector{ε}sG geometric stiffener strain vector

xxii LIST OF SYMBOLS

εx, εy, γxy bending strainsν Poisson´ s ratio∂

∂x,

∂

∂ypartial derivatives with respect tox andy

ρ mass density of the materialθn, θt slopes normal and transverse to the boundaryλ stiffener direction in mapped domainλ1 boundary line direction in mapped domain{ψ} normalized vectorω frequency of vibration

Subscripts

u for bare plateG for geometric stiffness matrixb for boundaryp for the plate element of the stiffened plates for the stiffener element of the stiffened plate

Operators

( ) first derivative with respect to time( ) second derivative with respect to time[ ]−1 inverse of the matrix[ ]−T transpose of the matrix

List of Tables

3.1 Cubic Serendipity Shape Function . . . . . . . . . . . 50

5.1 Numerical factors α, β and β1 for uniformly loadedsimply supported rectangular plates . . . . . . . . . . 122

5.2 Numerical factorsα for simply supported rectangularplates with central concentrated load . . . . . . . . . 122

5.3 Deflection and moments at the centre of the all edgesclamped skew rhombic plates under UDL . . . . . . . 123

5.4 Deflection and moments at the centre of the all edgessimply supported skew rhombic plates under UDL . . 126

5.5 Deflection and moments along the mid-span radial line(x-axis) of the annular sector plate . . . . . . . . . . . 128

5.6 Deflection and moments at the centre of the circularplate under different loading and boundary conditions 129

5.7 Frequency parametersλ = ωa2(ρ/D)1/2 for rectan-gular plate . . . . . . . . . . . . . . . . . . . . . . . .131

5.8 Convergence of frequency parametersλ = ωa2(ρ/D)1/2

for all edges simply supported square plate . . . . . . 132

5.9 Frequency parametersλ = ωa2(ρt/D)1/2 of skew platesfor different skew angles (φ) and for a/b=1.0, ν =0.3 133

5.10 Frequency parametersλ = ωa2(ρ t/D)1/2 of skewplates for different skew angles (φ) and for a/b = 2.0,ν =0.3 . . . . . . . . . . . . . . . . . . . . . . . . . . .134

5.11 Frequency parametersλ = ωa2(ρ/D)1/2 for all edgessimply supported trapezoidal plate . . . . . . . . . . . 136

xxiii

xxiv LIST OF TABLES

5.12 Frequency parametersλ =ωa2

2π

√ρ

Dfor all edges clamped

trapezoidal plate . . . . . . . . . . . . . . . . . . . . . 137

5.13 Frequency parametersλ = ωa2

√ρ

Dfor triangular

plates . . . . . . . . . . . . . . . . . . . . . . . . . . .139

5.14 Values ofω for annular sector plates . . . . . . . . . . 141

5.15 Frequency parametersλ=ωa2(ρh/D)1/2 for ellipticaland circular plates . . . . . . . . . . . . . . . . . . . . 143

5.16 Buckling parameter k = λb2/π2D for uniformly com-pressed all edges simply supported rectangular plates145

5.17 Buckling parameter k = λb2/π2D for uniformly com-pressed all edges clamped rectangular plates. . . . . 146

5.18 Buckling parameter k = λb2/π2D for all edges clampedrectangular plates with biaxial uniform compression . 147

5.19 Buckling parameter k = λb2/π2D for uniaxially com-pressed all edges simply supported rectangular plateswith triangular load i.e; α = 1 in the expressionNx =

N0(1− αy

b) . . . . . . . . . . . . . . . . . . . . . . . .148

5.20 Buckling parameter k = λb2/π2D for uniaxially com-pressed all edges simply supported and clamped skewplates (Aspect ratio = 1.0, ν = 0.3) . . . . . . . . . . . 149

5.21 Buckling parameter k = (Nr)cra2/D for uniformly

compressed simply supported and clamped circularplates (ν = 0.3) . . . . . . . . . . . . . . . . . . . . . . 150

5.22 Deflection at the centre of simply supported squarestiffened plate(×104 mm.) . . . . . . . . . . . . . . . 154

5.23 Convergence of deflection(w), plate moment(My) andplate stress(σx) of the eccentrically stiffened squareplate at its centre with different mesh divisions.. . . . 154

5.24 Central deflection of rectangular cross-stiffened plate(×103 mm.) . . . . . . . . . . . . . . . . . . . . . . . .155

LIST OF TABLES xxv

5.25 Geometrical and material properties of the specimensof the rectangular slab with edge beams. . . . . . . . 166

5.26 Deflection and stress at the beam soffit of the rectan-gular slab with edge beams . . . . . . . . . . . . . . . 169

5.27 Convergence of deflection at outer girder and outeredge . . . . . . . . . . . . . . . . . . . . . . . . . . . .175

5.28 Convergence of moments at the centre. . . . . . . . . 175

5.29 Deflection (mm.) at inner edge and inner girder . . . 176

5.30 Deflection (mm.) at outer girder and outer edge . . . 177

5.31 Frequency in Hz of a clamped square plate with a cen-tral concentric stiffener . . . . . . . . . . . . . . . . . 181

5.32 Frequency in Hz of a clamped square plate with a cen-tral eccentric stiffener . . . . . . . . . . . . . . . . . . 183

5.33 Frequency parameter[ω(a/π)2√

ρt/D] of square cross-stiffened plate with concentric stiffeners having all edgesclamped . . . . . . . . . . . . . . . . . . . . . . . . . .184

5.34 Frequency in Hz of a simply supported rectangularplate with a central L-shaped eccentric stiffener in theshorter span direction . . . . . . . . . . . . . . . . . . 184

5.35 Frequency parameter[ω(a/R)2√

ρt/D] of simply sup-ported multi-stiffened rectangular plate with concen-tric stiffeners in one direction . . . . . . . . . . . . . . 186

5.36 Frequency parameter [ω(a/R)2√

ρh/D] of a simplysupported multi-stiffened skew plate having concen-tric stiffeners in one direction . . . . . . . . . . . . . . 189

5.37 Frequency in Hz of all edges clamped trapezoidal platewith a central concentric stiffener . . . . . . . . . . . 192

5.38 Frequency parameter [ωa2√

ρt/D] of annular sectorplate with concentrically placed circumferential edgestiffeners . . . . . . . . . . . . . . . . . . . . . . . . .193

xxvi LIST OF TABLES

5.39 Frequency parameter [ωa2√

ρt/D] of annular sectorplate with eccentrically placed circumferential edgestiffeners . . . . . . . . . . . . . . . . . . . . . . . . .196

5.40 Frequency (Hz/Parameter) of all edges clamped cir-cular stiffened plates . . . . . . . . . . . . . . . . . . . 198

5.41 Frequency in Hz of a simply supported elliptical platewith a central eccentric stiffener in the shorter spandirection . . . . . . . . . . . . . . . . . . . . . . . . .201

5.42 Buckling parameter k = λb2/π2D for square platewith a central concentric stiffener subjected to uniax-ial and uniform compression in the stiffener direction 203

5.43 Buckling parameter k = λb2/π2D for uniformly com-pressed all edges simply supported rectangular stiff-ened plates . . . . . . . . . . . . . . . . . . . . . . . .204

5.44 Buckling parameter k = λb2/π2D for rectangular platewith a central concentric stiffener subjected to uniax-ial and uniform compression in the stiffener direction 205

5.45 Buckling parameter k = λb2/π2D for skew stiffenedplate . . . . . . . . . . . . . . . . . . . . . . . . . . . .206

5.46 Buckling parameter k = (Nr)cr a2/D for uniformlycompressed circular plates with concentric stiffenersalong the diameter with varying flexural and torsionalstiffness parameters of the stiffener . . . . . . . . . . 207

List of Figures

3.1 Mapping of the arbitrarily shaped plate . . . . . . . . 50

3.2 Mapping of the element . . . . . . . . . . . . . . . . . 51

3.3 Coordinate axes at a typical point of a curved boundary 62

3.4 Stretching of an element. . . . . . . . . . . . . . . . . 73

3.5 Coordinate axes at any point of a curved stiffener . . 81

3.6 Sectional view of a typical stiffener. . . . . . . . . . . 83

3.7 Stiffener orientation in the mapped domain . . . . . . 86

4.1 Basic Elements of the Computer Programmes . . . . 105

4.2 Preprocessor unit of the computer codes. . . . . . . . 108

4.3 Processor unit of the computer codes . . . . . . . . . 112

4.4 Flowchart for free vibration and buckling analysis . . 117

5.1 Location of the boundary nodal points of a rectangu-lar plate . . . . . . . . . . . . . . . . . . . . . . . . . .121

5.2 Location of the boundary nodal points of a skew plate 125

5.3 Annular sector plate showing boundary nodal points. 127

5.4 Circular plate with boundary nodal points . . . . . . 130

5.5 A typical skew plate . . . . . . . . . . . . . . . . . . . 134

5.6 Trapezoidal plate for simple supports showing the bound-ary nodal points . . . . . . . . . . . . . . . . . . . . . 135

5.7 Trapezoidal plate for clamped supports . . . . . . . . 136

5.8 Right triangular plate with boundary nodal points . . 138

5.9 Annular sector plate of sector angle90◦ . . . . . . . . 140

5.10 Simply supported square plate with a central stiffener 152

xxvii

xxviii LIST OF FIGURES

5.11 Variation of deflection along centrelines of simply sup-ported square plate with a central stiffener . . . . . . 153

5.12 Simply supported rectangular plate with a central stiff-ener in each direction . . . . . . . . . . . . . . . . . . 156

5.13 Deflection at x = 190.5 mm. and x = 381.0 mm. forplate with two concentric stiffeners, under distributedload . . . . . . . . . . . . . . . . . . . . . . . . . . . .157

5.14 Deflection at x = 190.5 mm. and x = 381.0 mm. forplate with two eccentric stiffeners, under distributedload . . . . . . . . . . . . . . . . . . . . . . . . . . . .157

5.15 Moment Mxx at y = 381.0 mm. and y = 762.0 mm. forplate with two concentric stiffeners, under distributedload . . . . . . . . . . . . . . . . . . . . . . . . . . . .158

5.16 Moment Mxx at y = 381.0 mm. and y = 762.0 mm. forplate with two eccentric stiffeners, under distributedload . . . . . . . . . . . . . . . . . . . . . . . . . . . .158

5.17 Moment Myy at x = 190.5 mm. and x = 381.0 mm. forplate with two concentric stiffeners, under distributedload . . . . . . . . . . . . . . . . . . . . . . . . . . . .159

5.18 Moment Myy at x = 190.5 mm. and x = 381.0 mm. forplate with two eccentric stiffeners, under distributedload . . . . . . . . . . . . . . . . . . . . . . . . . . . .159

5.19 Deflections at x = 190.5 mm. and x = 381.0 mm. forplate with two stiffeners under concentrated load. . . 160

5.20 Moment Mxx at y = 381.0 mm. and y = 762.0 mm.for plate with two concentric stiffeners under concen-trated load . . . . . . . . . . . . . . . . . . . . . . . .160

5.21 Moment Mxx at y = 381.0 mm. and y = 762.0 mm. forplate with two eccentric stiffeners under concentratedload . . . . . . . . . . . . . . . . . . . . . . . . . . . .161

LIST OF FIGURES xxix

5.22 Moment Myy at x = 190.5 mm. and x = 381.0 mm.for plate with two concentric stiffeners under concen-trated load . . . . . . . . . . . . . . . . . . . . . . . .161

5.23 Moment Myy at x = 190.5 mm. and x = 381.0 mm. forplate with two eccentric stiffeners under concentratedload . . . . . . . . . . . . . . . . . . . . . . . . . . . .162

5.24 Rectangular multi-stiffened plate . . . . . . . . . . . . 163

5.25 Variation of deflection along the centre line of the rect-angular multi-stiffened plate . . . . . . . . . . . . . . 164

5.26 Variation of plate moment Mx along the centre line ofthe rectangular multi-stiffened plate . . . . . . . . . . 164

5.27 Rectangular slab with two edge beams. . . . . . . . . 167

5.28 Deflection along A-A of the slab with edge beams. . . 167

5.29 Deflection along B-B of the slab with edge beams. . . 168

5.30 Deflection along C-C of the slab with edge beams. . . 168

5.31 Stress at the beam soffit of the slab with edge beams. 169

5.32 Skew bridge deck with beams in both directions . . . 171

5.33 Deflection along A-A of the stiffened skew bridge deck172

5.34 Deflection along B-B of the stiffened skew bridge deck172

5.35 Curved bridge deck with two circumferential girders 174

5.36 Simply supported circular plate with a stiffener alongthe diameter . . . . . . . . . . . . . . . . . . . . . . .178

5.37 Deflection along diameters of a simply supported cir-cular plate under distributed load . . . . . . . . . . . 179

5.38 Clamped square plate with a central eccentric stiffener182

5.39 Simply supported rectangular plate with a central ec-centric stiffener . . . . . . . . . . . . . . . . . . . . . 187

5.40 Simply supported rectangular plate with concentricstiffeners in one direction . . . . . . . . . . . . . . . . 188

5.41 Simply supported skew plate with concentric stiffen-ers in one direction . . . . . . . . . . . . . . . . . . . . 190

xxx LIST OF FIGURES

5.42 All edges clamped trapezoidal plate with a concentricstiffener in one direction . . . . . . . . . . . . . . . . 191

5.43 Annular sector plate with concentrically placed cir-cumferential stiffeners . . . . . . . . . . . . . . . . . . 194

5.44 Annular sector plate with eccentrically placed circum-ferential stiffeners . . . . . . . . . . . . . . . . . . . . 195

5.45 Circular plate with a central stiffener . . . . . . . . . 1995.46 Elliptical plate with a central stiffener . . . . . . . . . 200

Chapter -1

INTRODUCTION

1.1 Introduction

Plates used as structural elements take different shapes due to their

functional or structural requirements as well as from the aesthetic con-

sideration. These arbitrarily shaped, elastic thin plates are widely used

in civil, marine, aeronautical and mechanical engineering applications.

Various engineering structures consisting of these thin plates of differ-

ent shapes are often stiffened with stiffening ribs for achieving greater

strength with relatively less material and thus making the structure cost

effective. While the stiffening elements add negligible weight to the over-

all structure, their influence on strength and stability is enormous. In this

process the strength/weight ratio is improved dramatically which is vital

in some specific structures like ship, aircrafts and similar other types.

These plates of arbitrary geometries are subjected to the static lat-

eral load, the dynamic load and the inplane load for which three types

of analysis such as static, free vibration and stability are to be carried

out. In these analyses, the geometry of the plate as well as its boundary

conditions play a major role in the choice of the methods of the solution.

Exact solutions for plates are available only for certain shapes, bound-

aries and loading conditions. An attempt to have an analytical solution

1

2 INTRODUCTION

of the arbitrarily shaped plates with complex boundary conditions may

lead to an extremely tedious though not impossible task because of the

complex nature of the problem arising out of a curved boundary. As a

result, various methods such as Rayleigh-Ritz method, Galerkin method

and the likes have been used by several investigators depending on the

suitability of the problem. Some investigators [83] have attempted the

conformal mapping [166] for solving plates of regular polygonal shape

whereas some have used the finite strip method [34] and the spline finite

strip method [37] for solving problems relating to plates of arbitrary ge-

ometry apart from the popular finite element method. Investigators from

various fields have contributed to the study of bare plates and stiffened

plates making the library of literature rich in the area of static, dynamic

and stability analyses of these plates.

The stiffened plates consist of a skin and a varying number of ribs.

The skin is termed asplate throughout this thesis and the terms such

as rib, stiffener, girder, beam and stringer are used interchangeably to

indicate the ribs. When the rib centroid is coincident with the plate mid-

dle surface, no inplane stresses are developed due to the bending of the

stiffener and this class of stiffened plates is identified asconcentrically

stiffened plates. In the other case, when the rib centroid and the plate

middle surface are eccentric, the inplane stresses developed in the plate

due to the stiffener bending have to be considered and this class of stiff-

ened plates is designated aseccentrically stiffened plates.

The optimum design of stiffened plate structures demands an effec-

tive analytical procedure. But the stiffening arrangements pose another

difficulty in addition to the inherent problem due to the diverse geometri-

cal configurations, loading and boundary conditions encountered in case

of bare plates for obtaining a suitable theoretical solution. Hence, the

earlier investigators modelled the stiffened plated system into a simpler

1.1 Introduction 3

structural form such as an orthotropic plate or a grid system which were

amenable to the solution procedure developed at that time. As the or-

thotropic plate model is applicable to closely equispaced stiffeners of

equal size only and the grid model can perform well only in case of

orthogonal stiffeners, the applicability of the models to a generalized

problem is severely restricted because of the simplicity inherent in the

approximations.

The emergence of the digital computers with their enormous com-

puting speed and core memory capacity has changed the outlook of the

structural analysts and caused the evolution of various numerical meth-

ods such as the finite element, the finite difference, the finite strip and the

boundary element method. These numerical tools allow the researchers

to model the structure in a more realistic manner with simpler mathemat-

ical forms.

In view of the availability of the computational facility, the orthotropic

and the grillage models can be replaced by the plate beam idealization

where the plate and the stiffeners can be modelled separately maintaining

the monolithic connection between the two and then one of the numeri-

cal methods may be applied for their analysis. Among all the numerical

methods, the finite element method has been found to be a powerful, ver-

satile and accurate one in the analysis of complex structures. But, in the

finite element analysis of plates with arbitrary configurations, the main

problem arises in the choice of a suitable element, as many of the present

elements are unable to cater to the arbitrary plate geometry.

In the past the most common approach to the finite element analysis

of arbitrarily shaped plates has been to approximate the curved bound-

aries with a large number of straight-edged triangular elements [7] [9]

[63] [108] or developing special purpose elements permitting the exact

representation of curved boundaries [141] or using a triangular element

4 INTRODUCTION

with one of the sides being modified to include a curved edge [35]. But

these elements being developed to accommodate a particular plate geom-

etry, none of them can be generalized to represent an arbitrary edge such

as straight, skew or curved.

Another successful approach in this pursuit is the application of the

isoparametric element because of its generality to model a curved bound-

ary successfully. Unfortunately, this element which is having the shear

strain term based on the Mindlin’s theory becomes very stiff when used

to model thin structures, resulting inexact solutions. This effect is termed

as shear-locking which makes this otherwise successful element unsuit-

able. Much effort has been put to identify and eliminate the source of

this shear-locking effect. The most successful technique for alleviating

the problem associated with this shear-locking is through evaluating cer-

tain transverse shear coefficients of the element stiffness matrix using

a lower order numerical integration rule than that which is required to

evaluate the coefficients exactly as discussed by Zienkiewicz and Tay-

lor [199]. This technique which is known as reduced or selective inte-

gration has been used on elements which shear-lock when exact integra-

tion is performed. However, an inexact integration scheme results in a

rank deficient element stiffness matrix, which in turn, generates addi-

tional zero strain deformation modes in a solution known as zero-energy

modes, other than the rigid body movements and which must be sup-

pressed through stabilization techniques. It has been found that all the

displacement-based shear deformable plate elements of this kind fail on

many occasions either by shear-locking or singular behaviour.

Thus it is felt that in spite of vast number of elements present in the

literature [67] since the inception of the finite element method in the early

1960s, still there is a need of development of suitable elements which can

model the thin plates of arbitrary geometry successfully.

1.2 The Objective and Scope of Present Investigation 5

1.2 The Objective and Scope of PresentInvestigation

The objective of the present investigation is to formulate simple and ef-

ficient finite elements for static, free vibration and buckling analyses of

the bare and stiffened plates of arbitrary geometrical shapes under diverse

loading and boundary conditions and demonstrate the performance of the

proposed elements through the numerical examples in the related fields.

In this thesis, a new four-noded plate bending element is proposed

for the analysis of the bare plates, which is derived, though from the sim-

plest rectangular basic plate bending element having 12 degrees of free-

dom largely known asACM Element[1], but it has all the advantages of

the isoparametric element to model an arbitrary plate shape and without

the disadvantage of the shear locking problem. Further, for the analysis

of the stiffened plates, a stiffened plate bending element is formulated

by combining the four-noded rectangular plane stress element having 8

degrees of freedom with the 12 degrees of freedomACM Plate Bending

Element. The incorporation of boundary conditions is made in the most

general manner to cater to the need of the curved boundary as well as to

the more practical mixed boundary conditions.

As the element developed for the bare plate analysis is capable of

modelling an arbitrary plate geometry, a large number of static, dynamic

and stability problems in the bare plate domain of square, rectangular,

skew, trapezoidal, triangular, circular, elliptical, annular sector geome-

tries are considered and the results are presented showing the elegance

and efficiency of the proposed element.

The element developed for the analysis of the stiffened plates has the

same feature of accommodating the arbitrary shape of the plate geome-

tries and the stiffener modelling is done for a general one. The stiffener

6 INTRODUCTION

is modelled in such a way as to lie anywhere within the plate element and

need not follow the nodal lines. Further, in the formulation, their orien-

tation is kept arbitrary which makes the analysis more flexible and the

mesh division independent of their location and orientation. The same

displacement interpolation functions as used for the plate elements are

adopted in the formulation of the stiffener element. This facilitates to

express the stiffness and the mass matrices of the stiffener in terms of the

nodal parameters of the plate element thus ensuring the compatibility of

the stiffener with the plate.

Similar to the bare plate; static, dynamic and stability analyses of

various stiffened plate configurations such as square, rectangular, skew,

trapezoidal, triangular, circular, elliptical, annular sector etc. with various

stiffener positions have been carried out.

The implementation of the methodology to different types of analysis

described in the investigation is made through the development of com-

puter programmes inC++. To make the analysis more cost effective, the

global elastic stiffness, mass and geometric stiffness matrices are stored

using the skyline storage technique. No standard or general software

package is used for these analyses and as such the computer programmes

developed here are general and complete in themselves. The computer

programmes have been run in theHP - UX 9000/819work station avail-

able at the Computer Centre of the Institute and theORIGIN 200 of the

Departmental Computer Laboratory.

The present investigation comprises the following topics:

1. Analysis of Arbitrary Bare Plates

(a) Static Analysis of Arbitrary Bare Plates : The static anal-

ysis is carried out for different geometrical plate shapes such

as square, rectangular, skew, annular sector and circular one

1.2 The Objective and Scope of Present Investigation 7

for various boundary and loading conditions and the results

are compared with the published ones.

(b) Free Vibration Analysis of Arbitrary Bare Plates : The

proposed element is tested by considering the free flexural

vibration analysis of bare plates of various shapes having var-

ious boundary conditions and the first few natural frequencies

are compared with those from open literature.

(c) Stability Analysis of Arbitrary Bare Plates : In the stabil-

ity analysis, bare plates of rectangular, skew and circular con-

figurations with different boundary and inplane loading con-

ditions are considered and the results are validated by com-

paring the buckling parameters obtained with those available

ones.

2. Analysis of Arbitrary Stiffened Plates

(a) Static Analysis of Arbitrary Stiffened Plates : A large num-

ber of stiffened plates of straight and curved edges with con-

centric as well as eccentric stiffeners are studied. The results

are presented in terms of stresses/stress resultants. Some new

results are also presented.

(b) Free Vibration Analysis of Arbitrary Stiffened Plates :

The first few natural frequencies of a large number of stiff-

ened plates having various planforms are presented. In the

analysis, eccentric as well as concentric stiffeners are con-

sidered and various boundary conditions are implemented.

Some new examples are also attempted.

(c) Stability Analysis of Arbitrary Stiffened Plates : Stability

analysis is carried out for rectangular, skew and circular stiff-

8 INTRODUCTION

ened plates with various boundary conditions and buckling

parameters are presented for various flexural and torsional

stiffness of the stiffeners. Few new results have been pre-

sented for this category of analysis.

Hence, in summary, a large number of numerical examples have been

considered in this investigation for static, dynamic and stability analyses

of bare plates and stiffened plates of various geometrical configurations.

Various loading and boundary conditions as well as concentric and eccen-

tric stiffeners are considered in the analysis. In addition to the examples

presented for the validation of the proposed method some new results are

also put forward.

Chapter -2

REVIEW OF LITERATURE

2.1 Introduction

Many exact solutions for thin elastic bare plates for bending and buck-

ling along with a few stiffened plate buckling analysis are well docu-

mented in Timoshenko’s monographs [184] and [185]. Leissa [86] has

presented free vibration analytical results for a number of cases of bare

plates. In the stiffened plate domain the analytical solutions have been

presented by Troitsky [186] for static, dynamic and stability analysis.

However, the analytic solutions in the open literature are incomplete be-

cause they become extremely tedious for complex problem definitions.

The advent of digital computer along with its capability of exponentially

increasing computing speed has made the analytically difficult problems

amenable through the various numerical methods and thus making the

literature rich in this area.

2.2 Review on Bare Plates

In the context of the present investigations, the following areas of analysis

pertaining to the bare plates are covered in the review of literature:

9

10 REVIEW OF LITERATURE

1. Static Analysis of Bare Plates

2. Free Vibration Analysis of Bare Plates

3. Stability Analysis of Bare Plates

2.2.1 Static Analysis of Bare Plates

The static analysis of bare plates using the finite element method is well

documented by Zienkiewicz and Taylor in their two volume of books [198]

and [199]. In this review, an attempt has been made to include the more

recent publications.

Sawko and Merriman [165] have proposed a curved element for the

analysis of plates with curved boundaries. They have represented the lat-

eral deflection over the element in terms of polar coordinates and have

considered four degrees of freedom at each node of the element. They

have presented results for circular plates having various boundary condi-

tions under uniform and concentrated loads. They have also analyzed the

simply supported curved bridge deck of Coull and Das [41] and Cheung

et al. [36] and results have been compared.

Chernuka et al. [35] have developed a triangular element in which

one of the edges is modified to a curved one in an effort to minimize

the error inherent in representing the shape of a curved boundary by a

series of straight segments. They have presented two different versions

of the element; one incorporating a quadratic curved edge and the other

a quartic one. They have analyzed circular and elliptical plates under

different loading and boundary conditions.

Mukhopadhyay [124] has studied the bending of skew plates for dif-

ferent skew angles by using semianalytic finite difference method fol-

lowing the philosophy of Kantorovich and the finite strip method. In this

2.2 Review on Bare Plates 11

method, a displacement function satisfying boundary conditions along

two opposite edges of the plate is assumed. The displacement function

is then substituted in the differential equation of the plate, which in turn

is reduced to an ordinary differential equation by some transformation

method. This resulting equation is then solved by the finite difference

method. Isotropic rhombic plates for various skew angles for all edges

simply supported case have been analyzed using this method. Results

have not agreed well for greater skew angles.

Bapu Rao et al. [14] have developed an annular and annular sector

elements associated with six and twenty degrees of freedom respectively

based on Reissener’s thick plate theory which includes the shear defor-

mation. The displacement field, which has been represented by the lat-

eral deflection and shear rotations, has been expressed in terms of inter-

polation polynomial functions of the radial and angular coordinates. In

this proposed method, the sector element can produce satisfactory results

only when the element size is small in the angular direction. Here, the

shear coefficient has been expressed as a free constant. They have also

presented the free vibration results for annular and sector plates.

Mukhopadhyay [126] has extended the semianalytical method for

analyzing bending of radially supported curved plates under different

boundary and loading conditions. He has presented extensive results for

annular sector plates subjected to concentrated and uniformly distributed

load.

Barve and Dey [17] have proposed a method incorporating the con-

cept of isoparametry in finite difference energy method by defining the

plate geometries and the displacement functions in curvilinear coordi-

nate system. They have presented the results for the plates of square,

circular, skew geometries along with the perspex model curved bridge

deck of Coull and Das [41]. Their method can accommodate plates of


various geometrical configurations where somewhat rectangular type of

discretization is possible, it is felt that it will be difficult to model some

of the plate geometries like triangular ones where the triangular shaped

discretization is inevitable.

Yang and Chong [192] have presented an alternative method of fi-

nite strip by replacing the cubic X-spline functions in place of usual

trigonometric series in the y-direction. Using this method they have an-

alyzed square plates having simply supported and fixed boundary con-

ditions subjected to uniformly distributed and concentrated load. They

have also analyzed a trapezoidal plate with simply supported edges un-

der uniformly distributed load.

Bhat [24] has determined the plate deflections under static loading by

generating an orthogonal set of beam characteristic polynomials using the

Gram-Schmidt process and applying the Rayleigh-Ritz method. Results

are presented for rectangular plates with all edges clamped and those with

three edges clamped and one edge free.

Tham et al. [181] have used the spline-finite-strip method to analyze

skew plates with different loading and support conditions. They have

mapped the original parallelogram plate in Cartesian coordinates to a unit

square plate in natural coordinates by a simple transformation relation-

ship and discretized the mapped square plate into strip elements. The

interpolation function for the out of plane displacement in the square re-

gion is expressed as a product ofB-3 splines in the strip directions and the

conventional Hermite cubic polynomial is considered in the other direc-

tion. They have presented results for skew plates for various skew angles

under different boundary and loading conditions. They have also ana-

lyzed stepped and continuous plates using this method. They have further

extended this work to include the bending of arbitrarily shaped general

plates [96]. They have analyzed the circular, elliptical, fan-shaped and


circular sector plate along with the parallelogram plate.

Dey and Malhotra [45] have analyzed orthotropic curved bridge decks

using a higher order finite strip method. They have employed a quintic

polynomial in the radial direction along with a basic function series in

the angular direction satisfying the boundary conditions along the radial

edgesa priori. Thus, a two dimensional plate bending problem has been

reduced to a one-dimensional one resulting great reduction in size and

bandwidth of the global stiffness matrix. They have presented results for

curve-edged plates using the first five harmonic components and com-

pared them with the available ones.

GangaRao and Chaudhary [53] have developed converging series so-

lutions for rectangular, skew and triangular plate configurations under

different boundary and loading conditions. For the rectangular and the

skew plates they have represented the deformed shape of the structure

by a combination of trigonometric and polynomial functions, the coef-

ficients of which are determined by using Galerkin technique. For the

triangular plate problems they have selected suitable shape functions for

the deformed shape representation. This method is applied to the straight-

edged plate configurations.

Butalia et al. [29] have presented a critical analysis of parallelogram-

shaped plates under bending using Mindlin nine-node Heterosis element.

To make ease of the specification of the boundary conditions, they have

transformed the element matrix corresponding to global axes to the local

ones. They have presented results for rhombic skew plates for differ-

ent skew angles and various boundary conditions and have compared the

results with a large number of published ones.

Peng-Cheng and Hong-Bo [146] have proposed a multivariable spline

element analysis for plate bending problems. They have used bicubic

B-spline functions to construct the bending moments and transverse dis-


placement fields. The spline element equations with multiple variables

have been derived based on Hellinger-Reissner principle. They have pre-

sented square plate results with simply supported and clamped edge con-

ditions subjected to uniformly distributed and concentrated loads.

Liew [97] has presented pb-2 Ritz function to study the static analysis

of arbitrarily shaped plates using the principle of minimum potential en-

ergy. The pb-2 Ritz function consists of the product of a two-dimensional

polynomial function and a basic function. The basic function is again a

product of the specified boundary equations. He has presented the de-

flections and moments of trapezoidal, skew and elliptical plates. Though

the computational effort is less in this method, the process of choosing

an appropriate basic function that changes with the configuration of the

plates as well as the boundary conditions of the edges complicates the

method. The complication is further added as the support conditions are

to be satisfieda priori in the basic function itself.

Au and Cheung [10] have developed isoparametric spline finite strip

method for the analysis of plane structures. They have used cubicB-

spline curves in the modelling of the geometry and the representation of

the displacement field as well. In this method, the plate is first discretized

into strips bounded by spline curves and then Mindlin plate formulation

is used to solve it. The investigators have used this method for bending,

plane stress and plane strain analyses. They have presented results for

a curved bridge model under its dead load, an S-shaped slab bridge un-

der uniformly distributed load and concentrated load, and a thick hollow

cylinder under internal pressure. This method yields a relatively narrow

band matrix and requires less computational effort.

Liu and Lin [102] have proposed a four-noded sixteen degrees of free-

dom conforming quadrilateral plate bending element in which the ele-

ment geometry is in bilinear polynomials representation, while the dis-


placement functions are in terms of modified bicubic polynomials satis-

fying energy orthogonality. In this formulation, the usual approach of ex-

pressing transverse displacements and rotations by separate expansions is

not allowed. They have presented results for square, rhombic and circu-

lar plates. It is reported that though they have obtained improved results

for the deflections and moments at the mid point of the square plates, but

the clamped-edge moments are worse.

Spline element method to analyze the bending of skew plates with

arbitrary boundary conditions has been presented by Mizusawa [113]. In

this method the displacement functions have been expressed in terms of

the two-wayB-spline functions. He has used a non-dimensional skew

coordinate system for the analysis. Deflections and bending moments

of rectangular plates and skew plates having various skew angles with

arbitrary boundary conditions have been presented and compared with

the published results.

Ng and Chen [139] have analyzed fan-shaped bridge decks by using

the spline finite strip method and the Mindlin plate theory. They have

used the reduced integration technique to eliminate the shear locking and

utilized the penalty function method to impose boundary condition at the

end of strips and intermediate supports.

Sengupta [169] has developed a three-noded element to study the

bending behaviour of skew rhombic plates of various boundary condi-

tions subjected to uniformly distributed as well as concentrated loads.

The study is limited to the skew plate configuration.

Hamouche et al. [58] have used a spectral solution methodology to

solve the biharmonic equation for analyzing the bending of thin plates

of arbitrary geometric shapes. They have investigated the use of Fourier

and Chebyshev expansions of the dependent variables to eliminate the

mathematical difficulties which arise in the fulfillment of the boundary


conditions. Based on these expansions and the Fast Fourier Transform, a

numerical methodology has been developed to solve plate bending prob-

lems. They have presented solutions for problems of triangular, circular,

annular and truncated circular sector plate configurations having various

boundary conditions.

Ayad et al. [11] have developed two hybrid-mixed finite elements,

MiSP3 (3-node triangular element) and MiSP4 (4-node quadrilateral el-

ement), which require C◦ continuity for kinematic variables and C−1 or

L2 continuity for bending moments and Shear forces following Mixed

Shear Projected (MiSP) approach based on the Hellinger-Reissner vari-

ational principle. In order to control the shear locking they have cho-

sen an independent shear strain approximation and edge projection for

strain-displacement relations. The approximations of the shear forces

have been derived from those of the bending moments using the corre-

sponding equilibrium relations. They have derived the modified MiSP

models (MMiSP) by defining the shear strains as projected shear strains

in place of defining it from a linear approximation of the nodal degrees

of freedom. They have analyzed skew and circular plates in addition to

the rectangular plates.

Saadatpour and Azhari [159] have presented a theoretical formula-

tion for the static analysis of simply supported plates of general shape.

The procedure is based on the Galerkin method and it uses the natural

coordinates to express the plate geometry. They have expressed the dif-

ferential equation of moment function in terms of distributed load and

that of out-of-plane deflection in terms of moment. For the above formed

two equations, they have assumed expanded form of solution in terms

of two types of basic functions (one trigonometric and the other poly-

nomial) and unknown generalized coordinates satisfying the simply sup-

ported straight geometric boundary conditions. The unknown general-


ized coordinates have been determined by the Galerkin method using the

basic functions as the weight functions. They have presented the deflec-

tion and the bending moment results of trapezoidal, parallelogram and

sector plates. Though it has been claimed, the triangular plate results are

not present in the paper. This method is suitable for straight-edged plates

of simply supported boundary conditions. This method requires more

analytical and computational effort for the plates of curved geometry.

2.2.2 Free Vibration Analysis of Bare Plates

The literature in free vibrations of bare plates is vast which is discussed in

a series of excellent review articles by Leissa [86], [88], [89], [90], [91],

[92], [93] and by Yamada and Irie [191]. Extensive free vibration study

of rectangular plates has also been carried out by Gorman [56]. Hence

the review here is limited to those more recent ones and relevant to the

present investigation.

Orris and Petyt [142] have used two conforming plate bending ele-

ments, one a quadrilateral and the other a triangular, and investigated the

free vibration characteristics of triangular and trapezoidal plates. They

have obtained the quadrilateral element by combination of the cubic de-

flection fields in each of the four triangular regions defined by the edges

of the quadrilateral and its diagonal. The deflection fields have been ex-

pressed in terms of the area coordinates of the triangular regions. They

have presented free vibration results of rectangular, triangular and trape-

zoidal plates by varying the ratio of top to bottom parallel chords of the

trapezoidal plate. The mode shapes and natural frequencies for trape-

zoidal plates with height to base ratios of 6:1 typifying the dimensions of

control surface ribs have also been presented. However, the elements are

not designed to accommodate the plates with curved boundaries.


Mizusawa et al. [116] have dealt with the free vibration of skew plates

for various boundary conditions using the Rayleigh-Ritz method with

B-spline functions as its coordinate functions. They have applied the

method of artificial springs to deal with the arbitrary boundary condi-

tions of the plates corresponding to deflection and the two slopes at each

of the edges. By assigning zero or infinite values to these spring con-

stants they have obtained free or fixed boundary conditions for the corre-

sponding restraint. They have also studied the convergence of the results

with respect to changes in the degree of theB-spline functions and in the

number of knots, for different skew angles. They have obtained the total

potential energy by adding the energy due to the springs corresponding

to boundary conditions, to that of the skew plate.

Leissa and Narita [94] have presented the free vibration natural fre-

quencies of simply supported circular plates for different values of Pois-

son’s ratio and number of internal nodal circles using the ordinary and

modified Bessel functions. They have considered the deflected shape of

the vibrating plate in polar coordinates and have derived the relationship

to obtain additional values of frequency parameters for large number of

internal nodal circles.

Maruyama and Ichinomiya [105] have described an experimental study

of the low frequency transverse vibration modes of wedge-shaped and

ring-shaped sector plates with all edges clamped which are carried out

by using the real time method of holographic interferometry. They have

studied the effects of the sector angle and the radii ratio on the natural fre-

quencies and the corresponding mode shapes of the sector plates with all

edges clamped and have compared their experimental results with those

of analytically obtained ones by the other investigators.

Bhat [23] has investigated the vibration problem of rectangular plates

by using a set of characteristic orthogonal polynomials in the Rayleigh-


Ritz method. The orthogonal polynomials have been generated by using

Gram-Schmidt process so as to satisfy the geometric boundary conditions

of the accompanying beam problems. He has presented vibration results

for rectangular plates of different boundary conditions.

Laura et al. [84] have analyzed transverse vibrations of a trapezoidal

cantilever plate of variable thickness using energy techniques. They have

described the structural deflections by characteristics orthogonal poly-

nomials in two variables and have applied the Rayleigh-Ritz method to

obtain the natural frequencies. They have presented the frequency param-

eters for cantilever trapezoidal plates for different aspect ratios and have

compared them with the results obtained by using the Rayleigh method

with an optimized exponent in the deflection expression.

Kim and Dickinson [78] have determined the free flexural vibrations

of a large number of right triangular plates for various combination of

free, simply supported or clamped boundary conditions by carrying out

the Rayleigh-Ritz analysis. They have used simple polynomials as the

admissible functions, which, through the use of a recurrence relationship

for the evaluation of the necessary integrals, lead to a simpler analysis.

Though the analysis presented is for specifically orthotropic plates, but

the majority of the numerical examples presented are for the isotropic

case of the right triangular plates.

Liew et al. [100] have extended the Gram-Schmidt procedure of Bhat [23]

and have studied the plate vibration by Rayleigh-Ritz method. Their se-

lection of the starting function in the set of orthogonal plate functions is

same as that of Bhat, but the higher terms are generated using a procedure

leading to faster convergence. They have presented frequency parameter

results for different boundary conditions of the rectangular plates.

Lam et al. [82] have proposed an approximate method based on the

Rayleigh-Ritz principle for vibration analysis of circular and elliptical


plates. A set of new starting functions has been proposed which satisfies

the geometrical boundary conditions of circular and elliptical plates with

clamped, simply supported or free circumferential peripheries. Further,

the Gram-Schmidt process has been used to generate the higher terms in

the set of plate functions. The use of these functions for the determination

of natural frequencies and mode shapes has been presented. They have

analyzed circular and elliptical plates for free, simply supported or fully

clamped boundary conditions.

Prasad et al. [149] have presented approximate formulae for the free

vibration of simply supported and clamped elliptical plates. They have

used the Rayleigh-Ritz method with a three-term deflection function. In

this method a deflection has to be chosen satisfying each boundary con-

dition which makes the analysis difficult.

Young and Dickinson [194] have used the Rayleigh-Ritz method for

the free vibrations study of the plates having one or more edges defined

by general polynomials, the admissible functions employed being prod-

ucts of simple polynomials. They have presented the free vibration re-

sults for the isotropic as well as rectangularly orthotropic plates and con-

sidered various plate geometries such as circular, elliptical, annular and

hypocycloidal. However, the incorporation of the boundary conditions is

somewhat complex and imposes the restrictions on the choice of deflec-

tion function.

Ding [46] has proposed the use of a fast converging series consisting

of static beam functions under point load as admissible functions in the

Rayleigh-Ritz method to study the problem of the flexural vibration of

rectangular plates. The admissible sets of displacement functions have

been obtained by varying the location of the point load applied to the

beam. Numerical examples of rectangular plates with various aspect ra-

tios and boundary conditions have been presented. In this method, the


boundary conditions of the plate dictate the type of admissible function

to be selected.

Mirza and Alizadeh [111] have idealized a cracked plate as a partially

supported one with varying support length and analyzed the triangular

plates for free vibration using eight-noded isoparametric quadrilateral el-

ement based on Mindlin plate theory. They have studied the effects of

the detached base length on vibration of these types of structures. They

have attempted to eliminate the transverse shear effects in thin plates by

employing a reduced Gaussian integration. However, it is well known

that these types of integration schemes lead to numerical complexities

and spurious mechanisms. Though the isoparametric element is capable

of addressing some non-conventional plate geometries, the authors have

presented only the triangular plate results.

Singh and Chakraverty [177] have analyzed rectangular and skew

plates for free vibration under different boundary conditions by using

boundary characteristic orthogonal polynomials in two variables. They

have first mapped the given plate into a square plate over which a set

of orthogonal polynomials satisfying the essential boundary conditions

is generated by using the Gram-Schmidt process. The Rayleigh-Ritz

method is then used to determine the frequencies for all possible combi-

nations of the boundary conditions and with different skew angles. This

is the extension of their previous work [176] where they have applied

the same method for studying the free vibration of annular circular and

elliptic plates. In another publication [175] they have presented results

for vibration of simply supported elliptical and circular plates using the

same method. Though the method in all the works is same but they differ

considerably when the orthogonal polynomials are generated for differ-

ent plate shapes. Hence it needs the reformulation of the problem for

each category of plates thereby lacking in its generality.


Geannakakes [54] has presented a theoretical formulation for the free

vibration analysis of bare plates of various shapes using natural coor-

dinate regions for defining the plate geometry in conjunction with nor-

malized characteristic orthogonal polynomials for defining the deflection

function and the Rayleigh-Ritz method to set up the eigenproblem. How-

ever, he used five different kinds of shape functions for the definition of

the plate geometry for five different regions; namely, the linear region,

the cubic region, the incomplete and complete quartic regions and the

quartic-linear region depending on the complexity of the plate boundary

definition. Thus the method adopts different definitions for different plate

geometries and thereby incurs loss in generality. Moreover, the numeri-

cal results obtained by this method depends on the number of integration

points and normalized characteristic orthogonal polynomials in each di-

rection. In the analysis the accuracy of the results was improved when

the number of integration points was increased. However, too many in-

tegration points (11 number of points in his analysis) tend to degrade the

accuracy of the computed integral results.

Saliba [161] has proposed a superposition technique for the solution

of free vibration problems of right triangular thin plates. He has modified

the six building block arrangement introduced by Gorman [57]. In his

modified version Saliba has used only two building blocks instead of six.

He has obtained the superposition of these blocks by first determining the

contributions of each individual building block to the relevant boundary

conditions and, second, due to the linear nature of the individual building

block problems, their total contribution to a given boundary condition has

been found by adding together their individual contributions. The Levy-

type solutions have been proposed where the support conditions can be

forced by adjusting the Fourier coefficients. Numerical results and mode

shapes for right-angled triangular plates with all possible combinations


of simple and clamped boundary conditions have been presented. Great

precaution is to be exercised while applying the method even for other

general straight boundary problems like general triangles, trapezoidal and

parallelogram plates.

Ghazi et al. [55] have used the Lagrange’s equations of motion cou-

pled with the finite element technique and analyzed the free vibration of

plates of pentagonal and heptagonal shapes. First, they have proposed an

18 degrees of freedom triangular plate bending element without consider-

ing the transverse shear effect and have presented results for isotropic as

well as orthotropic plates for various edge conditions. In the second part,

considering the transverse shear effect, they have formulated a higher or-

der 36 degrees of freedom finite element and results for some complex

combinations of rigidly clamped, simply supported and free edge condi-

tions for isotropic, orthotropic and laminated plates have been presented.

Houmat [65] has presented a method known as trigonometric hierar-

chical finite element method after Bardell [16] for the free flexural vi-

bration analysis of bare plates which is formulated in terms of a fixed

number of quintic polynomial shape functions plus a variable number of

trigonometric hierarchical shape functions. In this method a structure is

modelled as just one finite element and the number of hierarchical terms

is varied to obtain the results to a desired degree of accuracy. Here the

satisfaction of internalC◦ and/or C1 continuity along element interfaces

is avoided and the problems of stress singularities are overcome. Results

are presented for the square and rectangular plates with different bound-

ary conditions. This method because of its simplicity can only be applied

to simple and uniform structures.

Radhakrishnan et al. [151] have developed an approximate method to

estimate the fundamental frequency of a plate through the finite element

solution of its static deflections under a uniformly distributed load using


a frequency-static deflection relation without the associated eigenvalue

problem. Since a four-noded quadrilateral isoparametric plate element

has been chosen for the static deflection of the plate, they have been able

to present results for rectangular and circular plates having holes at the

centre. This method is useful to assess the approximate natural frequen-

cies of plates of arbitrary shape. However, the obvious problem of shear

locking associated with the isoparametric elements used for the static

analysis of thin plates in this method has not been discussed.

2.2.3 Stability Analysis of Bare Plates

Kapur and Hartz [76] have derived the stability coefficient matrices for

plates under different loading conditions for use with the stiffness ma-

trix following Bolotin [26]. These stability coefficient matrices allow the

modification of the conventional plate stiffness matrices to include the

effect of the in-plane stresses. They have applied this method for rectan-

gular plates under different in-plane loadings.

Durvasula [48] has used the Rayleigh-Ritz method with deflection

expressed in double Fourier sine series in oblique coordinates to study

the stability of the simply supported skew plates under uniform system

of inplane stresses which have been represented in terms of orthogonal

components. Buckling coefficients for simply supported skew plates of

various aspect ratios having varying skew angles have been presented.

This method limits its application to the rectangular and the skew plate

geometries.

Fried and Schmitt [52] have applied the gradient iterative techniques

to refine the finite element mesh at the obtuse corner of the skew plate

where stress singularity occurs, thus regaining the full rate of conver-

gence of the finite element vibration and stability results of skew plate


configurations. They have considered a general skew element having

four nodal points, each associated with four degrees of freedom and have

considered pure boundary conditions such as free on all sides. Free vibra-

tion results for equilateral plates having various skew angles and buckling

results for plates of different aspect ratios along with various skew angles

under compression and shear have been presented.

Prabhu and Durvasula [148] have considered the buckling problems

of clamped skew plates using oblique components of stress for oblique

geometry of the plate. They have expressed the deflection as a dou-

ble series of beam characteristic functions of a clamped-clamped beam

and have used energy method to obtain the critical buckling coefficients.

They have also made the convergence study under both direct and shear

loadings and have found that at least 18-term solution for the skew angle

up to30◦ and 32-term solution for the skew angle up to60◦ are required

for the fairly converged estimates of the buckling coefficients. They have

presented buckling coefficients for plates subject to inplane direct and

shear loadings for several combination of side ratios and skew angles.

Reddy and Tsay [155] have formulated a mixed rectangular element

having three degrees of freedom per node: the transverse deflection, and

the two normal moments for analyzing bare plates. They have presented

free vibration and stability results of square and rectangular plates. They

have also analyzed plates with uniform uniaxial compression for free vi-

bration. An attempt to derive non-rectangular elements to accommodate

the plates of general shape in this method is mathematically more in-

volved and hence it is confined to the square and rectangular plate con-

figurations.

Rubin [158] has presented an analytical method to obtain critical

loads and buckling mode shapes for polar-orthotropic sector plate with

radial edges simply supported and having arbitrary boundary conditions


along the circular edges. He has expressed the differential equations of

the plates in terms of polar coordinates. The method can be applied to

the buckling analysis of plates of pie-shaped and ring-shaped sectors of

orthotropic as well as isotropic ones.

Mukhopadhyay [128] has extended the static [127] and vibration [125]

analysis of plates to analyze the stability of ship plating and allied plated

structures using the semi-analytical method. He has presented results for

various edge conditions of the rectangular plates and different inplane

loading combinations.

Mizusawa et al. [117] have presented the bending, vibration and buck-

ling analyses of skew plates by using the modified Rayleigh-Ritz method

usingB-spline functions with Lagrange multipliers to deal with both ge-

ometric and natural boundary conditions. They have presented bending,

vibration and buckling results for various skew angles and boundary con-

ditions of the skew plates.

Mizusawa and Kajita [114] have dealt with vibration and buckling

analyses of skew plates with edges elastically restrained against rotation.

They have used the spline strip method where the skew plate has been

idealized by discrete strip elements. In this method, the displacement

function is expressed by the product of basic function series in the lon-

gitudinal direction satisfying the boundary conditions at the ends and the

B-spline functions known as piecewise polynomials whose higher deriva-

tives are continuous in the discretized subregions. They have studied the

effect of rotational stiffnesses, skew angles and aspect ratios on the vibra-

tion and buckling of skew plates and have presented characteristic charts

for them.

Liu and Chen [103] have extended the work of Harik [59] on stabil-

ity of annular plates by incorporating the elastically restrained boundary

conditions and by considering both radial and tangential in-plane stresses


using the semianalytic technique. In this method, the governing differen-

tial equation for the deflection is expressed in polar coordinates and its

solution is assumed as the product of a radial function and a beam func-

tion corresponding to identical elastically restrained boundary conditions

at the ends.

Mukhopadhyay [130] has presented a numerical method in which two

characteristic functions satisfying the boundary conditions along the op-

posite edges are assumed and then the displacement function is substi-

tuted into the differential equation of the plate for free vibration and buck-

ling which in turn is converted into an eigenvalue problem through some

transformations. Rectangular uniform isotropic plates of various aspect

ratios having varying degrees of rotational restraints along the edges have

been analyzed using various number of total harmonic terms in each di-

rection. Frequency and buckling parameters have been presented for

various boundary conditions which include the classical as well as the

elastically restrained in rotation and have been compared with the results

obtained by the semi-analytic finite difference method.

Mermertas and Belek [109] have developed a sector finite element

model with the wave propagation technique of cyclic symmetry to study

the static and dynamic stability of annular plates of variable thickness.

They have used the Mindlin plate finite element in conjunction with

Bolotin’s approach, with an isoparametric sector element. The effects

of thickness, various boundary conditions and loading have been investi-

gated.

Tham and Szeto [182] have applied the spline finite strip method to

the buckling analysis of arbitrarily shaped plates by the subparametric

transformation of the plates into the natural coordinate plane and express-

ing the displacements of the strip in terms of natural coordinate variables.

The displacements of each strip are described by interpolation functions


which are the products of piecewise polynomials andB-3 spline func-

tions. The formulated eigenvalue matrix equation for the buckling anal-

ysis has been solved following the procedure of standard finite element

method. They have presented the first and second buckling load factors

for the rectangular, parallelogrammic, triangular, circular and elliptical

plate configurations.

Singh and Venkateswara Rao [178] have presented design formulae

for the fundamental frequencies and critical buckling loads estimation of

elliptical plates for simply supported and clamped edges. Cortinez and

Laura [40] have proposed simple approximate formulae for a quick and

sufficiently accurate estimate of the fundamental frequencies and critical

loads of clamped plates by simplifying the conformal mapping method.

They have determined the fundamental frequencies and critical buckling

loads of regular polygonal plates, circular plate with two flat sides and a

square plate having rounded corners. This method can be applied to the

plate having clamped edges only.

Wang et al. [188] have used the pb-2 Rayleigh-Ritz method proposed

by Liew and Lam [98] and have analyzed the buckling of skew plates.

In this method, the Ritz functions consist of the product of a basic func-

tion and orthogonal polynomials, the degree of which may be increased

until the desired accuracy is reached. The basic function is formed by

taking the product of the equations expressing the boundary shape, with

each equation having the power of either 0, 1, or 2 corresponding to free,

simply supported, or clamped supporting edges respectively. As such,

the kinematic boundary conditions are automatically satisfied at the out-

set without using Lagrangian multipliers. This form of Ritz function has

an advantage over the trigonometric series, as the analyst need not search

for the appropriate trigonometric series because the boundary expressions

are already given. Using this method they have presented buckling results


for skew plates of various boundary conditions.

Jønsson et al. [72] have derived a hybrid displacement four-noded

rectangular plate element where the potential energy is modified by adding

Lagrange multiplier terms and thereby introducing independent field pa-

rameters. The boundary displacements and internal displacements have

been interpolated by the nodal parameters, while the curvatures and mo-

ments by locally condensed internal parameters. They have presented re-

sults for bending and stability of rectangular plates with various boundary

and loading conditions.

Zhang and Kratzig [196] have presented a four-noded rectangular

element of the Mindlin displacement model for thin plate bending and

buckling analysis. The element properties have been derived using dis-

crete Kirchhoff constraint according to an eight-node interpolatory pat-

tern specified to consist of bilinear Lagrangian and Serendipity bubble

functions. To satisfy the real Kirchhoff conditions in the thin plate limit, a

different discrete Kirchhoff constraint has been proposed. The element’s

bending behaviour has been shown only by a single element test. Buck-

ling analysis of square and rectangular plates have been presented.

Zhou et al. [197] have presented a semianalytical-seminumerical method

of solution for the buckling problem of simply supported annular sector

plates subjected to inplane pressure along the straight-edges. They have

first generated the analytical solution for the pre-buckling inplane stresses

of the plates based on the elastic theory of the plane-stress problem and

then used a seminumerical technique to obtain the critical buckling load.

The basic functions in the angular direction are chosen as the eigenfunc-

tions for the simply supported beam and the resulting ordinary differen-

tial equation is solved by a one-dimensional finite difference technique.

Wanji and Cheung [189] have proposed a refined triangular discrete

Kirchhoff thin plate bending element by improving the original DKT el-


ement. They have imposed a relaxed continuity condition of∂θx

∂y=

∂θy

∂xin the element, but satisfying in the strict sense the C1 continuity re-

quirement on the element boundary. They have proposed formulation of

the mass matrix and the geometric stiffness matrix by linear combina-

tion of the interpolations of the element displacement functions. They

have presented a clamped circular plate example for the bending analysis

and simply supported as well as clamped square plates for vibration and

buckling analyses.

Yuan and Jin [195] have employed the multi-term trial functions in

place of earlier used single-term trial function to the extended Kantorovich

method for the eigenvalue solution of elastic stability of rectangular thin

plates subjected to different inplane forces and boundary conditions. They

have derived the ordinary differential equations and the boundary condi-

tions through the associated variational principle reducing the problem

to a linear eigenvalue problem in ordinary differential equations in each

iteration step, which in turn has been solved using general-purpose or-

dinary differential equation solvers. They have presented a number of

numerical examples of rectangular plates.

2.3 Various Methods of Analysis of Stiff-ened Plates

For many years the research on stiffened plated structures has been a

subject of interest. Extensive efforts by many researchers have been put

into the investigation of the response of the stiffened plates under various

loading conditions. Due to its complexity and the number of parameters

involved being many, a complete understanding of all aspects of its be-

haviour is yet to be fully realized. In order to facilitate a solution to the

2.3 Various Methods of Analysis of Stiffened Plates 31

problem the researchers have made several assumptions leading to the

various methods of analysis.

A simplified approach for the analysis of stiffened plates has been

proposed by Huber [68] which is known as orthotropic plate theory. The

philosophy of the method is to convert the stiffened plate into an equiv-

alent plate with constant thickness by smearing out the stiffeners. If the

stiffeners are closely spaced then only this model is justified. When the

stiffeners are not identical in both directions or not equally spaced the

resulting thickness becomes non-uniform and the analysis becomes com-

plex.

Another model known as grillage model has been proposed by some

investigators. In this model the stiffened plate is considered as a plane

structure containing intersecting beams and carrying a lateral load through

the action of beam bending. The centroidal planes of the beams in the

two orthogonal directions are assumed to be same which affects the ac-

curacy in the stress computation. The beam properties are determined

considering the effective width of the plate, calculation of which is math-

ematically involved. It is not popular because of the drawbacks inherent

in the methodology.

The advent of the digital computer along with its exponentially in-

creasing computational speed as well as core memory capacity has given

the investigators a new direction to the analysis of the complicated struc-

tures thereby evolving simpler and more efficient methodologies. Among

these, the various numerical methods are:

• Finite difference method

• Dynamic relaxation method

• Finite element method


• Finite strip method

• Boundary element method

Among all the existing numerical methods, the finite element method

is undoubtedly the most versatile and accurate one specially for structures

having irregular geometry, material anisotropy, nonhomogeniety and any

type of loading and boundary conditions.

2.4 Review on Stiffened Plates

The vast amount of literature available in the area of stiffened plates

is rich in contributions by the researchers and scientists from diverse

fields. Many investigators have studied the static, dynamic and stability

behaviour of the stiffened plates and the developments of them up to mid-

eighties are well documented in [186],[132],[162], [164],[121],[122],[172]

and [134]. Troitsky [186] has extensively reviewed the literature per-

taining to rectangular stiffened plates for static, dynamic and stability

analyses which are based on orthotropic plate idealization. Different

methods of analysis and idealization techniques employed in the static

analysis of stiffened plates have been reviewed by Satsangi [162] and

Satsangi and Mukhopadhyay [164]. An extensive review on static, dy-

namic and stability analyses of bare and stiffened plates has been pre-

sented by Mukhopadhyay [132]. The dynamic behaviour of stiffened

plates comprising free, transient and random vibration analyses has been

reviewed by Mukherjee [121] and Mukherjee and Mukhopadhyay [122].

Mukhopadhyay and Mukherjee [134] have further extended their earlier

review to include the later developments on dynamic characteristics of

the stiffened plates. In the present investigation an attempt has been made

2.4 Review on Stiffened Plates 33

to supplement the earlier reviews on stiffened plates and in the context of

the present work they are limited to the following areas of analysis:

1. Static Analysis of Stiffened Plates

2. Free Vibration Analysis of Stiffened Plates

3. Stability Analysis of Stiffened Plates

2.4.1 Static Analysis of Stiffened Plates

Mukhopadhyay and Satsangi [136] have formulated an isoparametric stiff-

ened plate bending element following an approach where the stiffener

can be placed anywhere inside the element. This has added a distinct im-

provement over the earlier lumped stiffener and orthotropic plate models

for the stiffened plate analysis. Though this model is capable of accom-

modating various plate geometries, they have analyzed only rectangular

plates with various stiffener shape and position along both the directions.

They have also presented a scheme for obtaining stiffener stresses for the

lumped model.

O’Leary and Harari [140] have proposed a finite element method in

which the constraint between stiffener and the member is imposed by

means of Lagrange multipliers. This imposition has been performed at

the functional level, forming augmented variational principles. In order

to simplify the initial development and implementation of the proposed

method, two-dimensional stiffened beam finite elements are developed.

They have conducted numerical tests on several such elements and ob-

tained monotonic convergence. In the development of stiffened plate fi-

nite elements, they have treated the bending and membrane behaviours

separately. The stiffness matrix of a standard plate element has been


modified to account for an added beam element and additional terms im-

posing the constant between the two. They have presented results for the

deflection of the centre of the stiffened plate and have compared with the

series solution of Timoshenko and Krieger [185].

Al-Shawi and Mardirosian [4] have proposed an improved dynamic

relaxation method for the analysis of cantilever plates stiffened with edge

beams. The plate skin is modelled using an improved rectangular plate

bending element and the edge stiffener is modelled using the grillage

beam element. Different weighting parameters are multiplied with the

mass and damping factor of the structure, the optimum values of which

are obtained for different cantilever plates with edge stiffeners of different

sizes. The method is applicable to the rectangular plates only and the

stiffeners should lie on the nodal lines. The inplane displacements due to

the eccentricity of the stiffeners are not considered in the analysis.

Harik and Haddad [61] and Harik and Salamoun [62] have applied the

analytical strip method to the analysis of stiffened plates having annular

sector and rectangular planforms respectively, modelling the plate and

the stiffener separately. The bending, torsional and warping rigidities of

the stiffener have been considered in the formulation, but the inplane dis-

placements produced by the eccentricity of the stiffeners have not been

considered. The behaviour of the system is derived by imposing the edge

and continuity conditions on the closed form solutions of the individual

plate strips and beam elements. In this approach the stiffener and the line

loads along the strip must follow the nodal lines which imposes restric-

tions on the mesh division.

Petrolito and Golley [147] have proposed a variable degree of free-

dom macro plate bending element where the displacement function within

an element satisfies the governing thin plate equations, substantially re-

ducing the number of equations to be solved. In this method, large ele-


ments corresponding to structural units bounded by beams may be used

resulting in a minimum of data preparation. Additionally, a modified

version of the ACM element with conforming displacements is shown to

be a sub-element of the proposed element. Though the number of total

equations generated in this method is small, it cannot be generalized for

plates other than the rectangular ones and it demands the placement of

the stiffeners to lie along the nodal lines.

Bhimaraddi et al. [25] have presented the finite element static and

free vibration analysis of an orthogonally stiffened annular sector plate

by combining annular sector plate and curved beam elements. They have

incorporated the shear deformation and the rotary inertia in both of the

elements. Additionally, both the elements are based on higher order the-

ories to include the analysis of arbitrarily laminated structures. How-

ever, the formulation imposes restrictions when applied to the arbitrarily

shaped structures since it is based on the polar coordinate system.

Chong [38] has used the principle of minimum potential energy for

the analysis of stiffened plates with arbitrarily oblique and equally spaced

eccentric stiffeners which are smeared over the entire plate. The equiva-

lent rigidities for orthogonally and symmetrically oblique stiffened plates,

which are used in the Huber-type equilibrium equations are determined

by assuming that the gradients of the inplane stress resultants are zero.

The placement of the stiffeners along any arbitrarily oblique direction is

the major improvement in this technique. The other limitations of the or-

thotropic plate idealization such as the stiffeners should be light, identical

and closely spaced are still present in the formulation.

Basak [18] has presented an analytical method for the static and dy-

namic analyses of rectangular orthotropic plate with irregularly spaced

stiffened ribs, simply supported at its four edges and subjected to dis-

tributed lateral load having triangular load variation. The integral ac-


tion of the plate and the stiffeners is included in the modified differential

equation through Dirac delta function and the Heaviside lambda function.

Being an analytical method, the computational involvement is less.

Chan et al. [31] have proposed an exact solution procedure using the

U-transformation method for the static analysis of stiffened plates. In

this approach, only the rectangular plates with stiffeners concentrically

and periodically placed can be analyzed. Based on the energy principle,

Kukreti and Cheraghi [81] have proposed a method for the analysis of

a stiffened plate system consisting of a plate supported on a network of

steel girders. The deflection function is considered as a product of a

polynomial and a trigonometric series. The method is applied to stiffened

plates of rectangular configurations for various loading conditions and

results are compared with those obtained by the finite element method.

A semianalytical method has been proposed by Mukhopadhyay [133]

for bending analysis of stiffened plates. In this method a displacement

function satisfying the boundary conditions along two opposite edges is

assumed. This displacement function is then substituted in the differ-

ential equation of the plate which in turn is reduced to an ordinary dif-

ferential equation having constant coefficients by some transformations.

He has presented results for rectangular stiffened plates having a vary-

ing number of location of stiffeners and possessing different boundary

conditions and loadings.

2.4.2 Free Vibration Analysis of Stiffened Plates

Aksu [2] has used variational principle in conjunction with the finite dif-

ference method for the analysis of free vibration of stiffened plates where

the strain energy expressions for the plate and the stiffeners have been

developed which is reduced to an eigenvalue problem through the use of


energy principles. He has analyzed the cross-stiffened plates neglecting

the inplane inertia and inplane displacement. He has further extended his

work [3] considering the inplane inertia and inplane displacement in both

directions and has studied the effects of these inclusion for unidirection-

ally and cross-stiffened plates.

Ramakrishnan and Kunukkasseril [152] have presented an analyti-

cal method in which the deck is considered as a combination of annular

sector plates and circular ring segments. They have formulated the fre-

quency equations by matching the continuity condition at the junction of

the plate and the ring segments after obtaining the close-form solution for

both of them and have compared the results with the experimental ones.

Shastry and Venkateswara Rao [170] have studied the free vibration

of plates with arbitrarily oriented stiffeners using the triangular plate

bending element of Cowper et al. [42] in conjunction with a stiffener

element developed by Shastry et al. [171]. The formulation incorporates

the arbitrary orientation of the stiffeners. They have presented results for

rectangular stiffened plates.

Bapu Rao et al. [15] have reported their work on experimentally deter-

mined frequencies with real-time holographic interferometric techniques

for skew stiffened cantilever plates. These experimental results have been

verified by the analytical results obtained using three-noded plate element

with three-degrees of freedom per node along with the compatible beam

element. They have neglected the inplane displacements and inertia in

their theoretical formulation.

Bhandari et al. [21] have computed the natural frequencies of rectan-

gular and skew stiffened plates using the energy method and Lagrange’s

equation. The effect of torsion of the stiffener has been ignored in the

analysis. As they have used an oblique coordinate system, the applica-

tion is limited to the skew plates.


Mizusawa et al. [115] have studied the effect of the arrangement of

stiffening beams, skew angles and stiffness parameters on the vibration

characteristics of the skew stiffened plates by using the Rayleigh-Ritz

method withB-spline functions as the coordinate functions. To accom-

modate the skew plate shape they have used the skew coordinate system.

Eishakoff et al. [50] have analyzed a finite row of skin stringer panels

using modified Bolotin method for dynamic behaviour. All panels and

interior stringers have been assumed to be identical. The natural frequen-

cies for a five bay all edges clamped panel have been presented. This

method demands that the stiffeners should be equally spaced and they

should be of equal size.

Bhat [22] has studied the effect of the spacing of stiffeners on the nat-

ural frequencies of the plate using the Rayleigh-Ritz method and equa-

tions of optimization technique. He has presented only a square stiffened

plate results in his analysis.

Irie et al. [69] have analyzed free vibration of trapezoidal cantilever

stiffened plate by using a continuous coordinate transformation of an ar-

bitrarily shaped plate to a square one of unit length. They have used

deflection function of a cantilever and free beam in conjunction with the

approximate mode shape to model the stiffened plate. The natural fre-

quencies of the stiffened and the unstiffened cantilever trapezoidal plates

have been presented by them using the Ritz method which develops the

assumed deflected shapes satisfactorily with different functions. How-

ever, the requirement of choosing the deflected shape limits the general-

ization of this method.

Balendra and Shanmugam [12] have applied the grillage method to

rib stiffened plates of cellular construction. A matrix approach has been

adopted to solve for the free vibration problems. Comparison of theo-

retical results with experimental ones has been presented [13]. However,


the grillage approximation approach has not become very popular for dy-

namic analysis of stiffened plates.

Srinivasan and Thiruvenkatachari [180] have applied the concept of

spreading the properties of the stiffeners over the area of the plate and

have solved the problem of curved eccentrically stiffened plates with tor-

sionally soft stiffener using the integral equation technique. This method

has been applied to the all edges clamped annular sector plates with ec-

centric stiffeners for static and vibration analysis.

Plates reinforced with regular orthogonal array of uniform beams

have been analyzed by Mead et al. [107] using a method developed for

the study of wave propagation in two dimensional periodic structures. A

motion of plane wave type characterized by different propagation phase

constants in both the directions is considered. The governing equations of

free wave motion are set up using the hierarchical finite element method

and they are solved as an eigenvalue problem for the frequencies at which

particular waves will propagate. Though a large structure can be analyzed

by this method with minimal effort, but the application is limited to the

periodic structures only.

Mukhopadhyay [131] has applied the semi-analytic finite difference

method to the vibration and stability analysis of rectangular stiffened

plates based on the plate beam idealization. The displacement func-

tions satisfying the boundary conditions along the two opposite edges

are substituted in the governing equations and they are reduced to or-

dinary differential equations by suitable transformations. Though the

method has attractive features from the economic point of view, it has

all the drawbacks inherent in the semi-analytic finite difference method.

In this method, a separate formulation is needed for each different struc-

tural configuration. It is difficult to handle complex boundary conditions,

concentrated load application and similar other situations.


Using the Rayleigh-Ritz method Michimoto and Zubaydi [110] have

analyzed the free vibration of trapezoidal stiffened plates applying a tech-

nique of mapping the plate into a rectangular domain and evaluating the

plate skin energy on the basis of the mapped domain imparting the nec-

essary transformations. The placement of the stiffeners are parallel and

perpendicular to the two parallel sides of the trapezoidal plate which are

treated with respect to their actual configurations. Though this approach

has extended the application of the Rayleigh-Ritz method from the con-

ventional rectangular to trapezoidal plate shapes, the stiffener placement

restrictions along the orthogonal directions still persist. Some of the theo-

retical results have been verified with the experimental ones and a method

to determine the dimensions of a rectangular stiffened plate whose natu-

ral frequencies are equal to the trapezoidal one has been proposed.

Seinosuke and Aritomi [168] have studied the free vibration of stiff-

ened plates with a small initial curvature. They have used the Galerkin

method combined with the multiple mode approximation and verified the

natural frequency and mode shape results with those obtained experimen-

tally using laser holography. The formulation is applicable to rectangular

plate configurations and the stiffeners must lie parallel to either of the

edges.

Mizusawa [112] has used the spline finite element method to study

the free vibration analysis of stiffened annular sector plates having ar-

bitrary boundary conditions. The structure is idealized as a system of

annular sector plate and curved beams rigidly connected to each other.

The formulation is based on the lateral displacement only and the inplane

displacements due to the eccentricity of the stiffeners are not considered.

The bending and torsional effects are incorporated in the stiffener formu-

lation. The formulation is based on the polar coordinate system.

Koko and Olson [80] have applied the plate beam idealization tech-


nique to the finite element method for analyzing the free vibration of

stiffened plates using a super element which consists of a macro element

having analytical as well as the usual finite element shape functions. The

lateral and inplane effects as well as the beam torsion and its horizontal

bending have been incorporated in the formulation. The method limits

its application to the rectangular plate configurations and the stiffeners

placement demands that they should be placed on the nodal lines.

Palani et al [143] have studied the performance of an eight-noded and

an nine-noded isoparametric finite element models for static and vibra-

tion analysis of eccentrically stiffened plates/shells. They have derived

the models by combining serendipity or Lagrangian plate/shell elements

with the three-noded isoparametric beam element employing suitable

transformations for the eccentricity of the stiffeners. Numerical studies

have been made for the concentrically and eccentrically stiffened plates

using four mass lumping schemes. Though they have used the isopara-

metric elements which is capable to model the arbitrary shape of a plate,

they have not addressed any such plate configurations other than the rect-

angular ones. Moreover, their formulation demands the placement of the

stiffeners along the element boundaries of the plates/shells. However, in

another publication [144] they have extended their formulation to accom-

modate the arbitrary placement of the stiffeners and the application areas

to the skew and annular stiffened plates. They have considered the nine-

noded element to be superior to the eight-noded one which locks in shear

for thin plates.

Harik and Guo [60] have developed a compound finite element model

to investigate the eccentrically stiffened plates in free vibration where

they have treated the beam and the plate elements as integral parts of

a compound section, and not as independent bending components. In

their formulation, the neutral surface may not coincide in the orthogo-


nally stiffened directions of the compound section. They have presented

results for orthogonally stiffened rectangular plates.

Sheikh and Mukhopadhyay [173] applied the spline finite strip method

to the free vibration analysis of stiffened plates of arbitrary shapes. In

their formulation, the stiffeners can be placed anywhere within the plate

strip, and need not be placed along the nodal lines. They analyzed the

plates of rectangular, skew and annular shapes with concentric as well as

eccentric stiffeners.

Chen et al. [33] have presented a spline compound strip method for

the free vibration analysis of stiffened plates in which the plate has been

discretized and modelled as strip elements. The displacement function

of the strip element has been expressed as the product of the conven-

tional transverse shape functions and longitudinal cubicB-splines. The

flexural, torsional and axial effects of the stiffeners in the formulation

have been incorporated. The analysts have presented vibration results for

one-directional and cross-stiffened rectangular plates.

Holopainen [64] has proposed a finite element model for free vibra-

tion analysis of eccentrically stiffened plates. The formulation allows the

placement of any number of arbitrarily oriented stiffeners within a plate

element. He has modelled the behaviour of the plating by employing a

plate bending element consistent with the Reissner-Mindlin thick plate

theory and the stiffener element formulation is made consistent with the

plate element. To avoid spurious shear locking and to guarantee good

convergence behaviour, the plate and the stiffener elements are based on

mixed interpolation of tensorial components. He has applied the method

to analyze the rectangular and orthogonally stiffened plates.

Lee and Ng [85] have studied the free vibrations of rectangular stiff-

ened plates using the Rayleigh-Ritz method. In their formulation they

have incorporated the effects of torsional restraint in addition to the bend-


ing restraint of the stiffeners. They have considered the characteristic

beam functions as the shape functions and have studied the effects of the

location and orientation of the stiffener and its relative stiffness to the

plate.

Bedair [20] has studied the free vibration characteristics of stiffened

plates due to plate/stiffener proportions. He has considered the plate

and the stiffener as the discrete elements rigidly connected at their junc-

tions and the nonlinear strain energy function of the assembled struc-

ture has been transformed into an unconstrained optimization problem

to which Sequential Quadratic Programming has been applied to deter-

mine the magnitudes of the lowest natural frequency and the associated

mode shape. The formulation is restricted to identical and equally spaced

stiffeners thereby loosing the generality of accommodating the stiffeners

arbitrarily. Moreover, the method can predict only the lowest frequency

and its mode shape.

2.4.3 Stability Analysis of Stiffened Plates

Bryan [28] was the first investigator to deal with the stability problems

of plates stiffened with equispaced longitudinal stiffeners where he made

important suggestions about the placement of the stiffeners in order to

achieve maximum strength. Timoshenko [183] has investigated the prob-

lem of minimum stiffness of the stiffeners to prevent overall buckling.

Cox and Riddel [43] have extended the concept of Timoshenko by

including the torsional effects of the stiffener using a strain energy for-

mulation. They have also studied a multiple stiffener case.

Seide [167] has introduced the effect of eccentrically positioned stiff-

eners in his formulation through the effective moment of inertia of the

stiffeners. The effective moment of inertia has been varied with the pro-


portion of the plate area and with the stiffener area. Klitchieff [79] has

presented an expression for the minimum dimensions of the stiffeners to

withstand a predetermined critical load.

Sherbourne et al. [174] have used the orthotropic plate approach for

the behaviour of the simply supported stiffened and corrugated plates

under uniform axial compression. This method has all the drawbacks of

the orthotropic plate modelling.

Dean and Abdel-Malek [44] have presented a discrete field approach

to compute the elastic buckling of stiffened plates subjected to uniform

longitudinal compression. As they have used the orthotropic stiffened

plate modelling, the formulation is restricted to the equally spaced and

equally sized stiffeners. Also it is confined to the simply supported plates

of rectangular geometries.

Allman [6] has carried out the analysis for buckling loads of square

and rectangular plates using triangular element. He has presented the re-

sults both by including and neglecting the torsional stiffness of the stiff-

eners and has considered three cross-sections of the stiffeners such as

square, rectangular and circular.

Shastry et al. [171] have solved the problem of buckling analysis of

stiffened plates with arbitrarily oriented stiffeners using finite element

method. In their modelling, they have used triangular plate bending ele-

ment and compatible beam element and applied the method to the buck-

ling analysis of square and rectangular stiffened plates.

Hovichitr et al. [66] have presented an analytical approach to analyze

orthogonally, equally spaced, simply supported stiffened plates for bend-

ing and stability. They have treated the stiffener sections, which were

assumed to be identical, and a portion of the plate as single unit. Using a

variational method, they have generated governing differential equations

of order of ten which in turn were reduced to eight and then to four using


simplifying assumptions. Using Fourier series approximations, numeri-

cal solutions were obtained for simply supported panels and comparisons

were made between the tenth, eighth and fourth order solutions. This

method has many limitations such as stiffeners should be identical and

equally spaced and the edges are to be simply supported.

Mizusawa et al. [118] have applied the Rayleigh-Ritz method withB-

spline functions as the coordinate functions for analyzing skew stiffened

plates. TheB-spline functions used are continuous at nodal points for

higher order derivatives. They have studied the effect of various stiffness

parameters of the stiffener on the buckling load. The method has been

applied to the rectangular and the skew stiffened plate buckling.

Brown and Yettram [27] have proposed a conjugate load/displacement

method of analysis for the determination of the elastic buckling loads of

stiffened plates under various loading and support conditions. They have

highlighted the significance of the torsional rigidity of the stiffeners on

the overall behaviour of the complete structure. The method demands the

placement of the stiffeners to be oriented parallel to the x or y-coordinate

axes. They have analyzed buckling loads for rectangular stiffened plates.

Peng-Cheng et al. [145] have presented semianalytical approach us-

ing Rayleigh-Ritz method withB-spline functions as coordinate func-

tions to analyze static, vibration and stability behaviour of stiffened plates.

They have followed an alternative semianalytical approach and a compu-

tational scheme suitable for various types of boundary conditions. The

displacement components of the stiffened plate are defined in the form

of B-3 spline functions and the ribs are arranged parallel to x and y-

directions. They have analyzed rectangular plates with orthogonal orien-

tation of the stiffeners.

Mukhopadhyay and Mukherjee [135] have used an isoparametric stiff-

ened plate bending element for the buckling analysis of stiffened plates


in which the stiffener can be positioned anywhere within the plate ele-

ment and need not necessarily be placed on the nodal lines. They have

presented buckling results for square and skew stiffened plates and have

studied the effect of stiffener rigidity, torsional stiffness and eccentricity

of the stiffener on the buckling load. Though the element can readily ac-

commodate curved boundaries they have considered only the rectangular

and skew plates in their analysis.

Recently, an extensive review on the stability of stiffened plates has

been carried out by Bedair [19]. He has also presented a numerical

method for the prediction of the buckling load of multi-stiffened plates

under uniform compression following the philosophy of plate beam ide-

alization. He has employed the sequential quadratic programming to the

strain energy components of the plate and the stiffener elements which

are in terms of the out-of-plane and in-plane displacement functions. A

number of examples pertaining to the straight-edged orthogonally stiff-

ened plates buckling are presented. However, this method lacks in ana-

lyzing the curved boundary stiffened plates buckling.

Chapter -3

MATHEMATICALFORMULATION

3.1 The Basic Problems

This chapter presents the mathematical formulation for static, free vi-

bration and stability analyses of the bare and stiffened plates of various

shapes. The analysis techniques are applied to the plates and the stiffen-

ers. Since the displacement functions chosen for the formulation of the

bare and the stiffened plates are different, they are presented in separate

sections.

The equations of equilibrium for an elastic system undergoing small

displacements in matrix form are:

1. Static Analysis

[K]{δ} = {P} (3.1.1)

2. Free Vibration Analysis

[K]{δ}+ [M ]{δ} = {0} (3.1.2)

3. Stability Analysis

[K]{δ} − λ [KG] {δ} = {0} (3.1.3)

47

48 MATHEMATICAL FORMULATION

where[K], [M ] and[KG] are the global elastic stiffness, consistent mass

and geometric stiffness matrices respectively,{δ} and{δ} are the dis-

placement and acceleration vectors in the global coordinate system and

{P} is the load vector acting at the nodes. The global matrices used

in the Eqs.(3.1.1)-(3.1.3) are obtained by assembling the corresponding

element matrices which are derived in the forthcoming sections of this

Chapter.

3.2 Proposed Analysis

In the proposed method of analysis two types of basic structures such as

bare plates and stiffened plates are considered. The bare plate consists of

only a flat plate skin of arbitrary shape whereas the stiffened plate com-

prises the stiffening ribs in addition to the arbitrarily shaped flat plate

skin. In the present formulation the plates and the stiffeners are modelled

as discrete elements and the compatibility between them is maintained

by expressing the element stiffness matrix of the stiffener in terms of

the nodal degrees of freedom of the plate element in which the stiffener

is free to assume an arbitrary orientation, disposition and location. The

middle plane of the plate is taken as the reference plane. The formulation

is done for the plates having both the concentrically and eccentrically

placed stiffeners. Under the action of lateral loads the eccentrically stiff-

ened plates are having in-plane deformations in addition to the lateral

ones because of the eccentric position of the stiffeners. Hence the formu-

lation takes into account both the lateral and the in-plane displacements.

The boundary conditions are incorporated in the most general manner to

cater to the need of the curved boundary as well as to the more practical

mixed boundary conditions.

3.2 Proposed Analysis 49

3.2.1 The Basic Assumptions

The formulation is based on the following assumptions:

1. The normal to the middle plane of the plate before bending remains

straight and normal to the middle plane of the plate after bending.

2. The common normal to the plate and the stiffener system before

bending remains straight and normal to the deflected middle plane

of the plate after bending.

3. The horizontal bending of the stiffener is not taken into account.

4. The stress in the z-direction is small compared to the other stress

components and is thus neglected.

5. The material of the plate and the stiffener is same and follows

Hooke’s law.

6. The lateral deflection is small compared to the thickness of the

plate.

3.2.2 The Transformation of the Coordinate

The arbitrary shape of the wholeplate is mapped into aMaster Plateof

square region [-1,+1] in thes-t plane with the help of the relationship

given by (Zienkiewicz and Taylor [198]):

x =12∑i=1

Ni(s, t) xi y =12∑i=1

Ni(s, t) yi (3.2.1)

where(xi, yi) are the coordinates of the i-th node on the boundary of the

plate in thex-y plane andNi(s, t) are the corresponding cubic serendipity


Table 3.1:Cubic Serendipity Shape Function

Node Ni(ξ, η) Value ofξi andηi

Corner 132

(1 + ξiξ)(1 + ηiη) ξi = ±1

(1,4,7,10) [9(ξ2 + η2)− 10] ηi = ±1

Mid-side 932

(1 + ξiξ)(1− η2) ξi = ±1

(2,3,8,9) (1 + 9ηiη) ηi = ±13

Mid-side 932

(1 + ηiη)(1− ξ2) ξi = ±13

(5,6,11,12) (1 + 9ξiξ) ηi = ±1

x

y

1

2

3

4 5 6 7

8

9

10

1112

6 754

3

2

t

8

9

1011121

s

(-1/3,1) (1/3,-1)

(1,-1/3)

(1,1/3)

(1,-1)

(1,1)(-1,1)

(-1,-1)

(b) Master plate(a) Original plate

Figure 3.1:Mapping of the arbitrarily shaped plate

shape functions presented in the Table 3.1. The mapping of the original

plate to the Master plate is as shown in the Fig. 3.1. In a mapping based

on the serendipity shape function the interior opening at a corner node

should not be greater than180◦ (Zienkiewicz and Taylor [198]). This

3.2 Proposed Analysis 51

s

t

ξ

η

1

2

3

45 6

7

8

9

101112(−1,1) (1,1)

(−1,−1) (1,−1)

(b) Master elementin ξ−η plane

(a) Element in a 4x4 meshin s-t plane

Figure 3.2:Mapping of the element

angle is maximum in the case of the circular plate problems considered

in this investigation and is just equal to180◦. TheMaster Platein the s-t

plane, which is a square one instead of an arbitrary one, is divided into a

number of rectangular elements. For each rectangular element in thes-t

plane, twelve number of suitable nodes on its periphery are chosen and

their (x, y) coordinates are determined by using the Eq.(3.2.1) which is

based on the mapping of the whole arbitrary plate to theMaster Plateof

the Fig. 3.1. Thus the process of division of theMaster Platein the s-

t plane into the rectangular elements and the determination of the(x, y)

coordinates of the twelve nodes on the boundary of the elements becomes

simple because of its square geometry which rather would have been

more complex and tedious for the arbitrary geometry in thex-y plane.

Now the(x, y) coordinates of the twelve nodes on the boundary of each

rectangular element being known, eachelementis mapped to aMaster

Elementof square region [-1,+1] in theξ-η plane as shown in the Fig. 3.2


in a similar way as the original plate is mapped into theMaster Platein

the s-t plane using the same cubic serendipity shape functions given in

Eq.(3.2.1), but now the variables being changed from(s, t) to (ξ, η).

3.3 Arbitrary Bare Plate Bending Formu-lation

This section consists of the formulation of the elastic stiffness matrix, the

mass matrix and the geometric stiffness matrix of the bare plate element

which are[Ku]e, [Mu]e and[KuG]e respectively and assembling them into

the respective global matrices[Ku], [Mu] and [KuG]. Additionally, the

stiffness matrix[Ku]b for the general curved boundary line is consistently

formulated (straight and skew ones being the special cases of them) and

added to the respective elastic stiffness matrix of the element. A single

general element is used for the solution of the static, free vibration and

stability problems of plates of arbitrary configurations.

3.3.1 The Displacement Function

Each square element in theξ-η plane is considered for the generation

of the element matrices. For the proposed element, the four-noded rect-

angular non-conformingACM plate bending element with 12 degrees of

freedom (Adini and Clough [1]) is taken as the basic element. As the ele-

ment is in theξ-η plane, the shape functions and the nodal parameters for

the displacements and slopes are expressed in terms of the coordinatesξ

andη unlike thex andy coordinates of the parentACM element. Thus

the displacement field can be written as

w = [Nw] {δu} (3.3.1)

3.3 Arbitrary Bare Plate Bending Formulation 53

where

[Nw] = [N1w N1

θξN1

θηN2

w N2θξ

N2θη

N3w N3

θξN3

θηN4

w N4θξ

N4θη

]

(3.3.2)

{δu} =

[w1

(∂w

∂ξ

)

1

(∂w

∂η

)

1

. . . . . . w4

(∂w

∂ξ

)

4

(∂w

∂η

)

4

]T

(3.3.3)

The shape functions for the displacement field for thejth node are given

as (Zienkiewicz and Taylor [198]):

[N jw, N j

θξ, N j

θη] =

1

8[(ξ0 + 1)(η0 + 1)(2 + ξ0 + η0 − ξ2 − η2),

ξj(ξ0 + 1)2(ξ0 − 1)(η0 + 1), ηj(ξ0 + 1)(η0 + 1)2(η0 − 1)](3.3.4)

ξ0 = ξξj η0 = ηηj (3.3.5)

3.3.2 Elastic Stiffness Matrix Formulation

3.3.2.1 Stress-Strain Relationship

The generalized stress-strain relation in matrix form is given by

{σu} = [Du] {εu} (3.3.6)

where{σu} is the stress resultant vector given by

{σu} = [Mx My Mxy]T (3.3.7)

and[Du] is the rigidity matrix given by

[Du] =

DX D1 0

D1 DY 0

0 0 DXY

(3.3.8)


When isotropic material is considered;

DX = DY =Et3

12(1− ν2)

D1 = νDX

DXY =1− ν

2DX

(3.3.9)

The generalized strains are given by

{εu} =

[−∂2w

∂x2− ∂2w

∂y22

∂2w

∂x ∂y

]T

(3.3.10)

3.3.2.2 Strain-Displacement Relationship

The displacement functions of the plate element is expressed in terms of

the localξ-η coordinate system whereas the strains are in terms of the

derivatives of the displacements with respect to thex andy coordinates.

Hence before establishing the relationship between the strain and the dis-

placement the first and second order derivatives of the displacementw

with respect to thex-y coordinates are expressed in terms of those of the

ξ-η coordinates using the chain rule of differentiation and are obtained as

below:

∂w

∂x

∂w

∂y

= [J ]−1

∂w

∂ξ

∂w

∂η

(3.3.11)


−∂2w

∂x2

−∂2w

∂y2

2∂2w

∂x∂y

= [J2]−1

∂2w

∂ξ2

∂2w

∂η2

∂2w

∂ξ∂η

− [J2]−1[J1][J ]−1

∂w

∂ξ

∂w

∂η

(3.3.12)

where

[J ] =

∂x

∂ξ

∂y

∂ξ

∂x

∂η

∂y

∂η

(3.3.13)

[J1] =

∂2x

∂ξ2

∂2y

∂ξ2

∂2x

∂η2

∂2y

∂η2

∂2x

∂ξ∂η

∂2y

∂ξ∂η

(3.3.14)

[J2] =

−(

∂x

∂ξ

)2

−(

∂y

∂ξ

)2∂x

∂ξ

∂y

∂ξ

−(

∂x

∂η

)2

−(

∂y

∂η

)2∂x

∂η

∂y

∂η

−(

∂x

∂ξ

∂x

∂η

)−(

∂y

∂ξ

∂y

∂η

)1

2

(∂x

∂ξ

∂y

∂η+

∂x

∂η

∂y

∂ξ

)

(3.3.15)

From the above equations the strain vector of Eq.(3.3.10) can be ex-


pressed as

−∂2w

∂x2

−∂2w

∂y2

2∂2w

∂x∂y

=[

[TF1] [TF2]

]

∂w

∂ξ

∂w

∂η

∂2w

∂ξ2

∂2w

∂η2

∂2w

∂ξ ∂η

(3.3.16)

or

{ε(x, y)u} = [Tu] {ε(ξ, η)u} (3.3.17)

where

[TF1] = −[J2]−1[J1][J ]−1 [TF2] = [J2]−1 (3.3.18)

and{ε(x, y)u} and{ε(ξ, η)u} denote the strain vectors in the respective

coordinate systems, the expression for{ε(ξ, η)u} being given by;

{ε(ξ, η)u} =

[∂w

∂ξ

∂w

∂η

∂2w

∂ξ2

∂2w

∂η2

∂2w

∂ξ∂η

]T

(3.3.19)

Using Eqs. (3.3.1) and (3.3.2), the Eq.(3.3.19) can be rewritten as

{ε(ξ, η)u} =[Bu

] {δu} (3.3.20)

where

[Bu

]=

[∂Nw

∂ξ

∂Nw

∂η

∂2Nw

∂ξ2

∂2Nw

∂η2

∂2Nw

∂ξ ∂η

]T

(3.3.21)


Hence the combination of Eqs.(3.3.17) and (3.3.20), yields

{ε(x, y)u} = [Bu]{δu} (3.3.22)

where

[Bu] = [Tu][Bu] (3.3.23)

The stress-strain relationship from Eq.(3.3.6) can be expressed with

the help of the Eq.(3.3.22) as

{σu} = [Du][Bu]{δu} (3.3.24)

3.3.2.3 Stiffness Matrix of the Bare Plate Bending Element

Total potential energy of the plate element is given by

Πp =1

2

∫ ∫ ({ε(x, y)}T {σ(x, y)}

)dx dy −

∫ ∫wT q dx dy (3.3.25)

Applying the principle of minimum potential energy and making appro-

priate substitutions for{ε(x, y)} and{σ(x, y)}, Eq.(3.3.25) reduces to

[Ku]e{δu} = {P}e (3.3.26)

where{δu} is the vector of nodal displacements and{P}e is the vector

of nodal forces and[Ku]e is the plate element stiffness matrix given by

[Ku]e =

∫ ∫[Bu]

T [Du][Bu] dx dy (3.3.27)

Since the[Bu] matrix is a function ofξ andη, the Eq.(3.3.27) can be

rewritten as

[Ku]e =

∫ ∫[Bu]

T [Du][Bu] |J | dξ dη (3.3.28)

where|J | is the determinant of the Jacobian matrix [J] given by Eq.(3.3.13).

The integration of the Eq.(3.3.28) is carried out numerically by adopting

2× 2 Gaussian quadrature formula.


3.3.3 Consistent Mass Matrix of the Bare PlateElement

A consistent mass matrix of the plate element is formulated on the basis

of the lateral displacementw. The acceleration of a point in the middle

plane of the plate in terms of the interpolation function given in Eq.(3.3.1)

can be expressed as

¨{f} = ¨{w} = [Nw] ¨{δu} (3.3.29)

Hence the inertia force of a small element of volumedV at that point is

given by

{fI} = ρ dV ¨{w} = ρ dV [Nw] ¨{δu} (3.3.30)

whereρ is the mass density of the plate material.

If {FI} is the nodal inertia force parameter, then the contribution of

the inertia in the equation of motion can be obtained from the principle

of virtual work and can be expressed as

{dδT

} {FI} =

∫

v

{dfT

} {fI} (3.3.31)

The above equation with the help of the Eq.(3.3.30) can be rewritten as

{dδT

} {FI} =

∫

v

{dδT

}[Nw]T ρ dV [Nw] ¨{δu} (3.3.32)

from which

{FI} = ρ

∫

v

[Nw]T [Nw] dV ¨{δu} = [Mu]e ¨{δu} (3.3.33)

where[Mu]e is the mass matrix of the bare plate element and for constant

thicknesst it is given by

[Mu]e = ρ

∫

v

[Nw]T [Nw] dV = ρt

∫ ∫[Nw]T [Nw] |J | dξ dη (3.3.34)


3.3.4 Geometric Stiffness Matrix of the Bare PlateElement

To formulate the geometric stiffness matrix, the action of the in-plane

loads causing bending strains is considered. The membrane strains asso-

ciated with the small rotations∂w

∂xand

∂w

∂yof the plate mid-surface are

given by

{εuG} =

εx

εy

γxy

=

1

2

(∂w

∂x

)2

1

2

(∂w

∂y

)2

(∂w

∂x

)(∂w

∂y

)

(3.3.35)

If the stressesσx, σy andτxy are assumed to remain constant during

the occurrence of the strains{εuG}, the associated work is given by the

equation

W =

∫ ∫ ∫{εuG}T {σ} dx dy dz (3.3.36)

where

{σ} = [σx σy τxy]T (3.3.37)

Substituting the value of{εuG} from Eq.(3.3.35) in Eq.(3.3.36), yields

W =

∫ ∫ ∫ [1

2

(∂w

∂x

)21

2

(∂w

∂y

)2 (∂w

∂x

)(∂w

∂y

)]{σ} dx dy dz

=

∫ ∫ ∫1

2{θu}T [σu]{θu} dx dy dz

(3.3.38)

where

{θu} =

[∂w

∂x

∂w

∂y

]T

(3.3.39)


and

[σu] =

σx τxy

τxy σy

(3.3.40)

The Eq.(3.3.39) can be expressed in terms ofξ andη and can be rewritten

as

{θu} =

∂w

∂x

∂w

∂y

= [TuG]

∂w

∂ξ

∂w

∂η

(3.3.41)

where

[TuG] = [J ]−1 (3.3.42)

and

∂w

∂ξ

∂w

∂η

=[BuG

] {δu} (3.3.43)

where[BuG

]=

[[∂Nw

∂ξ

] [∂Nw

∂η

]]T

(3.3.44)

Hence combining Eq.(3.3.41) and Eq.(3.3.43){θu} can be expressed

as

{θu} = [TuG][BuG

] {δu} = [BuG] {δu} (3.3.45)

where

[BuG] = [TuG][BuG

](3.3.46)

Substituting the value of{θu} from Eq.(3.3.45) the Eq.(3.3.38) be-

comes

W =

∫ ∫ ∫1

2{δu}T [BuG]T [σu] [BuG] {δu} dx dy dz

=t

2

∫ ∫{δu}T [BuG]T [σu] [BuG] {δu} dx dy

(3.3.47)


The external work done by the nodal forces is given by

W =1

2{δu}T [KuG] {δu} (3.3.48)

From Eqs.(3.3.47) and (3.3.48) the geometric stiffness matrix of the bare

plate element can be written as

[KuG]e = t

∫ ∫[BuG]T [σu] [BuG] dx dy = t

∫ ∫[BuG]T [σu] [BuG] |J | dξ dη

(3.3.49)

where the subscriptedenotes that the matrix is for theplate element.

3.3.5 Boundary Conditions for the Bare Plate

As a general case the stiffness matrix for a curved boundary supported on

elastic springs continuously spread in the directions of possible displace-

ments and rotations along the boundary line is formulated from which

specific boundary conditions can be obtained by incorporating the ap-

propriate value of the spring constants. Considering a local axis system

x1-y1 at a point P on a curved boundary along the direction of the normal

to the boundary at that point as shown in the Fig. 3.3 the displacement

components along it can be found.

Let the angle made by the local axisx1-y1 with the global axisx-y beβ.

Hence a relationship between the two axes can be established as given

below.

x

y

=

cosβ −sinβ

sinβ cosβ

x1

y1

(3.3.50)


x

y

Px 1

y 1

β

Figure 3.3: Coordinate axes at a typical point of a curvedboundary

The displacements atP which may be restrained can be expressed as

{fbu} =

w

θn

θt

=

w

∂w

∂x1

∂w

∂y1

(3.3.51)

whereθn andθt represent the slopes which are normal and transverse to

the boundaries respectively. Substituting from Eqs.(3.3.50), the Eq.(3.3.51)

can be written as


{fbu} =

1 0 0

0 cosβ sinβ

0 −sinβ cosβ

w

∂w

∂x

∂w

∂y

(3.3.52)

Expressing Eq.(3.3.52) in terms of the shape functions;

{fbu} = [Nbu] {δu} (3.3.53)

where

[Nbu] =

1 0 0

0 cosβ sinβ

0 −sinβ cosβ

[Nw]

∂[Nw]

∂x

∂[Nw]

∂y

(3.3.54)

The reaction components per unit length along the boundary line due

to the elastic springs corresponding to the possible boundary displace-

ments given in the Eq.(3.3.51) can be expressed as

{fku} =

fkw

fkα

fkβ

=

kww

kαθn

kβθt

(3.3.55)


wherekw, kα and kβ are the spring constants or restraint coefficients

corresponding to the direction ofw, θn andθt respectively.

The Eq.(3.3.55) can be rewritten by combining the Eqs.(3.3.51), (3.3.52)

and (3.3.53) as

{fku} = [Nku]{δu} (3.3.56)

where

[Nku] =

kw 0 0

0 kα cosβ kα sinβ

0 −kβ sinβ kβ cosβ

[Nw]

∂[Nw]

∂x

∂[Nw]

∂y

(3.3.57)

Using Equations (3.3.52) and (3.3.55) the stiffness matrix can be ob-

tained by the virtual work principle and it can be expressed as

[Kbu] =

∫[Nbu]

T [Nku] |Jb| dλ1 (3.3.58)

whereλ1 is the direction of the boundary line in theξ-η plane and the

Jacobian|Jb| = ds1

dλ1

.

The value of the Jacobian along a boundary line is considered as a

constant quantity and is evaluated by the ratio of the actual length to the

length on the mapped domain considering any segment of the boundary

line.

A classical boundary condition can be attained by substituting a high

value of the restraint coefficients corresponding to the restraint direction.

3.4 Arbitrary Stiffened Plate Element Formulation 65

3.3.6 Stresses at the Nodes of the Bare Plate

Once the element nodal degrees of freedom{δu} are known the bend-

ing moments[Mx My Mxy] at the nodes are calculated by using the

Eqs.(3.3.6), (3.3.7) and (3.3.24) which becomes

{σu} = [Mx My Mxy]T = [Du]{εu} = [Du][Bu]{δu} (3.3.59)

3.4 Plate Element Formulation for Ec-centrically Stiffened Arbitrary Plate

The eccentrically stiffened plates consist of the stiffeners whose positions

are not symmetric with respect to the reference plane (plate mid-plane).

In such a situation there exists a coupling between the axial and the flex-

ural effects. Hence as a general case, a curved stiffener with eccentricity

with respect to the plate mid-plane is considered for the formulation. The

matrices for the concentric stiffeners can be obtained from those of the

eccentric ones by excluding the axial effects in the formulation. For solv-

ing the static, free vibration and stability problems of arbitrarily shaped

plates stiffened with the arbitrarily oriented stiffeners the elastic stiffness,

mass and geometric stiffness matrices of the plate element[Kp]e, [Mp]eand[KpG]e respectively and those of stiffener element[KS]e, [MS]e and

[KSG]e respectively are derived and they are assembled into the respec-

tive global matrices. The stiffness matrices[Kp]b for the general curved

boundaries are also derived following the similar procedure as in the case

of the boundaries of bare plate formulation.


3.4.1 The Displacement Function

For the proposed stiffened plate bending element, the bending deforma-

tion has been represented combining the four-noded rectangular non-

conforming ACM plate bending element with 12 degrees of freedom

(Adini and Clough [1]), already used in the formulation of the bare plate,

and the four-noded rectangular plane stress element with 8 degrees of

freedom for the in-plane deformations. As before, the element is in the

ξ-η plane, and the shape functions as well as the nodal parameters for the

displacements and slopes are expressed in terms of the coordinatesξ and

η instead ofx andy coordinates of the parentACM element. Thus the

displacement field can be written as:

{f} =

u

v

w

=

[Nu]

[Nv]

[Nw]

{δp} (3.4.1)

where[Nu], [Nv] and[Nw] are the vectors of the respective shape func-

tions out of which[Nu] and[Nv] are given as:

[Nu] = [N1u 0 0 0 0 N2

u 0 0 0 0

N3u 0 0 0 0 N4

u 0 0 0 0]

[Nv] = [0 N1v 0 0 0 0 N2

v 0 0 0

0 N3v 0 0 0 0 N4

v 0 0 0]

(3.4.2)


and[Nw] is given by the Eq.(3.3.2) and the displacement vector{δp} for

the stiffened plate is expressed as:

{δp} = [u1 v1 w1

(∂w

∂ξ

)

1

(∂w

∂η

)

1

. . . . . .

u4 v4 w4

(∂w

∂ξ

)

4

(∂w

∂η

)

4

]T

(3.4.3)

The shape functions for the displacement field corresponding to a

particular node, say thejth node can be expressed as:

• for the inplane displacements:

N ju = N j

v =1

4(1 + ξ0)(1 + η0) (3.4.4)

• and for the out of plane displacements: the same expression as

given by Eq.(3.3.4).

whereξ0 andη0 have their usual meanings as before.

3.4.2 The Plate Element Formulation


Considering the middle plane of the plate as the reference plane and tak-

ing the lateral and the in-plane displacements into account, the general-

ized stress-strain relationship can be obtained following the procedure of

section (3.3.2.1), the expressions for the stress and strain vectors and the

rigidity matrix being given by:

{σp} = [Fx Fy Fxy Mx My Mxy]T (3.4.5)


[Dp] =

DXA D1A

D1A DY A 0

DXY A

DXF D1F

0 D1F DY F

DXY F

(3.4.6)

where the elements of the matrix for isotropic material are given by:

DXA = DY A =Et

1− ν2DXF = DY F =

Et3

12(1− ν2)

D1A = νDXA D1F = νDXF

DXY A =1− ν

2DXA DXY F =

1− ν

2DXF

(3.4.7)

{εp} =

∂u

∂x

∂v

∂y

∂u

∂y+

∂v

∂x

−∂2w

∂x2

−∂2w

∂y2

2∂2w

∂x∂y

(3.4.8)



Using the relationship between thex-y and theξ-η coordinate systems

as obtained in the section (3.3.2.2), the first order derivatives of the dis-

placements with respect to thex andy coordinates in the expressions of

Eq.(3.4.8) can be written in terms of theξ andη coordinates such as:

∂u

∂x

∂v

∂y

∂u

∂y+

∂v

∂x

=

∂ξ

∂x

∂η

∂x0 0

0 0∂ξ

∂y

∂η

∂y

∂ξ

∂y

∂η

∂y

∂ξ

∂x

∂η

∂x

∂u

∂ξ

∂u

∂η

∂v

∂ξ

∂v

∂η

= [TA]

∂u

∂ξ

∂u

∂η

∂v

∂ξ

∂v

∂η

(3.4.9)

and the expression for the second order derivatives is same as that given

by the Eq.(3.3.16). Hence the strain vector of Eq.(3.4.8) can be expressed

as:

∂u

∂x

∂v

∂y

∂u

∂y+

∂v

∂x

−∂2w

∂x2

−∂2w

∂y2

2∂2w

∂x∂y

=

[TA] 0 0

0 [TF1] [TF2]

{ε(ξ, η)p} (3.4.10)


or,

{ε(x, y)p} = [Tp] {ε(ξ, η)p} (3.4.11)

where

[Tp] =

[TA] 0 0

0 [TF1] [TF2]

(3.4.12)

{ε(ξ, η)p} =

[∂u

∂ξ

∂u

∂η

∂v

∂ξ

∂v

∂η

∂w

∂ξ

∂w

∂η

∂2w

∂ξ2

∂2w

∂η2

∂2w

∂ξ ∂η

]T

(3.4.13)

Following the same procedure as in the case of the bare plate, the

strain-displacement relationship can be written as:

{σp} = [Dp] [Bp] {δp} (3.4.14)

where

[Bp] = [Tp][Bp

](3.4.15)

and

[Bp

]=

[∂Nu

∂ξ

∂Nu

∂η

∂Nv

∂ξ

∂Nv

∂η

∂Nw

∂ξ

∂Nw

∂η

∂2Nw

∂ξ2

∂2Nw

∂η2

∂2Nw

∂ξ ∂η

]T

(3.4.16)

3.4.2.3 Elastic Stiffness Matrix of the Plate Element of theStiffened Plate

Applying the principle of minimum potential energy and following the

same procedure of the bare plate, the stiffness matrix of the plate skin

can be written as:

[Kp]e =

∫ ∫[Bp]

T [Dp] [Bp] |J | dξ dη (3.4.17)


where the integration of the above equation is carried out numerically by

2× 2 Gaussian quadrature formula.

3.4.2.4 Consistent Mass Matrix of the Plate Element of theStiffened Plate

A consistent mass matrix for the plate element is formulated on the basis

of lateral as well as in-plane displacements.

The displacement components of a point at a depthz from the middle

plane of the plate can be expressed in terms of those at the plate mid-

plane as

{f} =

u

v

w

=

u− z∂w

∂x

v − z∂w

∂y

w

(3.4.18)

Using the displacement interpolation function from Eq.(3.4.1), the Eq.(3.4.18)

can be written as

{f} = [G][L][N ]{δp} = [Np] {δp} (3.4.19)

where

[G] =

1 0 0 −z 0

0 1 0 0 −z

0 0 1 0 0

(3.4.20)


[L] =

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0∂ξ

∂x

∂η

∂x

0 0 0∂ξ

∂y

∂η

∂y

(3.4.21)

[N ] =

[[Nu] [Nv] [Nw]

∂[Nw]

∂ξ

∂[Nw]

∂η

]T

(3.4.22)

The mass matrix of the plate element for constant thicknesst and

constant mass densityρ as derived earlier for the bare plate is given by:

[Mp] = ρ

∫

v

[Np]T [Np] dv (3.4.23)

which can be rewritten as

[Mp] = ρt

∫ ∫[N ]T [L]T [P ][L][N ] |J | dξ dη (3.4.24)

where

[P ] =

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0t2

120

1 0 0 0t2

12

(3.4.25)


3.4.2.5 Geometric Stiffness Matrix of the Plate Element ofthe Stiffened Plate

For the analysis of the buckling behaviour, the action of the in-plane loads

causing bending strains is considered by which the stiffness matrix is

modified by another matrix[KpG

](geometric stiffness matrix) and then

the eigenvalue problem as mentioned in the Eq.(3.1.3) is solved to obtain

the buckling parameter.

The stretched length for the transverse and in-plane displacements in

an element of lengthdx as shown in the Fig. 3.4 is expressed as:

∂u

∂xdx ¾-

∂w

∂xdx

dx

dx′

6

?

Figure 3.4:Stretching of an element

dx′ =

√(dx +

∂u

∂xdx

)2

+

(∂w

∂xdx

)2

= dx

[(1 +

∂u

∂x

)2

+

(∂w

∂x

)2]1

2

= dx

[1 +

∂u

∂x+

1

2

(∂u

∂x

)2

+1

2

(∂w

∂x

)2

+ . . .

]

(3.4.26)


Neglecting the higher order terms, the expression for the axial strain

of the mid-plane of the plate in the x-direction is:

εx =∂u

∂x+

1

2

(∂u

∂x

)2

+1

2

(∂w

∂x

)2

. . . (3.4.27)

The presence of the quadratic terms in the mid-plane strain accounts

for the transverse displacement. The expression for the strain at the mid-

plane of the plate is written as:

{ε} = {εpE}+ {εpG} (3.4.28)

whereεpE andεpG are the elastic and the geometric plate strain respec-

tively, and are given by:

{εpE} =

∂u

∂x

∂v

∂y

∂u

∂y+

∂v

∂x

∂w

∂x

∂w

∂y

(3.4.29)


and

{εpG} =

1

2

(∂w

∂x

)2

+1

2

(∂u

∂x

)2

+1

2

(∂v

∂x

)2

1

2

(∂w

∂y

)2

+1

2

(∂u

∂y

)2

+1

2

(∂v

∂y

)2

(∂w

∂x

)(∂w

∂y

)+

(∂u

∂x

)(∂u

∂y

)+

(∂v

∂x

) (∂v

∂y

)

0

0

(3.4.30)

The contribution of in-plane displacements to the geometric stiffness

matrix being insignificant, it is not considered in the formulation. Hence

the displacement field for the plate element can be written as:

{f} =

u

v

w

=

−z∂w

∂x

−z∂w

∂y

w

(3.4.31)

Substituting the values ofu, v andw from Eq.(3.4.31) in Eq.(3.4.30)

yields,


{εpG} =

1

2

(∂w

∂x

)2

+z2

2

(∂2w

∂x2

)2

+z2

2

(∂2w

∂x ∂y

)2

1

2

(∂w

∂y

)2

+z2

2

(∂2w

∂x ∂y

)2

+z2

2

(∂2w

∂y2

)2

(∂w

∂x

)(∂w

∂y

)+ z2

(∂2w

∂x2

)(∂2w

∂x ∂y

)+ z2

(∂2w

∂y2

)(∂2w

∂x ∂y

)

0

0

=1

2[A]{θp}

(3.4.32)

where

[A] =

∂w

∂x0 −z

∂2w

∂x20 −z

∂2w

∂x ∂y

0∂w

∂y0 −z

∂2w

∂y2−z

∂2w

∂x ∂y

∂w

∂y

∂w

∂x−z

∂2w

∂x ∂y−z

∂2w

∂x ∂y−z

(∂2w

∂x2+

∂2w

∂y2

)

(3.4.33)

and

{θp} =

[∂w

∂x

∂w

∂y− z

∂2w

∂x2− z

∂2w

∂y2− z

∂2w

∂x ∂y

]T

(3.4.34)

Taking the variation of Eq.(3.4.32)

δ {εpG} =1

2δ [A]{θp}+

1

2[A] δ {θp} (3.4.35)


It can be shown that (Zienkiewicz and Taylor [198]):

δ [A]{θp} = [A] δ {θp} (3.4.36)

Hence

δ {εpG} = [A] δ {θp} (3.4.37)

or

{εpG} = [A] {θp} (3.4.38)

The strain vector{θp} can be rewritten as:

{θp} =

1 0 0 0 0

0 1 0 0 0

0 0 +z 0 0

0 0 0 +z 0

0 0 0 0 −1

2z

∂w

∂x

∂w

∂y

−∂2w

∂x2

−∂2w

∂y2

2∂2w

∂x ∂y

= [HpG] {εpG}

(3.4.39)

The vector{εpG} can be expressed in terms of theξ-η coordinates as:

{εpG(x, y)} = [TpG] {εpG(ξ, η)} (3.4.40)

where{εpG(x, y)} denotes the strain vector in thex-y coordinate system

and

[TpG] =

[TF 3] 0

[TF 1] [TF 2]

(3.4.41)


{εpG(ξ, η)} =

[∂w

∂ξ

∂w

∂η

∂2w

∂ξ2

∂2w

∂η2

∂2w

∂ξ ∂η

]T

(3.4.42)

[TF 1] = −[J2]−1[J1][J ]−1

[TF 2] = [J2]−1

[TF 3] = [J ]−1 (3.4.43)

The Eq.(3.4.42) can be rewritten as:

{εpG(ξ, η)} =[BpG

] {δp} (3.4.44)

where

[BpG

]=

[(∂Nw

∂ξ

) (∂Nw

∂η

) (∂2Nw

∂ξ2

) (∂2Nw

∂η2

) (∂2Nw

∂ξ ∂η

)]T

(3.4.45)

Hence combining Eq.(3.4.40) and Eq.(3.4.44) yields,

{εpG(x, y)} = [TpG][BpG

] {δp} = [BpG] {δp} (3.4.46)

Hence

{θp} = [HpG] {εpG} = [HpG] [BpG] {δp} (3.4.47)

The internal work done by the distributed internal stresses can be ex-

pressed as:

δW =

∫ ∫ ∫{εpG}T {σ} dx dy dz (3.4.48)

where

{σ} = [σx σy τxy]T (3.4.49)

Substituting the values of{εpG} from the Eq.(3.4.46) in the Eq.(3.4.48)


δW = δ

∫ ∫ ∫{δp}T [BpG]T [HpG]T [A]T{σ} dx dy dz (3.4.50)

[A]T{σ} =

∂w

∂x0

∂w

∂y

0∂w

∂y

∂w

∂x

−z∂2w

∂x20 −z

∂2w

∂x ∂y

0 −z∂2w

∂y2−z

∂2w

∂x ∂y

−z∂2w

∂x ∂y−z

∂2w

∂x ∂y−z

(∂2w

∂x2+

∂2w

∂y2

)

σx

σy

τxy

=

σx τxy 0 0 0

τxy σy 0 0 0

0 0 σx 0 τxy

0 0 0 σy τxy

0 0 τxy τxy (σx + σy)

∂w

∂x

∂w

∂y

−z∂2w

∂x2

−z∂2w

∂y2

−z∂2w

∂x ∂y

= [σp]{θp}(3.4.51)


Hence Eq.(3.4.50) can be rewritten as:

δW = δ

∫ ∫ ∫{δp}T [BpG]T [HpG]T [σp]{θp} dx dy dz

= δ

∫ ∫ ∫{δp}T [BpG]T [HpG]T [σp] [HpG] [BpG] {δp} dx dy dz

(3.4.52)

The external work done by the nodal forces is given by:

δW = δ{δp}T{R} (3.4.53)

Equating the external work and the internal work

{R} =

∫ ∫ ∫[BpG]T [HpG]T [σp] [HpG] [BpG] {δp} dx dy dz

= [KpG]e {δp}(3.4.54)

where[KpG]e is the geometric stiffness matrix given by:

[KpG]e =

∫ ∫ ∫[BpG]T [HpG]T [σp] [HpG] [BpG] dx dy dz (3.4.55)

As the reference plane is the mid-plane of the plate, only the inner-

most integral contains the termsz. Hence this integration can be per-

formed separately. Hence

[KpG]e =

∫ ∫[BpG]T [σ] [BpG] dx dy

=

∫ ∫[BpG]T [σ] [BpG] |J | dξ dη

(3.4.56)

where

[σ] =

∫ t/2

−t/2

[HpG]T [σp] [HpG] dz (3.4.57)


3.4.3 The Stiffener Element Formulation

The stiffener is modelled as a separate element and the formulation of

its stiffness matrix is carried out by considering the axial force, bending

moment and torsional moment. As a general case, a curved stiffener

having eccentricity with respect to the mid-plane of the plate and placed

arbitrarily within the plate element is considered. Since the stiffener is a

curved one, its axis changes its direction from point to point and hence its

deformation at a particular point, say P is to be considered in the direction

of the tangent to the stiffener at that point as shown in the Fig. 3.5. The

displacement field is based on the assumption that the common normal

to the plate and the stiffener system before bending remains straight and

normal to the middle plane of the plate after bending.

P

αx

x’

yy’

Figure 3.5: Coordinate axes at any point of a curved stiff-ener

3.4.3.1 Coordinate Transformation for the Stiffener

In the Fig. 3.5,x′-y′ is the local coordinate system at point P wherex′

is the direction of the tangent to the stiffener which is at an angleα in


anti-clockwise direction with respect to the globalx-axis direction.

The coordinate systemsx-y andx′-y′ are related as

x

y

=

cos α −sinα

sinα cos α

x′

y′

(3.4.58)

The derivatives ofx andy with respect tox′ andy′ are obtained as

∂x

∂x′= cos α

∂x

∂y′= −sin α

∂y

∂x′= sinα

∂y

∂y′= cos α

(3.4.59)

The relationship between the local and the global displacements at

point P in the reference plane which is the mid-plane of the plate is given

by

u′

v′

w′

=

cos α sin α 0

−sinα cos α 0

0 0 1

u

v

w

(3.4.60)

whereu′, v′, w′ are the displacements at the middle plane of the plate

alongx′, y′ andz′ (z′ = z) directions respectively.


The axial displacement of a point, sayP1 in the stiffener at a depthz

(Fig. 3.6) from the reference middle plane of the plate and normal to it is

expressed as

u′ = u′ − z∂w′

∂x′(3.4.61)


Pz

Figure 3.6:Sectional view of a typical stiffener

The corresponding strain in the stiffener is

εsl =∂u′

∂x′− z

∂2w′

∂x′ 2(3.4.62)

and the normal stress is

σsl = E

[∂u′

∂x′− z

∂2w′

∂x′ 2

](3.4.63)

The axial force in the stiffener is given by

Fs =

∫

As

σsl dAs = E∂u′

∂x′

∫

As

dAs − E∂2w′

∂x′ 2

∫

As

z dAs

= EAs∂u′

∂x′− ESs

∂2w′

∂x′ 2(3.4.64)

whereAs is the cross sectional area andSs is the first moment of area of

the stiffener with respect to the middle plane of the plate. The value of

Ss depends on the disposition of the stiffener.

The bending moment is

Ms =

∫

As

σsl z dAs (3.4.65)

Substituting the value ofσs from Eq.(3.4.63) in Eq.(3.4.65), it yields

Ms = ESs∂u′

∂x′− EIs

∂2w′

∂x′ 2(3.4.66)


whereIs is the second moment of area (reference plane being the mid-

plane of the plate) of the stiffener.

It is observed from Eqs.(3.4.64) and (3.4.66) that the eccentricity of

the stiffener produces coupling between the axial and the flexural effects.

The torsional moment is given by

Ts = −GJs∂2w′

∂x′ ∂y′(3.4.67)

whereG is the modulus of rigidity andJs is the torsional constant of the

stiffener.

Combining Eqs.(3.4.64), (3.4.66), and (3.4.67) the generalized stress-

strain relationship of the stiffener in the local axis system at the point P

is expressed as

{σs} = [Ds] {εs} (3.4.68)

where

{σs} = [Fs Ms Ts]T (3.4.69)

{εs} =

[∂u′

∂x′− ∂2w′

∂x′ 2− ∂2w′

∂x′ ∂y′

]T

(3.4.70)

and

[Ds] =

E As E Ss 0

E Ss E Is 0

0 0 G Js

(3.4.71)


The local displacements and coordinate parameters in the generalized

strain vector of the stiffener given in Eq.(3.4.70) are replaced by the

global parameters using Eqs.(3.4.58) and (3.4.60) and it becomes

{εs} = [Ts] {εs} (3.4.72)


where

[Ts] =

cos2α sin2α1

2sin2α 0 0 0

0 0 0 cos2α sin2α −1

2sin2α

0 0 0 −1

2sin2α

1

2sin2α −1

2cos2α

(3.4.73)

and

{εs} =

[∂u

∂x

∂v

∂y

(∂u

∂y+

∂v

∂x

)− ∂2w

∂x2− ∂2w

∂y22

∂2w

∂x ∂y

]T

(3.4.74)

Once the strain vector of the stiffener is expressed in terms of the dis-

placement components at the mid-plane of the plate, the same displace-

ment shape function of the plate element is used which yields the stiff-

ness matrix of the stiffener in terms of the nodal parameters of the plate

element and by this process, the compatibility between the plate and the

stiffener element is retained.

It may be observed from Eqs.(3.4.74) and (3.4.10) that

{εs} = {ε(x, y)p} (3.4.75)

Hence using the same interpolation functions given in Eq.(3.4.4) and

Eq.(3.3.4), the Eq.(3.4.74) can be expressed with the help of Eq.(3.3.22)

as

{εs} = [Bp]{δp} (3.4.76)

Combining Eqs.(3.4.72) and (3.4.76), yields

{εs} = [Ts] [Bp]{δp} = [Bs] {δp} (3.4.77)

where

[Bs] = [Ts][Bp] (3.4.78)


ξ

η(-1,+1)

(-1,-1)

(+1,+1)

(+1,-1)

λ

Figure 3.7:Stiffener orientation in the mapped domain

3.4.3.4 Elastic Stiffness Matrix of the Stiffener Element

Following the steps mentioned earlier for the plate element, the elastic

stiffness matrix of the stiffener element is given by

[Ks]e =

∫[Bs]

T [Ds][Bs] dl (3.4.79)

Herel is taken along the stiffener axis inx-y plane. This can be rewritten

as

[Ks]e =

∫[Bs]

T [Ds][Bs] |Jst| dλ (3.4.80)

whereλ is in the direction of the stiffener axis in theξ-η plane as shown

in the Fig. 3.7 and the Jacobian|Jst| is given by

|Jst| = dl

dλ(3.4.81)

The Jacobian is calculated by the ratio of the actual length to the length

on the mapped domain considering any segment of the stiffener and is


constant when a straight line or a circular arc in thex-y plane is mapped

into a straight line. But in case of a complex mapping, the ratiodl

dλmay change from point to point. The integration is carried out along the

stiffener axis which isλ in the ξ-η plane by taking Gauss points on the

mapped stiffener axis.

3.4.3.5 Consistent Mass Matrix of the Stiffener Element

The consistent mass matrix for the arbitrarily oriented stiffener element

is formulated following the steps similar to that of the plate element and

it can be written as:

[Ms] = ρ

∫[N ]T [L]T [Ts]

T [Ps][Ts][L][N ] |Jst| dλ (3.4.82)

where

[Ps] =

Ascos2α Assinαcosα 0 −Sscosα 0

Assinα cosα Assin2α 0 −Sssinα 0

0 0 As 0 0

Sscosα −Sssinα 0 Is 0

0 0 0 0 Js

(3.4.83)


[Ts] =

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 cosα sinα

0 0 0 −sinα cosα

(3.4.84)

and the matrices[L] and [N ] are given by the Eqs.(3.4.21) and (3.4.22)

respectively.

3.4.3.6 Geometric Stiffness Matrix of the Stiffener Element

The stiffener strain for an x-directional stiffener is expressed as:

{εs} =

∂U

∂x

∂V

∂x

∂W

∂x

+

1

2

(∂W

∂x

)2

+1

2

(∂U

∂x

)2

0

0

= {εsE}+ {εsG}

(3.4.85)

The field variables are expressed as:

{f} =

U

V

W

=

−z∂w

∂x

−z∂w

∂y

w

(3.4.86)


Substituting the values of U, V and W in the expression for{εsG}, yields

{εsG} =

z2

2

(∂2w

∂x2

)2

+1

2

(∂w

∂x

)2

0

0

=1

2{θs}T {θs}

(3.4.87)

where

{θs}T =

{−z

∂2w

∂x2

∂w

∂x

}(3.4.88)

It can be shown that

δ {εsG} = {θs}T δ {θs} (3.4.89)

{θs} can be written as:

{θs} = [HsG] {εsG} (3.4.90)

where

[HsG] =

1 0

0 −z

(3.4.91)

and

{εsG}T =

{∂w

∂x

∂2w

∂x2

}(3.4.92)

{εsG} can be expressed as:

{εsG} =

∂w

∂x

∂2w

∂x2

= [BsG] {δp} (3.4.93)


Hence

{θs} = [HsG] [BsG] {δp} (3.4.94)

The internal work done by the distributed internal stresses can be ex-

pressed as:

δW =

∫∫∫{εsG}T σx dx dy dz

= δ

∫∫∫{δp}T [BsG]T [HsG]T σx [HsG] [BsG] {δp} dx dy dz

(3.4.95)

In the case of the stiffener along the x-direction, the integration with

respect toy and z can be performed with the innermost integral only,

puttingy-coordinates in the[BsG] matrix.

[HsG]T σx [HsG] =

1 0

0 −z

σx

1 0

0 −z

= σx

1 0

0 z2

(3.4.96)

Hence,

∫∫[HsG]T σx [HsG] dy dz =

σxAs 0

0 σxIs

= [σs]

(3.4.97)

If the stiffener coordinate axis is at an angle ofα with respect to the

globalx-axis (Fig. 3.5) then with the help of the Eq.(3.4.58) the deriva-

tives with respect to thex-y and thex′-y′ coordinates are related as:


∂w

∂x′

∂2w

∂x′2

= [TsG1]

∂w

∂x

∂w

∂y

−∂2w

∂x2

−∂2w

∂y2

2∂2w

∂x ∂y

(3.4.98)

where

[TsG1] =

cos α sin α 0 0 0

0 0 − cos2 α − sin2 α sin α cos α

(3.4.99)

The derivatives with respect to thex-y coordinates can be expressed in

terms ofξ-η coordinates such as:

∂w

∂x

∂w

∂y

−∂2w

∂x2

−∂2w

∂y2

2∂2w

∂x ∂y

= [TsG2]

∂w

∂ξ

∂w

∂η

∂2w

∂ξ2

∂2w

∂η2

∂2w

∂ξ ∂η

(3.4.100)


where

[TsG2] =

[J ]−1 [0]

[TF1] [TF2]

(3.4.101)

and[J ] is the Jacobian and[TF1], [TF2] being given by the Eq.(3.3.18).

Hence from the Eq.(3.4.93)

{εsG} = [TsG1] [TsG2]

∂Nw

∂ξ

∂Nw

∂η

∂2Nw

∂ξ2

∂2Nw

∂η2

∂2Nw

∂ξ ∂η

{δp}

= [TsG1] [TsG2][BsG

] {δp}

= [BsG] {δp}

(3.4.102)

The geometric stiffness matrix is given by

[KsG]e =

∫[BsG]T [σs] [BsG] dx

=

∫[BsG

]T[TsG2]

T [TsG1]T [σs] [TsG1] [TsG2]

[BsG

] |Jst| dλ

(3.4.103)


3.4.4 Boundary Conditions for the Stiffened Plate

The boundary conditions for the arbitrary stiffened plate are considered

following the same procedure as adopted in the case of the arbitrary bare

plate. Referring to the Fig. 3.3, the relationship between the in-plane

displacements in the local and the global coordinates at the pointP is

given by:

u1

v1

=

cosβ sinβ

−sinβ cosβ

u

v

(3.4.104)

whereu1 andv1 are the displacements along the direction ofx1 andy1

respectively.

The displacements at the pointP which may be restrained can be

expressed as

{fbp} =

u1

v1

w

θn

θt

=

u1

v1

w

∂w

∂x1

∂w

∂y1

(3.4.105)

whereθn andθt represent the slopes which are normal and transverse to

the boundaries respectively as in the case of bare plate.

Substituting from Eqs.(3.3.50) and (3.4.104), the Eq.(3.4.105) can be


written as

{fbp} =

cosβ sinβ 0 0 0

−sinβ cosβ 0 0 0

0 0 1 0 0

0 0 0 cosβ sinβ

0 0 0 −sinβ cosβ

u

v

w

∂w

∂x

∂w

∂y

(3.4.106)

Expressing Eq.(3.4.106) in terms of the shape functions;

{fbp} = [Nbp] {δp} (3.4.107)

where

[Nbp] =

cosβ sinβ 0 0 0

−sinβ cosβ 0 0 0

0 0 1 0 0

0 0 0 cosβ sinβ

0 0 0 −sinβ cosβ

[Nu]

[Nv]

[Nw]

∂[Nw]

∂x

∂[Nw]

∂y

(3.4.108)


The reaction components per unit length along the boundary line due

to the elastic springs corresponding to the possible boundary displace-

ments given in Eq.(3.4.105) can be expressed as

{fkp} =

fku

fkv

fkw

fkα

fkβ

=

kuu1

kvv1

kww

kαθn

kβθt

(3.4.109)

whereku, kv, kw, kα andkβ are the spring constants or restraint coeffi-

cients corresponding to the direction ofu1, v1, w, θn andθt respectively.

The Eq.(3.4.109) can be rewritten by combining the Eqs.(3.4.105),

(3.4.106) and (3.4.107) as

{fkp} = [Nkp]{δp} (3.4.110)


where

[Nkp] =

ku cosβ ku sinβ 0 0 0

−kv sinβ kv cosβ 0 0 0

0 0 kw 0 0

0 0 0 kα cosβ kα sinβ

0 0 0 −kβ sinβ kβ cosβ

[Nu]

[Nv]

[Nw]

∂[Nw]

∂x

∂[Nw]

∂y

(3.4.111)

Following the procedure similar to the case of bare plate the stiffness

of the boundary for the stiffened plate can be expressed as

[Kbp] =

∫[Nbp]

T [Nkp] |Jb| dλ1 (3.4.112)

3.4.5 Stresses in the Stiffener

Once the nodal displacements of the stiffened plate are known, the stress

resultants of the stiffener as expressed in the Eq.(3.4.69) can be obtained

with the help of the Eq.(3.4.68) and Eq.(3.4.67).

The stresses in the stiffener at a depthz from the mid-plane of the

plate can be computed as:

σz =Fs

As

− Fse

Is

z +Ms

Is

z (3.4.113)

3.5 Consistent Load Vector 97

whereFs, Ms, As andIs denote the axial force, bending moment, cross-

sectional area and moment of inertia respectively of the stiffener ande is

the eccentricity of the stiffener with respect to the plate mid-plane.

3.5 Consistent Load Vector

The nodal load vector for an element when subjected to a uniformly dis-

tributed load of intensityq(x, y) can be obtained by the expression

{P}e =

∫ ∫[Nj]

T q dx dy

=

∫ +1

−1

∫ +1

−1

[Nj]T q |J | dx dy

(3.5.1)

where[Nj] is the displacement function for thej-th node and|J | is the

determinant of the Jacobian. The global load{P} can be obtained by

assembling the nodal load vector{P}e of each of the elements. When

concentrated load is present at any of the nodal points, the load value is

added to the corresponding degree of freedom of that particular node.

3.6 Solution Procedures

The solution procedures adopted in the analysis of the arbitrary stiffened

and bare plates for static, dynamic and stability analyses are presented in

this section.

3.6.1 Static Analysis

The elastic stiffness matrices of the plate as well as the stiffener elements

are computed and they are assembled into the global elastic stiffness ma-

trix [K] which is stored by adopting the skyline storage (Zienkiewicz and


Taylor [198]) technique. In this process of storage, the matrix is stored

in a single array eliminating the zero entries if any within the band thus

reducing the storage requirement of the computer. The equation of equi-

librium for the static analysis given by the Eq.(3.1.1) is solved following

the Cholesky decomposition method by adopting the algorithm of Corr

and Jennings [39].

3.6.2 Free Vibration Analysis

The equilibrium equation for undamped free vibration is given by the

Eq.(3.1.2).

Considering the motion as harmonic the solution of the equation (3.1.2)

is

{δ} = H{ψ}eiωt (3.6.1)

where{ψ} is a normalized vector of the order of{δ}, H is the weight-

ing parameter of{ψ} andω is the frequency of vibration in radians per

second. On substitution the equilibrium equation becomes

[K]{ψ} = ω2[M ]{ψ} (3.6.2)

This is a generalized eigenproblem and is solved by the simultaneous it-

eration algorithm of Corr and Jennings [39] and its solution is the eigen-

valueω2 and the eigenvector{ψ}. The same skyline storage scheme as

earlier is adopted for the global elastic stiffness matrix[K] and the mass

matrix [M ].

3.6 Solution Procedures 99

3.6.3 Stability Analysis

The equilibrium equation for the stability analysis is given by the Eq.(3.1.3).

Since the matrix[K] is positive definite, it can be decomposed as:

[K] = [L][L]T (3.6.3)

where[L] is a lower triangular matrix. Hence Eq.(3.1.3) can be rewritten

as:

[L]−1 [KG] [L]−T [L]T{δ} =1

λ[L]T{δ} (3.6.4)

The above equation represents a standard eigenvalue problem which is

solved by the simultaneous iteration algorithm of Corr and Jennings [39]

and the eigenvalues corresponding to the lowest buckling loads are ob-

tained.

Chapter -4

COMPUTERIMPLEMENTATION

4.1 Introduction

The finite element method has been established as a powerful numeri-

cal tool because of its broad spectrum of generality and its ease of appli-

cability to rather more complex and difficult problems showing greater

efficacy in its solution than that of any other existing similar techniques.

This advantage of the method over others has led various research orga-

nizations and modern industries to endeavour the development of general

purpose software packages and other in-house codes for solving practical

problems of more complex nature. In an effort to make the method more

powerful and to address more complicated problems, the finite element

analysis programmes themselves become extremely complex and com-

putationally involved. These programmes are available as black box

modules which are to be used with the help of CAD programmes. These

conventional programmes cannot easily be modified to perform a desired

task necessitating redesign and rebuild of finite element libraries to suit

one’s need. Hence there is a requirement for finite element analysis pro-

grammes to be easily modifiable to introduce new analysis procedures

101

102 COMPUTER IMPLEMENTATION

and new kinds of design of structural components or even emerging tech-

nology of new materials whenever needed. In the present investigation,

the computer codes have been generated with such modularity which is

amenable to easy modification whenever the need arises.

Throughout all these years the finite element codes have been de-

veloped employing procedural language such as FORTRAN which is

unstructured in its nature. Now there is a trend to pay attention to the

verification, portability and reusability of the computer programmes dur-

ing the process of their development and to the possibility of the use of

other software products . However, FORTRAN does not have the pro-

vision to meet all these requirements. TheC++ language, apart from its

object-oriented programming approach allows for more efficient software

development, because it includes the dynamic memory allocation, decla-

ration of datatypes, modularization and the pointer concept. Though,

the benefit of object-oriented programming has not been utilized in the

present investigation, but the other advantages of the language as men-

tioned above have been fully utilized. Few of the utilities are of utmost

importance which are provided by Press et al. [150] and are used exten-

sively in the codes generated for the present investigation. Apart from

these, for efficiency of the finite element programmes, advance features

like automatic mesh generation, automatic nodal connectivity and skyline

storage scheme have been implemented in the computer codes.

4.2 Application Domain

The Computer Programmes have been developed in the present investi-

gation by making use of theC++ programming language to include a

wide spectrum of application domain. They have the analytical modules

to solve the following types of problems:

4.2 Application Domain 103

1. Static analysis of arbitrary bare plates - to evaluate displace-

ments and stress resultants at salient points of the bare plates of

arbitrary shape.

2. Free vibration analysis of arbitrary bare plates - to extract the

natural frequencies (eigenvalues) and the corresponding mode shapes

(eigenvectors) of the structure.

3. Stability analysis of arbitrary bare plates - to estimate the elastic

buckling load and the buckled mode shapes of the structure from

the eigenvalue solution.

4. Static analysis of arbitrary stiffened plates - to determine the

displacements and stress resultants at various points of the plate

skin and to evaluate stiffener stresses at different sections of the

eccentric and concentric stiffeners.

5. Free vibration analysis of arbitrary stiffened plates - to deter-

mine the natural frequencies of the stiffened plated structures along

with its corresponding mode shapes.

6. Stability analysis of arbitrary stiffened plates - to assess the

elastic buckling load of the structures and their buckling mode

shapes.

Computer programme codes have been written to incorporate vari-

ous boundary and loading conditions of the structures. The modularity

of the programme development has been retained by employing differ-

ent modules in the shape of differentC++ functions performing specific

functionalities of the programmes.


4.3 Description of the Programme

The finite element procedure involves three basic steps in terms of the

computation carried out which may be termed as:

• Preprocessor

• Processor

• Postprocessor

The different functions of these steps have been elaborated in the Fig. 4.1.

4.3.1 Preprocessor

This module of the programme reads the necessary information about the

geometry and boundary conditions of the plate, material properties, load-

ing configuration and its magnitude, stiffener orientation and its prop-

erties etc. Also in this module, all the nodal coordinates and the nodal

connectivity are generated. The differentfunctions which are used in

this module are described briefly in the subsequent sections. A flowchart

of the preprocessor unit has been shown in the Fig. 4.2.

4.3.1.1 functioninput()

The following variables are used in the functioninput() to generate the

data required for the analysis of the bare and the stiffened plates.

4.3 Description of the Programme 105

Sole the Equations for Different Analyses

PREPROCESSORRead the Input DataGenerate the meshGenerate Nodal ConnectivityRead the Stiffener Position and Orientation

Echo the Input Data

POSTPROCESSOR

print the Output

END

Assemble the Matrices to Global Matrix

5. Free Vibration Analysis of Stiffened Plates

3. Stability Analysis of Bare Plates

1. Static Analysis of Bare Plates

4. Static Analysis of Stiffened Plates

Solve the Equations for different Analyses

PROCESSORGenerate Element Matrices for the PlateGenerate Element Matrices for the Stiffener if requiredGenerate Boundary Stiffness Matrices for the Boundaries

2. Free Vibration Analysis of Bare Plates

6. Stability Analysis of Stiffened Plates

START

Figure 4.1:Basic Elements of the Computer Programmes


bpoin : Number of points on the boundary for the mapping

of the plate geometries

bcord : Cartesian coordinates of the boundary points

nnode : Number of node in the element

ndofn : Number of degrees of freedom per node

ngaus : Number of gauss points

nxi : Number of mesh ins-direction in the mapped domain

neta : Number of mesh int-direction in the mapped domain

nelem : Number of elements

npoin : Number of nodal points generated for the mapping

of the elements

nodes : Number of nodes for the analysis

tdof : Number of total degrees of freedom

young : Young’s modulous of elasticity

poiss : Poisson’s ratio

thick : thickness of the plate

ntype : Shape of the plate geometry

=1 Square plate

=2 Rectangular plate

=3 Annular plate

=4 Circular plate

=5 Skew plate

=6 Sector plate

=7 Elliptical plate

=8 Trapezoidal plate

=9 Triangular plate


stif : Index for bare or stiffened plate

=0 Bare plate analysis

=1 Stiffened plate analysis

soln : Index for type of analysis

=1 Static analysis

=2 Free vibration analysis

=3 Buckling analysis

4.3.1.2 functionnodgen()

The different variables used in the functionnodgen()which is used to

generate the peripheral nodes in each of the element’s boundary are as

presented below.

mnods : Nodal numbers in the element boundary

ielem : Element counter

inode : Node counter

4.3.1.3 functionstcod()

The functionstcod() is used to generate the nodal coordinates in the

mappeds-t domain of the plate. The different variables used in the func-

tion are:coord : Coordinates in the mapped domain

xi-divn : Length of element ins-direction in mapped domain

eta-divn : Length of element int-direction in mapped domain

xi-small :1

3of xi-divn

eta-small :1

3of eta-divn


input()bpoin, bcord, nnode, ndofn, ngaus, nxi,

neta, nelem, npoin, nodes, tdof, young,

poiss, thick, ntype, stif, soln

nodgen()mnods, ielem, inode

xycod()xynod,xi,eta

rgdplt()dmatx1, young,poiss, thick

coord, xi-divn,

stcod()

eta-divn, xi-small,eta-small

connect()lnods

stifin()w, d, e

band()hband, sky

rgdstf()As, Ss, Is, GJs, dmatx2

shape1, deriv1

sfr1()

Figure 4.2:Preprocessor unit of the computer codes


4.3.1.4 functionconnect()

The functionconnect()generates the nodal connectivity in the elements.

The variables used along with others is:

lnods : Node numbers associated with the element

4.3.1.5 functionband()

The functionband() computes the half bandwidth of the matrix and the

skyline value for the skyline storage scheme. It has the following vari-

ables:hband : Half bandwidth of the matrix

sky : Skyline value for the skyline storage

4.3.1.6 functionxycod()

The functionxycod() generates all the nodalx-y coordinates of the ele-

ments. The variables used are:xynod : Cartesian coordinates of the node

xi : s-coordinate of the node in mapped domain

eta : t-coordinate of the node in mapped domain

4.3.1.7 functionsfr1()

The functionsfr1() calculates the cubic serendipity shape functions, their

derivatives and elements of the Jacobian matrix. The different variables

in this function are:shape1 : Cubic serendipity shape functions

deriv1 : Shape function derivatives


4.3.1.8 functionrgdplt()

The functionrgdplt() has been used for computing the rigidity matrix of

the plate element. It comprises the following variables:

dmatx1 : Elements of the plate rigidity matrix

young : Young’s modulous of elasticity

poiss : Poisson’s ratio

thick : Plate thickness

4.3.1.9 functionstifin()

The functionstifin() reads the necessary information of the stiffeners. It

has the following variables:

w : Width of the stiffener

d : Depth of the stiffener

e : Eccentricity of the stiffener

4.3.1.10 functionrgdstf()

The functionrgdstf() calculates the different elements of the rigidity ma-

trix of the stiffener. The following variables are used in the function:

As : Cross-sectional area of the stiffener

Ss : First moment of area of the stiffener

Is : Second moment of area of the stiffener

G : Modulus of rigidity of the stiffener

Js : Torsional constant of the stiffener

dmatx2 : Elements of the stiffener rigidity matrix


4.3.2 Processor

This module of the programmes performs the following tasks:

1. Generation of the element matrices.

2. Assembly of the element matrices into global matrices.

3. Imposition of the boundary conditions.

4. Solution of the algebraic equations for static analysis of plates to

obtain nodal unknowns and the computation of the stress resultants

for the skin and the stiffener at all the nodes.

5. Determination of eigenvalues and eigenvectors for the free vibra-

tion and buckling analyses using simultaneous vector iteration tech-

nique.

A flowchart showing the processor unit is presented in the Fig. 4.3.

The various modular functions which are used in this processor unit are

briefly presented herein.

4.3.2.1 functionform-stif-mass-geom()

The functionform-stif-mass-geom()calls the other functions in turn for

the processing of each of the elements.

4.3.2.2 functionelm-stif-mass-geom()

This function generates the elastic and geometric stiffness and mass ma-

trices for the plate element.


global() global-stif()b

nd

-sti

f()

1

1 2 3

end

sfr1() jacob1() sfr2() bmat()

Computes element matrices

for the plate

elm-stif-mass-geom() elm-stf-mass-geom()Computes element matrices

for the stiffener

global-stif-mass-geom() global-stf-mass-geom()

Bare PlateAnalysis

stif Stiffened PlateAnalysis

soln

AnalysisStatic Free Vibration

Analysis AnalysisBuckling

stop

0

stfin()

Figure 4.3:Processor unit of the computer codes


4.3.2.3 functionelm-stf-mass-geom()

This function generates the elastic and geometric stiffness matrices and

the mass matrices of the stiffener elements.

4.3.2.4 functionglobal-stif-mass-geom()

The assembly of all the element matrices of the plate elements into the

global ones are carried out through this function.

4.3.2.5 functionglobal-stf-mass-geom()

The assembly of all the element matrices of the stiffener elements into

the global mtrices are performed through this function.

4.3.2.6 functionglobal()

This is a common function called by the individual functions to assemble

all the element matrices into global matrix.

4.3.2.7 functionelm-load()

This function calculates the consistent element load vector and takes into

account any application of concentrated load on the plate.

4.3.2.8 functiongbl-load()

The generated element load vectors are assembled into global load vector

using this function.


4.3.2.9 functionbnd-stif()

The stiffness of the boundary lines of the plate element if it happens to

be one of the elements in the periphery is computed in this function.

4.3.2.10 functionsfr2()

This module calculates the displacement shape functions and their deriva-

tives.

4.3.2.11 functionbmat()

This function evaluates the matrix for the strain-displacement relation-

ship.

4.3.2.12 functiondbmat()

The stress resultants are computed in this function.

4.3.2.13 functionsolve()

The functionsolve()is used to solve the simultaneous algebraic equations

generated in the process of analysis. The equilibrium equations are in the

form of [A]{X} = {B}, where[A] is the global stiffness matrix,{B}is the global load vector and{X} is the nodal unknown vector whose

solution is sought. This has been solved using Choleski factorization by

performing decomposition, forward elimination and backward substitu-

tion with the help of the functionsdecomp(), forsol(), andbacksol().

4.3.2.14 functionr8usiv()

This function is used for the eigenvalue solution. Using this module, a

simultaneous iteration algorithm has been adopted for the free vibration


and buckling analyses. The input data to this function are the global elas-

tic stiffness matrixgstif, the global geometric stiffness matrixgbl-geom,

the global mass matrixgbl-massand the corresponding pointer vectors.

Through this function the eigenvalues and the corresponding eigenvec-

tors are extracted. The required number of modes of vibration or buckling

is to be specified by the user. The function requires three arraysu, v and

w, of size(n,m) wheren is the total degrees of freedom andm is a value

higher than the number of modes. The numerical value ofm has been

considered as 1.5 times the number of modes in the present programme.

The tolerance value has been set to10−6 and the maximum number of

iterations to 40. The initial trial vectors are generated from a random

number generator. Ther8usiv() module consists of a number of func-

tions which are presented below with brief descriptions of their function-

alities and sequence in which they are called inside the functionr8usiv().

functionr8ured() : decomposes a symmetric matrix into lower

triangular matrix

functionr8uran() : generates random trial vectors

functionr8uort() : orthonormilises the vectors by the Schmidt

process

functionr8ubac() : solves the equation[l]T{v} = {u} by

backward substitution

functionr8upre() : performs premultiplication in the form

{v} = {l}{u}functionr8ufor() : solves equation{l}{v} = {u} by forward

substitution


functionr8udec() : sorts the vectors{u} and{v} according to the

descending order of eigenvalue prediction

functionr8uran() : generates trial vectors in{w}functionr8uort() : orthonormalizes{w}functionr8uerr() : estimates the vector errors in successive trials

A flowchart of this module is shown in the Fig. 4.4

4.3.3 Postprocessor

In this part of the programme , all the input data are echoed to check for

their accuracy. The functionprint-disp() is used to print the output data

in terms ofdisplacements, moments, stresses, eigenvaluesetc. depend-

ing on the type of analysis carried out. The results are stored in a series

of separate output files for each category of problems analyzed and those

values are used to prepare tables and graphs etc.


Computation ofNatural frequenciesand buckling load

8us

vi

r

r8ufor()

enter

return

end

Geometric Stiffness MatrixElastic Stiffness Matrix

Mass Matrix

r8uerr()

r8udec()

r8upre()

r8ubac()

r8uort()

r8uran()

r8ured()

Figure 4.4: Flowchart for free vibration and buckling anal-ysis

Chapter -5

NUMERICAL EXAMPLES

5.1 Introduction

The stiffened as well as bare plates with arbitrary geometries have got

an important role to play as one of the structural elements in the modern

day structures. These plates having various boundary conditions are sub-

jected to varieties of loading for which the stress analysis is to be carried

out. The present day trend for the stress analysis is to use some software

packages for this type of analysis. But these commercial softwares have

got the limitation in the sense that they are inefficient to handle the arbi-

trary orientation of the stiffener as they demand the mesh division to be

along the stiffener. Moreover, they are unable to implement the bound-

ary conditions successfully for a plate of arbitrary configuration having

a curved edge. The element developed here is very much efficient to ad-

dress the problems pertaining to the arbitrarily oriented stiffeners as well

as the curved-boundary arbitrary plates. In this chapter a large number

of numerical examples for the stiffened and bare plates of arbitrary con-

figurations having various boundary conditions and subjected to various

loading conditions are presented as a rigorous test to study the perfor-

mance of the proposed element.

The following types of problems are considered in this chapter:

119

120 NUMERICAL EXAMPLES

1. Arbitrary Bare Plates

(a) Static Analysis

(b) Free Vibration Analysis

(c) Stability Analysis

2. Arbitrary Stiffened Plates

(a) Static Analysis

(b) Free Vibration Analysis

(c) Stability Analysis

The results obtained by this present formulation are compared with

the theoretical and/or experimental ones published by the other investi-

gators wherever possible. The computer programmes are written inC++

and have been run in theHP - UX 9000/819work station andORIGIN

200. Unless otherwise mentioned the mesh division used in the present

analysis is16×16 considering the whole plate for almost all the geomet-

rical configurations.

5.2 Arbitrary Bare Plates

The problems relating to plates of arbitrary configurations without any

stiffeners are considered when they are subjected to static, dynamic and

buckling loads and are presented in the subsequent sections.

5.2.1 Static Analysis of Arbitrary Bare Plates

The plates with various geometries such as square, rectangular, skew, an-

nular, circular are analyzed under the static load to test the accuracy of the

5.2 Arbitrary Bare Plates 121

b/3

b/3

b/3

a/3 a/3 a/3

x

y

2

3

4 5 6 7

8

9

1 12 11 10

Figure 5.1: Location of the boundary nodal points of a rect-angular plate

present method. The plates are subjected to uniformly distributed and/or

concentrated static load and the results obtained are compared with the

published ones wherever possible.

5.2.1.1 Rectangular Plates Under Uniformly Distributed Load

Rectangular plates of various aspect ratios subjected to uniformly dis-

tributed load are analyzed and the factorsα, β andβ1 for the maximum

deflection and the maximum moments inx andy directions are compared

with the analytically obtained results of Timoshenko and Woinowsky-

Krieger [185] in the Table 5.1. The agreement between the results is ex-

cellent. The location of the twelve number of the boundary nodal points

of the rectangular plate which are used for the mapping to theMaster

Plate is shown in the Fig. 5.1.


Table 5.1: Numerical factors α, β and β1 for uniformlyloaded simply supported rectangular plates

wmax = αqa4

D(Mx)max = β q a2 (My)max = β1 q a2

b

aα β β1

Present Ref. [185] Present Ref. [185] Present Ref. [185]

1.0 0.00407 0.00406 0.0480 0.0479 0.0480 0.0479

1.5 0.00773 0.00772 0.0813 0.0812 0.0499 0.0498

2.0 0.01015 0.01013 0.1020 0.1017 0.0464 0.0464

5.2.1.2 Rectangular Plates Under Concentrated Load

The factorsα for the maximum deflection of the rectangular plates of var-

ious aspect ratios subjected to a central concentrated loadP are computed

and compared with the analytical results of Timoshenko and Woinowsky-

Krieger [185] in the Table 5.2. The results are found to compare well.

Table 5.2: Numerical factors α for simply supported rectan-gular plates with central concentrated load

(wmax = α

Pa2

D

)

b

a1.0 1.2 1.6 2.0 3.0

Present 0.01164 0.01360 0.01575 0.01659 0.01704

Ref. [185] 0.01160 0.01353 0.01570 0.01651 0.01690


5.2.1.3 All Edges Clamped Rhombic Plates Under UDL

The skew rhombic plates with all the edges clamped and subjected to

uniformly distributed load are analyzed for different skew angles and the

obtained results for the deflection and the principal moments at the cen-

tre are compared with those of Iyengar and Srinivasan [70], Ramesh et

al. [153], Morley [120], GangaRao and Chaudhary [53] and Butalia et

al. [29] in the Table 5.3. Quite fair agreement has been obtained with all

the results even up to the skew angle of75◦ except those of Ramesh et

al. [153] which are comparatively less than all other values.

Table 5.3: Deflection and moments at the centre of the alledges clamped skew rhombic plates under UDL

skew Source of wmax Mplmax Mplmin

angle Results =qa4

D× 10−2 = qa2 × 10−2 = qa2 × 10−2

Present 1.8033 9.1711 8.1365

Iyengar et al. [70] 1.7968 9.2520 -

15o Ramesh et al. [153] 1.7950 - -

Morley [120] 1.7968 9.2520 -

Butalia et al. [29] 1.7948 9.2207 8.1785

Present 1.2360 7.9714 6.2284

Iyengar et al. [70] 1.2299 8.000 -

30o Ramesh et al. [153] 1.2258 - -

Morley [120] 1.2304 - -

GangaRao and

Chaudhary [53] 1.2304 - -

Butalia et al. [29] 1.2281 7.9906 6.2273

continued . . . . . .


. . . . . . continued from previous page


angle Results =qa4

D× 10−2 = qa2 × 10−2 = qa2 × 10−2

Present 0.6068 5.8443 3.9521

Iyengar et al. [70] 0.6018 - -

45o Ramesh et al. [153] 0.5952 - -

Morley [120] 0.6032 - -

Butalia et al. [29] 0.5997 5.827 3.8933

Present 0.1747 3.2903 1.8465

Iyengar et al. [70] 0.1717 - -

60o Ramesh et al. [153] 0.1638 - -

GangaRao and

Chaudhary [53] 0.1728 - -

Butalia et al. [29] 0.1704 3.2602 1.7639

Present 0.0147 1.0042 0.4065

Iyengar et al. [70] 0.0144 - -

75o Ramesh et al. [153] 0.0120 - -

Butalia et al. [29] 0.0143 0.9998 0.3942

5.2.1.4 All Edges Simply Supported Rhombic Plates UnderUDL

The same rhombic plates which are analyzed in the previous example are

considered here when subjected to uniformly distributed load but with the

edges being simply supported. The deflection and the principal moments

at the centre of the plates are compared with the results of GangaRao and


Chaudhary [53], Morley [119], Argyris [8], Rossow [156], Jirousek [71]

and Butalia et al. [29] in the Table 5.4. The results for the skew angles

45◦ and less have compared well with those of others. The results for

skew angle60◦ has deviated from others, but it is closer to that of Butalia

et al. [29]. Even the the75◦ angle result is close to the result of Butalia

et al. [29]. The location of the boundary nodal points of the skew plate is

shown in the Fig. 5.2.

The different values adopted for the analysis of these plates are:

Length of each side of the rhombic plate (2a) =8

Thickness of the plate (t) =0.08

Young’s modulus of elasticity (E) = 8.736× 107

Poisson’s ratio (ν) = 0.3

Uniformly distributed load (q) = 16.0

θ = Skew angle

θ

4 5 6 7

2

3

1

8

9

101112

a/3 a/3 a/3

b/3

b/3

b/3x

y

Figure 5.2: Location of the boundary nodal points of a skewplate


Table 5.4: Deflection and moments at the centre of theall edges simply supported skew rhombic platesunder UDL


angle Results =qa4

D× 10−2 = qa2 × 10−1 = qa2 × 10−1

Present 5.8238 1.9224 1.7063

GangaRao and

15o Chaudhary [53] 5.8240 - -

Butalia et al. [29] 5.8013 1.9207 1.7082

Present 4.0573 1.6962 1.3185

Morley [119] 4.0960 1.7000 1.3320

GangaRao and

30o Chaudhary [53] 4.0960 - -

Butalia et al. [29] 3.9832 1.6790 1.2980

Present 2.0008 1.2579 0.8207

GangaRao and

Chaudhary [53] 2.1120 - -

45o Argyris [8] 2.0787 1.2983 0.8570

Butalia et al. [29] 1.9125 1.2266 0.7803

Present 0.5538 0.6948 0.3401

Morley [119] 0.6528 0.7640 0.4320

GangaRao and

Chaudhary [53] 0.6496 - -

60o Argyris [8] 0.6158 0.7668 0.4028

Rossow [156] 0.6526 - -

Jirousek [71] 0.6526 0.7625 0.4343

Butalia et al. [29] 0.5194 0.6662 0.3166

Present 0.0414 0.1860 0.0583

75o Butalia et al. [29] 0.0422 0.1906 0.0639


5.2.1.5 Annular Sector Plate Under Concentrated Loads

A perspex model bridge slab presented by Coull and Das [41] is analyzed

and the results are presented in the Table 5.5. The model bridge as shown

in the Fig. 5.3 is having its radial edges simply supported and the curved

ones free. Three cases of loading corresponding to unit load placed at the

inner radius, outer radius and mid radius along the x-axis of the slab are

considered. The comparison of the present results with those of Coull

and Das [41] indicates excellent agreement.

� � � � ��

� � � � ��

1

2

3

α/3

α/3α/3

α = included angle

x

y

11123@

a/3 ea

ch

Ri

45

6

7

8

9

10

Figure 5.3: Annular sector plate showing boundary nodalpoints


Table 5.5: Deflection and moments along the mid-spanradial line (x-axis) of the annular sector plate

t = 0.168, ν = 0.35, E = 4.6× 105

wmax Mrmax

Load Radial Present Coull and Present Coull and

Position distance Analysis Das [41] Analysis Das [41]

Unit 7.0 0.01701 0.0169 - -

load 9.0 0.01517 0.01517 0.3278 0.312

at inner 11.0 0.01640 0.0163 0.2055 0.204

radius 13.0 0.01945 0.0195 0.1651 0.186

Unit 7.0 0.01945 0.0194 0.4834 0.465

load 9.0 0.03538 0.0353 0.5058 0.492

at outer 11.0 0.05786 0.0578 0.6239 0.540

radius 13.0 0.08816 0.0876 - -

Unit 7.0 0.01551 0.0155 0.4481 0.437

load 9.0 0.02410 0.0241 0.4884 0.493

at mid 11.0 0.03424 0.0342 0.4804 0.462

radius 13.0 0.04571 0.0457 0.3852 0.384

5.2.1.6 Circular Plate Under Different Loadings and Bound-ary Conditions

A circular plate of unit radius (Fig. 5.4) subjected to uniformly distributed

load as well as concentrated loads is considered for the analysis. The

results are obtained for the plate having all the edges simply supported

and clamped considering a mesh division of16 × 16 for the entire plate

and they are compared with the analytical results of Timoshenko and


Woinowsky-Krieger [185] and spline finite strip results of Li et al. [96].

The present results agree well with those obtained from different sources.

Table 5.6: Deflection and moments at the centre of thecircular plate under different loading and bound-ary conditions

radius (a) = 1.0, D = 1.0, UDL = 1.0, Point load = 1.0

Bounday conditions

Loading Method Simply supported Clamped

conditions w Mr w Mr

Uniformly Present 0.06359 0.2071 0.01549 0.08191

distributed Ref. [185] 0.063702 0.20625 0.015625 0.08125

load

Point Present 0.05066 0.01998

load at Ref. [185] 0.05050 - 0.01989 -

r = 0

Point Present 0.02929 0.00719

load at Ref. [96] 0.02934 - 0.00728 -

r = a/2


15o15

o

30o

x

y

1

5 6

7

8

9

10

1112

3

4

2

Figure 5.4:Circular plate with boundary nodal points

5.2.2 Free Vibration Analysis of Arbitrary BarePlate

Using the proposed element the bare plates of various shapes such as

square, rectangular, skew, trapezoidal, triangular, annular sector, circular

and elliptical are analyzed to test the performance of the present method

in the free flexural vibration analysis. The plates are tested for various

boundary conditions and the results are compared with the published

ones wherever possible. Usually, the results tabulated are obtained with

a mesh division of24× 24. The results are presented in tabular form and

in the presented tables the various abbreviations used are:

SC - Support condition

SS - All edges simply supported

CC - All edges clamped

M - Mode sequence numbers


5.2.2.1 Free Vibration of Rectangular and Square Bare Plates

Rectangular plates with all edges simply supported and clamped hav-

ing aspect ratios of 1 and 0.4 are analyzed and first few frequencies ob-

tained are presented in the Table 5.7. The results are compared with those

of Leissa [87] where the plates having opposite edges simply supported

were dealt by using existing well-known exact solutions and those with

clamped supports by using the Ritz method. The results are found to be

in excellent agreement. The convergence study for the different mesh

sizes for the simply supported rectangular plate of aspect ratio 1 (square

plate) is also presented in the Table 5.8 where excellent convergence of

the element with increasing mesh divisions of the plate is obtained.

Table 5.7: Frequency parametersλ = ωa2(ρ/D)1/2 for rect-angular plate

a/b SC M 1 2 3 4 5 6

1.0 SS A 19.7392 49.3480 49.3480 78.9568 98.6960 98.6960

B 19.7209 49.2753 49.2753 78.6671 98.5340 98.5340

CC A 35.992 73.413 73.413 108.27 131.64 132.24

B 35.922 73.221 73.221 107.63 131.26 131.91

0.4 SS A 11.4487 16.1862 24.0818 35.1358 41.0576 45.7950

B 11.4352 16.1403 23.9941 34.9980 41.0002 45.5792

CC A 23.648 27.817 35.446 46.702 61.554 63.100

B 23.605 27.669 35.158 46.290 60.997 62.981

A - Leissa [87]; B - Present


5.2.2.2 Free Vibration of Bare Skew Plates

Skew plates of different skew angles (Fig. 5.5) having aspect ratios 1:1

and 1:2 for all edges simply supported and all edges clamped are ana-

lyzed and the results are compared with those of Liew and Lam [99],

Durvasula [47], Mizusawa et al. [116] and Singh and Chakraverty [177]

in Table 5.9 and 5.10. These investigators used various methods like

two-dimensional orthogonal plate function, Galerkin method, B-spline

functions and boundary characteristic orthogonal polynomials to obtain

the solution. In the solution for skew angle equal to60◦, the mesh size

was increased to36 × 36. The comparison is reasonably good for this

case with Mizusawa et al. [116]. It may be seen that for this skew angle,

results of Singh and Chakraverty [177] have differed substantially with

those of Mizusawa et al. [116]. The results are in excellent agreement

except in a few cases of higher modes with higher skew angles.

Table 5.8: Convergence of frequency parametersλ =ωa2(ρ/D)1/2 for all edges simply supportedsquare plate

Mode Mesh divisions

4× 4 8× 8 12× 12 16× 16 20× 20 24× 24

1 19.1434 19.5785 19.6668 19.6983 19.7129 19.7209

2 and 3 47.2841 48.7307 49.0635 49.1859 49.2436 49.2753

4 71.7430 76.5736 77.8369 78.3142 78.5417 78.6671

5 and 6 94.8795 97.4219 98.0808 98.3394 98.4645 98.5340


Table 5.9: Frequency parametersλ = ωa2(ρt/D)1/2 of skewplates for different skew angles (φ) and for a/b =1.0, ν =0.3

SC φ M 1 2 3 4 5 6

A 25.069 52.901 72.344 84.780 - -

30o B 25.314 52.765 73.006 87.478 130.25 -

C 25.0219 52.5501 71.9398 83.5642 122.031 122.558

A 34.938 66.422 100.87 107.78 - -

SS 45o B 36.970 67.023 113.26 114.93 175.28 -

C 35.6320 66.1028 99.9479 108.844 139.403 167.678

60o B 73.135 112.64 209.84 233.52 323.51 -

C 66.3452 104.637 147.839 194.135 213.670 245.783

D 46.140 81.691 105.51 119.52 165.80 -

30o B 46.166 81.613 105.56 119.98 167.16 -

C 45.9824 81.3367 104.849 118.479 163.449 164.744

D 65.929 106.59 149.031 158.900 199.366 231.936

CC 45o B 66.330 106.77 156.34 160.25 213.58 -

C 65.4204 105.950 146.859 156.569 193.976 228.140

E 120.90 177.75 231.74 292.54 301.81 357.58

60o B 127.06 185.00 282.94 322.61 385.49 -

C 121.274 176.750 229.394 287.224 303.618 347.786

A - Liew and Lam [99]; B - Singh and Chakraverty [177]; C - Present

D - Durvasula [47]; E - Mizusawa et al. [116]


Table 5.10: Frequency parametersλ = ωa2(ρ t/D)1/2 ofskew plates for different skew angles (φ) and fora/b=2.0, ν =0.3

SC φ M 1 2 3 4 5 6

B 64.069 96.558 153.76 218.69 237.12 -

30o C 63.633 95.779 146.907 209.653 226.004 252.081

B 93.772 132.09 209.83 302.31 341.10 -

SS 45o C 92.184 129.008 184.164 251.531 323.616 331.569

60o B 182.44 240.11 394.64 562.85 675.53 -

C 176.098 223.932 288.403 361.768 445.835 541.334

A 128.74 159.41 213.38 287.36 340.23 -

30o B 128.90 159.72 215.29 291.45 341.33 -

C 128.507 158.586 211.808 284.784 339.577 356.976

A 189.18 222.07 279.78 358.94 449.26 -

CC 45o B 190.00 223.90 294.67 385.53 509.03 -

C 188.820 220.701 276.869 353.380 439.970 503.396

A 369.28 405.44 470.19 563.36 681.00 -

60o B 372.52 416.35 552.09 707.17 1010.4 -

C 368.474 401.809 460.590 543.175 642.036 749.318

A - Mizusawa et al. [116]; B - Singh and Chakraverty [177]; C - Present

x

y

b

a

φ

Figure 5.5:A typical skew plate


5.2.2.3 Free Vibration of Trapezoidal Bare Plates

The bare trapezoidal plates having symmetrical geometry for different

values of the ratioa

b(Fig. 5.6) are analyzed for simple supports for a

mesh division of16 × 16 and compared with the results of Saliba [160]

and Geannakakes [54] in the Table 5.11 and the results of the clamped

support conditions of the plates having differenta

band

c

bratios (Fig. 5.7)

are compared with those of Liew and Lim [101] in the Table 5.12. Sal-

iba [160] used the superposition techniques whereas Geannakakes [54]

used Rayleigh-Ritz method together with natural coordinate regions and

normalized characteristic orthogonal polynomials. Liew and Lim [101]

solved the problem by using pb-2 Rayleigh-Ritz method. The results of

simply supported plates are in good agreement; those of Geannakakes [54]

are marginally higher. The clamped plate results are also in good agree-

ment except the higher frequencies of higherc/b ratio because of the high

distortion of the elements.

� ��

� � � ��

��

��

��

� � � � ��

��

��

� � � ��

x

y

b

1

2

3

4 5 6

8

9

10

7

12 11

αa

Figure 5.6: Trapezoidal plate for simple supports showingthe boundary nodal points


Table 5.11: Frequency parametersλ = ωa2(ρ/D)1/2 for alledges simply supported trapezoidal plate

(Ref. Fig. 5.6)

(α = 15o, ν = 0.3)

Mode sequence number

a/b Ref

.

1 2 3 4 5 6

A 0.9512 1.6737 2.8344 3.1078 3.7711 4.5410

1/2 B 0.9517 1.6746 2.8353 3.1090 3.7724 4.5495

C 0.9522 1.6746 2.8323 3.1052 3.7591 4.5380

A 0.7601 0.9147 1.1719 1.5311 1.9912 2.5490

1/4 B 0.7603 0.9150 1.1726 1.5323 2.0620 2.7350

C 0.7600 0.9140 1.1700 1.5272 1.9844 2.5382

A 0.7205 0.7563 0.8159 0.8994 1.0067 1.1377

1/8 B 0.7206 0.7564 0.8163 0.8999 1.0185 1.1729

C 0.7198 0.7535 0.8100 0.8896 0.9925 1.1189

A - Saliba [160]; B - Geannakakes [54]; C - Present

��

��

� � � ��

x

y

b

a

c

Figure 5.7:Trapezoidal plate for clamped supports


Table 5.12: Frequency parametersλ =ωa2

2π

√ρ

Dfor all

edges clamped trapezoidal plate

(Ref. Fig. 5.7)


a/b c/b Ref

.

1 2 3 4 5 6

1.0 0.2 A 11.34 19.89 23.31 30.34 36.22 39.10

B 11.38 19.93 23.34 30.37 36.19 39.06

0.4 A 9.224 15.58 19.89 24.51 29.19 34.22

B 9.177 14.79 19.82 20.24 27.37 29.02

0.6 A 7.560 13.35 16.71 22.42 23.26 29.64

B 7.164 9.84 14.77 16.64 22.32 23.46

0.8 A 6.444 12.27 13.89 19.44 21.50 25.15

B 5.209 7.43 13.05 13.43 15.99 21.27

1.5 0.2 A 19.21 30.57 42.30 43.70 58.72 60.67

B 19.16 30.44 42.08 43.46 58.32 60.14

0.4 A 16.38 24.45 33.73 37.55 45.30 51.11

B 16.34 24.33 33.09 37.39 41.95 50.75

0.6 A 13.74 19.54 27.93 32.93 39.76 42.05

B 13.69 18.47 22.74 30.12 32.83 41.46

0.8 A 11.44 16.60 25.27 28.29 34.02 37.40

B 11.00 13.11 17.97 26.41 28.19 32.47

2.0 0.2 A 29.12 43.43 59.34 66.76 77.08 90.87

B 29.02 43.17 58.88 66.31 76.36 89.82

0.4 A 25.55 35.91 47.02 59.27 60.59 73.47

B 25.48 35.73 46.69 58.53 60.25 70.86

0.6 A 22.10 28.98 37.03 48.01 54.44 62.53

B 22.05 28.74 34.96 40.99 50.82 54.21

0.8 A 18.72 23.58 31.48 42.99 48.05 55.06

B 18.60 21.49 25.48 33.16 44.41 47.92

A - Liew and Lam [101]; B - Present


74 5 6

89

1011

12

3

2

1

b

a

x

y

Figure 5.8: Right triangular plate with boundary nodalpoints

5.2.2.4 Free Vibration of Triangular Bare Plates

The results for the simply supported and clamped right triangular plates

(Fig. 5.8) with different ratios of height (b) to base (a) are compared in the

Table 5.13 with those of Kim and Dickinson [78] and Geannakakes [54].

Kim and Dickinson [78] used Rayleigh-Ritz method but Geannakakes [54]

used Rayleigh-Ritz method along with normalized characteristic orthog-

onal polynomials to obtain the results. The results have compared very

well except few cases of higher frequencies for plates with higherb/a

ratio where the results of Geannakakes [54] are having higher values.


Table 5.13: Frequency parametersλ = ωa2

√ρ

Dfor trian-

gular plates


SC b/a Ref

.

1 2 3 4 5 6

A 49.35 99.76 128.4 169.1 200.3 249.8

1.0 B 49.34 98.69 128.30 167.80 197.46 246.86

C 49.23 98.30 127.51 166.75 195.53 243.80

A 34.28 65.69 91.99 108.0 140.9 161.9

1.5 B 34.28 65.59 91.86 107.48 139.39 162.42

C 34.21 65.35 91.35 106.78 137.85 157.87

A 27.76 49.91 74.85 81.84 107.4 122.2

SS 2.0 B 27.76 49.88 74.88 81.51 108.43 121.65

C 27.70 49.69 74.15 80.91 105.05 118.88

A 24.15 41.15 60.65 72.28 84.92 104.2

2.5 B 24.14 41.14 61.145 71.99 86.49 103.66

C 24.09 40.94 60.06 71.45 82.33 101.31

A 21.85 35.63 51.27 66.73 71.03 92.84

3.0 B 21.84 35.65 52.15 66.67 73.97 94.15

C 21.78 35.42 50.60 65.70 68.35 85.93

A 93.78 157.79 194.77 242.80 277.67 335.77

1.0 B 93.78 157.78 194.76 242.81 277.71 335.84

C 93.49 157.44 193.65 241.65 275.54 332.22

A 53.44 82.43 113.51 121.92 152.09 168.68

CC 2.0 B 53.44 82.44 113.70 122.03 153.32 168.77

C 53.30 81.89 112.41 121.17 149.99 166.38

A 42.75 61.00 80.80 99.90 104.12 129.69

3.0 B 45.75 61.05 81.26 100.27 107.33 132.72

C 42.50 60.27 79.18 98.10 101.22 121.99

A - Kim and Dickinson [78]; B - Geannakakes [54]; C - Present


5.2.2.5 Free Vibration of Bare Annular Sector Plates

Annular sector plates of sector angle90◦ (Fig. 5.9) having different ratios

of inside to outside radii are analyzed for various boundary conditions.

Mukhopadhyay [129] solved the problems by using semianalytic finite

difference method. The results are compared in the Table 5.14 and they

are in good agreement. At higher radii ratio, however, the results have

differed at higher modes for plates having edges other than simply sup-

ported. The different abbreviations used in the table are:

CCCC - All edges clamped

CSSS - One radial edge clamped and all others simply supported

SSCC - Radial edges simply supported and curved edges clamped

SSSS - All edges simply supported

SSFF - Radial edges simply supported and curved edges free

y

Ri

Ro

90o

x

� � ��

� � ��

� � � � ��

� � � � ��

90o

Figure 5.9:Annular sector plate of sector angle90◦


Table 5.14:Values ofω for annular sector plates

(D=1, ρh=1, ν =0, Ro=1, α=90o)

SC Ri/Ro M 1 2 3 4 5

0 A 48.20 86.89 103.02 135.89 161.86

B 48.74 87.39 104.17 136.14 162.93

0.25 A 52.01 87.60 121.18 136.01 166.65

CC

CC

B 52.58 87.72 123.43 136.38 167.21

0.5 A 94.08 113.98 149.27 199.21 246.96

B 95.17 114.26 149.40 199.69 253.01

0 A 31.01 64.24 78.80 108.37 131.91

B 30.94 63.87 78.71 106.76 131.12

CS

SS 0.25 A 32.18 64.41 88.35 107.91 133.91

B 32.21 64.11 88.50 107.19 133.22

0.5 A 48.74 73.34 111.82 166.90 193.20

B 48.69 72.81 111.03 161.22 167.49

0 A 31.01 64.24 78.80 108.37 131.91

B 30.94 63.87 78.71 106.76 131.12

SS

CC 0.25 A 32.18 64.41 88.35 107.91 133.91

B 32.21 64.11 88.50 107.19 133.22

0.5 A 48.74 73.34 111.82 166.90 193.20

B 48.69 72.81 111.03 161.22 167.49

continued in the next page. . . . . .


continued from the previous page. . . . . .

SC Ri/Ro M 1 2 3 4 5

0 A 25.148 56.425 69.105 97.624 120.680

B 25.176 56.273 69.382 96.926 120.539

SS

SS 0.25 A 28.345 56.719 84.687 97.653 123.949

B 28.407 56.507 85.077 97.225 123.669

0.5 A 47.142 68.264 103.224 150.747 165.734

B 47.182 67.986 102.732 149.938 166.556

0 A 6.634 25.221 36.423 54.045 77.609

B 6.150 24.079 35.334 51.142 74.861

SS

FF 0.25 A 5.856 24.916 34.428 53.383 76.348

B 5.664 24.029 33.554 51.236 74.077

0.5 A 4.740 23.355 35.076 52.072 71.832

B 4.642 22.777 33.970 50.632 69.507

A - Mukhopadhyay [129]; B - Present

5.2.2.6 Free Vibration of Bare Elliptical and Circular Plates

The free vibration results of elliptical plates with simply supported and

clamped boundary conditions having different ratios (a/b) of major to

minor axis (a/b = 1; for circular plate) are compared with those of Lam

et al. [82], Leissa and Narita [94], Kim and Dickinson [77] and Gean-

nakakes [54] in the Table 5.15. Lamet al. [82] used two-dimensional or-

thogonal polynomials whereas Leissa and Narita [94] used Bessel func-

tions and their asymptotic expansions. Rayleigh-Ritz method with or-

thogonally generated polynomial admissible functions was used by Kim

and Dickinson [77] and Geannakakes [54] used Rayleigh-Ritz method


Table 5.15: Frequency parametersλ=ωa2(ρh/D)1/2 for el-liptical and circular plates

SC a/b M 1 2 3 4 5 6

A 4.935 13.898 13.898 25.613 25.613 29.724

B 4.935 13.898 13.898 25.613 25.613 29.720

1 C 4.935 13.898 13.898 25.613 25.613 29.720

D 4.938 13.910 13.910 25.623 25.647 29.785

E 4.962 13.890 13.890 25.495 25.579 29.642

A 13.213 23.641 38.325 46.149 57.616 62.764

2 D 13.254 23.648 38.370 46.214 57.948 62.991

E 13.733 24.215 40.555 46.954 61.375 62.060

A 27.080 40.114 56.908 78.315 98.515 104.549

SS 3 D 27.164 40.123 57.099 79.279 98.730 110.496

E 28.793 41.274 63.767 89.222 101.305 118.383

A 45.916 61.953 81.536 106.454 135.629 170.688

4 D 46.076 61.999 82.245 109.023 148.809 171.261

E 49.015 64.332 94.829 125.100 161.641 174.742

continued in the next page . . . . . .


continued from the previous page . . . . . .

SC a/b M 1 2 3 4 5 6

A 10.216 21.260 21.260 34.878 34.878 39.771

1 C 10.215 21.260 21.260 34.877 34.877 39.771

D 10.215 21.258 21.258 34.872 34.892 39.792

E 10.206 21.218 21.218 34.675 34.847 39.639

A 27.477 39.497 55.977 69.855 77.044 88.047

2 D 27.374 39.497 55.995 69.849 77.211 88.183

E 27.604 39.713 57.458 70.471 80.306 87.300

A 56.899 71.590 90.238 113.266 140.746 150.088

CC 3 D 56.792 71.602 90.417 114.169 145.350 150.111

E 57.598 71.667 94.961 123.575 152.369 156.630

A 97.598 115.608 137.268 164.324 195.339 255.094

4 D 97.589 115.714 138.234 167.321 207.443 261.530

E 99.005 115.557 147.038 182.376 222.159 264.272

A - Lam et al. [82]; B - Leissa and Narita [94];

C - Kim and Dickinson [78]; D - Geannakakes [54]; E - Present


Table 5.16:Buckling parameter k = λb2/π2D for uniformlycompressed all edges simply supported rectan-gular plates

Aspect Ratio (a/b)

Sou

rce

0.2 0.3 0.5 0.6 0.8 0.9 1.0 1.2 1.5 2.0

A 26.97 13.17 6.24 5.13 4.19 4.04 3.99 4.13 4.32 3.98

B 27.0 13.2 6.25 5.14 4.20 4.04 4.00 4.13 - -

C - - - - - - - - 4.34 -

D - - - - - - - - - 4.00

A - Present Analysis; B - Timoshenko and Gere [184];

C - Fried and Schmitt [52]; D - C. R. C. of Japan [30]

along with normalized characteristic orthogonal polynomials. The cor-

relation of the results is very good, though results of Geannakakes are

marginally higher.

5.2.3 Stability Analysis of Arbitrary Bare Plates

Stability analysis for the plates with various configurations such as rect-

angular, skew, annular, circular with various boundary conditions is car-

ried out and the buckling parameters are tabulated and compared with the

available results of the other investigators wherever possible.

5.2.3.1 Buckling of Uniaxially Compressed Simply Sup-ported bare Rectangular Plates

Buckling loads for the simply supported bare rectangular plates having

different aspect ratios (a/b) are computed and the results are presented in


Table 5.17:Buckling parameter k = λb2/π2D for uniformlycompressed all edges clamped rectangular plates

Aspect Ratio (a/b)

Sou

rce

0.4 0.5 0.75 1.0 1.25 1.5 1.75 2.0

A 27.76 19.20 11.67 9.99 9.17 8.22 7.96 7.68

B - 19.76 - 10.33 - - - -

C - - 11.40 10.08 - 8.30 - 8.15

D - - - 10.05 9.02 8.11 8.00 7.76

E - - - 10.07 9.25 8.33 8.11 7.88

F - - - 10.48 9.38 8.45 8.17 8.06

G - - - 9.66 9.20 8.30 8.18 7.87

A - Present Analysis; B - Mukhopadhyay [129];

C - C. R. C. of Japan [30];

D - Zhang and Kratzig [196]; E - Levy [95];

F - Maulbetsch [106]; G - Faxen [51]

the Table 5.16 in the form of buckling parameterk = λb2/π2D wherea

andb are the length and width of the plate andD is the flexural rigidity

of the plate given byD = Et3/12(1 − ν2). The results are compared

with the analytical results of Timoshenko and Gere [184]. Few of the

buckling parameters are compared with the finite element results of Fried

and Schmitt [52] and the handbook of structural stability [30]. The results

are found to be in excellent agreement.


Table 5.18: Buckling parameter k = λb2/π2D for all edgesclamped rectangular plates with biaxial uniformcompression

Aspect Ratio (a/b)

Sou

rce

0.5 0.8 1.0 1.5 2.0

A 15.56 6.91 5.26 4.09 3.89

B - - 5.33 4.11 3.87

C - - 5.33 4.14 3.94

D 15.76 - 5.30 - 3.65

A - Present Analysis;

B - Zhang and Kratzig [196];

C - Ng [138]; D - Mukhopadhyay [129]

5.2.3.2 Buckling of Uniaxially Compressed Clamped BareRectangular Plates

Rectangular bare plates of different aspect ratios with all edges clamped

are considered for the buckling load analysis and the results are compared

in the Table 5.17 with the semianalytical results of Mukhopadhyay [129]

and finite element results of Zhang and Kratzig [196] and series solution

of Levy [95], Maulbetsch [106] and Faxen [51].

5.2.3.3 Buckling of Biaxially Compressed Clamped BareRectangular Plates

The buckling load results of the biaxially compressed clamped rectan-

gular plates having various aspect ratios are presented in the Table 5.18.

The results obtained by Zhang and Kratzig [196] (finite element method),


Table 5.19:Buckling parameter k = λb2/π2D for uniaxiallycompressed all edges simply supported rectan-gular plates with triangular load i.e; α = 1 inthe expressionNx = N0(1− α

y

b)

Aspect Ratio (a/b)

Sou

rce

0.4 0.6 0.8 1.0 1.5 2.0

A 15.05 9.70 8.10 7.78 8.30 7.73

B 15.1 9.7 8.1 7.8 8.4 -

A - Present Analysis;

B - Timoshaenko and Gere [184];

Ng [138] (collocation least square method) and Mukhopadhyay [129]

(semianalytical method) are compared in the table along with the present

ones which are found to agree well.

5.2.3.4 Buckling of Simply Supported Bare RectangularPlates Uniaxially Compressed by Triangular Load

The buckling parameters for the bare rectangular plates with varying as-

pect ratios subjected to uniaxial compressive triangular load i.e;α = 1 in

the expressionNx = N0(1 − αy

b) are presented in the Table 5.19 along

with the analytical results of Timoshenko and Gere [184] where excellent

agreement is obtained.

5.2.3.5 Buckling of Uniaxially Compressed Bare Skew Plates

Numerical results for the buckling under uniaxial compression of the

skew rhombic (aspect ratio = 1) plates having angles of skew varying


Table 5.20: Buckling parameter k = λb2/π2D for uniaxi-ally compressed all edges simply supported andclamped skew plates (Aspect ratio = 1.0, ν =0.3)

BC φ A B C D E F G

0 3.98 4.00 4.00 4.00 4.00 - 4.00

15 4.37 4.53 4.412 4.245 4.38 - 4.32

30 5.82 6.17 5.91 5.12 5.61 - 5.55

45 9.64 11.00 10.22 10.22 - - 9.07

Sim

ply

Sup

port

ed

60 20.98 - 24.56 - - - -

0 9.99 10.06 10.07 10.06 - 10.06 -

15 10.72 10.93 - 10.44 - 10.60 -

30 13.30 14.00 13.53 13.51 - 13.39 -

45 19.30 21.70 20.05 20.08 - 20.07 -Cla

mpe

d

60 36.32 - 42.38 - - - -

BC - Boundary Conditions; φ = Skew Angle;

A - Present Analysis; B - Mukhopadhyay and Mukherjee [135];

C - Mizusawa et al. [118]; D - Fried and Schmitt [52];

E - Yoshimura and Iwata [193]; F - Wittrick [190];

G - Durvasula and Nair [49]


Table 5.21: Buckling parameter k = (Nr)cra2/D for

uniformly compressed simply supported andclamped circular plates (ν = 0.3)

Boundary Mesh Present Timoshenko

Condition Size Analysis and Gere [184]

4× 4 16.0513

6× 6 15.0061

8× 8 14.7807

10× 10 14.7091

12× 12 14.6803

14× 14 14.666

Cla

mpe

d

16× 16 14.656 14.68

4× 4 3.83696

6× 6 4.02363

8× 8 4.09947

10× 10 4.13905

12× 12 4.16403

14× 14 4.1824Sim

ply

Sup

port

ed

16× 16 4.19765 4.20

from 0◦ to 60◦ with all edges simply supported and clamped are pre-

sented in the Table 5.20 along with the results of other investigators such

as Mukhopadhyay and Mukherjee [135], Mizusawa et al. [118], Fried

and Schmitt [52], Yoshimura and Iwata [193], Wittrick [190], Durvasula

and Nair [49]. The results are in good agreement. Best agreement is

obtained with Durvasula.

5.3 Arbitrary Stiffened Plates 151

5.2.3.6 Buckling of Uniformly Compressed Bare CircularPlates

The buckling loads for the simply supported and the clamped bare circular

plates are computed and the results are presented in the form of the pa-

rameterk = (Nr)cra2/D where(Nr)cr is the critical compressive force

uniformly distributed around the edge of the plate,a is the radius of the

circular plate andD is the flexural rigidity of the plate. The results are

presented in the Table 5.21 for various mesh divisions of the whole plate

to study the convergence of the buckling parameter and they are com-

pared with the analytical values of Timoshenko and Gere [184]. There is

excellent agreement between the results.

5.3 Arbitrary Stiffened Plates

In this section the problems relating to the arbitrary stiffened plates are

analyzed when they are subjected to static, dynamic and buckling loads

and are presented in the subsequent subsections.

5.3.1 Static Analysis of Arbitrary Stiffened Plates

Stiffened plates of different shapes with eccentric as well as concentric

stiffeners with different boundary conditions and loadings are analyzed

and results are compared with those available. The results are presented

in tabular or graphical forms depending on the suitability for the purpose

of comparing them with those of others.


5.3.1.1 Square Plate with a Central Stiffener

A simply supported square plate with a central stiffener as shown in

the Fig. 5.10 is considered. The problem is solved both for eccentri-

cally and concentrically placed stiffeners. The plate and the stiffeners

have the same material properties with elastic modulusE = 1.1721 ×105 N/mm2(17× 106 psi) and Poisson’s ratioν = 0.3. The plate is sub-

jected to a uniformly distributed load of6.89476×10−3 N/mm2(1.0 psi).

The plate is analyzed by the present method using various mesh divisions

and the analysis is carried out with the mesh division of16 × 16 for the

whole plate. The deflection curves along the two centre lines are com-

pared in Table 5.22 and Fig. 5.11 with the results obtained by Rossow

and Ibrahimkhail [157] who applied the constraint method of the finite

element analysis. In Table 5.22, results from NASTRAN and STRUDL

E = 11721 x 10 N/mm5 2

ν = 0.3

0.254

0.2542

.54

Y

X

25.4

A A

25

.4

All dimensions are in mm.

SECTION AT A-A

Figure 5.10: Simply supported square plate with a centralstiffener


Figure 5.11:Variation of deflection along centrelines of sim-ply supported square plate with a central stiff-ener

are also indicated. The agreement between the results is excellent.

To test the convergence of the results obtained by the present method,

the deflection, moment and top fiber stress of the plate skin at the central

location of the eccentrically stiffened square plate for varying number of

mesh divisions are computed and presented in the Table 5.23 from which

the attainment of good convergence for all is evident.


Table 5.22: Deflection at the centre of simply supportedsquare stiffened plate(×104 mm.)

Distributed LoadSource

Eccentric Concentric

Rossow and Ibrahimkhail [157] 34.722 115.722

NASTRAN 37.846 -

STRUDL - 115.291

Present Method 34.696 115.875

Table 5.23: Convergence of deflection(w), plate moment(My) and plate stress(σx) of the eccentricallystiffened square plate at its centre with differentmesh divisions.

Mesh Deflection(w) Moment(My) Stress(σx)

Division ×104(mm.) ×103(N −mm/mm) (N/mm2)

2× 2 32.614 6.005 4.532

4× 4 34.163 21.943 9.339

6× 6 34.493 24.412 9.992

8× 8 34.595 25.275 10.189

10× 10 34.646 25.697 10.271

12× 12 34.671 25.933 10.3097

14× 14 34.671 26.075 10.3297

16× 16 34.696 26.169 10.3394


Table 5.24:Central deflection of rectangular cross-stiffenedplate (×103 mm.)

Concentrated Load Distributed LoadSource

Eccentric Concentric Eccentric Concentric

Rossowand

Ibrahimkhail[157]

32.258 87.986 224.790 611.505

NASTRAN 31.496 - 221.336 -

Chang [32] 31.648 87.986 228.498 611.556

STRUDL - 87.960 - 612.648

Presentmethod

31.445 87.986 226.873 611.454

5.3.1.2 Cross Stiffened Rectangular Plate

A simply supported rectangular plate with a central stiffener in each di-

rection shown in Fig. 5.12 is analyzed for a uniform pressure of6.89 ×10−2 N/mm2 (10.0 psi) as well as for a concentrated load of4.448 kN

(1.0 kip) at the centre of the plate with a mesh division of16 × 16. The

material properties areE = 2.0684 × 105 N/mm2 (30 × 106 psi) and

ν = 0.3 for both the plate and the stiffener. The same problem is solved

by Rossow and Ibrahimkhail [157] by applying the constraint method to

the finite element analysis and by Chang [32] using conventional finite

element method. The deflection and bending moments along the differ-

ent lines for eccentrically as well as concentrically stiffened plates sub-

jected to uniformly distributed load as well as concentrated load obtained

by different methods are compared in Figs. 5.13-5.23. Additionally a


6.35

12.7

12

7

Section at A-A

E = 2.0684 x 10 N/mm5 2

ν = 0.3

6.35

12.7

76

.2

Section at B-B

Sti

ffen

ers

A

762

y

B B

A

15

24

x


Figure 5.12:Simply supported rectangular plate with a cen-tral stiffener in each direction

comparison of the central transverse displacements obtained by various

methods is made in the Table 5.24.


Figure 5.13: Deflection at x = 190.5 mm. and x = 381.0 mm.for plate with two concentric stiffeners, underdistributed load

Figure 5.14: Deflection at x = 190.5 mm. and x = 381.0 mm.for plate with two eccentric stiffeners, underdistributed load


Figure 5.15: Moment Mxx at y = 381.0 mm. and y = 762.0mm. for plate with two concentric stiffeners,under distributed load

Figure 5.16: Moment Mxx at y = 381.0 mm. and y = 762.0mm. for plate with two eccentric stiffeners,under distributed load


Figure 5.17: Moment Myy at x = 190.5 mm. and x = 381.0mm. for plate with two concentric stiffeners,under distributed load

Figure 5.18: Moment Myy at x = 190.5 mm. and x = 381.0mm. for plate with two eccentric stiffeners,under distributed load


Figure 5.19:Deflections at x = 190.5 mm. and x = 381.0 mm.for plate with two stiffeners under concentratedload

Figure 5.20: Moment Mxx at y = 381.0 mm. and y = 762.0mm. for plate with two concentric stiffenersunder concentrated load


Figure 5.21: Moment Mxx at y = 381.0 mm. and y = 762.0mm. for plate with two eccentric stiffenersunder concentrated load

Figure 5.22: Moment Myy at x = 190.5 mm. and x = 381.0mm. for plate with two concentric stiffenersunder concentrated load


Figure 5.23: Moment Myy at x = 190.5 mm. and x = 381.0mm. for plate with two eccentric stiffenersunder concentrated load

5.3.1.3 Rectangular Multi-Stiffened Plate

A rectangular steel plate stiffened with nine number of equispaced T-

stiffeners along the short span as shown in the Figure 5.24 is analyzed

by the present method with a mesh division of16 × 16 for the whole

plate. The plate is subjected to a concentrated load of 10 kN at its centre.

The long edges are considered as simply supported and the short ones

free. The deflections and bending moments along the central line in the

longitudinal direction are computed and the results are compared graph-

ically in the Figures 5.25-5.26 with those of Smith [179] who solved the

problem based on generalized slab beam analysis.


E = 2.06843 x 10 N/mm

ν = 0.3

5 2

y

A A

25

40

x

76.2

Section at A-A

14

.22

4

15

2.4


4.7752

508

Figure 5.24:Rectangular multi-stiffened plate


Figure 5.25: Variation of deflection along the centre line ofthe rectangular multi-stiffened plate

Figure 5.26:Variation of plate moment Mx along the centreline of the rectangular multi-stiffened plate


5.3.1.4 Rectangular Slab with Two Edge Beams

Three numbers of specimens of slabs having edge beams at two opposite

edges made of Araldite as shown in the Figure 5.27 are considered for the

analysis using the present method employing a mesh division of16× 16

for the whole plate. Each of the specimens is considered to be free at the

edges where the beams are placed and simply supported along the edges

transverse to the beams. Two concentrated loads of equal magnitude are

applied at the centre of each of the beams. The Table 5.25 describes the

details of the geometrical and the material properties of the specimens

along with the magnitude of the concentrated loads applied to each one

of them. The deflections along different sections and the normal stresses

at the bottom of the beam along its length are compared graphically with

the finite element results of Mukhopadhyay and Satsangi [136] in the

Figures 5.28-5.31. Additionally, the numerical values of the deflections

and the stresses at some typical points are compared with the theoretical

and experimental results of Allen [5] along with the finite element results

of Mukhopadhyay and Satsangi [136] and the spline finite strip results

of Sheikh [172] in the Table 5.26. The agreement between the results is

found to be reasonably close.


Table 5.25: Geometrical and material properties of thespecimens of the rectangular slab with edgebeams

(Ref. Figure 5.27)

Specimen a b d L

No. (mm) (mm) (mm) (mm)

SPEC1 138.58 4.52 19.10 131.78

SPEC2 138.58 4.52 15.24 131.78

SPEC3 138.58 4.52 11.43 131.78

Specimen t E ν P

No. (mm) (N/mm2) (N )

SPEC1 4.445 2977 0.35 176.59

SPEC2 4.445 2977 0.35 161.03

SPEC3 4.445 2977 0.35 61.39


t

C

A

A

B

B

C

d

a

L

x

y

b

Figure 5.27:Rectangular slab with two edge beams

Figure 5.28: Deflection along A-A of the slab with edgebeams


Figure 5.29: Deflection along B-B of the slab with edgebeams

Figure 5.30: Deflection along C-C of the slab with edgebeams


Figure 5.31: Stress at the beam soffit of the slab with edgebeams

Table 5.26: Deflection and stress at the beam soffit of therectangular slab with edge beams

Source Specimen No.

of Result 1 2 3

Deflection Present Method 0.4216 0.7214 0.5842

(mm.) Sheikh [172] 0.4191 0.7290 0.6045

at 25.4 mm. Experimental [5] 0.4521 0.6477 0.5232

from the Theoretical [5] 0.4064 0.7087 0.5918

beam centre Mukhopadhyay 0.4572 0.6477 0.6756

and Satsangi [136]

Stress Present Method 14.5410 20.1534 12.3485

(N/mm2) Sheikh [172] 15.5615 21.7323 13.5758

at the Experimental [5] 15.3339 25.0568 12.9759

beam centre Theoretical [5] 13.3000 18.3883 11.4798

Mukhopadhyay 14.8858 20.1396 12.5071

and Satsangi [136]


5.3.1.5 Stiffened Skew Bridge Deck

A perspex scale model of a450 skew bridge deck (Fig. 5.32) having five

equispaced longitudinal beams and two transverse edge beams supported

at the ends of each of the equally spaced longitudinal beams, the vertical

deflection and movement in the direction of the axes of the transverse

beams at these points being restrained and subjected to a concentrated

vertical load of 100N acting at the midpoint of one of the longitudinal

free edges is analyzed with a mesh of16× 16 applied to the entire plate.

The transverse and the longitudinal beams are having depths and widths

of 22.0mm and 6.0mm respectively. The Young’s modulus and Pois-

son’s ratio for the 3-mm thick slab are 3354.0N/mm2 and0.390 respec-

tively and those for the beams are 3176.0N/mm2 and 0.335 respectively.

The vertical deflections along the loaded free edge and along the centre

line in the transverse direction are compared in Fig. 5.33-5.34 with the

experimental results and also with those of Just [74]. The agreement is

reasonably good.


Y

jY

j

A A645o

3

3

300

>

ULongitudinal Beams

:yTransverse Beams

¸

®

¸

¸

¸

®

®

®

®¸

®¸

¸¸X

X-

-

Y

Y

?

?

6

¾- 6

3

22

Section at X-X

6

?

?

¾- 6

Section at Y-Y

3

22


9090

9090

45

45

B

B

Figure 5.32: Skew bridge deck with beams in both direc-tions


Figure 5.33: Deflection along A-A of the stiffened skewbridge deck

Figure 5.34: Deflection along B-B of the stiffened skewbridge deck


5.3.1.6 Stiffened Curved Bridge Deck

A perspex model of a horizontally curved bridge deck (Fig. 5.35) with

two curved girders having free curved boundaries and simply supported

on straight edges subjected to a concentrated load of100 N applied at dif-

ferent points of the deck is analyzed with different mesh sizes. The con-

vergence study for this problem for the load placed at position 4 and 7 for

the deflection at various positions on the outer girder and the outer edge

and that of the moments at the centre are presented in Table 5.27-5.28

respectively. The convergence for the deflection and moment is found to

be good. The deflections obtained with a mesh division of16 × 16 at

the inner edge, inner girder, outer girder and outer edge are compared

with the theoretical and experimental results of Joshipara [73], finite

element results of Kalani et al. [75] and Satsangi and Mukhopadhyay

[163] in Table 5.29-5.30. The elastic modulus and Poisson’s ratio are

3.6 × 104Kg/cm2 and 0.38 respectively. At all places except the inner

edge, present results compare favorably with the experimental ones.


Figure 5.35: Curved bridge deck with two circumferentialgirders


Table 5.27: Convergence of deflection at outer girder andouter edge

Load Mesh Outer Girder Outer Edge

at B C D B C D

4× 8 1.4453 2.0520 1.4453 1.5897 2.2318 1.5897

8× 8 1.4464 2.0536 1.4464 1.5905 2.2327 1.5905

4 12× 8 1.4464 2.0536 1.4464 1.5899 2.2315 1.5899

12× 12 1.4528 2.0633 1.4528 1.5985 2.2436 1.5985

16× 16 1.4552 2.0669 1.4552 1.6015 2.2475 1.6015

4× 8 2.7119 3.4075 2.2145 3.6265 4.4724 2.8711

8× 8 2.7147 3.4098 2.2158 3.6301 4.4752 2.8726

7 12× 12 2.7338 3.4288 2.2273 3.6525 4.4989 2.8870

16× 16 2.7412 3.4361 2.2317 3.6614 4.5080 2.8926

Table 5.28:Convergence of moments at the centre

Load at Mesh Mx My Mxy

4× 8 0.3191 1.4521 0.2795

4 8× 8 0.3469 1.4614 0.2749

12× 12 0.3415 1.4583 0.2908

16× 16 0.3377 1.4501 0.2975

4× 8 2.6735 0.0706 0.0542

7 8× 8 2.7075 0.1086 0.0965

12× 12 2.6294 0.0941 0.0263

16× 16 2.6199 0.0882 0.0269


Table 5.29: Deflection (mm.) at inner edge and inner girder

Load Source of Inner Edge Inner Girder

at Results B C D B C D

Experimental [73] 2.12 1.54 0.84 1.16 1.22 0.70

Theoretical [73] 1.99 1.47 0.83 1.08 1.20 0.72

1 FEM [75] 2.03 1.46 0.89 1.13 1.24 0.74

FEM [163] 2.31 1.37 0.81 1.04 1.10 0.62

Present 2.03 1.42 0.80 1.02 1.14 0.67

Experimental [73] 1.28 1.92 1.30 1.38 2.04 1.39

Theoretical [73] 1.20 1.79 1.20 1.30 1.93 1.30

4 FEM [75] 1.32 1.88 1.32 1.38 1.99 1.38

FEM [163] 1.22 1.84 1.22 1.31 1.96 1.51

Present 1.14 1.70 1.14 1.24 1.85 1.24

Experimental [73] 0.40 0.65 0.42 1.07 1.57 1.03

Theoretical [73] 0.44 0.67 0.49 1.11 1.49 1.01

7 FEM [75] 0.49 0.72 0.58 1.26 1.68 1.16

FEM [163] 0.22 0.34 0.25 0.87 1.13 0.72

Present 0.35 0.55 0.41 1.07 1.45 0.99

Experimental [73] 0.46 0.60 0.43 1.54 2.12 1.56

Theoretical [73] 0.47 0.65 0.47 1.63 2.29 1.63

10 FEM [75] 0.52 0.69 0.52 1.86 2.69 1.86

FEM [163] 0.22 0.32 0.22 1.55 2.17 1.55

Present 0.30 0.43 0.30 1.60 2.24 1.60


Table 5.30: Deflection (mm.) at outer girder and outer edge

Load Source of Outer Girder Outer Edge

at Results B C D B C D

Experimental[6] 0.38 0.64 0.44 0.16 0.34 0.36

Theoretical[6] 0.44 0.67 0.50 0.25 0.47 0.41

1 FEM[8] 0.43 0.61 0.47 0.21 0.41 0.43

FEM[13] 0.21 0.33 0.23 0.04 0.03 0.07

Present 0.35 0.55 0.41 0.13 0.30 0.30

Experimental[6] 1.60 2.16 1.60 1.74 2.42 1.74

Theoretical[6] 1.49 2.12 1.49 1.63 2.29 1.63

4 FEM[8] 1.69 2.29 1.69 1.83 2.59 1.88

FEM[13] 1.42 2.03 1.42 1.54 2.17 1.54

Present 1.45 2.06 1.45 1.60 2.24 1.60

Experimental[6] 2.87 3.48 2.27 3.81 4.41 2.92

Theoretical[6] 2.63 3.28 2.12 3.46 4.24 2.70

7 FEM[8] 3.09 3.84 2.57 4.09 4.95 3.23

FEM[13] 2.53 2.97 1.74 3.51 4.00 2.31

Present 2.74 3.43 2.23 3.66 4.51 2.89

Experimental[6] 4.45 6.62 4.24 6.16 9.82 6.06

Theoretical[6] 4.24 6.16 4.24 5.66 8.94 5.66

10 FEM[8] 4.98 7.03 4.98 6.82 10.43 6.82

FEM[13] 4.77 7.01 4.77 6.58 10.91 6.58

Present 4.50 6.54 4.50 6.09 9.66 6.09


0.254

0.254

2.5

4

Section at A-A

25.4

Stiffener

A A


Figure 5.36:Simply supported circular plate with a stiffeneralong the diameter

5.3.1.7 Circular Plate with a Central Stiffener

A simply supported circular plate of diameter 25.4 mm. (1.0in.) with a

central stiffener (both eccentrically and concentrically placed) as shown

in the Fig. 5.36 subjected to a uniform pressure of 6.8947 kPa (1.0psi)

keeping all the other material and the geometric properties of the plate

and stiffener same as that of the square stiffened plate solved by Rossow

and Ibrahimkhail [157] is analyzed using a mesh of16×16 by the present

method and the deflection curves along the two centre lines are plot-

ted in the Figure 5.37. The central deflections of the circular stiffened


plate are found to be1.1496 × 10−4 and3.6068 × 10−4 whereas those

of Rossow and Ibrahimkhail [157] for the square stiffened plate being

1.367× 10−4 and4.556× 10−4 for the eccentric and the concentric stiff-

eners respectively. Hence the central deflection of a centrally stiffened

plate is reduced by approximately 16% and 21% for eccentric and con-

centric stiffeners respectively when the plate configuration is changed

from the square one to the circular one keeping the side of the square

plate and the diameter of the circular one same, the reduction in the plate

area being about 21%. This result is presented for the first time.

Figure 5.37: Deflection along diameters of a simply sup-ported circular plate under distributed load


5.3.2 Free Vibration Analysis of Arbitrary StiffenedPlates

Free vibration analysis of arbitrary stiffened plates having various shapes

and boundary conditions are carried out and results are compared with

those available. Results are presented in tabular form. Results tabulated

are obtained with a mesh division of16 × 16 for the whole plate unless

otherwise mentioned.

5.3.2.1 Free Vibration of Concentrically Stiffened ClampedSquare Plate

A square plate clamped in all edges having a centrally placed eccentric

stiffener as presented by Nair and Rao [137] using the package STIFPT1

has been analyzed by the present method using various mesh divisions

ranging from8 × 8 to 24 × 24 for the whole plate. The problem has

also been solved by Mukherjee and Mukhopadhyay [122], Mukhopad-

hyay [131], and Sheikh and Mukhopadhyay [173] using the finite element

method, the semi-analytical method and the spline finite strip method re-

spectively. The first six frequencies are compared with those of previous

investigators in Table 5.31. The agreement is excellent with Nair and

Rao [137] and Sheikh and Mukhopadhyay [173]. The results of Mukher-

jee and Mukhopadhyay [122] differ from present ones because they used

a very coarse mesh (6 × 6 for the entire plate) in their finite element

analysis. Similarly slightly varying results are obtained by Mukhopad-

hyay [131] because the inplane displacement is not considered in the

analysis. The Table 5.31 also presents the convergence study showing

good convergence of the results.

Mukherjee and Mukhopadhyay [122] analyzed this plate problem to

investigate the possible weaknesses of the isoparametric element which


Table 5.31:Frequency in Hz of a clamped square plate witha central concentric stiffener

plate size = 600 mm× 600 mm, plate thickness = 1.0 mmν = 0.34, ρ = 2.78× 10−6kg/mm3, E = 6.87× 107 N/mm2

As = 67.0 mm2, Is = 2290 mm4, Js = 22.33 mm4

Met

hod

MeshMode sequences

size 1 2 3 4 5 6

8× 8 49.453 62.656 71.857 81.373 107.896 113.321

12× 12 49.960 63.199 73.492 83.416 110.791 116.821

Pre

sent

16× 16 50.152 63.410 74.132 84.243 111.990 118.338

20× 20 50.244 63.513 74.442 84.649 112.585 119.103

24× 24 50.295 63.571 74.614 84.877 112.919 119.537

A 50.45 63.71 75.16 85.50 113.69 120.89

B 51.30 65.53 80.473 95.28 122.90 141.70

C 48.54 60.80 72.26 82.60 110.77 117.93

D 50.43 63.72 75.07 85.46 113.96 120.82

A - Nair and Rao [137]; B - Mukherjee and Mukhopadhyay [122]

C - Mukhopadhyay [131]; D - Sheikh and Mukhopadhyay [173]

has the undesirable locking effects for very thin plates. Since the width to

thickness ratio of this plate is600 the plate can be considered as very thin.

They observed that the results did not converge until the shear stiffness

of the very thin plate was reduced artificially to that of a thin plate which

clearly depicts the effect of shear locking inherent in the element based

on Mindlin plate bending theory.


3-6ρ = 2.78 10 x Kg/mm

600 mm

E = 6.87 x 10 N/mm

ν = 0.34

A A

1 mm

3 mm

SECTION AT A-A

60

0 m

m

20

mm

7 2

Figure 5.38: Clamped square plate with a central eccentricstiffener

5.3.2.2 Free Vibration of Eccentrically Stiffened ClampedSquare Plate

Rao et al. [154] studied the effect of the eccentricity of the stiffener on the

frequencies by investigating the same square plate as in the previous ex-

ample but with a stiffener of20mm×3mm size (Fig. 5.38). Sheikh [173]

solved this problem by the spline finite strip method. Using the present

element, the problem is analyzed for the free vibration and the results are

compared with those of Sheikh [173] in the Table 5.32 where the results


are found to agree well.

Table 5.32:Frequency in Hz of a clamped square plate witha central eccentric stiffener

Method Mode sequences

1 2 3 4 5 6

Present 54.371 65.101 79.808 84.584 116.531 118.676

Sheikh [173] 54.759 65.435 80.805 85.745 118.521 120.966

5.3.2.3 Free Vibration of Cross Stiffened Square Plate

A square plate having central concentric stiffeners placed in both direc-

tions with all its edges clamped is analyzed by the present method using

a mesh division of16 × 16. Mizusawa et al. [116] solved this problem

using the B-spline functions in both directions. The same problem was

analyzed by Shastry and Rao [170] using high precision elements for the

plate and the stiffener and by Sheikh [173] employing the spline finite

strip method. The results obtained by the present method are presented

in the Table 5.33 and compared with those of others and found to be in

good agreement.

5.3.2.4 Free Vibration of Eccentrically Stiffened Rectangu-lar Plate

A rectangular plate with simply supported edges having a central L-

shaped eccentric stiffener in the shorter span direction as presented by

Madsen [104] is analyzed. The results are compared with those of Mad-

sen [104] and Sheikh and Mukhopadhyay [173] who used Rayleigh-Ritz


Table 5.33: Frequency parameter [ω(a/π)2√

ρt/D] ofsquare cross-stiffened plate with concentric stiff-eners having all edges clamped

(EIs/Da = 10.0, As/at = 0.1)


1 2 3 4 5 6 7

A 11.35 11.70 12.65 12.65 17.42 24.81 24.96

B 10.97 11.79 12.61 12.61 17.74 24.58 24.68

C 10.97 11.80 12.62 12.62 17.67 24.64 24.81

D 10.97 11.87

A - Present; B - Mizusawa et al. [116]; C - Sheikh [173];

D - Shastry and Rao [170]

Table 5.34: Frequency in Hz of a simply supported rect-angular plate with a central L-shaped eccentricstiffener in the shorter span direction

ν = 0.3, ρ = 7825 Kg/m3, E = 2.051× 1011 N/m2


1 2 3

Madsen (Raleigh-Ritz Method) [104] 147.92 175.04 338.71

Madsen (Differential Equations) [104]144.06 175.16 335.93

Sheikh and Mukhopadhyay [173] 143.03 175.02 336.37

Present 142.76 174.85 334.83


method and the spline finite strip method respectively in Table 5.34.

Madsen [104] in his analysis neglected the in-plane displacements whereas

Sheikh and Mukhopadhyay [173] analyzed by both neglecting and in-

cluding the in-plane displacements. The present investigation has been

carried out by including the in-plane displacements. The results agree

well with those of Sheikh and Mukhopadhyay [173] and with the solu-

tion of the differential equations of Madsen [104]. It appears that for this

problem inplane displacements do not influence the natural frequencies

significantly.

5.3.2.5 Free Vibration of Rectangular Multi-stiffened Plates

Rectangular plates with unidirectional concentric stiffeners for varying

number of panels and different aspect ratios (Fig. 5.40) are analyzed us-

ing a mesh division of16 × 16 by the present method and the first four

number of frequencies are compared with the finite difference results of

Wah [187] and the finite element results of Mukherjee and Mukhopad-

hyay [122] in the Table 5.35. The results are having good co-relation

between them.

5.3.2.6 Free Vibration of Multi-stiffened Skew Plates

Simply supported skew plates having concentric stiffeners in one direc-

tion presented by Bhandari et al. (1979) are analyzed for different skew

angles using mesh divisions of20 × 20 and the first four natural fre-

quencies of plates upto45o skew angles are compared with those of

other investigators in Table 5.36 which is found to agree well. Bhan-

dari et al. [21] analyzed the problem by using Lagrange’s equations and


Table 5.35: Frequency parameter[ω(a/R)2√

ρt/D] of sim-ply supported multi-stiffened rectangular platewith concentric stiffeners in one direction

N Ma

b= 1

a

b=

1

2

A B C A B C

1 2.602 2.697 2.611 1.345 1.363 1.348

3 2 5.375 5.833 5.378 2.602 2.678 2.610

3 8.043 7.989 8.062 4.346 4.440 4.343

4 10.340 10.103 10.346 5.375 5.110 4.878

1 1.464 1.493 1.470 0.757 0.767 0.754

2 3.026 3.170 3.027 1.464 1.488 1.460

4 3 4.556 4.700 4.569 2.447 2.486 2.443

4 5.847 6.143 5.798 3.026 3.000 2.739

1 0.163 0.166 0.164 0.084 0.086 0.085

12 2 0.336 0.349 0.337 0.163 0.168 0.152

3 0.508 0.521 0.510 0.272 0.277 0.271

4 0.651 0.670 0.647 0.336 0.318 0.281

N - No. of panels; M - Mode sequence number;

A - Wah [187]; B - Mukherjee and Mukhopadhyay [122];

C - Present


E = 2.051 x 10 N/mm11 2

ρ = 7825ν = 0.3

Kg/m3

650 mm

AA

4.98 mm

3 mm

SECTION AT A-A

3 mm

10 mm

45

0 m

m

20

mm

Figure 5.39:Simply supported rectangular plate with a cen-tral eccentric stiffener


b

a = 609.6 mm x no. of panels (R)

A A

25.4 mm

25.4 mm

10

1.6

mm

SECTION AT A-A

609.6 mm

Figure 5.40: Simply supported rectangular plate with con-centric stiffeners in one direction


Table 5.36: Frequency parameter [ω(a/R)2√

ρh/D] of asimply supported multi-stiffened skew plate hav-ing concentric stiffeners in one direction

a/b = 1 a/b = 2

S M A B C D A B C D

1 1.447 1.493 1.493 1.489 0.747 0.767 0.767 0.761

0o 2 2.990 3.170 3.170 3.065 1.447 1.488 1.488 1.473

3 4.562 4.700 4.700 4.590 2.437 2.486 2.486 2.453

4 5.780 6.143 6.143 5.843 2.727 3.000 3.000 2.755

1 1.580 1.581 1.564 0.810 0.813 0.815 0.858

15o 2 3.263 3.372 3.221 1.525 1.549 1.552 1.426

3 4.922 4.934 4.789 2.659 2.651 2.659 2.626

4 6.102 6.416 5.800 2.789 2.842 2.849 2.787

1 1.986 1.895 1.979 0.989 0.995 0.986 1.050

30o 2 3.890 4.029 4.064 1.736 1.800 1.775 1.900

3 5.809 5.736 5.605 2.992 3.139 3.108 3.185

4 6.923 7.316 6.592 3.303 3.255 3.276 3.246

1 2.964 2.649 2.828 1.445 1.431 1.409 1.508

45o 2 5.342 5.431 5.560 2.295 2.403 2.310 2.320

3 8.016 7.866 7.793 3.613 3.950 3.786 3.661

4 8.925 9.376 8.457 4.903 4.735 4.791 4.828

1 4.888 3.547

60o 2 8.507 4.360

3 12.353 5.854

4 13.857 7.833

A - Bhandari et al. [21];

B - Mukherjee and Mukhopadhyay [122]

(Slope along the boundaries not restrained);

C - Mukherjee and Mukhopadhyay [122]

(Slope along the boundaries restrained); D - Present Analysis;


employing beam characteristic functions in oblique coordinates whereas

Mukherjee and Mukhopadhyay [122] solved the problem by finite element

method. The results for plates with60o skew angle being new the same

could not be compared.

a = 609.6 mm x no. of panels (R)

25.4 mm

25.4 mm

10

1.6

mm

SECTION AT A-A

609.6 mm

A

A

b

θ

Figure 5.41: Simply supported skew plate with concentricstiffeners in one direction


5.3.2.7 Free Vibration of Trapezoidal Stiffened Plates

The clamped square plate with the central concentric stiffener solved in

the previous example (Section 5.3.2.1) has been extended here to the

trapezoidal plate and analyzed for differentc/b ratios wherec andb are

the two parallel top and bottom sides of the plate (Fig. 5.42) respectively.

The first six natural frequencies are presented in the Table 5.37. It may

be observed that all the frequencies tend to increase as thec/b ratio de-

creases and this increase is more prominent in the case of the lower fre-

quencies. This result is presented for the first time.

c

600 mm

600

mm

(b)

Figure 5.42:All edges clamped trapezoidal plate with a con-centric stiffener in one direction


Table 5.37: Frequency in Hz of all edges clamped trape-zoidal plate with a central concentric stiffener

plate thickness = 1.0 mmν = 0.34, ρ = 2.78× 10−6kg/mm3, E = 6.87× 107 N/mm2

As = 67.0 mm2, Is = 2290 mm4, Js = 22.33 mm4

c/b Mode sequences

ratio 1 2 3 4 5 6

0.2 100.297 118.819 155.765 161.276 214.771 216.551

0.4 85.471 103.048 125.777 139.179 169.577 185.725

0.6 71.729 88.506 100.451 117.019 137.554 153.006

0.8 59.620 75.086 83.927 97.513 121.217 130.518

5.3.2.8 Free Vibration of Concentrically Stiffened AnnularSector Plates

Two sets of annular sector plates having stiffeners concentrically placed

along the circumferential directions (Fig. 5.43) and supported on two

radial edges are considered for the free vibration analysis by the present

method. In one of the sets the stiffeners are placed along the two circum-

ferential edges of the plate and in the other set an additional circumfer-

ential stiffener is placed along the central line of the plate. The problems

are studied by Ramakrishnan and Kunukkaseril [152] using the classical

plate theory and by Sheikh and Mukhopadhyay [173] using the spline fi-

nite strip method. The present results are compared with those of others

in the Table 5.38 and good agreement in the results is obtained.


Table 5.38: Frequency parameter [ωa2√

ρt/D] of annularsector plate with concentrically placed circum-ferential edge stiffeners

Two stiffeners Three stiffeners

M A B C A B C

1 7.837 7.981 7.871 8.057 8.600 8.662

2 19.358 19.484 19.399 17.807 20.723 20.998

3 32.531 32.908 32.379 35.291 36.874 36.616

4 41.318 42.235 41.247 39.882 41.933 40.793

5 48.799 49.676 48.624 56.735 58.349

6 67.306 67.249 66.982 74.843 72.726

7 73.339 74.799 72.598 76.549 76.892

8 78.273 78.264 77.780 78.910 79.646

A - Ramakrishnan and Kunukkaseril [152];

B - Sheikh and Mukhopadhyay [173]:

C - Present


22.5o

A

A

a = 25

0.5

2

0.5

SECTION AT A-A

b = 50

Figure 5.43:Annular sector plate with concentrically placedcircumferential stiffeners

5.3.2.9 Free Vibration of Eccentrically Stiffened AnnularSector Plate

Annular sector plates simply supported on two radial edges having stiff-

eners along the two circumferential edges (Fig. 5.44) presented by Mukher-

jee and Mukhopadhyay [123] are solved and the first six natural frequen-

cies obtained are compared in Table 5.39. Stiffeners of two sizes (S1 and

S2) as shown in the Fig. 5.44 are considered in the analysis. Mukher-

jee and Mukhopadhyay [123] analyzed the problem using finite element


method and Sheikh and Mukhopadhyay [173] used the spline finite strip

method. The results agree well for all the frequencies.

� � � � � � � � � � � � � � � � � � � � � � � ��

� � � � � � � � � � � � � � � � � � � � � � � ��

� � � � � � � ��

��

A

A

a = 25

b = 50

A

A

a = 25

b = 50

0.5

8 x 0.25

0.5 4 x 0.5

4 x 0.5

4 x 0.5

SECTION AT A-A


(Stiffener S2)(Stiffener S1)

22.5o

Figure 5.44: Annular sector plate with eccentrically placedcircumferential stiffeners


Table 5.39: Frequency parameter [ωa2√

ρt/D] of annularsector plate with eccentrically placed circumfer-ential edge stiffeners

Mode sequences

Stif

fene

r

Met

hod

1 2 3 4 5 6

A 23.445 30.041 44.436 56.250 77.176 81.324

S1 P 23.172 29.276 42.981 55.975 75.013 81.895

A 19.466 26.978 41.382 51.955 74.442 77.792

S2 B 19.011 27.514 40.779 50.196 73.203 76.433

P 19.041 26.359 40.671 49.283 72.866 75.635

P - Present;

A - Mukherjee and Mukhopadhyay [123];

B - Sheikh and Mukhopadhyay [173]


5.3.2.10 Free Vibration of Circular Stiffened Plates

Three different square stiffened plates attempted in the earlier sections

are extended to circular stiffened plates keeping all the properties of the

corresponding plate and the stiffener same but the geometry of the plate

being changed from a square to a circular one keeping the diameter equal

to one of the sides of the square stiffened plate. In the first case the

clamped square stiffened plate having the central concentric stiffener pre-

sented in the section 5.3.2.1 is extended to the circular one (Fig. 5.45).

Similarly in the second case, the same square plate but having an eccen-

tric rectangular stiffener (20mm×3mm) considered by Rao et al. [154]

and presented in the section 5.3.2.2 is extended to the circular stiffened

plate. Lastly, a clamped square cross-stiffened plate with concentric

stiffeners analyzed by Mizusawa et al. [116] and attempted in the sec-

tion 5.3.2.3 is extended to the desired circular one. The results of the first

seven natural frequencies of all the circular stiffened plates (C1, C2, C3)

are presented in Table 5.40 along with the corresponding original square

stiffened plate results (S1, S2, S3) for comparison of the effects of the

curved boundaries on the frequencies/frequency parameters. There is in-

crease in the frequencies in all the cases for all the modes because of the

curved boundaries which is expected, as the circular plate having the di-

ameter equal to the side of the square plate is stiffer than the square one.

The circular stiffened plate results are presented for the first time.

5.3.2.11 Free Vibration of Elliptical Stiffened Plate

The simply supported rectangular plate with a central eccentric L-shaped

stiffener presented in the previous example in the Section 5.3.2.4 is ex-

tended to an elliptical stiffened plate by retaining all its properties but


Table 5.40:Frequency (Hz/Parameter) of all edges clampedcircular stiffened plates

C1: Diameter = 600 mm, plate thickness = 1.0 mm,

ν = 0.34 ρ = 2.78× 10−6 Kg/mm3,

E = 6.87× 107 N/mm2, Is = 2290 mm4,

As = 67.0 mm2, Js = 22.35 mm4

(Nair and Rao [137]) (concentric stiffener)

C2: Material properties and plate dimensions same as

CIR-1 but with an eccentric rectangular stiffener

of size20 mm× 3 mm

(Rao et al. [154])

C3: Diameter = a, EIs/Da = 10.0, As/ah = 0.1

(Mizusawa et al. [116])

T Mode sequences

1 2 3 4 5 6 7

S1 H 50.152 63.410 74.132 84.243 111.990 118.338

C1 58.475 72.467 96.724 111.189 139.549 149.083

S2 H 54.371 65.101 79.808 84.584 116.531 118.676

C2 63.216 74.660 104.865 111.880 145.922 156.153

S3 P 11.35 11.70 12.65 12.65 17.42 24.81 24.96

C3 13.22 14.84 16.58 16.58 23.25 28.71 30.08

T - Type of plate

H - Frequency in Hz

P - Frequency parameter[ω(a/π)2√

ρt/D]


A A

600 mm

1mm

3 mm

20

mm

SECTION AT A-A

xρ = 2.78 10 ν = 0.34

Kg/mm3

E = 6.87 x 10 N/mm7 2

-6

Figure 5.45:Circular plate with a central stiffener

changing the geometrical shape of the plate to an ellipse by keeping the

major and minor axes of the ellipse equal to the length and width of the

rectangular plate respectively (Fig. 5.46). The results of the first six fre-

quencies for the elliptical stiffened plate (ELP) as well as the correspond-

ing rectangular stiffened plate (REC) are presented in the Table 5.41.

While comparing the results of the elliptical stiffened plate with those of

the rectangular stiffened plate an increase in the frequencies is observed

in the second, fourth and the fifth mode of the elliptical plate but the first


45

0 m

m

650 mm

3 mm

10 mm

20

mm

3 mm4.98 mm

SECTION AT A-A

11 2E = 2.051 x 10 N/mm

ρ = 7825 Kg / m3

ν = 0.3

Figure 5.46:Elliptical plate with a central stiffener

and the third mode frequencies are found to be lower whereas there is

hardly any change in the frequency of the sixth mode. The results are

presented for the first time.


Table 5.41: Frequency in Hz of a simply supported ellipti-cal plate with a central eccentric stiffener in theshorter span direction

ν = 0.3, ρ = 7825 Kg/m3, E = 2.051× 1011 N/m2

Plate Mode sequences

Type 1 2 3 4 5 6

REC 142.76 174.85 334.83 352.82 367.67 517.48

ELP 129.10 234.75 241.30 383.25 423.16 518.72

5.3.3 Stability Analysis of Arbitrary Stiffened Plates

Stability analysis for the stiffened plates with various configurations and

boundary conditions is carried out and the buckling parameters are tabu-

lated and compared with the published results of the other investigators

wherever possible.

5.3.3.1 Buckling of Square Stiffened Plates

A number of square stiffened plates with a concentric central stiffener

have been analyzed for various stiffener sizes and flexural rigidities, and

the buckling parameters are presented in Table 5.42 for plates with dif-

ferent boundary conditions. The ratio of the cross-sectional area of the

stiffener to that of the plate (As/bt) is varied from0.05 to 0.20 and the

ratio of the bending stiffness of the stiffener to that of the plate (EIs/bD)


is varied from5 to 25. The torsional inertia of the stiffener has been ne-

glected in the analysis. The results are compared with the semi-analytic

finite difference results of Mukhopadhyay [131] and they agree fairly

well.

5.3.3.2 Buckling of Simply Supported Rectangular StiffenedPlates

A series of simply supported rectangular stiffened plates with a concen-

tric central stiffener have been analyzed for various proportions of the

plate and of the stiffener and the buckling parameters are presented along

with those of other investigators in Table 5.43. The ratio of the cross-

sectional area of the stiffener to that of the plate (As/bt) is varied from

0.05 to 0.20 and the ratio of the bending stiffness of the stiffener to that

of the plate (EIs/bD) is varied from5 to 20. The torsional inertia of

the stiffener has been neglected in the analysis. To analyze this prob-

lem Mukhopadhyay [131] used the semi-analytic finite difference method

whereas Mukhopadhyay and Mukherjee [135] used the finite element

method. Good agreement of the results is obtained when compared with

the results of Timoshenko and Gere [184] and those of others.

5.3.3.3 Buckling of Rectangular Stiffened Plates with Dif-ferent Boundary Conditions

Rectangular stiffened plates with a concentric central stiffener have been

analyzed for two aspect ratios of the plate and for different proportions

and rigidities of the the stiffener and the buckling parameters are pre-

sented in Table 5.44. As before the ratio of the cross-sectional area of the

stiffener to that of the plate (As/bt) is varied from0.05 to 0.20 and the

ratio of the bending stiffness of the stiffener to that of the plate (EIs/bD)


Table 5.42: Buckling parameter k = λb2/π2D for squareplate with a central concentric stiffener sub-jected to uniaxial and uniform compression inthe stiffener direction

ν = 0.3

Boundary Condition

CCCC CSSC

EI s

/bD

As/b

t

Present [131] Present [131]

0.05 24.25 25.46 17.35 17.32

5 0.10 24.25 25.46 17.15 17.05

0.20 24.25 25.46 16.41 16.27

0.05 24.25 - 17.94 -

10 0.10 24.25 - 17.93 -

0.20 24.25 - 17.90 -

0.05 24.25 25.46 18.03 18.36

15 0.10 24.25 25.46 18.03 18.36

0.20 24.25 25.46 18.02 18.34

0.05 24.25 25.46 18.070 -

20 0.10 24.25 25.46 18.068 -

0.20 24.25 25.46 18.064 -

0.05 24.25 - 18.09 18.46

25 0.10 24.25 - 18.09 18.46

0.20 24.25 - 18.09 18.46


Table 5.43:Buckling parameter k = λb2/π2D for uniformlycompressed all edges simply supported rectan-gular stiffened plates

ν = 0.3

Aspect Ratio (a/b)

1 2

EI s

/bD

As/b

t

Present [184] [135] Present [184] [131]

0.05 11.84 12.0 11.72 7.93 7.96 7.93

5 0.10 11.02 11.1 10.93 7.27 7.29 7.28

0.20 9.64 9.72 9.70 6.24 6.24 6.24

0.05 15.73 16.0 16.0 10.16 10.20 10.16

10 0.10 15.73 16.0 16.0 9.33 9.35 9.33

0.20 15.49 15.8 15.44 8.02 8.03 8.02

0.05 15.73 16.0 16.0 12.36 12.4 -

15 0.10 15.73 16.0 16.0 11.36 11.4 -

0.20 15.73 16.0 16.0 9.77 9.80 -

0.05 15.73 16.0 - 14.52 14.6 -

20 0.10 15.73 16.0 - 13.36 13.4 -

0.20 15.73 16.0 - 11.51 11.6 -


Table 5.44: Buckling parameter k = λb2/π2D for rectangu-lar plate with a central concentric stiffener sub-jected to uniaxial and uniform compression inthe stiffener direction

ν = 0.3

Boundary Condition

CSCS SCSC

Aspect Ratio Aspect Ratio

EI s

/bD

As/b

t

1 2 1 2

0.05 18.98 13.54 21.77 18.03

5 0.10 18.98 12.61 21.40 16.41

0.20 18.98 11.03 16.47 13.85

0.05 18.98 16.64 21.76 21.18

10 0.10 18.98 16.64 21.76 21.25

0.20 18.98 16.66 21.76 19.63

0.05 18.98 16.66 21.76 21.17

15 0.10 18.98 16.66 21.76 21.16

0.20 18.98 16.66 21.76 21.26

0.05 18.98 16.66 21.76 21.15

20 0.10 18.98 16.66 21.76 21.15

0.20 18.98 16.66 21.76 21.22

is varied from5 to 20. The torsional inertia of the stiffener has been

neglected in the analysis. These results are presented for the first time.


Table 5.45: Buckling parameter k = λb2/π2D for skewstiffened plate

(Aspect Ratio = 1.0)EIs/bD = 10.0; GJs/bD = 0.0;

As/bt = 0.1; ν = 0.3

Boundary Condition Skew Angle Present [118] [135]

0 16.0 16.0 16.0

All Edges Simply Supported 30 19.96 20.28 20.90

45 27.68 28.68 29.89

0 24.24 24.89 30.8

All Edges Clamped 30 32.41 33.74 36.9

45 47.97 51.62 56.3

5.3.3.4 Buckling of Skew Stiffened Plates with DifferentBoundary Conditions

Skew stiffened plates with a concentric central stiffener and having dif-

ferent boundary conditions have been analyzed for different skew angles

and the buckling parameters are presented in Table 5.45. The present

results agree well with the finite element results of Mukhopadhyay and

Mukherjee [135] and those of Mizusawa et al. [118] who analyzed the

problem usingB-spline functions.

5.3.3.5 Buckling of Uniformly Compressed DiametricallyStiffened Circular Plates

The buckling loads for the all edges simply supported (SS) and clamped

(CC) circular plates with concentric stiffeners along the diameters are

computed with varying flexural and torsional stiffness parameters of the


Table 5.46: Buckling parameter k = (Nr)cr a2/D for uni-formly compressed circular plates with concen-tric stiffeners along the diameter with varyingflexural and torsional stiffness parameters of thestiffener

As/at = 0.1; ν = 0.3

EIs

aD

GJs

aDk

Single Stiffener Cross Stiffeners

SS CC SS CC

0.0 0.0 4.20 14.72 4.20 14.72

2.5 0.0 7.09 26.64 10.79 44.34

5.0 0.0 9.66 26.65 16.67 44.35

7.5 0.0 11.92 26.65 22.21 44.35

10.0 0.0 13.19 26.65 27.22 44.35

0.0 0.0 4.20 14.72 4.20 14.72

0.0 2.5 4.27 14.97 4.63 16.28

0.0 5.0 4.27 14.97 4.63 16.29

stiffener and the results are presented in the form of buckling parameter

k = (Nr)cra2/D where(Nr)cr is the critical compressive force uniformly

distributed around the edge of the plate,a is the radius of the circular plate

andD is the flexural rigidity of the plate . The results are presented in

the Table 5.46. These results are presented for the first time.

Chapter -6

CONCLUSIONS

6.1 Summary

The investigation carried out in this thesis may be summarized as fol-

lows:

• A new four-noded plate bending element has been proposed in the

manner of isoparametric element for the analysis of the bare plates,

which is derived from the simplest rectangular basic plate bend-

ing element having 12 degrees of freedom largely known asACM

element. The new element has all the advantages of an isoparamet-

ric plate bending element by which it is capable of accommodating

the arbitrary geometrical configuration of a plate but without the

disadvantages of the shear-locking problem inherent in the isopara-

metric element.

• A stiffened plate bending element has been proposed for the anal-

ysis of the stiffened plates by combining the 12 degrees of free-

dom rectangular basicACM element and a four-noded rectangu-

lar plane stress element of 8 degrees of freedom. For this stiffened

plate element, a general curved stiffener element has been proposed

209

210 CONCLUSIONS

as a discrete element whose direction and orientation are arbitrary

inside the plate element facilitating its placement and the mesh

generation in the plate independent of the nodal lines. The com-

patibility between the stiffener element and the plate skin is main-

tained by expressing the stiffness matrix of the stiffener element

in terms of the nodal degrees of freedom of the plate element.

Through this compatibility, the contribution of the stiffener to the

stiffness of the plate bending element is truly reflected by its lo-

cation and orientation inside the element, contributing more to the

nodes at the vicinity and less to the far ones.

• A consistent formulation has been carried out for the derivation of

the stiffness matrix for a curved boundary assuming it to be sup-

ported on elastic springs continuously spread along the boundary

line. The springs are considered in the directions of the possible

displacements and rotations of the boundary line. Any specific

boundary condition can be attained from this general one by as-

signing an appropriate value to the spring constants.

• Using these elements, static, free vibration and stability analyses

of bare plates and the stiffened plates have been carried out. In

the extensive numerical examples presented, an attempt has been

made to include various plate geometries such as square, rectangu-

lar, skew, trapezoidal, triangular, circular, elliptical, annular sector

etc. in these three categories of analysis both for bare as well as

stiffened plates having eccentric and concentric stiffeners. Conver-

gence studies for few typical plate geometry problems have been

carried out and results have been compared with the published ones

to validate the proposed method of analysis. Few of the new results

have also been presented.

6.2 Conclusions 211

6.2 Conclusions

Based on the present investigation, the following concluding remarks can

be made:

• Since the starting of the pioneering work on the finite element

method in the early 1960s, a huge number of elements for the plate

analysis have been proposed by various researchers. In this pur-

suit, many of the elements have been found by the analysts to be

inadequate in some way or other when attempt has been made to

make use of them in certain category of problems, thereby intro-

ducing further new elements. In the present investigation, it was

felt that there was hardly a common successful element to address

to the problem of thin plates having arbitrary plate geometries in

spite of the whole wealth of plate elements available in the litera-

ture. This has prompted the present investigation to the proposition

of new elements for bare and stiffened plate analyses of arbitrary

geometrical configurations.

• The elements proposed for the bare and stiffened plate analyses are

well equipped to model the arbitrary shape of the plate, though they

are based on simplest rectangular plate bending element.

• The number of examples dealt for the static, free vibration and sta-

bility analyses of bare and stiffened plates using these elements

have encountered no numerical difficulties during the computation

showing their versatility in the analysis of arbitrary plate domain.

• The stiffener in the stiffened plate bending element has been mod-

elled as a general curved discrete element whose position, orien-

tation and disposition are independent of the nodal networks of

212 CONCLUSIONS

the plate and thus has a distinct advantage over the other model of

stiffened plate element. Any number of stiffeners having different

shape and size can be accommodated anywhere inside the element

making its use attractive enough.

• Using a single stiffened plate bending element, analysis of a large

number of plates of various geometrical configurations with vari-

ous stiffener orientations and positions is possible.

• The various loading conditions have been incorporated by consid-

ering the consistent load vector. The boundary conditions along

the curved boundary have been dealt with properly by consistently

formulating the stiffness matrix of the curved boundary line.

• The various classes of problems attempted using these elements

and the correlation of the results obtained by them with those of

published ones show the versatility and the efficacy of the proposed

method.

6.3 Further Scope of Research

The possible extensions to the present investigation are as presented be-

low:

• The present study has been confined to the linear range of analysis.

With little effort, this proposition can be easily extended to include

the works relating to large deflection and large amplitude vibration

analyses of bare and stiffened plates of arbitrary shape.

• The post buckling behaviour of the arbitrary stiffened plates can

be included as further extension to the present study of buckling

behaviour.

6.3 Further Scope of Research 213

• Response analysis of arbitrary bare and stiffened plates is another

area of extension which can be attempted.

• The present formulation can be extended to include the random

vibration analysis of arbitrary stiffened plates.

• Material nonlinearity may be taken into account in the formulation

for further extension of the arbitrary stiffened plate configurations.

• The modern age structures have extensive use of composite mate-

rials. To the present investigation these composite materials can be

added as a further extension.

• The plates studied here are of uniform thickness. The elements can

be modified to incorporate the plates of varying thickness. In a

similar manner, the study can be extended to the stiffeners having

varying depth.

• For the thick plate analysis, the inclusion of shear is inevitable.

The present model can be extended to include the shear effect and

thereby making the thick plate analysis of arbitrary shape possible.

List of Publications

International Journal

1. Manoranjan Barik andMadhujit Mukhopadhyay , “Finite element

free flexural vibration analysis of arbitrary plates”,Finite Elements

in Analysis and Design, 29, pp. 137-151, 1998.

2. Manoranjan Barik andMadhujit Mukhopadhyay , “Free flexu-

ral vibration analysis of arbitrary plates with arbitrary stiffeners”,

Journal of Vibration and Control, (Accepted for publication).

3. Madhujit Mukhopadhyay , Manoranjan Barik andAbdul Hamid

Sheikh, “Bending analysis of arbitrary plates with arbitrary stiff-

eners”,Structural Engineering and Mechanics(Communicated for

publication).


stability analysis of arbitrary stiffened plates”,Computers and Struc-

tures, (Communicated for publication).


buckling of arbitrary plates”,Journal of Engineering Mechanics,

ASCE, (Communicated for publication).

215

216 LIST OF PUBLICATIONS

Conference

1. Madhujit Mukhopadhyay ,Y. V. Satish Kumar andManoranjan

Barik , “A novel analysis of grillage structures using stiffened plate

bending element”,Trends in Structural Engineering Towards the

21st Century: Structural Engineering Convention 1997, Indian In-

stitute of Technology, Madras, Chennai, pp.55-62, 1997.

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