plastic buckling of mindlin plates
TRANSCRIPT
PLASTIC BUCKLING OF MINDLIN PLATES
TUN MYINT AUNG
NATIONAL UNIVERSITY OF SINGAPORE
2006
PLASTIC BUCKLING OF MINDLIN PLATES
TUN MYINT AUNG
B. Eng, (Yangon Institute of Technology)
A THESIS SUBMITTED
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
DEPARTMENT OF CIVIL ENGINEERING
NATIONAL UNIVERSITY OF SINGAPORE
2006
i
ACKNOWLEDGMENT
The author wishes to express his sincere gratitude to Professor Wang Chien Ming, for
his guidance, patience and invaluable suggestions throughout the course of study. His
extensive knowledge, serious research attitude and enthusiasm have been extremely
valuable to the author. Also special thanks go to Professor J. Charabarty for his
valuable discussions and help in the research work.
The author is grateful to the National University of Singapore for providing the
research scholarship during the four-year study. The author also would like to express
his gratitude to his girlfriend, Ms Chuang Hui Ming, and to his friends; Mr. Kyaw
Moe, Mr. Sithu Htun, Ms. Thida Kyaw, Ms. Khine Khine Oo, Mr. Kyaw Myint Lay,
Mr. Vo Khoi Khoa, Mr. Tran Chi Trung, Mr. Vu Khac Kien for their kind help and
encouragement.
Finally, the author wishes to express his deep gratitude to his family, for their love,
understanding, encouragement and continuous support.
ii
TABLE OF CONTENTS
ACKNOWLEDGEMENT ................................................................................... i
TABLE OF CONTENTS ……………………………………………………… ii
SUMMARY …………………………………………………………………….. vi
NOMENCLATURE ……………………………………………………………. ix
LIST OF FIGURES ……………………………………………………………. xii
LIST OF TABLES ……………………………………………………………… xviii
CHAPTER 1. INTRODUCTION ....................................................................... 1
1.1 Literature Review …………………..……………………………………. 3
1.1.1 Plastic Buckling Theories of Plates ……………………………… 3
1.1.2 Inclusion of Effect of Transverse Shear Deformation …………... 7
1.2 Objectives and Scope of Study ……….………………………….……... 8
1.3 Organization of Thesis ……..……………………………………….……. 11
CHAPTER 2. GOVERNING EQUATIONS FOR PLASTIC BUCKLING
ANALYSIS …………………………………………………….. 13
2.1 Mindlin Plate Theory …………………………………………………..… 13
2.1.1 Assumptions ……………………………………………………… 14
2.1.2 Displacement Components ………………………………………. 15
2.1.3 Strain-Displacement Relations .…………………………………... 16
2.2 Stress-Strain Relations in Plastic Range …................................................. 17
2.2.1 Derivation of Stress-Strain Relations Based on Prandtl-Reuss
Equation ………………………………………………………….. 18
Table of Contents
iii
2.2.2 Derivation of Stress-Strain Relations Based on Hencky’s
Deformation Theory ……………………………………………... 23
2.2.3 Material Modeling …………….….………….…………………... 25
2.3 Derivation of Energy Functional and Governing Equations …………….. 29
CHAPTER 3. RITZ METHOD FOR PLASTIC BUCKLING ANALYSIS .. 33
3.1 Introduction ………………………………………………………………. 34
3.2 Boundary Conditions …………………………………………………….. 35
3.3 Ritz Formulation in Cartesian Coordinate System ………………………. 37
3.4 Ritz Formulation in Skew Coordinate System …………………………... 43
3.5 Ritz Formulation in Polar Coordinate System …………………………… 49
3.6 Solution Method for Solving Eigenvalue Problem ………………………. 59
3.7 Computer Programs ……………………………………………………… 60
CHAPTER 4. RECTANGULAR PLATES …………………………………. 61
4.1 Introduction ……………………………………………………….…........ 61
4.2 Uniaxial and Biaxial Compression ………………………………………. 63
4.3 Pure Shear Loading ………………………………………………………. 88
4.4 Combined Shear and Uniaxial Compression …………………………..... 97
4.5 Concluding Remarks……………………………………………………… 103
CHAPTER 5. TRIANGULAR AND ELLIPTICAL PLATES ……………… 104
5.1 Introduction ………………………………………………………………. 104
5.2 Triangular plates ………………………………………………………… 104
5.3 Elliptical Plates …………………………………………………………... 112
Table of Contents
iv
5.4 Concluding Remarks …………………………………………………….. 117
CHAPTER 6. SKEW PLATES ……………………………………………….. 118
6.1 Introduction ………………………………………………………………. 118
6.2 Uniaxial and Biaxial Loading ……………………………………………. 119
6.3 Shear Loading ……………………………………………………………. 133
6.4 Concluding Remarks .……………………………………………………. 145
CHAPTER 7. CIRCULAR AND ANNULAR PLATES …………………….. 146
7.1 Introduction ………………………………………………………..……... 146
7.2 Analytical Method ……………………………………………………….. 148
7.3 Circular Plates ……………………………………………………………. 153
7.4 Annular Plates …………………………………………………………… 157
7.5 Circular and annular plates with intermediate ring supports …………….. 168
7.6 Concluding Remarks ……………………………………………………... 178
CHAPTER 8. PLATES WITH COMPLICATING EFFECTS …………….. 180
8.1 Introduction ………………………………………………………………. 180
8.2 Internal line/curved/loop supports ……………………………………….. 181
8.3 Point supports ……………………………………………………………. 185
8.4 Elastically restrained edges and mixed boundary conditions ……………. 191
8.5 Elastic foundation ………………………………………………………... 194
8.6 Internal line Hinges ………………………………………………………. 199
8.7 Intermediate in-plane loads ………………………………………………. 203
8.8 Concluding Remarks ……………………………………………………... 205
Table of Contents
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CHAPTER 9. CONCLUSIONS AND RECOMMENDATIONS …………… 207
9.1 Conclusions……………………………………………………………….. 207
9.2 Recommendations for Future Studies ……………………………….…… 210
REFERENCES ……………………………...………………………………….. 212
Appendix A: Ritz Program Codes ……………………………………………... 222
Appendix B: Elements of Matrices …………………………………………….. 249
List of Author’s Publications ………………………………………………….. 256
vi
SUMMARY
This thesis is concerned with the plastic bifurcation buckling of Mindlin plates. So far,
most of the limited studies on plastic buckling of plates have been based on the
classical thin plate theory (which neglects the effect of transverse shear deformation).
In view of this, the first-order shear deformation theory proposed by Mindlin is
adopted in this study to model the plates so as to take into consideration the effect of
transverse shear deformation that becomes significant when the plate thickness-to-
width ratio exceed 0.05. Moreover, earlier studies have been confined to simple
rectangular and circular plate shapes. The present study considers, for the first time,
the plastic buckling of arbitrary shaped plates that include skew, triangular, elliptical
and annular shapes in addition to the commonly treated rectangular and circular
shapes. The plates may also be subjected to in-plane normal and/or shear stresses and
the edges may take any combination of boundary conditions.
In order to capture the plastic behaviour of the plates, two widely used plasticity
theories are considered. These two theories are the incremental theory (IT) of plasticity
and deformation theory (DT) of plasticity. The explicit expressions of stress-strain
relations in elastic/plastic range are derived for the first time based on the adopted
plasticity theories for arbitrary in-plane loading. On the basis of these stress-strain
relations, the total potential energy functional is formulated and the Ritz method is
used for the plastic buckling analysis of plates.
It is worth noting that the Ritz method is automated for the first time for such plastic
buckling plate analysis. This automation is made possible by employing Ritz functions
comprising the product of mathematically complete two-dimensional polynomials and
boundary equations raised to appropriate powers that ensure the satisfaction of the
Summary
vii
geometric boundary conditions a priori. The Ritz formulations are coded for use in
MATHEMATICA for three types of coordinate systems, namely Cartesian, skew and
polar coordinate systems so as to better suit the plate shapes considered. The
employment of MATHEMATICA enables the differentiation, integration and algebraic
manipulations to be done in a symbolic mode and permits executions of the
mathematical operations in an exact manner.
It should be remarked that the presented Ritz program codes can be used to study
not only the plastic buckling of plates but also the elastic buckling of plates. One can
calculate the elastic buckling stress parameter based on the classical thin plate theory
by setting the thickness-to-width ratio to a small value. If one wish to calculate the
elastic buckling stress parameter based on Mindlin plate theory, it can be easily done
by setting the tangent modulus and secant modulus equal to Young’s modulus.
The results obtained using the developed Ritz method were compared with some
limited plastic buckling solutions found in the open literature. The good agreement of
the results provides verification of the formulations and program codes. The effects of
transverse shear deformation, geometrical parameters such as aspect ratios, thickness-
to-width ratios and material parameters on the plastic buckling stress parameter are
investigated for various plate shapes, boundary and loading conditions. Moreover, the
vast plastic buckling data presented in this thesis should serve as a useful reference
source for researchers and engineers who are working on analysis and design of plated
structures.
Based on comparison and convergence studies, the Ritz method is found to be an
efficient and accurate numerical method for plastic buckling of plates. It can be used to
study not only the plates with traditional boundary conditions and loading conditions
but also the plates with complicating effects such as the presence of internal
Summary
viii
line/curved supports, elastically restrained boundary conditions, elastic foundations,
internal line hinges, intermediate in-plane loads. The treatments of these complicating
effects in the formulations and solutions technique are also presented.
In addition to the Ritz method, a new analytical method is featured for handling the
asymmetric buckling problems of circular and annular Mindlin plates. This method is
based on the Mindlin (1951) approach where the governing equations of the plates
were transformed into standard forms of partial differential equations by using three
potential functions. The general solutions of the governing equations are then
substituted into the boundary equations and the resulting homogeneous equations are
solved for the exact plastic buckling stress parameters. The exact solutions should be
useful to researchers for verifying their numerical solutions.
ix
NOMENCLATURE a length of plates
b width of plates
c, k Ramberg-Osgood parameters
D flexural rigidity of plates
DT deformation theory of plasticity
E Young’s modulus
G effective shear modulus
h thickness of plates
IT incremental theory of plasticity
N number of polynomial terms
n number of nodal diameters
nnM bending moment normal to the plate edge
nsM twisting moment at the plate edge
p degree of polynomial
nQ transverse shear force
R Radius of circular or annular plate
ijs stress deviator tensor
wvu ,, displacement components
U strain energy
V potential energy
X contact function
ρδµχγβα ,,,,,, parameters used in stress-strain relations
x
β loading ratio xy σσ /
yzxzxy γγγ ,, shear strain components
ijδ Kronecker delta
zyx εεε ,, normal strain components
ε total effective strain
eε effective elastic strain
pε effective plastic strain
ζ aspect ratio of plates (a/b or b/R )
1ζ ratio of the radius of first internal ring support to the radius of
circular or annular plate (b1/R)
2κ Mindlin’s shear correction factor
λ buckling stress parameter
eλ elastic buckling stress parameter
Rλ plastic buckling stress parameter for R shear
Sλ plastic buckling stress parameter for S shear
0σ nominal yield stress
2.0σ 2% offset yield stress
85.07.0 ,σσ secant yield stresses
cσ buckling stress
mσ hydrostatic stress 3/)( zyx σσσ ++
zyx σσσ ,, normal stresses
σ effective stress
Nomenclature
xi
yzxzxy τττ ,, shear stresses
ν Poisson’s ratio
τ thickness-to-width ratio h/b or thickness-to-radius ratio h/R
θφφφφ ,,, ryx rotations about y-, x-, θ - and r- axes
ψ skew angle of skew plate
xii
LIST OF FIGURES
Fig. 2.1 Deformation of cross-section of Mindlin plates …………………… 16
Fig. 2.2 Ramberg-Osgood stress-strain relation ……………………………. 26
Fig. 2.3 Illustration of 7.0σ and 85.0σ on the stress-strain curve ……………. 27
Fig. 2.4 Offset yield stress ………………………………………………….. 29
Fig. 3.1 Forces and moments on a plate element …………………………… 37
Fig. 3.2 Rectangular plate under in-plane compressive stresses ……………. 38
Fig. 3.3 (a) Skew coordinates and plate dimensions (b) Directions of stresses in the infinitesimal element ……………………………….. 44
Fig. 3.4 Definitions of ζ and Ω …………………………………………… 51
Fig. 4.1 FSCS rectangular plate under in-plane compressive stresses ……… 62
Fig. 4.2 Normalized plastic buckling stress 2.0/σσ c versus b/h ratio for SSSS square plate ………………………………………………….. 71
Fig. 4.3 Normalized plastic buckling stress 2.0/σσ c versus b/h ratio for CCCC square plate ………………………………………………… 72
Fig. 4.4 Normalized plastic buckling stress 2.0/σσ c versus b/h ratio for SCSC square plate …………………………………………………. 72
Fig. 4.5 Normalized plastic buckling stress 2.0/σσ c versus b/h ratio for SSSF square plate ………………………………………………….. 73
Fig. 4.6 Effect of transverse shear deformation on plastic buckling stress (IT) of a square plate ………………………………………………. 73
Fig. 4.7 Effect of transverse shear deformation on plastic buckling stress (DT) of a square plate ……………………………………………… 74
Fig. 4.8 Plastic buckling stress parameter λ versus aspect ratio a/b for SSSS plate based on IT …………………………………………….. 76
Fig. 4.9 Plastic buckling stress parameter λ versus aspect ratio a/b for SSSS plate based on DT …………………………………………… 76
Fig. 4.10 Plastic buckling stress parameter λ versus aspect ratio a/b for CCCC plate based on IT …………………………………………… 77
xiii
Fig. 4.11 Plastic buckling stress parameter λ versus aspect ratio a/b for CCCC plate based on DT ………………………………………….. 77
Fig. 4.12 Plastic buckling stress parameter λ versus aspect ratio a/b for SCSC plate based on IT ……………………………………………. 78
Fig. 4.13 Plastic buckling stress parameter λ versus aspect ratio a/b for SCSC plate based on DT …………………………………………... 78
Fig. 4.14 Plastic buckling stress parameter λ versus aspect ratio a/b for SSSF plate based on IT ……………………………………………. 79
Fig. 4.15 Plastic buckling stress parameter λ versus aspect ratio a/b for SSSF plate based on DT …………………………………………… 79
Fig. 4.16 Contour and 3D plots of simply supported rectangular plate under uniaxial loading ( 05.0=τ , 1.1/ =ba ) for (a) IT, (b) Elastic and (c) DT ……….................................................................................... 80
Fig. 4.17 Plastic buckling stress parameter λ versus stress ratio β for SSSS square plate ………………………………………………………… 82
Fig. 4.18 Plastic buckling stress parameter λ versus stress ratio β for CCCC square plate ………………………………………………… 82
Fig. 4.19 Plastic buckling stress parameter λ versus stress ratio β for SCSC square plate ………………………………………………………… 83
Fig. 4.20 Plastic buckling stress parameter λ versus stress ratio β for SSSF square plate ………………………………………………………… 83
Fig. 4.21 Mode switching with respect to stress ratio β for SSSS square plate ………………………………………………………………... 84
Fig. 4.22 Effect of material properties on normalized plastic buckling stress 2.0/σσ c for SSSS square plate (IT) ……………………………….. 85
Fig. 4.23 Effect of material properties on normalized plastic buckling stress 2.0/σσ c for SSSS square plate (DT) ………………………………. 85
Fig. 4.24 Normalized plastic buckling stress 2.0/σσ c versus modified slenderness ratio for SSSS square plate ……………………………. 86
Fig. 4.25 Comparison study of buckling stresses obtained by DT and IT with test results (Al-7075-T6) 87
Fig. 4.26 Comparison study of buckling stresses obtained by DT and IT with test results (Al-2014-T6) 87
List of Figures
xiv
Fig. 4.27 Normalized plastic buckling stress 2.0/στ xy versus b/h ratio for SSSS square plate ………………………………………………….. 92
Fig. 4.28 Normalized plastic buckling stress 2.0/στ xy versus b/h ratio for CCCC square plate ………………………………………………… 93
Fig. 4.29 Normalized plastic buckling stress 2.0/στ xy versus b/h ratio for SCSC square plate …………………………………………………. 93
Fig. 4.30 Normalized plastic buckling stress 2.0/στ xy versus b/h ratio for SSSF square plate ………………………………………………….. 94
Fig. 4.31 Plastic buckling stress parameter xyλ versus aspect ratio a/b for SSSS rectangular plate …………………………………………. 95
Fig. 4.32 Plastic buckling stress parameter xyλ versus aspect ratio a/b for CCCC rectangular plate ………………………………………... 95
Fig. 4.33 Plastic buckling stress parameter xyλ versus aspect ratio a/b for SSSF rectangular plate ………………………………………..... 96
Fig. 4.34 Interaction curves for SSSS rectangular plates under combined uniaxial compression and shear stresses (τ = 0.05) ……………… 101
Fig. 4.35 Normalized interaction curves for simply supported rectangular plate (a/b = 4) ……………………………………………………… 102
Fig. 4.36 Normalized interaction curves for clamped rectangular plate )4/( =ba …………………………………………………………... 102
Fig. 5.1 CS*F isosceles triangular plate and coordinate system ……………. 105
Fig. 5.2 CS*F right-angled triangular plate and coordinate system ………... 105
Fig. 5.3 Variation of λ with respect to a/b ratio for S*S*S* isosceles triangular plates ……………………………………………………. 111
Fig. 5.4 Variation of λ with respect to a/b ratio for S*S*S* right angled triangular plates ……………………………………………………. 111
Fig. 5.5 Elliptical plate under uniform compression and coordinate system .. 112
Fig. 5.6 Variation of λ with respect to a/b ratio for simply supported elliptical plates ……………………………………………………... 116
Fig. 5.7 Variation of λ with respect to a/b ratio for clamped elliptical plates ……………………………………………………………... 117
List of Figures
xv
Fig. 6.1 Skew plate under biaxial compression stresses ………………….. 120
Fig. 6.2 Normalized plastic buckling stress 2.0/σσ c versus b/h ratio for simply supported skew plate (ψ = 30˚) …………………………... 130
Fig. 6.3 Normalized plastic buckling stress 2.0/σσ c versus b/h ratio for clamped skew plate (ψ = 30˚) …………………………………….. 130
Fig. 6.4 Plastic buckling stress parameter λ versus aspect ratio a/b for simply supported skew plate (ψ = 30˚) …………………………... 131
Fig. 6.5 Plastic buckling stress parameter λ versus aspect ratio a/b for clamped skew plate (ψ = 30˚) ……………………………………. 131
Fig. 6.6 Plastic buckling stress parameter λ versus skew angle ψ for simply supported skew plate (a/b = 1) …………………………… 132
Fig. 6.7 Plastic buckling stress parameter λ versus skew angle ψ for clamped skew plate (a/b = 1) …………………………………….. 132
Fig. 6.8 Two kinds of shear loading conditions on skew plates (a) R shear loading and (b) S shear loading …………………………………... 133
Fig. 6.9 Normalized buckling stress 2.0/στ xy versus b/h ratio for simply supported skew plate ( 1/ =ba ) ………………………………….. 141
Fig. 6.10 Normalized buckling stress 2.0/στ xy versus b/h ratio for clamped skew plate ( 1/ =ba ) ……………………………………………... 142
Fig. 6.11 Variation of plastic buckling stress parameters with respect to aspect ratio for SSSS skew plate with ψ = 30˚ …………………… 142
Fig. 6.12 Variation of plastic shear buckling stress parameters with respect to aspect ratio for CCCC skew plate with ψ = 30˚ ……………….. 143
Fig. 6.13 Variation of plastic (R shear) buckling stress parameters with respect to skew angle of SSSS skew plate ……………………….. 143
Fig. 6.14 Variation of plastic (S shear) buckling stress parameters with respect to skew angle of SSSS skew plate ……………………….. 144
Fig. 6.15 Variation of plastic (R shear) buckling stress parameters with respect to skew angle of CCCC skew plate ……………………… 144
Fig. 6.16 Variation of plastic (S shear) buckling stress parameters with respect to skew angle of CCCC skew plate ……………………… 145
Fig. 7.1 Clamped circular plate under in-plane compressive stress ………. 153
Fig. 7.2 C-S annular plate under in-plane loading ………………………… 158
List of Figures
xvi
Fig. 7.3 Variation of buckling stress parameter λ with respect to ζ ratio for C-C annular plate ……………………………………………... 164
Fig. 7.4 Variation of buckling stress parameter λ with respect to ζ ratio for C-S annular plate ……………………………………………... 164
Fig. 7.5 Variation of buckling stress parameter λ with respect to ζ ratio for C-F annular plate ……………………………………………... 165
Fig. 7.6 Variation of buckling stress parameter λ with respect to ζ ratio for S-S annular plate ……………………………………………... 165
Fig. 7.7 Variation of buckling stress parameter λ with respect to ζ ratio for S-C annular plate ……………………………………………... 166
Fig. 7.8 Variation of buckling stress parameter λ with respect to ζ ratio for S-F annular plate ……………………………………………... 166
Fig. 7.9 Variation of buckling stress parameter λ with respect to ζ ratio for F-C annular plate ……………………………………………... 167
Fig. 7.10 Variation of buckling stress parameter λ with respect to ζ ratio for F-S annular plate ……………………………………………... 167
Fig. 7.11 Critical buckling mode shapes for C-C annular plates of various ζ ratios (DT, τ = 0.05) ……………………………………….. 168
Fig. 7.12 Simply supported (a) annular plate (b) circular plate with multiple concentric ring supports ………………………………………….. 169
Fig. 7.13 Variation of buckling stress parameter λ with respect to ring support location 1ζ for annular plate ( ,2.0=ζ IT) ………………. 175
Fig. 7.14 Variation of buckling stress parameter λ with respect to ring support location 1ζ for annular plates ( ,2.0=ζ DT) ……………. 175
Fig. 7.15 Variation of buckling stress parameter λ with respect to ring support location 1ζ for circular plates …………………………… 176
Fig. 7.16 Critical Buckling mode shapes for C-C annular plates with various ring support locations (b/R = 0.2,DT, h/R = 0.05) ……….. 177
Fig. 8.1 Plates with internal (a) line, (b) curved and (c) loop supports …… 182
Fig. 8.2 Simulation of internal supports using point supports …………… 182
Fig. 8.3 Rectangular plate with an internal line support ………………….. 183
List of Figures
xvii
Fig. 8.4 Critical buckling mode shapes of square plates with internal line support ……………………………………………………………. 185
Fig. 8.5 Four point supported square plate ………………………………... 190
Fig. 8.6 Square plates with mixed boundary conditions under uniaxial compression ………………………………………………………. 193
Fig. 8.7 Typical buckling mode shapes for bilateral and unilateral buckling of clamped rectangular plates …………………………... 199
Fig. 8.8 A rectangular plate with an internal hinge under in-plane compressive stress ………………………………………………... 201
Fig. 8.9 Rectangular plate under edge and intermediate compressive stress …………………………………………………………….. 204
xviii
LIST OF TABLES
Table. 2.1 Values of 85.07.0 ,σσ and n for various aluminum alloys under room temperature (From Bruhn, 1965) …………………………... 28
Table. 4.1 Sample basic functions for rectangular plates with various boundary conditions ……………………………………………… 63
Table. 4.2 Comparison and convergence study of λ for square plates under uniform uniaxial loading with various boundary conditions …….. 66
Table. 4.3 Comparison and convergence study of λ for rectangular plates under uniform uniaxial loading with various boundary conditions (a/b = 2) …………………………………………………………... 67
Table. 4.4 Convergence and comparison studies of plastic buckling stress parameters λ for square plates under uniform biaxial loading ….. 68
Table. 4.5 Convergence and comparison studies of plastic buckling stress parameters λ for rectangular plates under uniform biaxial loading (a/b = 2) …………………………………………………………... 69
Table. 4.6 Parametric study on the value of transverse shear correction factor for simply supported square plate under biaxial loading .………... 71
Table. 4.7 Convergence and comparison studies of plastic buckling stress parameters for SSSS rectangular plates under shear ………. 91
Table. 4.8 Convergence and comparison studies of plastic buckling stress parameters for CCCC rectangular plates under shear ……... 92
Table. 4.9 Buckling mode shapes for Rectangular plate under pure shear )03.0( =τ ………………………………………………………… 96
Table. 4.10 Convergence study of plastic buckling shear stress parameters xyλ for simply supported plates under combined loadings (τ = 0.03) ... 100
Table. 4.11 Convergence study of plastic shear buckling stress parameters xyλ for clamped square plates under combined loadings. (τ = 0.03) …. 100
Table. 4.12 Comparison study of elastic shear buckling stress parameters xyλ for simply supported plates under combined loadings. ( ,001.0=τ ν = 0.3) ……………………………………………….. 101
List of Tables
xix
Table. 5.1 Convergence and comparison studies of plastic buckling stress factors λ for equilateral triangular plates )2/3/( =ba under uniform compression ……………………………………………... 108
Table. 5.2 Convergence and comparison studies of plastic buckling stress factors λ for right angled triangular plates under uniform compression (a/b = 1) …………………………………………….. 108
Table. 5.3 Effect of transverse shear deformation on λ for S*S*S* isosceles triangular plates …………………………………………………... 110
Table. 5.4 Effect of transverse shear deformation on λ for CCC isosceles triangular plates …………………………………………………... 110
Table. 5.5 Convergence and comparison studies of plastic buckling stress factors λ for circular plates under uniform compression ..…….. 114
Table. 5.6 Convergence and comparison studies of plastic buckling stress factors λ for elliptical plates under uniform compression (a/b=2) ………………………………………………………….. 114
Table. 5.7 Effect of transverse shear deformation on λ for simply supported circular (a/b = 1) and elliptical plates (a/b = 3) ……... 115
Table. 5.8 Effect of transverse shear deformation on λ for clamped circular (a/b = 1) and elliptical plates (a/b = 3) ………………... 115
Table. 6.1 Convergence study of plastic buckling stress parameters λ for simply supported skew plate under uniaxial compression ( 0=β , a/b=1) ………………………………………………….. 121
Table. 6.2 Convergence study of plastic buckling stress parameters λ for simply supported skew plate under biaxial compression
σσσ == yx ( 1=β , a/b=1) ……………………………………. 122
Table. 6.3 Convergence study of plastic buckling stress parameters λ for clamped skew plate under uniaxial compression ( 1/,0 == baβ ) 123
Table. 6.4 Convergence study of plastic buckling stress parameters λ for clamped skew plate under biaxial compression
σσσ == yx ( 1=β , a/b = 1) ……………………………………. 124
Table. 6.5 Comparison study against existing elastic solutions for simply supported skew plates (τ = 0.001) ……………………………... 125
Table. 6.6 Comparison study against existing elastic solutions for clamped skew plates (τ = 0.001) ………………………………………… 126
List of Tables
xx
Table. 6.7 Effect of transverse shear deformation on λ for simply supported skew plates under uniaxial compression (a/b = 1) ……………….. 127
Table. 6.8 Effect of transverse shear deformation on λ for clamped skew plates under uniaxial compression (a/b = 1) ……………………... 128
Table. 6.9 Convergence study of plastic buckling parameters for simply supported skew plates under shear loading (ψ = 45˚) ……………. 135
Table. 6.10 Convergence study of plastic buckling parameters for simply supported skew plates under shear loading (ψ = -45˚) …………… 135
Table. 6.11 Convergence study of plastic buckling parameters for clamped skew plates under shear loading (ψ = 45˚) ……………………….. 136
Table. 6.12 Convergence study of plastic buckling parameters for clamped skew plates under shear loading (ψ = -45˚) ………………………. 136
Table. 6.13 Comparison study of elastic shear buckling stress parameter of skew plates under shear (a/b = 1) ………………………………… 137
Table. 6.14 Effect of transverse shear deformation on Rλ for simply supported skew plates (a/b = 1) ………………………………….. 138
Table. 6.15 Effect of transverse shear deformation on Sλ for simply supported skew plates (a/b = 1) ………………………………….. 138
Table. 6.16 Effect of transverse shear deformation on Rλ for clamped skew plates (a/b = 1) …………………………………………………… 139
Table. 6.17 Effect of transverse shear deformation on Sλ for clamped skew plates (a/b = 1) …………………………………………………… 139
Table. 7.1 Convergence and comparison study of plastic buckling stress parameter λ for circular plates under uniform radial compression …………………………………………………….. 157
Table. 7.2 Convergence study of plastic buckling stress parameter λ for annular plates ( 4.0/ == Rbζ ) ………………………………….. 160
Table. 7.3 Comparison study of plastic buckling stress parameter λ for annular plates ( 4.0/ == Rbζ ) ………………………………….. 161
Table. 7.4 Convergence study of plastic buckling stress parameter λ for annular plates with one concentric ring support ( 6.0,2.0 1 == ζζ ) ……………………………………………… 172
Table. 7.5 Convergence study of plastic buckling stress parameter λ for circular plates with one concentric ring support ( 6.01 =ζ ) ……… 173
List of Tables
xxi
Table. 7.6 Comparison study between the optimal ring support radius and radius of nodal circle of second mode shape …………………… 178
Table. 8.1 Convergence and comparison studies of plastic buckling stress parameter λ for rectangular plate with internal line supports …. 184
Table. 8.2 Convergence and comparison study of plastic buckling stress parameter λ for four points supported square plate ……………. 190
Table. 8.3 Comparison studies of buckling stress parameter λ square plates with mixed boundary conditions ………………………… 194
Table. 8.4 Comparison study of unilateral buckling stress parameter λ for rectangular plates under shear …………………………………... 197
Table. 8.5 Unilateral plastic buckling stress parameters for rectangular plates under shear (IT) ………………………………………….. 198
Table. 8.6 Unilateral plastic buckling stress parameters for rectangular plates under shear (DT) ………………………………………… 198
Table. 8.7 Buckling stress parameter λ for square plate with an internal line hinge under uniaxial loading ………………………………. 203
Table. 8.8 Convergence and comparison of plastic buckling stress parameters 1λ of rectangular plates under edge and intermediate loads. ( 04.0,502 =−= τλ ) ……………………………………... 205
1
INTRODUCTION
Plates and plated structures are widely used in many engineering structures such as
aircrafts, ships, buildings, and offshore structures. Owing to their relatively small
thickness when compared to the length dimensions, the design strength of these
structures is commonly governed by their buckling capacities. Buckling is a
phenomenon in which a structure undergoes visibly large transverse deflection in one
of the possible instability modes. Buckling of a structural component may affect the
strength or stiffness of the whole structure and even triggers unexpected global failure
of the structure. Therefore, it is important to know buckling capacities of the structures
in order to avoid premature failure.
The first event that highlighted the importance of plate buckling was related to a
series of tests carried out in the laboratories of University of London on box beams for
Britannia Bridge in 1845 (Walker, 1984). A number of failure cases were due to local
instability, which is now referred to as local buckling. However, the first theoretical
CHAPTER 1
Introduction
2
examination of buckling of plates was attributed to Bryan (1891). He presented the
buckling analysis for rectangular plates under uniform uniaxial compression using the
energy criterion of stability. It should be noted that he was not only the first to analyze
the stability of plates, but also the first to apply the energy criterion of stability to the
buckling problem of plate. The application of the energy criterion was later proven to
be a powerful tool for the analyses of buckling problems which could not be solved by
conventional analytical methods because of the difficulty in the mathematical
treatment. Since then, many researchers have studied the buckling of plates of various
shapes, loading and boundary conditions using a variety of methods for analysis.
Research on the buckling of plates can be generally divided into two groups:
elastic buckling and plastic buckling. In the case of elastic buckling, it is assumed that
the buckling stress is below the proportional limit of the plate material and the linear
stress-strain relation is adopted. In many practical problems, however, the plates may
be stressed beyond the proportional limit before the buckling take place i.e. buckling
observed in the plastic range. This phenomenon may be likely to occur in the cases of
plates whose materials posses a low proportional limit when compared to the nominal
yield stress, for example aluminium alloy and stainless steel. In the plastic range, the
stress-strain relation is no longer linear and thus the actual buckling load is smaller
than the value determined by the elastic buckling analysis. Therefore, plastic buckling
analysis needs to be carried out in order to obtained more accurate buckling loads
when the aforementioned conditions hold. In the next section, a literature review on
the application of plasticity theories in plastic buckling problems of plates and the
success of these theories is presented.
Introduction
3
1.1 Literature Review
In this section, the development of plastic buckling theories and the paradox in the
plastic buckling of plates will be presented.
1.1.1 Plastic Buckling Theories of Plates
Various plasticity theories have been proposed in the literature for the plastic
buckling analysis of plates. The most commonly used theories are the deformation
theory of plasticity (DT) and the incremental theory of plasticity (IT). The deformation
theory of plasticity (DT) was pioneered by Ilyushin (1946) and later modified by
Stowell (1948) based on the assumption that no unloading take place during buckling
(Shanley, 1947). On the other hand, the incremental theory (IT) was first developed by
Handelman and Prager (1948). They pointed out that certain postulate of the theory of
plasticity which cannot be satisfied by using the stress-strain relation developed by
Ilyushin (1946). The main difference between the two theories is that IT relates plastic
strain increment to the stress while DT relates the total strain to the state of stress.
Pride and Heimerl (1949) performed experiments on simply supported plates in
order to assess the various plastic buckling theories in predicting plastic buckling
stress. They tested square tubes of 14S-T6 aluminum alloy, for which each of the four
plates may be considered simply supported. The results obtained were in good
agreement with Stowell’s unified theory for the plastic buckling of columns and plates
(1948). It was found that the results obtained from deformation theory of plasticity
(Ilyushin 1948) gave a close correlation with the test results while the incremental
Introduction
4
theory of plasticity (Handelman and Prager 1948) yielded results that do not agree with
experimental results. Another experimental investigation on the plastic buckling of
plate was studied by Dietrich et al. (1978). Their experimental results are slightly
lower than the critical buckling loads as predicted by Stowell’s theory (DT).
Despite the fact that the deformation theory is mathematically less correct
especially in nonproportional loading (Hill, 1950), many researchers have continued to
use it for solving plastic buckling problems since the results show good agreement
with experimental results. For example, Bijlaard (1949) assumed that plastic
deformation is governed only by the amount of elastic shearing energy. He derived the
stress-strain relations by writing the infinitely small excess strains as total differentials
and computing the partial derivatives of the strains with respect to the stresses. El-
Ghazaly and Sherbourne (1986) proved that the deformation theory can be used for the
elastic-plastic buckling analysis of plates under nonproportional external loading and
nonproportional stresses. Loading, unloading, and reloading situations have been
conveniently considered via conducting multistage analysis after casting the
constitutive relations of the deformation theory in an incremental form.
On the other hand, many researchers have tried to justify the theoretically more
acceptable incremental theory of plasticity, even though it does not yield results that
are in agreement with experimental results. Generally, the incremental theory proposed
by Handleman and Prager (1948) yield the results that are higher than experimental
results. Pearson (1950) modified the Handelman and Prager’s calculation by the
assumption of continuous loading. He solved the plastic buckling problem of
rectangular plate under compressive edge thrust and showed that results were in fair
agreement with experimental work by Pride and Heimerl (1949).
Introduction
5
As another attempt, Sewell (1964) showed that the bifurcation buckling stress is
rather sensitive to the direction of normal to the local yield surface. This cast doubt on
the validity of any experiment that does not give proper attention to the yield surface.
Dubey and Pindera (1977) also studied the effect of rotation of principal axes on the
shear modulus in elastic-plastic solids. They concluded that, by taking proper account
of the rotation of principal axes, the effective shear modulus of an elastic-plastic solid
can be considerably reduced from the elastic-shear modulus which, in turn, lower the
plastic buckling load obtained from the incremental theory. However, Hutchinson
(1974) pointed out that basic experiments on the multiaxial stress-strain behaviour of
typical metals failed to show the occurrence of corners on yield surfaces. Another
drawback for the corner theory is that although one can fit the appropriate corner to
observe experimental results in order to obtain good agreement, one cannot estimate
the appropriate corner a priori.
Moreover, based on imperfection sensitivity, Onat and Drucker (1953) tried to
justify the inaccuracy of incremental theory in the plastic buckling of plates. They
considered the case of an axially loaded cruciform and showed that unavoidable initial
imperfections may significantly lower the plastic buckling load predicted by the
incremental theory of plasticity. In addition, using the general von Karman non-linear
plate equations, Hutchinson and Budiansky (1974) confirmed analytically Onat-
Drucker’s conclusion concerning imperfection sensitivity. Neale (1975) also studied
the effect of imperfections on the plastic buckling of simply supported rectangular
plates. He showed that plastic buckling load obtained from incremental theory can be
reduced considerably in the presence of small initial imperfections in geometry. He
concluded that a proper consideration of imperfections in geometry provides an
explanation for the apparent plastic buckling paradox for axially compressed
Introduction
6
rectangular plates. Although this justification seems to be logical, it leaves the value of
the buckling load dependent on the amount and distribution of imperfections. This
lends credibility to the use of the deformation theory in the prediction of bifurcation
buckling load for engineering purpose.
Similarly, as another justification, Gjelsvik and Lin (1987) investigated the plastic
buckling of plate including the effect of friction acting on the loaded edges during
tests. The stresses due to friction, which die out away from the edges, are shown to
effectively reduce the incremental theory buckling loads to those predicted by the
deformation theory and the experimental results. It was argued that the amount of
friction required for such a reduction is less than normally expected at the loaded edges
of a real test. Recently, Tuğcu (1991) also showed that incremental theory predictions
were significantly reduced when small amounts of biaxial and in-plane shear stresses
are included in the prebuckling state as these stresses can be produced by the
experimental set-up. Unlike the analysis of Gjelsvik and Lin (1987), where the edges
effects were assumed to die out away from the edges, Tuğcu (1991) assumed that these
non-zero stress components exist uniformly in the prebuckling state.
In spite of the continuous efforts and progress made in the last few decades, plastic
buckling of plates continues to remain a “paradox” for researchers in the field.
Accordingly, some researchers studied elastic/plastic buckling problems of plates
based on both the deformation theory and the incremental theory (for e.g. Shrivastava,
1979; Shrivastava, 1995; Durban and Zuckerman, 1999; Wang et al. 2001). Durban
and Zuckerman (1999) carried out a detailed parametric study on the elasto-plastic
buckling of rectangular plates under biaxial compression by an analytical method.
Apart from confirming that the deformation theory furnishes lower buckling stresses
than those computed using incremental theory, they reported the existence of an
Introduction
7
optimal loading part for the deformation theory model. Shrivasta (1979) and Wang et
al. (2001) studied the plastic buckling problems of rectangular plates based on Mindlin
plate theory.
In this study, both plasticity theories (IT and DT) are adopted for plastic buckling
analysis of plates in order to compare the results from these two theories.
1.1.2 Inclusions of Effect of Transverse Shear Deformation
Most of the aforementioned studies on plastic buckling analysis of plates adopted
the classical thin plate theory. It is well known that the classical thin plate theory
neglects the effect of transverse shear deformation. This effect becomes significant
when the plate thickness increases. Since plastic buckling is most likely to occur in
relatively thick plates, the effect of transverse shear deformation should be taken into
account in the plastic buckling analysis.
One of the earlier studies on plastic buckling of thick plate was done by
Shrivastava (1979). Three cases of plastic buckling of rectangular plates were
considered based on the Mindlin plate theory. The thick plate results were compared
with thin plate results based on both incremental (IT) and deformation theory (DT) of
plasticity. It was found that the reduction due to the effect of transverse shear
deformation in plastic buckling stress parameter was generally larger for IT than that
for DT. However, this difference in the reduction was not large enough to compensate
for the differences between the results from two theories.
Another study on plastic buckling of thick plate was investigated by Wang et
al. (2001). They used an analytical method to solve for the plastic buckling of (a)
uniformly loaded rectangular plates with two opposite edges simply supported while
Introduction
8
the other edges may take any boundary conditions (b) uniformly loaded circular plate
with simply supported or clamped boundary condition.
Recently, Wang et al. (2004) developed an analytical relationship between the
plastic buckling loads of thick plate and its corresponding thin plate elastic buckling
loads. Since elastic buckling stresses of classical thin plates are well-known, plastic
buckling stresses of thick plates can be readily obtained without the tedious nonlinear
analysis to solve more complicated plastic buckling equations. However, the relation is
only valid for simply supported plates of polygonal shapes.
All the aforementioned studies on the plastic buckling of thick plates were solved
by analytical methods for simple plate shapes and boundary conditions. These
analytical methods can only be employed for only a limited number of buckling
problems of practical importance due to the complexity arising from material
nonlinearity and the difficulty in getting a function to satisfy the geometric and static
boundary conditions of non-simple plate shapes. Therefore, for plastic buckling
problems of plates with general shapes and boundary conditions, numerical methods
such as the finite element method, the finite difference method, the finite strip method
and the Ritz method have to be used.
1.2 Objectives and Scope of Study
There are few analytical expressions of plasticity correction factors (Stowell, 1948;
Bijlaard, 1949) in order to account for the effect of material nonlinearity. Plastic
buckling solutions can be calculated from corresponding elastic buckling solutions
through these plasticity correction factors. However, due to the material nonlinearity,
these calculations still result in transcendental equations which need iterative method
Introduction
9
for determining the plastic buckling solutions. Moreover, the analytical expressions of
plasticity correction factors are only available for limited loading and boundary
conditions. On the other hand, the numerical plastic buckling solutions exist for several
loading and boundary conditions for rectangular plates (El-Ghazaly et al., 1984;
Bradford and Azhari, 1994, 1995; Smith et al., 2003). It would be useful for the design
profession if a simple computing algorithm was available, that is readily adaptable to
specific problems in determining the plastic bifurcation buckling loads.
In this thesis, an algorithm is developed for the plastic buckling analysis of Mindlin
plates. The analysis makes the use of the Ritz method. The major advantage of the Ritz
method is that much less computer memory is needed when compared to the finite
element method (Leonard and Moon, 1990; Surendran, 2001). So far, the Ritz method
has been extensively used in the studies of elastic buckling and vibration problems of
plates (Liew et al., 1998; Wang et al., 1994; Liew and Wang, 1995). The scope of this
thesis is to extend the Ritz method for the plastic buckling analysis of Mindlin plates
and to examine the plastic behaviour of Mindlin plates with various shapes, boundary
conditions and loading conditions.
In addition to developing a simple algorithm, an analytical method is featured for
tackling the asymmetric plastic buckling problems of circular and annular Mindlin
plates. Hitherto, no analytical asymmetric buckling solutions are available in the
literature for circular and annular Mindlin plates, even for the elastic buckling case.
Therefore, this analytical method and the exact solutions will be useful for researchers
who wish to verify their numerical solutions and methods.
It should be noted that this thesis mainly focuses on the plastic bifurcation buckling
analysis of Mindlin plates. It has been assumed that all the plates considered are
initially perfectly flat. Moreover, the assumption of small deflection (effect of
Introduction
10
geometrical nonlinearity is neglected) needs to be adopted in order to obtained plastic
bifurcation buckling load. Therefore, the postbuckling analysis (which accounts the
effect of geometrical nonlinearity and initial imperfections) is beyond the scope of this
thesis.
In summary, the main contributions of this thesis include:
(1) Developing a Ritz formulation and program codes to tackle plastic buckling
problems of Mindlin plates for various shapes (sides are defined by polynomial
functions) and boundary conditions. The program was coded in
MATHEMATICA programming language which is a high level programming
language that allows readers not only to understand the program easily but also be
able to perform mathematical operations in a symbolic and exact manner. The
Ritz program code should be useful for engineers and designers who wish to
perform plastic buckling analysis of plates since the procedure is simple to
understand and easy to use.
(2) An analytical approach is developed for the first time in order to solve asymmetric
plastic buckling problems as well as elastic buckling problems of circular and
annular Mindlin plates.
(3) So far plastic buckling of thick plates has been studied only for rectangular plates
with two simply supported opposite edges. This study treats general plate shape
and investigates the plastic buckling behaviour of various shapes such as
rectangular, skew, triangular and elliptical, circular and annular plates with all
possible combinations of boundary conditions. Extensive plastic buckling results
are presented for the aforementioned plate shapes.
Introduction
11
(4) Modifications of the Ritz method for complicating effects such as the presence of
internal line/curved/point supports, internal line hinges, elastic restrained and
mixed boundary conditions, elastic foundation and intermediate in-plane loads.
In the next section, a brief overview of the organization of thesis is given.
1.3 Organization of Thesis
In this chapter, the background information of plate buckling and literature review
on the plastic buckling of plates is presented. Moreover, the objectives, scope of study
and organization of thesis are also presented.
In Chapter 2, the explicit expressions of stress-strain relations in the plastic range
are derived for the first time based on the Prandtl-Ruess and the Hencky equations.
Finally, adopting the Mindlin plate theory, the energy functional and the governing
equations are presented for plastic buckling of plates under arbitrary loading.
In Chapter 3, the detailed formulations of the Ritz method for plastic buckling of
Mindlin plate based on three coordinate systems, namely Cartesian, skew and polar
coordinate systems are presented.
In Chapters 4 and 5, the plastic buckling analyses are carried out by using the Ritz
method with the Cartesian coordinate system. Chapter 4 presents the parametric
studies on the plastic buckling of rectangular Mindlin plate under in-plane compressive
stresses, shear stress and the combinations of these stresses. In Chapter 5, the
applicability of the Ritz method to various shapes is illustrated by considering the
plastic buckling problems of isosceles triangular plates, right angled triangular plates
and elliptical plates.
Introduction
12
In Chapter 6, plastic buckling of skew Mindlin plates is studied by using the Ritz
method with the skew coordinate system. The effects of aspect ratio, skew angle and
thickness-to-width ratio on the plastic buckling stress parameter are investigated.
In Chapter 7, plastic buckling of circular and annular plates is studied by using the
Ritz method with the polar coordinate system. In addition to the Ritz method, the
development of an analytical method is presented for the first time for asymmetric
buckling of circular and annular plates. The plastic buckling analyses of circular and
annular plates with/without an internal ring support are performed by both methods.
The independent check of the results by both methods confirms the correctness of the
methods.
In Chapter 8, modifications to the Ritz formulation are presented for handling the
plastic buckling of Mindlin plates with complicating effects such as internal
line/curved/point supports, elastic foundation, internal line hinges and intermediate in-
plane loads. The results are verified using existing elastic buckling solutions.
Chapter 9 presents conclusions of the research and recommendations for future
studies.
13
GOVERNING EQUATIONS FOR PLASTIC
BUCKLING OF PLATES
This chapter presents the derivation of the energy functional and the governing
equations for the bifurcation plastic buckling of thick plates under in-plane loadings.
The significant effect of transverse shear deformation on the buckling stress for thick
plates is taken into consideration by adopting the Mindlin plate theory. In order to
capture the plastic buckling behaviour of the plate, the two commonly used plasticity
theories, namely the incremental theory of plasticity (IT) and the deformation theory
of plasticity (DT) were considered. Based on these plasticity theories, the derivation of
the stress-strain relations in the plastic range is presented in detail.
2.1. Mindlin Plate Theory
Generally, plate analyses are based on the classical thin plate theory (also called
Kirchhoff plate theory since it is first proposed by Kirchhoff, 1850) in which the effect
CHAPTER 2
Governing Equations for Plastic Buckling Analysis of Plates
14
of transverse shear is normally neglected. However, when dealing with thick plates,
the classical thin plate theory underpredicts the deflections and overpredicts the
buckling loads and natural frequencies due to the effect of transverse shear
deformation. As we are dealing with thick plates, it is necessary to adopt a more
refined plate theory such as the Mindlin plate theory that will allow for the effect of
transverse shear deformation.
2.1.1 Assumptions
The assumptions of the Mindlin plate theory are (Mindlin 1951):
• The plate thickness remains unchanged during deformation, i.e. transverse
inextensibility of the plate is maintained.
• No in-plane deformation in the middle surface of the plate after deformation.
• The normals during bending undergo constant rotations about the middle
surface while maintaining the straightness and thereby admitting a constant
shear strain through the plate thickness (Fig. 2.1). These constant rotations of
the normals to the middle surface about y- and x- axes now become unknown
independent variables and are denoted by xφ and yφ , respectively.
It should be noted that the first two assumptions are the same as their classical thin
plate counterparts. The third assumption that allows the constant rotation of normal is
the main difference between the Mindlin plate theory and the classical thin plate
theory. The allowance of constant rotation implies that transverse shear strain is
constant through the thickness of the plate. This, however, contradicts the fact that the
actual transverse shear strain distribution is parabolic through the thickness. As the
Governing Equations for Plastic Buckling Analysis of Plates
15
constant strain (stress) violates the statical requirement of vanishing shear stress at the
surface of the plate, a shear correction factor 2κ was proposed by Mindlin (1951) to
compensate for the error. He pointed out that for an isotropic plate, the shear
correction factor 2κ depends on Poisson’s ratio v and it may vary from 76.02 =κ for
v = 0.3 to 91.02 =κ for v = 0.5. On the other hand, by comparing the constitutive
Mindlin shear force with the one proposed by Reissner (1945), who assumed a
parabolic shear stress distribution at the outset of his plate theory formulation, the
implicit shear correction factor of Reissner (1945) becomes 6/52 =κ . This value of
shear correction factor has been commonly used for the analyses of Mindlin plates, for
example, Raju and Rao (1989), Liew et al. (2004) and Hou et al. (2005). Therefore, the
shear correction factor 6/52 =κ will be used in this study.
2.1.2 Displacement Components
Based on the Mindlin assumptions, the displacement fields are given by,
),(),,( yxzzyxu xφ= (2.1a)
),(),,( yxzzyxv yφ= (2.1b)
),(),,( yxwzyxw = (2.1c)
in which vu, are in-plane displacements in the x and y directions, respectively, w
the transverse displacement in the z direction and yx φφ , the rotations about the y- and
x- axes, respectively as shown in Fig. 2.1. Note that by setting xwx ∂−∂= /φ and
ywy ∂−∂= /φ , one recovers the displacement fields of the classical thin plate theory.
Governing Equations for Plastic Buckling Analysis of Plates
16
2.1.3 Strain-Displacement Relations
The normal strains yx εε , and shear strains yzxzxy γγγ ,, are given by (Mindlin,
1951).
xu
x ∂∂
=ε (2.2a)
yv
y ∂∂
=ε (2.2b)
xv
yu
xy ∂∂
+∂∂
=γ (2.2c)
xw
zu
xz ∂∂
+∂∂
=γ (2.2d)
xw∂∂
xφ
w
Undeformed shape
Deformed shape
x
z
h
Fig. 2.1 Deformation of cross-section of Mindlin plates
Governing Equations for Plastic Buckling Analysis of Plates
17
yw
zv
yz ∂∂
+∂∂
=γ (2.2e)
In view of Eqs. (2.1a-c), the strain-displacement relations can be expressed as
xz x
x ∂∂
=φ
ε (2.3a)
yz y
y ∂
∂=
φε (2.3b)
⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂+
∂∂
=xy
z yxxy
φφγ (2.3c)
xw
xxz ∂∂
+= φγ (2.3d)
yw
yyz ∂∂
+= φγ (2.3e)
2.2 Stress-Strain Relations in Plastic Range
While the strains are linearly related to the stresses by Hooke’s law in the elastic
range, the relations between stress-strain are nonlinear in the plastic range. It has been
pointed out in Chapter 1 that many plasticity theories have been proposed in order to
capture the nonlinear stress-strain relations. In this study, the two commonly used
plasticity theories (IT and DT) are considered. In the sequel, the detailed derivation of
stress-strain relations for commonly used plasticity theories of IT and DT are
presented.
Governing Equations for Plastic Buckling Analysis of Plates
18
2.2.1 Derivation of Stress-Strain Relations Based on Prandtl-Reuss Equation
Prandtl (1925) and Reuss (1930) assumed that the plastic strain increment is, at any
instant of loading, proportional to the instantaneous stress deviation (Mendelson,
1968); i.e.
λε dsd ijp
ij = 3,2,1, =ji (2.3)
where pijdε is the plastic strain increment, λd is a positive scalar which may vary
throughout the loading history and ijs is the stress deviator tensor which is given by
ijmijijs δσσ −= (2.4)
where ijδ is the Kronecker delta and mσ denotes the hydrostatic stress which is equal
to 3/)( zyx σσσ ++ . Equation (2.3) states that the increments of plastic strain depend
on the current values of the deviatoric stress state, not on the stress increment required
to reach this state. According to the assumption that the material remains isotropic
throughout the deformation, the state of hardening at any stage is therefore specified
by the current uniaxial yield stress denoted by σ . The quantity σ is known as the
equivalent stress or effective stress, which increases with increasing plastic strain, and
is given by von Mises yield criterion as follows
( )ijij ss23
=σ (2.5)
or
( ) ( ) ( ) ( ) 2/1222222 62
1zxyzxyxzzyyx τττσσσσσσσ +++−+−+−= (2.6)
Governing Equations for Plastic Buckling Analysis of Plates
19
The corresponding equivalent or effective plastic strain increment pdε is given by
( )pij
pij
p ddd εεε32
= (2.7)
or
( ) ( ) ( ) 222
32 p
xpz
pz
py
py
px
p ddddddd εεεεεεε −+−+−=
( ) ( ) ( ) 2/1222 666 pzx
pyz
pxy ddd εεε +++ (2.8)
In view of Eqs. (2.3) and (2.5), the Eq. (2.7) can be written for a work-hardening
Prandtl-Reuss material as
λσε dd p
32
= (2.9)
By using the value of λd from Eq. (2.9), the Prandtl-Reuss flow rule, i.e. Eq. (2.3),
may be written as (Chakrabarty, 2000)
ijt
ijij
pp
ij sEE
dsHdsdd ⎟⎟
⎠
⎞⎜⎜⎝
⎛−===
112
323
23
σσ
σσ
σεε (2.10)
since EEd
ddd
ddd
dd
H t
eep 111−=−=
−==
σε
σε
σεε
σε (2.11)
where tE is the tangent modulus that is equal to εσ dd / , εd is the total effective
strain increment which is equal to pe dd εε + and edε is the effective elastic strain
increment. It can be seen that Eq. (2.10) relates the increments of plastic strain to the
stress and thus such theory is called the incremental theory of plasticity.
According to the generalized Hooke’s Law, the elastic strain increment is given by:
Governing Equations for Plastic Buckling Analysis of Plates
20
kkijijeij d
Eds
Ed σδννε
3211 −
+⎟⎠⎞
⎜⎝⎛ +
= (2.12)
Therefore, the rate form of complete Prandtl-Reuss equation relating the stress rate to
the strain rate is given by adding Eqs. (2.10) and (2.12)
ijt
ijkkijij sEEsE ⎟⎟
⎠
⎞⎜⎜⎝
⎛−+⎟
⎠⎞
⎜⎝⎛ −
++= 123
321)1(
σσδσννε&
&&& (2.13)
during the continued loading of a plastically stressed element.
Let sτqp xyyx −=−=−= and ,σσ at the point of bifurcation. According to the
usual assumption that the plate is in a state of plane stress i.e. 0=== zxyzz ττσ , the
effective stress σ in Eq. (2.3) can be expressed as
2222 3 xyyyxx τσσσσσ ++−= (2.14)
A straightforward differentiation of Eq. (2.14) gives
226)2()2(
στσσ
σσ xyyx sddpqdqpd +−+−
−= (2.15)
on setting px −=σ , qy −=σ , sxy −=τ at the point of bifurcation. Using the
expression for σσ /d from Eq. (2.15) and the fact that 2222 3sqpqp ++−=σ at the
onset of buckling, the constitutive Eq. (2.13) therefore furnishes a system of
homogeneous equations which can be expressed in the following matrix form:
Governing Equations for Plastic Buckling Analysis of Plates
21
⎪⎪⎪
⎭
⎪⎪⎪
⎬
⎫
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
=
⎪⎪⎪
⎭
⎪⎪⎪
⎬
⎫
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
yz
xz
xy
y
x
yz
xz
xy
y
x
t
ddddd
csym
ccccccc
ddddd
E
τττσσ
κ
κγγγεε
255
244
33
2322
131211
0
000000
(2.16)
where
⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟
⎠⎞
⎜⎝⎛ −−= 2
2
2
2
11 4131
σσsq
EE
c t (2.17a)
⎭⎬⎫
⎩⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟
⎠⎞
⎜⎝⎛ −−−−−= 2
2
212 213)21(1
21
σσν spq
EE
EE
c tt (2.17b)
σσsqp
EE
c t ⎟⎠⎞
⎜⎝⎛ −⎟⎠⎞
⎜⎝⎛ −=
2123
13 (2.17c)
⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟
⎠⎞
⎜⎝⎛ −−= 2
2
2
2
22 4131
σσsp
EE
c t (2.17d)
σσspq
EE
c t ⎟⎠⎞
⎜⎝⎛ −⎟⎠⎞
⎜⎝⎛ −=
2123
23 (2.17e)
2
2
33 19)1(2σ
ν sEE
EE
c tt ⎟⎠⎞
⎜⎝⎛ −++= (2.17f)
EE
cc t)1(25544
ν+== (2.17g)
and 2κ is the Mindlin shear correction factor.
By using the process of matrix inversion, we finally obtain the following
constitutive equations:
Governing Equations for Plastic Buckling Analysis of Plates
22
⎪⎪⎪
⎭
⎪⎪⎪
⎬
⎫
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
=
⎪⎪⎪
⎭
⎪⎪⎪
⎬
⎫
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
yz
xz
xy
y
x
yz
xz
xy
y
x
EGsym
EGE
γγγεε
κ
κδµγχβα
τττσσ
&
&
&
&
&
&
&
&
&
&
2
2
0
000000
(2.18)
where
( )2233322
1 ccc −=ρ
α (2.19a)
( )331223131 cccc −=ρ
β (2.19b)
( )2133311
1 ccc −=ρ
γ (2.19c)
( )221323121 cccc −=ρ
χ (2.19d)
( )231113121 cccc −=ρ
µ (2.19e)
( )2122211
1 ccc −=ρ
δ (2.19f)
)1(244 ν+==
EcE
G t (2.19g)
and
33
2322
131211
csymccccc
EE
t
=ρ (2.20)
Governing Equations for Plastic Buckling Analysis of Plates
23
2.2.2 Derivation of Stress-Strain Relations Based on Hencky’s Deformation
Theory
Unlike the Prandtl-Reuss constitutive relations, Hencky (1924) proposed the total
stress-strain relation which relates the total strain component to the current stress.
Therefore, instead of Eq. (2.10), one has
ij
pp
ij sσεε23
= (2.21)
Since the strain rate vector in this case is not along the normal to the Mises yield
surface in the stress space, the yield surface must be supposed to have locally changed
in shape so that the normality rule still holds. The possibility of the formation of a
corner on the yield surface may also be included. The incremental form of the Hencky
equation Eq. (2.21) is easily found as:
ij
p
ij
ppp
ij dssdddd
σε
σε
σε
σσε
23
23
+⎟⎟⎠
⎞⎜⎜⎝
⎛−= (2.22)
where σε /p can be written as
EEs
ep 11−=
−=
σεε
σε (2.23)
By combining Eq. (2.12) and Eq. (2.22), and using Eqs. (2.11) and (2.23), one
obtains the following rate form of the complete stress-strain relation for DT:
ijst
kkijijs
ij sEE
EEs
EEE ⎟⎟
⎠
⎞⎜⎜⎝
⎛−+
−+⎟⎟
⎠
⎞⎜⎜⎝
⎛ −−=
σσσδννε
23
321
221
23 &
&&& (2.24)
Governing Equations for Plastic Buckling Analysis of Plates
24
In view of Eq. (2.15), the constitutive equation (Eq. 2.24) furnishes
⎪⎪⎪
⎭
⎪⎪⎪
⎬
⎫
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
=
⎪⎪⎪
⎭
⎪⎪⎪
⎬
⎫
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
yz
xz
xy
y
x
yz
xz
xy
y
x
t
dd
csym
ccccccc
ddddd
E
τττσσ
κ
κγγγεε
255
244
33
2322
131211
0
000000
(2.25)
where
⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛−−= 2
2
2
2
11 4131
σσsq
EE
cs
t (2.26a)
⎭⎬⎫
⎩⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛−−−−−= 2
2
212 213)21(1
21
σσν spq
EE
EE
cs
tt (2.26b)
σσsqp
EE
cs
t ⎟⎠⎞
⎜⎝⎛ −⎟⎟⎠
⎞⎜⎜⎝
⎛−=
2123
13 (2.26c)
⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛−−= 2
2
2
2
22 4131
σσsp
EE
cs
t (2.26d)
σσspq
EE
cs
t ⎟⎠⎞
⎜⎝⎛ −⎟⎟⎠
⎞⎜⎜⎝
⎛−=
2123
23 (2.26e)
( ) 2
2
33 19213σ
ν sEE
EE
EE
cs
tt
s
t⎟⎟⎠
⎞⎜⎜⎝
⎛−+−−= (2.26f)
⎟⎟⎠
⎞⎜⎜⎝
⎛−−==
EE
EE
cc t
s
t )21(35544 ν (2.26g)
It can be seen that Eq. (2.25) has the same form as Eq. (2.16). Therefore, by the
process of matrix inversion, the stress-strain relation for DT can be simply expressed
as in the matrix form as that given in Eq. (2.18). The corresponding parameters
Governing Equations for Plastic Buckling Analysis of Plates
25
ρδµχγβα and ,,,,,, G for DT can be obtained by substituting 4411 ,.....,cc of Eq.
(2.26a-g) into Eqs. (2.19) and (2.20).
2.2.3 Material Modeling
The strain hardening plastic behavior of most structural materials used in aerospace
and off-shore applications can be adequately described by the strain-stress relationship
proposed by Ramberg and Osgood (1943) as follow
c
Ek
E ⎟⎟⎠
⎞⎜⎜⎝
⎛+=
0
0
σσσσε (2.27)
where 0σ is a nominal yield stress depending on the value of k, c is a dimensionless
constant that describes the shape of the stress-strain relationship with c = ∞ for
elastic-perfectly plastic response and k is the horizontal distance between the knee of
c = ∞ curve and the intersection of the c curve with the 1/ 0 =σσ line as shown in
Fig. 2.2.
By differentiating both sides of Eq. (2.27), and noting that the tangent modulus
εσ ddEt /= , one obtains
1
0
1−
⎟⎟⎠
⎞⎜⎜⎝
⎛+=
c
t
kcEE
σσ ; 1>c (2.28a)
while after some manipulations and by noting that εσ /=sE , one obtains
1
0
1−
⎟⎟⎠
⎞⎜⎜⎝
⎛+=
c
s
kEE
σσ ; 1>c (2.28b)
Governing Equations for Plastic Buckling Analysis of Plates
26
Based on the materials in the experiments of Aitchison and Miller (1942), Ramberg
and Osgood assumed the expressions of k and c as:
73
=k (2.29a)
85.0
7.0ln
717ln
1
σσ
+=c (2.29b)
7.00 σσ = (2.29c)
0σσ
0σεE
Fig. 2.2 Ramberg-Osgood stress-strain relation
1 + k
c
Ek
E ⎟⎟⎠
⎞⎜⎜⎝
⎛+=
0
0
σσσσε
0
0.5
1
1.5
0 1 2 3 4
2=c 3=c
5=c
10=c
20=c
∞=c
Governing Equations for Plastic Buckling Analysis of Plates
27
where 7.0σ and 85.0σ are the secant yield stresses corresponding to the intersections of
the stress-strain curve and the lines of slope 0.7E and 0.85E respectively from the
origin as shown in Fig. 2.3.
With the above assumption of k = 3/7, only three parameters ( cE ,, 7.0σ ) are needed
to specify the Ramberg-Osgood stress-strain model. These parameters are given by
Bruhn (1965) for a wide range of materials and some of them are shown in Table 2.1.
The modified version of Eq. (2.27) was proposed by Hill (1944) by using a 2%
offset yield stress, 2.0σ , for 0σ and is given by
c
E ⎟⎟⎠
⎞⎜⎜⎝
⎛+=
2.0
002.0σσσε (2.30)
⎟⎟⎠
⎞⎜⎜⎝
⎛=
1.0
2.0log
301.0
σσ
c (2.31)
ε
σ E 0.85E 0.7E
7.0σ
85.0σ
Fig. 2.3 Illustration of 7.0σ and 85.0σ on the stress-strain curve
Governing Equations for Plastic Buckling Analysis of Plates
28
in which 1.0σ is the 1% offset stress as shown in Fig. 2.4.
Table 2.1. Values of 85.07.0 ,σσ and c for various aluminum alloys under room temperature (Bruhn, 1965)
Material E 106 psi
tuσ ksi
2.0σ ksi
7.0σ ksi
85.0σ ksi
c
Stainless Steel AISI 301 ¼ Hard Sheet
Transverse Compression 27.0 125 80 73 63 6.9 Longitudinal Compression 26.0 125 43 28.2 23 5.2
AISI 301 ¾ Hard Sheet Transverse Compression 27.0 150 118 116.5 105 9.2 Longitudinal Compression 26.0 150 58 48 37 4.4
Low Carbon & Alloy Steels AISI 4130, 4140, 4340 Heat Treated 29.0 125 113 111 102 10.9
Aluminum Alloys 2014-T6 Forgings
h ≤ 4 in 10.7 62 52 52.3 50 20 2024-T3 Sheet & Plate
Heat treated, h ≤ 0.25 in 10.7 65 40 39 36 11.5 2024-T4 Sheet & Plate
Heat treated, h ≤ 0.5 in 10.7 65 38 36.7 34.5 15.6 6061-T6, Heated & aged
h < 0.25 10.1 42 35 35 34 31 7075-T6, Bare Sheet & Plate
h ≤ 0.5 10.5 76 67 70 63 9.2
Magnesium Alloys AZ61A Extrusions
h ≤ 0.249 6.3 38 14 12.9 12.3 19
Governing Equations for Plastic Buckling Analysis of Plates
29
In the literature, both Ramberg-Osgood models, i.e. Eq. (2.27) and Eq. (2.30), have
been used. It can be seen that Eq. (2.27) is the more general form of Ramberg-Osgood
model since Eq. (2.30) can be deduced from it by equating
2.02.00 002.0 and
σσσ Ek == . Therefore, the more general Ramberg-Osgood formula
Eq. (2.27) will be used herein.
2.3 Derivation of Energy Functional and Governing Equations
The strain energy functional of the Mindlin plate is given by (Chakrabarty, 2000)
∫ ++++=V
yzyzxzxzxyxyyyxx dVU γτγτγτεσεσ &&&&&&&&&&21 (2.32)
2.0σ 1.0σ
0.001 0.002 ε
σ
Fig. 2.4. Offset yield stress
0
Governing Equations for Plastic Buckling Analysis of Plates
30
In view of Eqs. (2.2a-e) and the stress-strain relations given by either of Eq. (2.18) or
Eq. (2.25) and after the integration over the plate thickness, the strain energy
functional can be expressed as
2332323
126121221
⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂+
∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂⎟⎠⎞
⎜⎝⎛∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂+⎟
⎠⎞
⎜⎝⎛∂∂
= ∫ xyEh
yxEh
yEh
xEhU yxyxy
A
x φφδφφβφγφα
⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂
∂∂
+∂∂
∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂
∂∂
+∂
∂
∂
∂+
xxyxEh
yyyxEh yxxxyxyy φφφφχφφφφµ
66
33
dAyw
xwGh yx
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
++⎟⎠⎞
⎜⎝⎛
∂∂
++22
2 φφκ (2.33)
where A is the plate area and dA = dxdy
The potential energy V for the plate subjected to uniform in-plane compressive
stresses is given by
∫⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
⎟⎠⎞
⎜⎝⎛∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+⎟⎠⎞
⎜⎝⎛∂∂
−=A
xyyx dAyw
xwh
ywh
xwhV τσσ 2
21
22
(2.34)
The total potential energy functional can be expressed as
VU +=Π (2.35)
which can be expressed in the form:
dxdyyxyxy
wxwwyxF
A
yyy
yxx∫∫ ⎟⎟
⎠
⎞⎜⎜⎝
⎛∂
∂
∂
∂
∂
∂
∂∂
∂∂
∂∂
=Πφφ
φφφ
φ ,,,,,,,,,, (2.36)
Governing Equations for Plastic Buckling Analysis of Plates
31
Using calculus of variations, the Euler-Lagrange differential equations associated
with the minimization of the total potential energy functional with respect to arbitrary
variation of yxw φφ and , are respectively given by (Forray, 1968):
0,,
=⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
∂∂
−⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
∂∂
−∂∂
yx wF
ywF
xwF (2.37a)
0,,
=⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
∂∂
−⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
∂∂
−∂∂
yxxxx
Fy
Fx
Fφφφ
(2.37b)
0,,
=⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
∂∂
−⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
∂∂
−∂∂
yyxyy
Fy
Fx
Fφφφ
(2.37c)
where x,)(• and y,)(• refer to the differentiation with respect to x and y respectively. In
view of Eqs. (2.33), (2.34) and (2.35), the governing equations are given by
ywh
yxwh
xw
yxGh yx
yx
∂∂
∂∂
−∂∂
∂∂
−⎟⎟⎠
⎞⎜⎜⎝
⎛∇+
∂
∂+
∂∂
σσφφ
κ 22
0=⎟⎠⎞
⎜⎝⎛
∂∂
∂∂
−⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
∂∂
−xwh
yywh
x xyxy ττ (2.38a)
⎟⎠⎞
⎜⎝⎛
∂∂
+−⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂+
∂∂
+∂
∂+
∂∂
∂∂
xwGh
xyEh
yEh
xEh
x xyxyx φκ
φφχφβφα 2333
121212
0121212
333
=⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
∂∂
+∂
∂+⎟⎟
⎠
⎞⎜⎜⎝
⎛∂
∂+
∂∂
∂∂
+x
Ehy
Ehxy
Ehy
xyyx φχφµφφδ (2.38b)
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+−⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂+
∂∂
+∂∂
+∂
∂
∂∂
ywGh
xyEh
xEh
yEh
y yyxxy φκ
φφµφβφγ 2333
121212
0121212
333
=⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
∂∂
+∂
∂+⎟⎟
⎠
⎞⎜⎜⎝
⎛∂
∂+
∂∂
∂∂
+x
Ehy
Ehxy
Ehx
xyyx φχφµφφδ (2.38c)
Governing Equations for Plastic Buckling Analysis of Plates
32
in which the parameters δµχγβα ,,,,, are given by Eqs. (2.19a-g). By using the
boundary conditions at the plate edges, the aforementioned governing equations can be
solved for plastic buckling stress.
33
RITZ METHOD FOR PLASTIC BUCKLING
ANALYSIS
This chapter presents the formulations of the Ritz method for plastic buckling of
Mindlin plates under in-plane compressive loadings. In this version of Ritz method, it
is proposed that the displacement functions be approximated by the product of
mathematically complete, two-dimensional polynomial functions and basic functions
comprising boundary equations that are raised to appropriate powers so as to ensure
the satisfaction of the geometric boundary conditions at the outset. By minimizing the
total potential energy functional with respect to the parameterized displacement
functions, one obtains a system of homogeneous equations which forms the governing
eigenvalue equation. The Ritz formulations will be presented for three different types
of coordinate systems, namely the Cartesian, the skew and the polar coordinate
systems for use in handling plates of various shapes.
CHAPTER 3
Ritz Method for Plastic Buckling Analysis
34
3.1 Introduction
In 1909, Walter Ritz published a paper that demonstrates his famous method for
minimizing a functional, determining frequencies and mode shapes. Since then, the
Ritz method has been widely used because of its simplicity in implementation. Two
years after Ritz’s paper (1909), Rayleigh (1911) published a book where he
complained that Ritz had not recognized his similar work (Rayleigh, 1877). Therefore
it is sometime referred to as the Rayleigh-Ritz method. However, Leissa (2005)
investigated carefully the historical works of Rayleigh and Ritz and came to
conclusion that Rayleigh’s name should not be attached to the Ritz method.
In the Ritz method, the displacement function, ),( yxℜ is approximated by a finite
linear combination of the trial functions in the form
),(),(1
yxcyxm
iii∑
=
≈ℜ φ (3.1)
in which ),( yxiφ are the approximate functions which individually satisfies at least the
geometric boundary conditions to ensure convergence to the correct solutions. The
static boundary conditions need not be satisfied by these approximate functions. By
minimizing the energy functional Π with respect to each of the unknown coefficients
ci, a set of homogeneous equations is obtained as follows
;0=∂Π∂
ic mi ,......,2,1= (3.2)
For buckling and vibration problems, the above set of homogeneous equations is
reduced to eigenvalue and eigenvector problems.
Ritz Method for Plastic Buckling Analysis
35
The exact solution is obtained if infinite terms are adopted in Eq. (3.1). However, it
is impractical to use infinite number of terms and so the number of terms is usually
truncated to N terms in applications. The choice of the approximate functions is very
important in order to simplify the calculations and to guarantee convergence to the
exact solution. Some of the commonly used trial functions in the Ritz method for plate
analysis are the products of eigenfunctions of vibrating beams (Leissa, 1973;
Dickinson and Li, 1982; Narita and Leissa, 1990), the generated beam functions
(Bassily and Dickinson, 1975, 1978), the spline functions (Mizusawa, 1986), the beam
characteristic orthogonal polynomials (Bhat, 1985) and two dimensional polynomial
functions with appropriate basic functions (Bhat, 1987; Laura et al., 1989; Liew, 1990;
Liew and Wang, 1992, 1993; Geannakakes, 1995). Among them, the latter trial
functions can be used for the analysis of plates of general shape and boundary
conditions while the others may not be so convenient. Moreover, the computational
accuracy may also be increased since the polynomial functions admit exact
calculations of differentiation and integration of the functions (Liew et al., 1998).
Therefore, in this study, mathematically complete, two-dimensional polynomial
functions are adopted together with basic functions comprising boundary equations
that are raised to appropriate powers in order to ensure the satisfaction of the geometric
boundary conditions.
3.2 Boundary Conditions
The following boundary conditions are considered for the plastic buckling problems of
Mindlin plates;
(1) Simply Supported Edge (S and S*)
Ritz Method for Plastic Buckling Analysis
36
There are two kinds of simply supported edges for Mindlin plates. The first kind
(S), which is commonly referred to as the hard type simply supported edge, requires
0 ,0 ,0 === wM snn φ (3.3)
The second kind (S*), commonly referred to as the soft type simply supported edge,
requires
0 ,0 ,0 === wMM nsnn (3.4)
where nnM is the bending moment normal to the edge, nsM the twisting moment at
the edge as shown in Fig. 3.1, and sφ the rotation of the mid-plane normal in the
tangent plane sz to the plate edge.
(2) Clamped Edge (C)
The boundary conditions for a clamped edge are
0 ,0 ,0 === wsn φφ (3.5)
in which nφ is the rotation of the mid-plane to the edge.
(3) Free Edge (F)
For a free edge, the boundary conditions are
0 ,0 ,0 =∂∂
−==nwNQMM nnnnsnn (3.6)
where nQ is the transverse shear force in the normal direction n to the plate edge and
nnN is the inplane compressive force in the normal direction of the edge.
Ritz Method for Plastic Buckling Analysis
37
3.3 Ritz Formulation in Cartesian Coordinate System
Consider an isotropic, rectangular plate of uniform thickness h, length a, width b
and subjected to in- plane uniform compressive stresses of magnitudes xσ , yσ and
uniform shear stress xyτ as shown in Fig. 3.2. The edges of the plate may take on any
combination of simply supported, clamped or free edges.
For generality and convenience, the coordinates are normalized with respect to the
maximum length dimensions (a and b as shown in Fig. 3.2) and the following
nondimensional terms are introduced:
ηξζτηξ ddAba
bww
bh
by
ax
====== ;;2;;2;2 (3.7)
Fig. 3.1 Forces and moments on a plate element
Ritz Method for Plastic Buckling Analysis
38
)1(12;;; 2
3
2
2
2
2
2
2
νπτ
λπσ
λπσ
λ−
====EhD
Dhb
Dhb
Dhb xy
xyy
yx
x (3.8)
Then, in view of Eqs (3.2) and (3.3), the strain energy and potential energy functionals
in Eqs. (2.33) and (2.34) become respectively,
∫∫⎪⎩
⎪⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂+⎟⎟
⎠
⎞⎜⎜⎝
⎛∂∂
=A
yxyx EhEhEhUηφ
ξφ
ζβηφ
ζγξφα
ζ 6121221 32
2323
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂
∂
∂
∂+⎟⎟
⎠
⎞⎜⎜⎝
⎛∂
∂+
∂∂
+ηφ
ζξφ
ζηφµ
ξφ
ηφ
ζδ xyyyx EhEh612
33
⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂+
∂∂
∂∂
+ξφ
ηφ
ζξφχ yxxEh
6
3
ηξη
φξζ
φζκ ddwwGhbyx
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
++⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
++22222 1
4 (3.9)
∫∫⎥⎥⎦
⎤
⎢⎢⎣
⎡
∂∂
∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−−=
Axyyx ddwwwwEhV ηξ
ηξζλ
ηζλ
ξλ
νπ
ζ2
)1(1281
22
2
32
(3.10)
Fig. 3.2 Rectangular plate under in-plane compressive stresses
x, ξ
σx
a
y, η
σy
σy
b σx
τxy
τxy
Ritz Method for Plastic Buckling Analysis
39
Therefore, the total potential energy functional Π is given by
VU +=Π (3.11)
The adopted admissible p-Ritz functions for the deflection and rotations of the
plate are given by (Liew et al. 1998)
( ) ),(,0 0
ηξφηξ wm
p
q
q
imcw ∑∑
= =
= (3.12a)
( ) ),(,0 0
ηξφηξφ xm
p
q
q
imx d∑∑
= =
= (3.12b)
( ) ),(,0 0
ηξφηξφ ym
p
q
q
imy e∑∑
= =
= (3.12c)
where p is the degree of the complete polynomial space, ci, di, ei the unknown
coefficients to be varied with the subscript m given by
iqqm −++= 2/)2)(1( (3.13)
and the total numbers of mmm edc ,, are given by
2)2)(1( ++
=ppN (3.14)
The functions ym
xm
wm φφφ ,, are given as follow:
iqiwb
wm
−= ηξφφ (3.15a)
iqixb
xm
−= ηξφφ (3.15b)
iqiyb
ym
−= ηξφφ (3.15c)
Ritz Method for Plastic Buckling Analysis
40
In view of Eqs. (3.15a-c), the Eqs. (3.12a-c) give the mathematically complete two
dimensional polynomials which follow the Pascal-type of expansion, for example:
For p = 1, ( ) ( )ηξφηξ 310, cccw wb ++= (3.16a)
For p = 2 ( ) ( )265
24321, ηξηξηξφηξ ccccccw w
b +++++= (3.16b)
The functions yb
xb
wb φφφ and ,, are the basic functions which are the product the
boundary equations raised to appropriate power so the admissible Ritz functions
satisfy the geometric boundary conditions as follow.
[ ]∏=
ΩΓ=ne
jj
wb
wj
1
),( ηξφ (3.17a)
[ ]∏=
ΩΓ=ne
jj
xb
xj
1
),( ηξφ (3.17b)
[ ]∏=
ΩΓ=ne
jj
yb
yj
1
),( ηξφ (3.17c)
where ne is the number of edges, jΓ the boundary equation of the jth edge and
yj , , ΩΩΩ x
jwj are given by
⎩⎨⎧
=Ω ∗ )( clampedor ) and ( supportedsimply 1)(free0
CSSFw
j (3.18a)
⎩⎨⎧
=Ω)( clampedor direction - in the )( supportedsimply 1
direction- in the )(or )*( supportedsimply or )( free0CxS
ySSFxj (3.18b)
⎩⎨⎧
=Ω)( clampedor direction -y in the )( supportedsimply 1
direction- in the )(or )*( supportedsimply or )( free0CS
xSSFyj (3.18c)
On the basis of foregoing energy functional and p-Ritz functions, the application of
the Ritz method yields:
Ritz Method for Plastic Buckling Analysis
41
0,0,0,, =∂Π∂
∂Π∂
∂Π∂
mmm edc (3.19)
where m = 1,2,….,N. The substitution of Eqs. (3.11) and (3.12a-c) into Eq. (3.19)
yields the following homogeneous set of equations
[ ] 0=⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
edc
K (3.20)
where vectors c, d and e are given by
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
=
mc
cc
:2
1
c ;
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
=
md
dd
:2
1
d ;
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
=
me
ee
:2
1
e (3.21a-c)
The matrix [K] has a structure of
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
ee
dedd
cecdcc
KKKKKK
symmetric (3.22)
and its elements are given by
∫∫⎢⎢⎣
⎡
∂
∂
∂∂
−−⎟
⎟⎠
⎞⎜⎜⎝
⎛
∂
∂
∂∂
+∂
∂
∂∂
=A
wj
wix
wj
wi
wj
wicc
ij EGK
ξφ
ξφ
ζνλ
ηφ
ηφ
ζξφ
ξφ
ζτκ 1
)1(481
4 22
2
ηξξφ
ηφ
ηφ
ξφ
νλ
ηφ
ηφ
ζν
λdd
wj
wi
wj
wixy
wj
wiy
⎥⎥⎦
⎤⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂
∂∂
+∂
∂
∂∂
−−
∂
∂
∂∂
−−
)1(48)1(48 22 (3.23a)
Ritz Method for Plastic Buckling Analysis
42
∫∫ ⎥⎦
⎤⎢⎣
⎡∂∂
=A
xj
wicd
ij ddE
GK ηξφξφ
τκ
2
2
4 (3.23b)
∫∫ ⎥⎦
⎤⎢⎣
⎡∂∂
=A
yj
wice
ij ddE
GK ηξφηφ
ζτκ
2
2
4 (3.23c)
∫∫⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂
∂∂
+∂
∂
∂∂
+∂
∂
∂∂
+∂
∂
∂∂
=A
xj
xi
xj
xi
xj
xi
xj
xidd
ij ddK ηξξφ
ηφ
ηφ
ξφχ
ηφ
ηφδζ
ξφ
ξφ
ζα
12121
12
ηξφφτ
ζκ ddE
G xj
xi ⎥
⎦
⎤+ 2
2
4 (3.23d)
∫∫⎢⎢⎣
⎡
∂
∂
∂∂
+∂
∂
∂∂
=A
yj
xi
yj
xide
ijKξφ
ηφδ
ηφ
ξφβ
1212
ηξξφ
ξφ
ζχ
ηφ
ηφµζ dd
yi
xi
yi
xi
⎥⎦
⎤∂∂
∂∂
+∂∂
∂∂
+1212
(3.23e)
∫∫⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂
∂∂
+∂
∂
∂∂
+∂
∂
∂∂
+∂
∂
∂∂
=A
yj
yi
yj
yi
yj
yi
yj
yiee
ijKξφ
ηφ
ηφ
ξφµ
ξφ
ξφ
ζδ
ηφ
ηφγζ
121212
ηξφφτ
ζκ ddE
G yj
yi ⎥
⎦
⎤+ 2
2
4 (3.23f)
where i, j = 1, 2,….N and G,,,,,, χµδγβα are parameters depending on the material
properties and stress level as shown in Chapter 2. The plastic buckling stress parameter
can be obtained by setting the determinant of the [ ]K matrix in Eq. (3.20) to zero and
solving the characteristic equation for the roots (eigenvalues). The lowest eigenvalue
corresponds to the critical plastic bifurcation buckling stress.
Ritz Method for Plastic Buckling Analysis
43
3.4 Ritz Formulation in Skew Coordinate System
For the analysis of skew or parallelogram plates, it is more expedient to adopt the
skew coordinate system. For example, the hard type simply supported boundary
condition of oblique edge can be readily implemented in a skew coordinate system
while the Ritz formulation with the Cartesian coordinate system has to be modified in
order to implement that this kind of boundary condition.
Consider a skew plate of uniform thickness h with length a, oblique width b and
skew angle ψ as shown in Fig. 3.3a. The expression of energy functional in skew
coordinate system can be readily obtained by transforming of Cartesian coordinate
system into skew coordinate system. It can be seen from Fig. 3.3a that these
coordinates are related by (Liew et al. 1998)
ψtanyxx −= (3.24a)
ψsecyy = (3.24b)
The rotations yx φφ , can be expressed in the skew coordinate system as
ψφφ cos),(),( yxyx xx = (3.25a)
),(sin),(),( yxyxyx yxy φψφφ +−= (3.25b)
The relationship between the partial derivatives of rectangular and skew coordinates
are given by
xx ∂•∂
=∂•∂ )()( (3.26a)
ψψ sec)(tan)()(yxy ∂•∂
+∂•∂
−=∂•∂ (3.26b)
Ritz Method for Plastic Buckling Analysis
44
For generality and convenience, the following nondimensional terms are adopted:
ηξζτηξ ddAba
bww
bh
by
ax
====== ;;2;;2;2 (3.27)
In view of Eqs. (3.24) to (3.27), the strain energy functional in Eq. (2.33) become
⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−⎪⎩
⎪⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
⎢⎣
⎡= ∫∫ ξ
φξφ
ψξφ
ψβψξφαψ
ζyxxx
A
EhEhU sinsin6
cos12
cos21
22
3233
ξφ
ψξφ
ψγψηφ
ξφ
ζηφ
ξφ
ψζ∂
∂−
⎩⎨⎧
∂∂
+⎭⎬⎫
∂
∂
∂∂
+∂∂
∂∂
− yxyxxx Eh sinsin12
secsin 23
ξ, , xx
b/2
b/2
a/2 a/2
y
yy φφ , ψ
η,y
xφ xφ
Fig. 3.3. (a) Skew coordinates and plate dimensions (b) Directions of stresses in the infinitesimal element
σx
σy
τxy
(b)
(a)
Ritz Method for Plastic Buckling Analysis
45
232
sin212
cossin⎭⎬⎫
⎩⎨⎧
∂
∂−
∂∂
−∂∂
+⎭⎬⎫
∂
∂+
∂∂
−ξφ
ηφ
ζξφ
ψδψηφ
ζηφ
ψζ yxxyx Eh
⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂+
∂∂
∂
∂
⎪⎩
⎪⎨⎧
−⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂+
∂∂
−ξφ
ηφ
ζηφ
ζξφ
ηφ
ζψµ yxyyxEh23
sin6
⎪⎭
⎪⎬⎫⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂+
∂
∂−
∂∂
−∂∂
∂∂
+ηφ
ζξφ
ψηφ
ψζξφ
ψξφ
ψ yyxxx 2sin3sin3sin2sin 2
⎭⎬⎫
⎩⎨⎧
∂
∂−
∂∂
−∂∂
∂∂
−ξφ
ηφ
ζξφ
ψξφχψ yxxxEh sin2
6cos
32
⎜⎜⎝
⎛∂∂
−−+⎪⎩
⎪⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
++ξ
ψψφζζφξ
ψφζκψ wwGhbxyx tansincos
4cos
222
ηξη
ψζ ddw⎥⎥⎦
⎤
⎪⎭
⎪⎬⎫
⎟⎟⎠
⎞∂∂
+2
sec (3.28)
The potential energy functional, V, of the in-plane uniform loads is given by
⎪⎩
⎪⎨⎧
∂∂
∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−= ∫∫ ηξ
ψζη
ζψλξ
ψλν
πζ
wwwwEhV yA
x sin2seccos)1(128
12
22
2
32
ηξξ
ψηξ
ζλξ
ψ ddwwwwxy
⎥⎥⎦
⎤
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−∂∂
∂∂
+⎪⎭
⎪⎬⎫
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−22
2 sin2sin (3.29)
The total potential energy functional,Π , in the skew coordinate system is obtained
by summing the strain energy functional and the potential energy functional.
It should be noted that the plastic buckling stress parameters, xyyx λλλ ,, used in Eq.
(3.29) are based on the stress resultants on an infinitesimal rectangular element as
shown in Fig. 3.3b.
Ritz Method for Plastic Buckling Analysis
46
The admissible Ritz displacement functions for the skew coordinate system have
the same form as those for the rectangular coordinate system (Eqs. 3.12 to 3.18) except
for the notation of the coordinate system. However, for clarity and convenience, these
admissible functions for skew coordinate system are repeated as follows:
( ) ),(,0 0
ηξφηξ wm
p
q
q
imcw ∑∑
= =
= (3.30a)
( ) ),(,0 0
ηξφηξφ xm
p
q
q
imx d∑∑
= =
= (3.30b)
( ) ),(,0 0
ηξφηξφ ym
p
q
q
imy e∑∑
= =
= (3.30c)
where the functions ym
xm
wm φφφ ,, are given by:
iqiwb
wm
−= ηξφφ (3.31a)
iqixb
xm
−= ηξφφ (3.31b)
iqiyb
ym
−= ηξφφ (3.31c)
The functions yb
xb
wb φφφ and ,, are the basic functions which are the product the
boundary equations raised to appropriate power so the admissible Ritz functions
satisfy the geometric boundary conditions as follows:
[ ]∏=
ΩΓ=ne
jj
wb
wj
1
),( ηξφ (3.32a)
[ ]∏=
ΩΓ=ne
jj
xb
xj
1
),( ηξφ (3.32b)
[ ]∏=
ΩΓ=ne
jj
yb
yj
1
),( ηξφ (3.32c)
Ritz Method for Plastic Buckling Analysis
47
where ne is the number of edges, jΓ the boundary equation of the jth edge and
yj , , ΩΩΩ x
jwj are given by
⎩⎨⎧
=Ω ∗ )( clampedor ) and ( supportedsimply 1);(free0
CSSFw
j (3.33a)
⎩⎨⎧
=Ω)( clampedor direction - in the )( supportedsimply 1
direction;- in the )(or )*( supportedsimply or )( free0CxS
ySSFxj (3.33b)
⎩⎨⎧
=Ω)( clampedor direction -y in the )( supportedsimply 1
direction;- in the )(or )*( supportedsimply or )( free0CS
xSSFyj (3.33c)
By substituting the displacement functions given by Eqs. (3.30a-c) into Eq. (3.19),
the resulting eigenvalue equation has the same form as Eq. (3.20) and the elements of
the matrix [ ]K are given by
∫∫ ⎢⎢⎣
⎡
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂
∂∂
+∂
∂
∂∂
−∂
∂
∂∂
+∂
∂
∂∂
=A
wj
wi
wj
wi
wj
wi
wj
wicc
ij EGK
ξφ
ηφ
ηφ
ξφ
ψηφ
ηφ
ζξφ
ξφ
ζψ
τκ sin1sec4 2
2
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂
∂∂
+∂
∂
∂∂
⎪⎩
⎪⎨⎧
−∂
∂
∂∂
−−
∂
∂
∂∂
−−
ξφ
ηφ
ηφ
ξφ
ψηφ
ηφ
ζνψπλ
ξφ
ξφ
νζψπλ w
jwi
wj
wi
wj
wiy
wj
wix sin
)1(48sec
)1(48cos
2
2
2
2
⎪⎩
⎪⎨⎧
∂
∂
∂∂
+∂
∂
∂∂
−−
⎪⎭
⎪⎬⎫
∂
∂
∂∂
+ξφ
ηφ
ηφ
ξφ
νπλ
ξφ
ξφ
ζψ w
jwi
wj
wixy
wj
wi
)1(48sin
2
22
ηξξφ
ξφ
ζψ dd
wj
wi
⎥⎥⎦
⎤
⎪⎭
⎪⎬⎫
∂
∂
∂∂
−sin2 (3.34a)
∫∫ ⎥⎦
⎤⎢⎣
⎡∂∂
−∂∂
=A
xj
wix
j
wicd
ij ddE
GK ηξφηφ
ψζφξφ
τκ sin4 2
2
(3.34b)
∫∫ ⎥⎦
⎤⎢⎣
⎡∂∂
−∂∂
=A
yj
wiy
j
wice
ij ddE
GK ηξφξφ
ψφηφ
ζτ
κ sin4 2
2
(3.34c)
Ritz Method for Plastic Buckling Analysis
48
⎪⎩
⎪⎨⎧
∂
∂
∂∂
−∂
∂
∂∂
+⎢⎢⎣
⎡
∂
∂
∂∂
= ∫∫ ξφ
ηφ
ψξφ
ξφ
ζψψψγ
ξφ
ξφ
ζψα x
jx
ixj
xi
A
xj
xidd
ijK sinsinsintan12
1cos12
23
⎪⎩
⎪⎨⎧
∂
∂
∂∂
+⎪⎭
⎪⎬⎫
∂
∂
∂∂
+∂
∂
∂∂
−ξφ
ξφ
ψζ
ψψβηφ
ηφ
ζηφ
ξφ
ψxj
xi
xj
xi
xj
xi sin2cossin
12sin
⎪⎩
⎪⎨⎧
∂
∂
∂∂
+∂
∂
∂∂
+⎪⎭
⎪⎬⎫
∂
∂
∂∂
−∂
∂
∂∂
−ηφ
ηφ
ζξφ
ξφ
ψζ
ψδηφ
ξφ
ξφ
ηφ x
jx
ixj
xi
xj
xi
xj
xi 2sin4cos
12
⎪⎩
⎪⎨⎧
∂
∂
∂∂
+∂
∂
∂∂
+⎪⎭
⎪⎬⎫⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂
∂∂
+∂
∂
∂∂
−ηφ
ξφ
ξφ
ηφ
ψχηφ
ξφ
ξφ
ηφ
ψxj
xi
xj
xi
xj
xi
xj
xi 2cos
12sin
⎪⎩
⎪⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂
∂∂
+∂
∂
∂∂
+∂
∂
∂∂
−+⎪⎭
⎪⎬⎫
∂
∂
∂∂
ηφ
ξφ
ξφ
ηφψ
ηφ
ηφ
ζψµξφ
ξφ
ζψ x
jx
ixj
xi
xj
xi
xj
xi
2sin3sin
6sin4
ηξφψφζτ
κξφ
ξφ
ζψ dd
EG x
jx
i
xj
xi
⎥⎥⎦
⎤+
⎪⎭
⎪⎬⎫
∂
∂
∂∂
− cos4
sin22
22
(3.34d)
∫∫ ⎢⎢⎣
⎡
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
∂
∂
∂∂
−⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂
∂∂
+∂
∂
∂∂
+∂
∂
∂∂
−=A
yj
xi
yj
xi
yj
xi
yj
xide
ijKηφ
ηφ
ζηφ
ξφ
ξφ
ηφ
ψξφ
ξφ
ζψψγ sinsintan
12
2
⎪⎭
⎪⎬⎫
∂
∂
∂∂
−⎪⎩
⎪⎨⎧
∂
∂
∂∂
+⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
∂
∂
∂∂
−∂
∂
∂∂
+ξφ
ξφ
ζψ
ξφ
ηφ
ψδξφ
ξφ
ζψ
ηφ
ξφ
ψβ yj
xi
yj
xi
yj
xi
yj
xi sin2cos
12sincos
12
ηφ
ηφ
ζξφ
ξφ
ζψµ
ξφ
ξφ
ψζχ
∂
∂
∂∂
+⎪⎩
⎪⎨⎧
∂
∂
∂∂
+∂
∂
∂∂
+yj
xi
yj
xi
yj
xi
22 sin3
12cos
12
ηξφψφψτ
κξφ
ηφ
ηφ
ξφ
ψ ddE
G yj
xi
yj
xi
yj
xi
⎥⎥⎦
⎤−
⎪⎭
⎪⎬⎫⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂
∂∂
+∂
∂
∂∂
− cossin4
sin2 2
2
(3.34e)
∫∫⎢⎢⎣
⎡
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
∂
∂
∂∂
+∂
∂
∂∂
−∂
∂
∂∂
−∂
∂
∂∂
=A
yj
yi
yj
yi
yj
yi
yj
yiee
ijKηφ
ηφ
ζηφ
ξφ
ψξφ
ηφ
ψξφ
ξφ
ζψψγ sinsinsinsec
12
2
ηξξφ
ηφ
ηφ
ξφ
ξφ
ξφ
ζψµ
ξφ
ξφ
ζψδ dd
yj
yi
yj
yi
yj
yi
yj
yi
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
∂
∂
∂∂
+∂
∂
∂∂
+∂
∂
∂∂
+∂
∂
∂∂
+sin2
12cos
12
Ritz Method for Plastic Buckling Analysis
49
ηξφψφζτ
κ ddE
G yj
yi ⎥
⎦
⎤+ cos
4 2
2
(3.34f)
where i, j = 1, 2,….N.
3.5 Ritz Formulation in Polar Coordinate System
The energy functionals in polar coordinates can be obtained from the corresponding
Cartesian coordinates equations by using the relations θcosrx = and θsinry = .
However, the transformation procedure involves some lengthy mathematical
manipulations. Therefore, the procedure would be simpler if the energy functional is
derived directly from the strain-displacement relations in the polar coordinates. Based
on the assumptions of the Mindlin plate theory, the displacement components in the
polar coordinates can be expressed as
),(),,( θφθ rzzru rr = (3.35a)
),(),,( θφθ θθ rzzru = (3.35b)
),(),,( θθ rwzrw = (3.35c)
where θuur , are in-plane displacements in the r and θ directions, respectively and
θφφ ,r are rotation about θ and r axes respectively. The corresponding strain-
displacement relations are given by
rz r
rr ∂∂
=φ
ε (3.36a)
⎟⎠⎞
⎜⎝⎛ +∂∂
= rrz φ
θφ
ε θθθ (3.36b)
Ritz Method for Plastic Buckling Analysis
50
⎟⎠⎞
⎜⎝⎛ −
∂∂
+∂∂
=rrr
z rr
θθθ
φφθφ
γ 1 (3.36c)
rw
rrz ∂∂
+= φγ (3.36d)
θφγ θθ ∂
∂+=
wrz1 (3.36e)
In view of Eq. (3.36a-e), the strain energy functional and potential energy
functional can be expressed in polar coordinates as:
2
2
3323
1261221
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛
∂∂
++⎟⎠⎞
⎜⎝⎛
∂∂
+∂∂
+⎟⎠⎞
⎜⎝⎛∂∂
= ∫∫ θφ
φγθφ
φφβφα θθ
rrA
rr
rEh
rrEh
rEhU
⎩⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
∂∂
+⎟⎠⎞
⎜⎝⎛
∂∂
−∂∂
−θφ
ξφ
θφµ
θφφ
φδ θθθθ
rr
rrEh
rrEh 11
612
32
2
3
⎜⎜⎝
⎛∂∂
∂∂
+⎭⎬⎫⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+−⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
+ηφ
ξφχ
ηφ
φφ
ξφ
ηφ
φ θθθ rrr
rr r
Ehrrr
16
11 3
22
θθ
φφκφξφ
θθθ rdrdw
rrwGh r
⎥⎥⎦
⎤
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎟⎠⎞
⎜⎝⎛
∂∂
++⎟⎠⎞
⎜⎝⎛
∂∂
++⎟⎟⎠
⎞−
∂∂
+22
2 1 (3.37)
θθ
τθ
σσ θθθ rdrdwrwh
rw
rh
rwhV
Arrr∫∫
⎥⎥⎦
⎤
⎢⎢⎣
⎡
∂∂
∂∂
+⎟⎠⎞
⎜⎝⎛∂∂
+⎟⎠⎞
⎜⎝⎛∂∂
−=21
21 2
2
2
(3.38)
where θθσσ ,rr are the inplane stresses in the radial and tangential directions,
respectively and θτ r is the shear stress.
The polar coordinates (r, θ) can be transformed into a set of nondimensional
coordinates (ξ, η) by the following relations:
[ ]ξζζ )1()1(2
−++=Rr (3.39a)
Ritz Method for Plastic Buckling Analysis
51
ηθ Ω=21 (3.39b)
where the parameters (ζ, Ω) are defined in order to get the limits of integration from -1
to 1 as shown in Fig. 3.4. Note that, in order to avoid singularity problems in the case
of a circular plate, ζ needs to be set to a very small number (e.g. 10-7) and the inner
edge is made free.
θ R r
ζR
R
(a) Circular plate ζ = 0, Ω = 2π
(b) Annular plate ζ ≠ 0, Ω = 2π
Ω
R
Ω ζR
R
(c) Sectorial plate ζ = 0, Ω ≠ 0
(d) Annular sectorial plate ζ ≠ 0, Ω ≠ 0
Fig. 3.4 Definitions of ζ and Ω
Ritz Method for Plastic Buckling Analysis
52
For generality and convenience, the following nondimensional terms are adopted
ηξτ ddARww
Rh
Rrr ==== ;;; (3.40a)
DhR
DhR
DhR r
rrr
r 2
2
2
2
2
2
;;πτλ
πσλ
πσλ θ
θθθ
θ === (3.40b)
In view of aforementioned nondimensional terms, the strain energy functional can be
expressed as
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
Ω+
∂∂
−+
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−Ω−
= ∫∫ ηφ
φξφ
ζβ
ξφ
ζαζ θ2
)1(2
6)1(2
128)1( 323
rr
A
r
rEhEhU
2
2
2
2
3 2)1(
212
212 ⎟⎟
⎠
⎞⎜⎜⎝
⎛∂∂
Ω−
∂∂
−−+⎟⎟
⎠
⎞⎜⎜⎝
⎛∂∂
Ω++
ηφ
ξφ
ζφδ
ηφ
φγ θθ
θ rr
rrr
Eh
⎜⎜⎝
⎛∂∂
Ω+⎟⎟
⎠
⎞⎜⎜⎝
⎛∂∂
Ω+
∂∂
−⎩⎨⎧
∂∂
Ω+
ηφ
φηφ
ξφ
ζηφµ θθ r
rr
rrrEh
22
3 222)1(
226
⎭⎬⎫⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
Ω+−⎟⎟
⎠
⎞∂∂
−+
ηφ
φφξφ
ζθ
θθ 22
)1(2
2 rrr
⎟⎟⎠
⎞⎜⎜⎝
⎛−
∂∂
−+
∂∂
Ω∂∂
−+ θ
θ φξφ
ζηφ
ξφ
ζχ
)1(22
)1(2
6
3rr
rEh
ηξη
φξζ
φκ θ ddrwr
wGhR r⎥⎥⎦
⎤
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
Ω++⎟⎟
⎠
⎞⎜⎜⎝
⎛∂∂
−++
2222 2
)1(2 (3.41)
The potential energy functional, V, becomes
∫∫⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
Ω−
+⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−Ω
−−=
Ar
wr
wEhV22
2
32 1)1()1()1(24 η
ζλξζ
λν
πθ
ηξηξ
λ θ ddrwwrr ⎥
⎦
⎤∂∂
∂∂
+12 (3.42)
Ritz Method for Plastic Buckling Analysis
53
The Ritz displacement functions can be written as follows:
( ) ),(,0 0
ηξφηξ wm
p
q
q
imcw ∑∑
= =
= (3.43a)
( ) ),(,0 0
ηξφηξφ rm
p
q
q
imr d∑∑
= =
= (3.43b)
( ) ),(,0 0
ηξφηξφ θθ m
p
q
q
ime∑∑
= =
= (3.43c)
where p is the degree of the complete polynomial space, cm, dm, em the unknown
coefficients and the subscript m is given by Eq. (3.8).
The functions θφφφ mrm
wm ,, are given as follows:
iqiwb
wm
−= ηξφφ (3.44a)
iqirb
rm
−= ηξφφ (3.44b)
iqibm
−= ηξφφ θθ (3.44c)
in which the basic functions θφφφ brb
wb and ,, are the product of boundary equations with
appropriate power as follows:
[ ]∏=
ΩΓ=ne
jj
wb
wj
1
),( ηξφ (3.45a)
[ ]∏=
ΩΓ=ne
jj
rb
rj
1
),( ηξφ (3.45b)
[ ]∏=
ΩΓ=ne
jjb
j
1
),(θ
ηξφθ (3.45c)
Ritz Method for Plastic Buckling Analysis
54
where ne is the number of edges, jΓ the boundary equation of the jth edge and
θj , , ΩΩΩ r
jwj are given by
⎩⎨⎧
=Ω ∗ )( clampedor ) and ( supportedsimply 1);(free0
CSSFw
j (3.46a)
⎩⎨⎧
=Ω)( clamped1
;)(or )*( supportedsimply or )( free0C
SSFrj (3.46b)
⎩⎨⎧
=Ω)( clampedor )( supportedsimply 1
);*( supportedsimply or )( free0CS
SFjθ (3.46c)
The elements of matrix [ ]K are given by
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
∂
∂
∂∂
Ω−
+∂
∂
∂∂
−Ω
⎢⎣
⎡= ∫∫ η
φηφζ
ξφ
ξφ
ζτκ w
jwi
wj
wi
A
ccij r
rE
GK )1()1(2
2
ηφ
ηφζλ
ξφ
ξφ
ζλ
νπ θ
∂
∂
∂∂
Ω−
+⎪⎩
⎪⎨⎧
∂
∂
∂∂
−Ω
−−
wj
wi
wj
wir
rr )1()1()1(12 2
2
ηξξφ
ηφ
ηφ
ξφ
λ θ ddwj
wi
wj
wi
r⎥⎥⎦
⎤
⎪⎭
⎪⎬⎫⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂
∂∂
+∂
∂
∂∂
+ (3.47a)
∫∫ ∂∂Ω
=A
rj
wicd
ij ddrE
GK ηξφξφ
τκ
22
2
(3.47b)
∫∫ ∂∂−
=A
j
wice
ij ddE
GK ηξφηφζ
τκ θ
2)1(
2
2
(3.47c)
⎭⎬⎫
⎩⎨⎧
+−Ω
+∂
∂
∂∂
⎢⎣
⎡−Ω
= ∫∫ ri
ri
rj
ri
rj
ri
A
ddij E
rGr
rK φφτ
κφφγζξφ
ξφ
ζα
2
2
124)1(
)1(12
ηφ
ηφδζφ
ξφ
ξφ
φβ∂
∂
∂∂
Ω−
+⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
∂∂
+∂
∂Ω+
rj
rir
j
ri
rjr
i r12)1(
24
Ritz Method for Plastic Buckling Analysis
55
ηξξφ
ηφ
ηφ
ξφχ
ηφ
φφηφµζ dd
r
rj
ri
rj
ri
rjr
irj
ri
⎥⎥⎦
⎤⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂
∂∂
+∂
∂
∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂∂
⎟⎠⎞
⎜⎝⎛ −
+1224
)1( (3.47d)
θθθθ
φηφζ
ξφ
ηφδ
ηφ
φζγηφ
ξφβ
j
rij
rijr
iA
jride
ij rEG
rK
∂∂−
−∂
∂
∂∂
+∂
∂−+
⎢⎢⎣
⎡
∂
∂
∂∂
= ∫∫ 24)1(
1224)1(
12
⎟⎟⎠
⎞⎜⎜⎝
⎛ Ω−−
∂
∂Ω+
∂
∂
∂∂
Ω−
+ θθθ
φφζξφ
φηφ
ηφζµ
jri
jri
jri
rr 4)1(
2)1(
12
ηξφξφ
ξφ
ξφ
ζχ θ
θ
ddrj
rij
ri
⎥⎥⎦
⎤⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
−∂
∂
∂∂
−Ω
+)1(
224
(3.47e)
⎪⎩
⎪⎨⎧
∂
∂
∂∂
−Ω
+−Ω
+⎢⎢⎣
⎡
∂
∂
∂∂
Ω−
= ∫∫ ξφ
ξφ
ζφφ
τζκ
ηφ
ηφγζ θθ
θθθθ
jiji
A
jieeij
rE
GErG
rK
)1(124)1(
12)1(
2
2
⎪⎭
⎪⎬⎫⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂∂
−−
+ξφ
φφξφ
φφζ θθθ
θθθ j
iji
jir 21
4)1(
ηξηφ
φφηφζ
ξφ
ηφ
ηφ
ξφµ θ
θθθθθθθ
ddr
jij
ijiji
⎥⎥⎦
⎤
⎪⎭
⎪⎬⎫⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂∂−
⎪⎩
⎪⎨⎧
−∂
∂
∂∂
+∂
∂
∂∂
+2
)1(12
(3.47f)
where i, j = 1, 2,….N .
For the special cases of circular and annular plates, the deflection and rotations of
the plate may be written as (Irie et al., 1982)
)cos()(),( πηξηξ nww = (3.48a)
)cos()(),( πηξφηξφ nrr = (3.48b)
)sin()(),( πηξφηξφ θθ n= (3.48c)
where n is the arbitrary integer number which represents the number of nodal diameter.
Ritz Method for Plastic Buckling Analysis
56
It should be noted that the trigonometric functions which appear in the energy
integration can be calculated as follows:
⎩⎨⎧
>=
== ∫− 0 when 2
0 when 1)(cos
1
1
21 n
ndnS ηπη (3.49a)
⎩⎨⎧
>=
== ∫− 0 when 1
0 when 0)(sin
1
1
22 n
ndnS ηπη (3.49b)
...... 2, 1, 0, for 0)sin()cos(1
1
==∫−
ndnn ηπηπη (3.49c)
In views of Eqs. (3.48a-c) and (3.49a-c), the strain energy and potential energy
functionals can be expressed as
( )θφφξφ
ζβ
ξφ
ζαζ nS
rEhSEhU r
r
A
r +∂∂
−+
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−Ω−
= ∫∫ 1
32
1
3
)1(2
6)1(2
128)1(
( )2
222
12
3
)1(2
1212 ⎟⎟⎠
⎞⎜⎜⎝
⎛+
∂∂
−−+++ rr nrS
rnS
rEh φ
ξφ
ζφδφφγ θθθ
ξφξζ
φκ θ drwnr
SwSGhR r⎥⎥⎦
⎤
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎟⎠⎞
⎜⎝⎛ −+⎟⎟
⎠
⎞⎜⎜⎝
⎛∂∂
−++
2
2
2
122 1
)1(2 (3.50)
∫∫⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎟⎠⎞
⎜⎝⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛∂∂
−−Ω−
−=A
r ddrwnr
SwSEhV ηξλξζ
λν
πζθ
2
2
2
12
32 1)1(
2)1(128
)1( (3.51)
In this special case, the Ritz displacement functions may be assumed as:
∑=
=N
i
wiicw
1)( φξ (3.52a)
∑=
=N
i
riir d
1
)( φξφ (3.52b)
Ritz Method for Plastic Buckling Analysis
57
∑=
=N
iiie
1
)( θθ φξφ (3.52c)
where N is the degree of the one-dimensional polynomial and iii edc ,, are the
unknown coefficients. The functions θφφφ iri
wi ,, are the product of the one-dimensional
polynomial and basic functions as follows
iwb
wi ξφφ = (3.53a)
irb
ri ξφφ = (3.53b)
ibi ξφφ θθ = (3.53c)
where the functions θφφφ brb
wb and ,, are the basic functions given by
[ ]∏=
ΩΓ=ne
jj
wb
wj
1
)(ξφ (3.54a)
[ ]∏=
ΩΓ=ne
jj
rb
rj
1
)(ξφ (3.54b)
[ ]∏=
ΩΓ=ne
jjb
j
1
)(θ
ξφθ (3.54c)
where ne is the number of edges, jΓ the boundary equation of the jth edge and
θj , , ΩΩΩ r
jwj are given by Eqs. (3.46a-c).
The elements of matrix [ ]K in Eq. (3.20) become
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧ Ω−
+∂
∂
∂∂
−Ω
⎢⎣
⎡= ∫∫ w
jwi
wj
wi
A
ccij Sn
rSr
EGK φφζ
ξφ
ξφ
ζτκ
22
12
2
4)1(
)1(
Ritz Method for Plastic Buckling Analysis
58
ηξφφζλ
ξφ
ξφ
ζλ
νπ θ ddSn
rSr w
jwi
wj
wir
⎥⎥⎦
⎤
⎪⎭
⎪⎬⎫Ω−
+⎪⎩
⎪⎨⎧
∂
∂
∂∂
−Ω
−− 2
212
2
4)1(
)1()1(12 (3.55a)
∫∫ ∂∂Ω
=A
rj
wicd
ij ddSrE
GK ηξφξφ
τκ
12
2
2 (3.55b)
∫∫Ω−
=A
jwi
ceij ddnS
EGK ηξφφζτ
κ θ22
2
4)1( (3.55c)
⎭⎬⎫
⎩⎨⎧
+−Ω
+∂
∂
∂∂
⎢⎣
⎡−Ω
= ∫∫ ri
ri
rj
ri
rj
ri
A
ddij S
ErGS
rSrK φφ
τκφφγζ
ξφ
ξφ
ζα
12
2
11 124)1(
)1(12
ηξφφδζφξφ
ξφ
φβ ddSnr
S rj
ri
rj
ri
rjr
i⎥⎥⎦
⎤Ω−+
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
∂∂
+∂
∂Ω+ 2
21 48
)1(24
(3.55d)
θθθθ
φηφζ
ξφ
ηφδ
ηφ
φζγηφ
ξφβ
j
rij
rijr
iA
jride
ij rEGnSnS
rnSK
∂∂−
+∂
∂
∂∂Ω
−∂
∂Ω−+
⎢⎢⎣
⎡
∂
∂
∂∂Ω
= ∫∫ 24)1(
2448)1(
24 211
⎟⎟⎠
⎞⎜⎜⎝
⎛ Ω−−
∂
∂Ω+
∂
∂
∂∂
Ω−
+ θθθ
φφζξφ
φηφ
ηφζµ
jri
jri
jri
rr 4)1(
2)1(
12
ηξφξφ
ξφ
ξφ
ζχ θ
θ
ddrj
rij
ri
⎥⎥⎦
⎤⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
−∂
∂
∂∂
−Ω
+)1(
224
(3.55e)
⎪⎩
⎪⎨⎧
∂
∂
∂∂
−Ω
+−Ω
+⎢⎢⎣
⎡
∂
∂
∂∂
Ω−
= ∫∫ ξφ
ξφ
ζφφ
τζκ
ηφ
ηφγζ θθ
θθθθ
jiji
A
jieeij
rE
GErG
rK
)1(124)1(
12)1(
2
2
⎪⎭
⎪⎬⎫⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂∂
−−
+ξφ
φφξφ
φφζ θθθ
θθθ j
iji
jir 21
4)1(
ηξηφ
φφηφζ
ξφ
ηφ
ηφ
ξφµ θ
θθθθθθθ
ddr
jij
ijiji
⎥⎥⎦
⎤
⎪⎭
⎪⎬⎫⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂∂−
⎪⎩
⎪⎨⎧
−∂
∂
∂∂
+∂
∂
∂∂
+2
)1(12
(3.55f)
where i, j = 1, 2,….N .
Ritz Method for Plastic Buckling Analysis
59
3.6 Solution Method for Solving Eigenvalue Problem
For elastic buckling, the aforementioned eigenvalue problems can be expressed in
the standard form of generalized eigenvalue problem and a standard eigenvalue
routine, for e.g. EISPAC (Smith et al., 1974), can be used to solve the problem.
However, for plastic buckling, the matrix [ ]K depends nonlinearly on the stresses
since the parameters G,,,,,, δµχγβα contains the buckling stress term. In order to
solve this nonlinear eigenvalue problem, the Sturm sequence iteration method with
bisection strategy was adopted. Normally, the Sturm sequence procedure is done by
the Gaussian elimination which reduces the matrix in to an upper triangular matrix
form (Gupta, 1978). The number of sign changes in the diagonal elements of the
triangular matrix indicates the number of eigenvalues of the matrix that are smaller
than the trial stress level. It was found that the built-in function “Eigenvalues” in
MATHEMATICA (Wolfram, 1999) can be used instead of Gaussian elimination. The
use of built-in function “Eigenvalues” is found to be faster than using the Gaussian
elimination in the determination of number of eigenvalues that are smaller than the
trial stress. Therefore, the following procedure was adopted for the calculations of
plastic buckling stresses. A trial stress value of λ is first assumed and the eigenvalues
of matrix [ ])( trialλK are calculated. Since the number of eigenvalues of [ ]K , which are
less than trialλ is equal to the number of negative eigenvalues of [ ])( trialλK , the value
of trialλ which gives one negative eigenvalue is sought. Upon identifying the value of
trialλ that gives one negative eigenvalue, a simple bisection algorithm is used to solve
Ritz Method for Plastic Buckling Analysis
60
the characteristic equation of the matrix [ ]K until the desired accuracy of the solution
is achieved. In this way, the lowest eigenvalue of the matrix [ ]K will not be missed.
3.7 Computer Programs
Three computer programs have been written in MATHEMATICA language for the
foregoing Ritz formulations. One program, called BUCKRITZRE, is for determining
the plastic buckling stresses of rectangular, triangular, elliptical and trapezoidal plates
and it is based on the aforementioned rectangular coordinate formulations (Eqs. 3.23a-
f). Another program, called BUCKRITZSKEW, is written based on the skew
coordinate formulations (Eqs. 3.33a-f) and it is to compute the plastic buckling stress
of skew plates. The last program, called BUCKRITZPOLAR, is used to solve the
plastic buckling problems of circular and annular plates and it is based on the polar
coordinate system. The program codes and detailed explanations of the codes are given
in Appendix A.
61
CHAPTER 4
RECTANGULAR PLATES
4.1 Introduction
This chapter is concerned with the plastic buckling analysis of rectangular Mindlin
plates under uniaxial compressive stress ( xσ ), biaxial compressive stress ( yx σσ , ),
shear stress ( xyτ ) and combinations of these stresses as shown in Fig. 4.1. The
boundary conditions of the plate edges may take on any combination of simply
supported, clamped and free edges. The Ritz method associated the rectangular
Cartesian coordinate system is employed for the plastic bifurcation buckling analyses
of such plates. The Ritz program code is called BUCKRITZRE and it is given in
Appendix A. The program code is designed to handle Mindlin plates subjected to any
combinations of in-plane and shear stresses and arbitrary boundary conditions.
The preparation of the input file for the program code is so simple that only the
basic functions and desired degree of polynomial need to be defined, apart from the
usual information on the material and geometric properties of the plate such as aspect
Rectangular Plates
62
ratio and thickness-to-width ratio. The sample basic functions for rectangular Mindlin
plates with various boundary conditions are given in Table 4.1.
For brevity and convenience, the boundary conditions of the plate will be
represented by a four-letter designation. For example, a FSCS plate will have a free
boundary condition at 2/ax −= , a simply supported boundary condition at 2/by −= ,
a clamped boundary condition at 2/bx = and a simply supported boundary condition
at 2/by = as shown in Fig. 4.1.
In computing the buckling solutions, a typical aerospace material of Al 7075 T6 is
adopted. Its material properties are: Poisson’s ratio ν = 0.33, 6105.10 ×=E psi,
707.0 =σ ksi, 672.0 =σ ksi and the Ramberg-Osgood parameters k = 3/7 and 2.9=c .
In order to verify the results, comparison studies are carried out against existing plastic
buckling solutions wherever available. The effect of thickness-to-width ratio, aspect
ratio, loading ratio and material properties on the plastic buckling behaviour of
rectangular plates are investigated.
C
S
S
F x, ξ
σx
a
y, η
Fig. 4.1 FSCS rectangular plate under in-plane compressive stresses
σy
σy
b σx
τxy
Rectangular Plates
63
Table 4.1 Sample basic functions for rectangular plates with various boundary conditions
Boundary conditions Basic functions
wbφ 1111 )1()1()1()1( −−++ ηξηξ x
bφ 1010 )1()1()1()1( −−++ ηξηξ SSSS
ybφ 0101 )1()1()1()1( −−++ ηξηξ
wbφ 1111 )1()1()1()1( −−++ ηξηξ x
bφ 1111 )1()1()1()1( −−++ ηξηξ CCCC
ybφ 1111 )1()1()1()1( −−++ ηξηξ wbφ 1011 )1()1()1()1( −−++ ηξηξ x
bφ 1011 )1()1()1()1( −−++ ηξηξ CSFS
ybφ 0001 )1()1()1()1( −−++ ηξηξ
4.2 Uniaxial and Biaxial Compression
For generality, yσ is expressed in terms of xσ as xy σβσ = . Therefore, the loading
ratio 0=β represents an applied uniaxial stress while 1=β indicate an applied
uniform biaxial stress. For a given value of loading ratioβ , the plastic buckling stress
parameter )/( 22 Dhbc πσλ = can be determined by solving the eigenvalue equation
given in Eq. (3.20). Since 0=xyτ , the parameters µχδγβα ,,,,,, G used in stress-
strain relations become simpler and can now be expressed in an explicit form given by
for the deformation theory of plasticity (DT)
( ) ( ) ⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−−−−−+= 2
2
13212213σσβ
ννρ x
s
tt
s EE
EE
EE (4.1a)
Rectangular Plates
64
⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−−= 2
2
1341σσ
ρα x
s
t
EE
(4.1b)
( ) ⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−−−−= 2
2
1321221σσβ
νρ
β x
s
tt
EE
EE
(4.1c)
⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−−= 2
22
1341σσβ
ργ x
s
t
EE
(4.1d)
⎟⎟⎠
⎞⎜⎜⎝
⎛−+
==132
1
sEEE
G
νδ (4.1e)
0== µχ (4.1f)
21 ββσσ +−= x (4.1g)
for the incremental theory of plasticity (IT)
2
22 1)21(3)21()45(
σσβ
νννρ xtt
EE
EE
⎟⎠⎞
⎜⎝⎛ −−−−−−= (4.2a)
⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ −−= 2
2
1341σσ
ρα xt
EE
(4.2b)
( ) ⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ −−−−= 2
2
1321221σσβ
νρ
β xtt
EE
EE
(4.2c)
⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ −−= 2
22
1341σσβ
ργ xt
EE
(4.2d)
)1(21ν
δ+
==EG (4.2e)
Noted that by setting EEs = , the expressions of µχδγβα ,,,,,, G based on DT
reduced to those corresponding to IT.
Rectangular Plates
65
Rectangular plates with two opposite sides simply supported while the other two
sides take on any combination of boundary conditions are first considered. Such
supported rectangular plates allow us to verify the convergence of the Ritz solutions
because of the availability of analytical solutions (Wang et al. 2001a).
It is worth noting that, for the special case of uniformly stressed simply supported
rectangular plate (i.e. 1=β ), the analysis given in Wang (2001a) can be further
simplified by noting that ( EG /2−− βα ) = 0. After some simplifications, the exact
expression of plastic buckling stress parameter λ can be expressed in a neat
transcendental equation given by
)(12
))(1(122222222
22222
ζατπκζ
ζακλnm
GE
nmv
++
+−= (4.3)
where m and n are the integers depending on the buckling mode shape of the plate. For
a square plate ( 1=ζ ), the integers m and n become unity for the lowest plastic
buckling stress parameter cλ . Therefore Eq. (4.3) furnishes
222
22
6
)1(12
ατπκ
ακλ
GE
vc
+
−= (4.4)
The analytical solutions, which were derived by Handleman and Prager (1948),
Shrivastava (1979), Durban and Zuckerman (1999) and Wang et al. (2001a), are
presented in Tables 4.2 to 4.5 for comparison purposes.
Convergence and comparison studies for square (a/b = 1) and rectangular (a/b = 2)
plates under uniaxial and biaxial compression are shown in Tables 4.2 to 4.5. It can be
seen that the solutions for SSSS plates converge very fast while a higher polynomial
Rectangular Plates
66
degree is needed for plates with free edges. Generally, the solutions from the Ritz
method converge well when the degree of polynomial p = 12 is used. Hence, the
degree of polynomial p = 12 will be used for all subsequent calculations. It can be seen
that the converged results are found to be within 0.08% of analytical solutions given
by Wang et al. (2001). This shows that the accuracy of plastic buckling results
determined by the Ritz method is excellent. However, the results from Shrivasta
(1979) are slightly higher than the present results. This is attributed by the fact that
Shrivasta’s results shown in Tables 4.2 are recalculated from the truncated expressions
presented in his paper. It was found that the present results are slightly lower than the
corresponding results of Handleman and Prager (1948) and Durban and Zuckerman
(1999). This is due to the neglect of the effect of transverse shear deformation in their
analysis.
Table 4.2 Comparison and convergence study of λ for square plates under uniform uniaxial loading with various boundary conditions.
τ = 0.001 τ = 0.050 τ = 0.1 BC p IT DT IT DT IT DT
2 4.4581 4.4581 3.7718 2.8735 3.5950 0.9250 4 4.0021 4.0021 3.4969 2.7958 3.1990 0.9144 6 4.0000 4.0000 3.4955 2.7954 3.1908 0.9144 8 4.0000 4.0000 3.4955 2.7954 3.1908 0.9144
Wang et al. (2001a) 4.0000 4.0000 3.4955 2.7954 3.1908 0.9144
Shrivastava (1979) 4.0000 4.0000 3.5278 2.8058 3.4636 0.9205
SSSS
Handleman and Prager
(1948) 4.0000 __ 3.5740 __ 3.5193 __
8 1.9632 1.9632 1.8552 1.8388 1.0380 0.7627 10 1.9632 1.9632 1.8455 1.8299 1.0177 0.7619 12 1.9631 1.9631 1.8419 1.8265 1.0151 0.7619 14 1.9631 1.9631 1.8409 1.8256 1.0149 0.7619
FSFS
Wang et al. (2001a) 1.9616 1.9616 1.8407 1.8254 1.0149 0.7700
Rectangular Plates
67
Table 4.2 Continued τ = 0.001 τ = 0.050 τ = 0.1 BC p
IT DT IT DT IT DT 8 2.2829 2.2829 2.0479 2.0222 1.3933 0.7701 10 2.2828 2.2828 2.0404 2.0155 1.0339 0.7672 12 2.2828 2.2828 2.0378 2.0131 1.0334 0.7667 14 2.2828 2.2828 2.0371 2.0125 1.0334 0.7667
SSFS
Wang et al. (2001a) 2.2811 2.2811 2.0369 2.0124 1.0334 0.7667
8 2.3038 2.3038 2.0565 2.0303 1.0404 0.7672 10 2.3037 2.3037 2.0489 2.0235 1.0353 0.7667 12 2.3037 2.3037 2.0462 2.0212 1.0349 0.7667 14 2.3037 2.3037 2.0455 2.0206 1.0348 0.7667
CSFS
Wang et al. (2001a) 2.3019 2.3019 2.0454 2.0204 1.0348 0.7667
Table 4.3 Comparison and convergence study of λ for rectangular plates under uniform uniaxial loading with various boundary conditions (a/b = 2)
τ = 0.001 τ = 0.05 τ = 0.1 BC p IT DT IT DT IT DT
4 4.0114 4.0114 3.5001 2.7972 3.3423 0.9146 6 4.0001 4.0001 3.3397 2.7954 3.0655 0.9078
8 4.0000 4.0000 3.3225 2.7954 3.0506 0.906910 4.0000 4.0000 3.3222 2.7954 3.0503 0.9069SSSS
Wang et al. (2001a) 4.0000 4.0000 3.3222 2.7954 3.0503 0.9068
Handleman and Prager
(1948) 4.0000 __ 3.4759 __ 3.2839 __
8 2.1854 2.1854 2.0344 2.0064 1.1059 0.772910 2.1852 2.1852 2.0212 1.9948 1.0569 0.768712 2.1852 2.1852 2.0101 1.9849 1.0389 0.767114 2.1852 2.1852 2.0021 1.9778 1.0341 0.7667
FSFS
Wang et al. (2001a) 2.1835 2.1835 1.9944 1.9708 1.0318 0.7479
8 2.2311 2.2310 2.0461 2.0186 1.0914 0.771710 2.2310 2.2310 2.0341 2.0080 1.0507 0.768212 2.2310 2.2310 2.0251 2.0000 1.0366 0.767014 2.2310 2.2310 2.0198 1.9952 1.0333 0.7667
SSFS
Wang et al. (2001a) 2.2293 2.2293 2.0155 1.9914 1.0326 0.7666
Rectangular Plates
68
Table 4.3 Continued τ = 0.001 τ = 0.05 τ = 0.075 BC p
IT DT IT DT IT DT 8 2.2314 2.2314 2.0462 2.0187 1.0907 0.771610 2.2314 2.2314 2.0342 2.0081 1.0503 0.768212 2.2314 2.2314 2.0253 2.0001 1.0365 0.767014 2.2314 2.2314 2.0199 1.9953 1.0332 0.7667
CSFS
Wang et al. (2001a) 2.2297 2.2297 2.0157 1.9915 1.0326 0.7666
Table 4.4. Convergence and comparison studies of plastic buckling stress parameters λ for square plates under uniform biaxial loading.
τ = 0.001 τ = 0.050 τ = 0.1 BC p IT DT IT DT IT DT
2 2.2291 2.2291 2.0073 1.9969 0.8484 0.7445 4 2.0010 2.0010 1.8720 1.8656 0.8086 0.7300 6 2.0000 2.0000 1.8713 1.8649 0.8084 0.7300
8 2.0000 2.0000 1.8713 1.8649 0.8084 0.7300 Wang et al.
(2001a) 2.0000 2.0000 1.8713 1.8649 0.8084 0.7300 SSSS
Durban and Zuckerman
(1999) 2.0000 2.0000 1.8713 1.8649 0.6661 0.7346
8 0.9159 0.9159 0.9054 0.9054 0.6401 0.6249 10 0.9159 0.9159 0.9041 0.9041 0.6401 0.6249 12 0.9159 0.9159 0.9035 0.9035 0.6401 0.6249 14 0.9159 0.9159 0.9033 0.9033 0.6401 0.6249
FSFS
Wang et al. (2001a) 0.9158 0.9158 0.9033 0.9032 0.6401 0.6250
8 1.0287 1.0287 1.0073 1.0072 0.6865 0.6567 10 1.0287 1.0287 1.0054 1.0053 0.6865 0.6567 12 1.0287 1.0287 1.0047 1.0047 0.6865 0.6567 14 1.0287 1.0287 1.0046 1.0045 0.6865 0.6567
SSFS
Wang et al. (2001a) 1.0284 1.0284 1.0045 1.0044 0.6865 0.6567
8 1.1104 1.1104 1.0816 1.0815 0.7236 0.6775 10 1.1104 1.1104 1.0790 1.0789 0.7236 0.6775 12 1.1104 1.1104 1.0781 1.0780 0.7236 0.6775 14 1.1104 1.1104 1.0779 1.0778 0.7236 0.6775
CSFS
Wang et al. (2001a) 1.1099 1.1099 1.0777 1.0777 0.7236 0.6775
Rectangular Plates
69
Table 4.5. Convergence and comparison studies of plastic buckling stress parameters λ for rectangular plates under uniform biaxial loading (a/b = 2).
τ = 0.001 τ = 0.050 τ = 0.1 BC p
IT DT IT DT IT DT
2 1.4388 1.4388 1.4158 1.4151 0.7180 0.6835 4 1.2508 1.2508 1.2371 1.2369 0.6869 0.6623 6 1.2500 1.2500 1.2363 1.2361 0.6868 0.6622 8 1.2500 1.2500 1.2363 1.2361 0.6868 0.6622
SSSS
Wang et al. (2001a) 1.2500 1.2500 1.2363 1.2361 0.6868 0.6622
8 0.9390 0.9390 0.9297 0.9297 0.6402 0.6254 10 0.9390 0.9390 0.9286 0.9285 0.6402 0.6254 12 0.9390 0.9390 0.9274 0.9274 0.6402 0.6254 14 0.9390 0.9390 0.9266 0.9265 0.6402 0.6254
FSFS
Wang et al. (2001a) 0.9389 0.9389 0.9257 0.9257 0.6402 0.6257
8 0.9753 0.9753 0.9632 0.9632 0.6527 0.6351 10 0.9753 0.9753 0.9615 0.9615 0.6527 0.6351 12 0.9753 0.9753 0.9601 0.9601 0.6527 0.6351 14 0.9753 0.9753 0.9593 0.9593 0.6527 0.6351
SSFS
Wang et al. (2001a) 0.9752 0.9752 0.9587 0.9586 0.6527 0.6351
8 0.9833 0.9833 0.9704 0.9704 0.6574 0.6385 10 0.9833 0.9833 0.9685 0.9685 0.6574 0.6384 12 0.9833 0.9833 0.9670 0.9670 0.6574 0.6384 14 0.9833 0.9833 0.9661 0.9661 0.6574 0.6384
CSFS
Wang et al. (2001a) 0.9831 0.9831 0.9654 0.9654 0.6574 0.6384
Effect of thickness-to-width ratio and transverse shear deformation
The variations of normalized plastic buckling stress with respect to b/h ratios are
shown in Figs. 4.2 to 4.5 for square plates under unaxial ( 0=β ) and biaxial
compressive stresses ( 1=β ). It can be seen that the buckling stresses associated with
Rectangular Plates
70
DT are consistently lower than their IT counterparts. The difference between the
buckling stresses obtained based on IT and DT increases with the b/h ratio decreases,
i.e. plate thickness increases. Moreover, the difference is more significant for more
restrained boundary conditions for the same b/h ratio. On the other hand, all the plastic
buckling curves (i.e. IT and DT) converge to the elastic buckling curve as b/h ratio
increases since buckling takes place in the elastic range for large b/h ratios.
In Figs. 4.2 to 4.5, the buckling curves based on the classical thin plate theory are
also presented with the view to examine the effect of transverse shear deformation.
The buckling stress, which is equivalent to that of the classical thin plate theory (CPT),
can be readily obtained by setting a large number 10002 =κ for the shear correction
factor (see Table 4.6). Note that, in Figs. 4.2 to 4.5, the dashed lines refer to the
classical thin plate theory and the solid lines represent the Mindlin plate theory. It can
be seen that the effect of transverse shear deformation lowers the buckling stress and
the effect is more significant when the plate is relatively thick. The effect is not
significant for plastic buckling when compared to that of the elastic buckling
counterparts. For example, the reductions for SSSS square plate under biaxial
compression (b/h = 6) are about 13%, 4.8% and 1% in the cases of elastic theory, IT
and DT, respectively. Generally, the reductions are larger for the buckling stress
associated with IT when compared with the corresponding reductions in their DT
counterparts. This is because the effect of transverse shear deformation depends on the
value of plastic buckling stress as shown in Figs. 4.6 and 4.7. It can be seen that the
influence of the effect of transverse shear deformation increases when the plastic
buckling stress increases.
Rectangular Plates
71
Table 4.6 Parametric study on the value of shear correction factor for simply supported square plate under biaxial loading.
2κ τ = 0.001 0.05 0.10 0.15 0.20
Incremental theory of plasticity 5/6 2.0000 1.8713 0.8084 0.6429 0.6210 Mindlin plate
theory 1000 2.0000 1.8896 0.8168 0.6710 0.6700 Classical thin plate theory - 2.0000 1.8896 0.8168 0.6711 0.6700
Deformation theory of plasticity 5/6 2.0000 1.8649 0.7300 0.3694 0.2239 Mindlin plate
theory 1000 2.0000 1.8830 0.7346 0.3729 0.2270 Classical thin plate theory - 2.0000 1.8830 0.7346 0.3729 0.2270
0
0.25
0.5
0.75
1
1.25
1.5
5 10 15 20 25 30 35 40
Fig. 4.2 Normalized plastic buckling stress σc/σ0.2 versus b/h ratio for SSSS square plate
b/h
0=β
1=β
Elastic IT DT
σ c/σ
0.2
a
b
ba σ
σβ
σ
Rectangular Plates
72
0.25
0.5
0.75
1
1.25
1.5
5 10 15 20 25 30 35 40
ElasticITDT
b/h
Fig. 4.3 Normalized plastic buckling stress σc/σ0.2 versus b/h ratio for CCCC square plate
σ c/σ
0.2
0=β
1=β
a b
ba
σ
σ
σβ
0.25
0.5
0.75
1
1.25
1.5
5 10 15 20 25 30 35 40
ElasticITDT
b/h
Fig. 4.4 Normalized plastic buckling stress σc/σ0.2 versus b/h ratio for SCSC square plate
σ c/σ
0.2
0=β
1=β
a
a
b
b
σ
σ
σβ
Rectangular Plates
73
0
0.25
0.5
0.75
1
1.25
1.5
5 10 15 20 25 30 35 40
a
ab
b σσ
σ
σβ
0.00
0.03
0.06
0.09
0.12
0.15
0 0.03 0.06 0.09 0.12 0.15
Fig. 4.6 Effect of transverse shear deformation on plastic buckling stress (IT) of a square plate
Dhc /3σ (Mindlin)
Dhc /3σ (CPT)
CPT SSSS plate (Mindlin)
CCCC plate (Mindlin)
Elastic IT DT
Fig. 4.5 Normalized plastic buckling stress σc/σ0.2 versus b/h ratio for SSSF square plate
b/h
σ c/σ
0.2
0=β
1=β
Rectangular Plates
74
Effect of aspect ratio a/b
The variations of plastic buckling stress parameters with respect to aspect ratios a/b
are presented in Figs. 4.6 to 4.13 for various boundary conditions. Aspect ratios
between 0.5 and 4.0 are considered since the buckling capacity has the most significant
variation over this range of aspect ratios. Generally, it is found that the plastic buckling
stress parameters are almost constant for aspect ratios greater than 4. The thickness-to-
width ratios in these figures are chosen in order to capture the plastic buckling
behaviour when the normalized buckling stress ( 2.0/σσ c ) is in the vicinity of unity.
Therefore, according to Figs. 4.2 to 4.5, τ = 0.05 is used for SSSS, SCSC, CCCC plates
while τ = 0.1 is used for SSSF plates. In order to observe the effect of loading ratio, we
consider β = 0, 0.25, 0.5, 0.75 and 1.
0
0.003
0.006
0.009
0.012
0.015
0 0.003 0.006 0.009 0.012 0.015
Fig. 4.7 Effect of transverse shear deformation on plastic buckling stress (DT) of a square plate
Dhc /3σ (Mindlin)
Dhc /3σ (CPT)
CPT SSSS plate (Mindlin)
CCCC plate (Mindlin)
Rectangular Plates
75
For the cases of SSSS, CCCC, SCSC, it can be seen that some of the buckling
curves are composed of individual curve segments that correspond to different
buckling modes. This is due to the fact that the number of half-sine waves in the
buckling mode shape of the plates increases as the aspect ratio increases. The locations
of cusps, at the aspect ratios of coincident buckling modes, are not the same for
different plate thicknesses because of the effect of transverse shear deformation. It was
also found that these cusps locations depend on the plasticity theories used. For
example, the buckling mode comprises two half sine waves for IT while the plate
buckles in the shape of one half sine wave in the cases of elastic buckling and plastic
buckling based on DT for h/b = 0.05 and a/b = 1.1 (see Fig 4.14). This information
would be useful when one needs to assume the distribution pattern of initial
imperfection that yield the lowest buckling load in a postbuckling analysis. The
phenomena of mode switching disappear at a certain value of β . For example, a
smooth buckling curve is obtained when β = 0.5 in the case of SSSS plates. In
contrast to the aforementioned cases, the plastic buckling stress parameters decrease
monotonically with increasing aspect ratios in the case of SSSF plates, regardless of
the value of β , i.e. no mode switching phenomenon is observed.
When the buckling curves of different values of β are compared, the buckling
stress parameters are observed to decrease as the value of β increases except for some
cases of 1/ <ba . For a/b <1, an unexpected behaviour of buckling stress parameter
with respect to β value is observed. This behaviour is due to mode switching which
will be proven later.
Rectangular Plates
76
1
1.5
2
2.5
3
3.5
4
0.5 1 1.5 2 2.5 3
λ =
σxh
b2 /(π2 D
)
Aspect ratio a/b
Fig. 4.8 Plastic buckling stress parameter λ versus aspect ratio a/b for SSSS based on IT
0=β
25.0=β
5.0=β
75.0=β
0.1=β
1
1.5
2
2.5
3
3.5
0.5 1 1.5 2 2.5 3
λ =
σxh
b2 /(π2 D
)
Aspect ratio a/b
Fig. 4.9 Plastic buckling stress parameter λ versus aspect ratio a/b for SSSS based on DT
0=β
25.0=β
5.0=β
75.0=β
0.1=β
Rectangular Plates
77
0=β
25.0=β 5.0=β 75.0=β
0.1=β 2
3
4
5
6
7
8
0.5 1 1.5 2 2.5 3
Fig. 4.10 Plastic buckling stress parameter λ versus aspect ratio a/b
Aspect ratio a/b
λ =
σxh
b2 /(π2 D
)
2.25
2.5
2.75
3
3.25
3.5
3.75
0.5 1 1.5 2 2.5 3
Aspect ratio a/b
Fig. 4.11 Plastic buckling stress parameter λ versus aspect ratio a/b
0=β 25.0=β
5.0=β
75.0=β
0.1=β
λ =
σxh
b2 /(π2 D
)
Rectangular Plates
78
2
2.5
3
3.5
4
4.5
5
5.5
0.5 1 1.5 2 2.5 3
λ =
σxh
b2 /(π2 D
)
Aspect ratio a/b
Fig. 4.12 Plastic buckling stress parameter λ versus aspect ratio a/b for SCSC based on IT
0=β
25.0=β
5.0=β
75.0=β
0.1=β
2.25
2.5
2.75
3
3.25
3.5
0.5 1 1.5 2 2.5 3
λ =
σxh
b2 /(π2 D
)
Aspect ratio a/b
Fig. 4.13 Plastic buckling stress parameter λ versus aspect ratio a/b for SCSC based on DT
0=β 25.0=β 5.0=β
75.0=β
0.1=β
Rectangular Plates
79
0
0.25
0.5
0.75
1
1.25
0.5 1 1.5 2 2.5 3
λ =
σxh
b2 /(π2 D
) 0=β
25.0=β 5.0=β
75.0=β
1=β
Aspect ratio a/b
Fig. 4.14 Plastic buckling stress parameter versus aspect ratio a/b for SSSF based on IT
0
0.25
0.5
0.75
1
1.25
0.5 1 1.5 2 2.5 3
λ =
σxh
b2 /(π2 D
)
Aspect ratio a/b
Fig. 4.15 Plastic buckling stress parameter versus aspect ratio a/b for SSSF based on IT
0=β 25.0=β 25.0=β 25.0=β
5.0=β
75.0=β
0=β
Rectangular Plates
80
Effect of loading ratio β
Figures 4.15 to 4.18 show the variations of plastic buckling stress parameter λ over
a range of loading ratio β for square plates with various boundary conditions. Since
the thickness-to-width ratio τ has a strong influence on the plastic buckling behaviour,
four different thickness-to-width ratios, namely =τ 0.001, 0.05, 0.075, and 0.1 are
considered.
In the case of 001.0=τ , it can be seen that the IT curves coincide with their
corresponding DT curves for all boundary conditions. Moreover, the plastic buckling
stress parameter decreases steadily with the increasing value of loading ratio β .
For large thickness-to-width ratios, the plastic buckling stress parameter may
decrease or increase with increasing β . The optimal value of β for the maximum
buckling capacity in DT curves is observed. For example, the maximum buckling
strength (DT) is obtained when the value of 5.0≈β in CCCC and SCSC plates for
1.0 and 075.0=τ . In contrast to DT buckling curves, the concave buckling curves of
(a) (b) (c)
Fig. 4.16. Contour and 3D plots of simply supported rectangular plate under uniaxial loading (τ = 0.05, a/b = 1.1) for (a) IT, (b) Elastic and (c) DT
Rectangular Plates
81
IT are observed except for SSSF plate. In the case of SSSF, the IT and DT buckling
curves are close to each other and the optimal value of β is found to be about 0.4.
These unexpected behaviours were also observed in Figs. 4.7 to 4.9 and Fig. 4.11 for
small a/b ratios. Similar observations were also reported in Durban and Zuckerman
(1999). It is believed that these unexpected buckling behaviours are observed due to
the mode switching.
In order to verify this hypothesis, we perform an analytical analysis where the
numbers of half wave of buckling shapes were predetermined. The analysis is carried
out for SSSS square plate and the results are shown in Fig. 4.19. In this figure, the
solid lines refer to the buckling mode shape with m = 1 and n = 1 while the dashed
lines represent the buckling mode shape with m = 2 and n = 1 (m and n are numbers of
half wave in the x and y direction, respectively). It can be seen that the plate always
buckles with a mode shape associated with m = 1 and n = 1 for small thickness-to-
width ratios, regardless of the loading ratio. However, for large thickness-to-width
ratios, the plate first buckles in a m = 2, n = 1 mode shape up to certain value of
loading ratio and then the critical mode shape changes to a m = 1, n = 1 mode shape as
the loading ratio increases. This m = 2 mode shape is not evident in the case of elastic
buckling for a square plate. Only in the plastic buckling, such a two half sine waves
buckling mode shape is possible depending on the loading ratio for a square plate.
Rectangular Plates
82
τ = 0.001τ = 0.05
τ = 0.075
τ = 0.1
0
1
2
3
4
5
0 0.2 0.4 0.6 0.8 1
IT DT
λ =
σxh
b2 /(π2 D
)
Stress ratio β
Fig. 4.17 Plastic buckling stress parameter versus stress ratio β for SSSS square plate
0
2
4
6
8
10
12
0 0.2 0.4 0.6 0.8 1
λ =
σxh
b2 /(π2 D
)
Stress ratio β
Fig. 4.18 Plastic buckling stress parameter versus stress ratio β for CCCC square plate
τ = 0.001
τ = 0.05τ = 0.075
τ = 0.1
IT DT
Rectangular Plates
83
0
2
4
6
8
0 0.2 0.4 0.6 0.8 1
λ =
σxh
b2 /(π2 D
)
Stress ratio β
Fig. 4.19 Plastic buckling stress parameter versus stress ratio β for SCSC square plate
τ = 0.001
τ = 0.05
τ = 0.001 τ = 0.1
IT DT
0
0.5
1
1.5
2
0 0.2 0.4 0.6 0.8 1
τ = 0.001τ = 0.05
τ = 0.075
τ = 0.1
IT DT
Stress ratio β
Fig. 4.20 Plastic buckling stress parameter versus stress ratio β for SSSF square plate
λ =
σxh
b2 /(π2 D
)
Rectangular Plates
84
Effect of material properties
The aforementioned discussions are based on the analysis with Al 7075-T6 and the
results may vary if a different material is used. Therefore, a parametric study is carried
out for SSSS square plate under uniaxial compression in order to examine the effect of
material properties on the buckling capacity. The normalized buckling curves are
presented in Figs 4.20 and 4.21 for various Ramberg-Osgood parameters c and 7.0/σE
ratios. It can be seen that the buckling capacity increases before a certain stress level
and after that it decreases as the value of c increases. This behaviour can be explained
by the trends of the stress-strain curves with respect to the c value which were shown
in Fig. 2.2. The buckling curves are found to be shifted to the right if the ratio 0/σE
increases. This effect can be incorporated in the slenderness ratio (b/h) as shown in
Fig. 4.22.
τ = 0.001
τ = 0.025 τ = 0.05 τ = 0.001 τ = 0.1
λ =
σxh
b2 /(π2 D
)
Fig. 4.21 Mode switching with respect to stress ratio β for SSSS square plate
Stress ratio β
0
1
2
3
4
5
6
7
0 0.25 0.5 0.75 1
Rectangular Plates
85
σ c/σ
0.2
b/h
Fig. 4.22 Effect of material properties on normalized plastic buckling stress 2.0/σσ c for SSSS square plate (IT)
0
0.25
0.5
0.75
1
1.25
1.5
15 20 25 30 35 40 45 50
c = 2c = 5
c = 10
c = 15
c = 2c = 5
c = 10
c = 15
E/σ0 = 150 E/σ0 = 500
b/h
Fig. 4.23 Effect of material properties on normalized plastic buckling stress 2.0/σσ c for SSSS square plate (DT)
0
0.5
1
1.5
2
10 15 20 25 30 35 40 45 50
c = 2
c = 5
c = 10
c = 15
E/σ0 = 150 E/σ0 = 500
σ c/σ
0.2
Rectangular Plates
86
Comparison with experiments
It has been shown in Fig. 4.22 that the effect of the variation of 7.0/σE can be
absorbed by multiplying E/7.0σ in the value of slenderness ratio. Therefore the
commonly used slenderness ratio eh
bλν )1(12 2− (where Dhbee /2σλ = is the elastic
buckling parameter) becomes eσσλ /7.0= , where eσ refers to the elastic buckling
stress. Since the plastic buckling stresses obtained by IT and DT differ greatly when
the thickness to width ratio is large, the comparison studies against test data for
rectangular plates under uniaxial compression are presented in Figs. 4.23 and 4.24. It
can be seen that the test results are in favour with the results obtained by DT which
violates the fundamental rule of plastic flow of metal. On the other hand the results
0
0.25
0.5
0.75
1
1.25
1.5
10 15 20 25 30 35 40 45 50
σ c/σ
0.2
)100/( 7.0σEhb
Fig. 4.24 Normalized plastic buckling stress 2.0/σσ c versus modified slenderness ratio for SSSS square plate
IT
DT
Rectangular Plates
87
given by analysis based on IT, which is regarded as most satisfactory theory for
treating plasticity problems, are significantly higher than test data for small slenderness
ratios.
0
0.25
0.5
0.75
1
1.25
1.5
0 0.5 1 1.5 2 2.5
ab
σc σc
2.0σσ c
λ Fig. 4.26 Comparison study of buckling stresses obtained by
DT and IT with test results (Al-2014-T6)
Anderson (1956)
Pride (1949)
IT
DT
Elastic
Anderson (1956)
λ
2.0σσ c
0
0.25
0.5
0.75
1
1.25
1.5
0 0.5 1 1.5 2 2.5
Fig. 4.25 Comparison study of buckling stresses obtained by DT and IT with test results (Al-7075-T6)
ab
σc σc Elastic
IT
DT
Rectangular Plates
88
4.3 Pure Shear Loading
In the case of pure shear, the in-plane stresses xσ and yσ are set to zero in Eqs.
(2.19a-g) and Eq. (2.20). Hence the effective stress σ and the parameters
µχδγβα ,,,,,, G become as follows
For deformation theory of plasticity (DT)
23 xyτσ = (4.5a)
( ) ( ) ⎥⎦⎤
⎢⎣⎡ −−−+=
EE
EE s
s
ννρ 212213 (4.5b)
ργα 4== (4.5c)
( ) ⎥⎦⎤
⎢⎣⎡ −−=
EEsν
ρβ 21221 (4.5d)
⎟⎟⎠
⎞⎜⎜⎝
⎛−+
=
132
1
tEEν
δ (4.5e)
⎟⎟⎠
⎞⎜⎜⎝
⎛−+
=132
sEE
EGν
(4.5f)
0== µχ (4.5g)
For incremental theory of plasticity (IT)
211ν
γα−
== (4.6a)
21 ννβ−
= (4.6b)
Rectangular Plates
89
⎟⎟⎠
⎞⎜⎜⎝
⎛−+
=
132
1
tEEν
δ (4.6c)
)1(2 ν+=
EG (4.6d)
where xyτ is the shear stress.
The convergence studies in Tables 4.7 and 4.8 show the convergence of the results
with respect to the degree of polynomial. It can be seen that p = 12 is sufficient for
reasonably converged results and therefore, p = 12 will be used for subsequent
calculations. For comparison purposes, the plastic buckling stress parameters are
computed from their corresponding elastic thin plate results by multiplying with the
plasticity reduction factors which were derived by Gerard (1948), Bijlaard (1949) and
Gerard and Becker (1957).
Gerard (1948) modified the secant modulus method for plates under shear by
replacing the secant modulus Es with the shear secant modulus Gs. The latter modulus
is equal to δE in the present study. In view of modified secant method, the plasticity
reduction factor η becomes Gs/G. It was found that the results obtained by using the
secant modulus method are consistently lower than the present results based on DT
and IT.
Bijlaard (1949), it was found that the results, which are obtained by Bijlaard’s
(1949) formula, are consistently higher than the present Ritz results for both thickness-
to-width ratios (τ = 0.001 and τ = 0.05). This is due to an error in the parameters used
in the plastic reduction factor formula. An erratum was reported by Peters (1954) who
pointed out that the symbols, A, B, and F appearing in the case 6 of Table 5 are not the
same symbols as A, B, and F as given in Eq. 21 of Bijlaard (1949). Instead, they are
Rectangular Plates
90
the same symbols as those given in Eq. 36 of Bijlaard (1947). Unfortunately, the paper
by Bijlaard (1947) is not accessible to the author and hence the corrected results cannot
be computed and verified. On the other hand, the present results agree well with the
solutions obtained from plasticity reduction factor reported in Table 2 of Gerard and
Becker (1957). When the thickness-to-width ratio τ = 0.025, the present solutions are
however slightly lower than Gerard’s solutions due to the effect the transverse shear
deformation.
Effect of transverse shear deformation and thickness-to-width ratio
The variations of normalized buckling stress with respect to b/h ratio are shown in
Figs. 4.23 to 4.26. The effect of transverse shear deformation can be observed by
comparing the solid lines (Mindlin plate theory) and dashed lines (classical thin plate
theory). In general, the effect of shear deformation is more pronounced in the case of
IT than DT. It is interesting to note that the IT buckling curves are parallel with the
elastic buckling curves for small b/h ratios.
Rectangular Plates
91
Table 4.7 Convergence and comparison studies of plastic buckling stress parameters for SSSS rectangular plates under shear
a/b 1 2 3 4 τ p
IT DT IT DT IT DT IT DT Elastic thin
plate results
9.3246 6.5460 5.8403 5.6246
0.001 8 9.3387 9.3387 6.5544 6.5544 5.8464 5.8464 5.7437 5.7437 10 9.3252 9.3252 6.5463 6.5463 5.8404 5.8404 5.6326 5.6326 12 9.3245 9.3245 6.5460 6.5460 5.8402 5.8402 5.6249 5.6249 14 9.3244 9.3244 6.5460 6.5460 5.8401 5.8401 5.6246 5.6246
Gerard (1946) __ 9.3246 __ 6.5460 __ 5.8403 __ 5.6246
Bijlaard (1949) __ 9.5327 __ 6.6923 __ 5.9706 __ 5.7503
Gerard and
Becker (1957)
__ 9.3246 - 6.5460 __ 5.8403 __ 5.6246
0.025 8 6.7903 6.2047 5.5982 5.4344 5.2718 5.1550 5.2176 5.1091 10 6.7820 6.2010 5.5951 5.4321 5.2673 5.1514 5.1385 5.0427 12 6.7815 6.2008 5.5949 5.4319 5.2671 5.1513 5.1327 5.0379 14 6.7814 6.2008 5.5949 5.4319 5.2670 5.1513 5.1325 5.0377
Gerard (1948) __ 5.3610 __ 4.8793 __ 4.6875 __ 4.6187
Bijlaard (1949) __ 6.7260 __ 5.8606 __ 5.4975 __ 5.3673
Gerard and
Becker (1957)
__ 6.4507 __ 5.5858 __ 5.2503 __ 5.1330
Rectangular Plates
92
Table 4.8 Convergence and comparison studies of plastic buckling stress parameters for CCCC rectangular plates under shear
a/b 1 2 3 4 τ p
IT DT IT DT IT DT IT DT Elastic thin plate results 14.6421 10.2480 9.5343 9.2952
0.001 8 14.659 14.659 10.578 10.578 9.8601 9.8601 10.325 10.325 10 14.643 14.643 10.262 10.262 9.5575 9.5575 9.4936 9.4936 12 14.642 14.642 10.248 10.248 9.5353 9.5353 9.3406 9.3405 14 14.642 14.642 10.248 10.248 9.5341 9.5341 9.2984 9.2984
0.025 8 11.1994 7.1910 8.3101 6.6143 7.9557 6.5070 8.6245 6.6516 10 11.1781 7.1890 8.0692 6.5635 7.6356 6.4280 7.6610 6.4269 12 11.1770 7.1889 8.0591 6.5615 7.6044 6.4211 7.4493 6.3723 14 11.1769 7.1889 8.0587 6.5615 7.6029 6.4208 7.4231 6.3658
Gerard (1948) __ 5.8463 __ 5.4709 __ 5.3875 __ 5.3573
Gerard and
Becker (1957)
__ 7.2690 __ 6.6432 __ 6.4973 __ 6.4440
b/h
τ xy/σ
0.2
Fig. 4.27 Normalized plastic buckling stress 2.0/στ xy versus b/h ratio for SSSS square plate
0
0.25
0.5
0.75
1
20 30 40 50 60 70
Elastic IT DT
xyτ a b
Rectangular Plates
93
b/h
Fig. 4.28 Normalized plastic buckling stress 2.0/στ xy versus b/h ratio for CCCC square plate
Elastic IT DT
τ xy/σ
0.2
xyτ a b
0.25
0.5
0.75
1
10 20 30 40 50 60 70
Elastic IT DT
b/h
Fig. 4.29 Normalized plastic buckling stress 2.0/στ xy versus b/h ratio for SCSC square plate
τ xy/σ
0.2
xyτ a b
0.25
0.5
0.75
1
10 20 30 40 50 60 70
Rectangular Plates
94
Effect of aspect ratio a/b
The variations of plastic buckling stress parameters with respect to aspect ratio a/b
are shown in Figs. 4.27, 4.28 and 4.29 for SSSS, CCCC and SSSF rectangular plates
respectively. It is observed that the buckling mode switches in the cases of SSSS and
CCCC but the locations of cusps cannot be seen clearly in those figures. On the other
hand, no mode switching phenomenon is observed in the case of SSSF. Typical
buckling mode shapes of rectangular plates under shear are shown in Table 4.9 for
various aspect ratios by using DT.
0
0.25
0.5
0.75
1
10 20 30 40 50 60 70
Fig. 4.30 Normalized plastic buckling stress 2.0/στ xy versus b/h ratio for SSSF square plate
b/h
Elastic IT DT
τ xy/σ
0.2
xyτ a
b
Rectangular Plates
95
0
5
10
15
20
25
0.5 1 1.5 2 2.5 3 3.5 4
Elastic
IT
DT
λ xy=τ xy
hb2 /(π
2 D)
Aspect ratio a/b
Fig.4.31 Plastic buckling stress parameter, λxy, versus aspect ratio, a/b, for SSSS rectangular plate
τ = 0.025 τ = 0.05
xyτ
a b
0
10
20
30
40
0.5 1 1.5 2 2.5 3 3.5 4
λ xy=τ xy
hb2 /(π
2 D)
Aspect ratio a/b
Fig.4.32 Plastic buckling stress parameter, λxy, versus aspect ratio, a/b, for CCCC rectangular plate under shear stress
Elastic
IT
DT
τ = 0.025 τ = 0.05
xyτ
a b
Rectangular Plates
96
Table 4.9 Buckling mode shapes for rectangular plates under pure shear. (τ = 0.03)
BC a/b 3D Plot Contour Plot
1
2
SSSS
4
0
10
20
0.5 1 1.5 2 2.5 3 3.5 4
λ xy=τ xy
hb2 /(π
2 D)
Fig.4.33 Plastic buckling stress parameter, λxy, versus aspect ratio, a/b, for SSSF rectangular plate under shear stress
Aspect ratio a/b
τ = 0.025 τ = 0.05
xyτ
a b
Rectangular Plates
97
Table 4.10 Continued.
1
2
CCCC
4
1
SSSF
2
4.4 Combined Shear and Uniaxial Compression
In practice, structural plate elements are often subjected to both inplane compressive
and shear loads simultaneously. Elastic buckling of plates under such combined in-
plane loads has been studied by many researchers and well documented in text books
such as Timoshenko and Gere (1961). In the case of plastic buckling, Peters (1954)
conducted experiments on long square tubes and compared the results with plastic
buckling stresses predicted by the flat plate theory. The results were found to agree
Rectangular Plates
98
well with the theoretical results obtained by Stowell (1949) and an interaction formula
was proposed. Another theoretical study was performed by Tugcu (1996) based on the
incremental theory of plasticity and the deformation theory of plasticity with the view
to understand the effect of axial force on the shear buckling of long plate strips. It was
found that the shear buckling results based on IT were highly sensitive to the presence
of axial loads while those based on DT were less affected. Recently, Smith et al.
(2003) studied the plastic buckling of rectangular plate under various loading
conditions. However, their study was confined to the classical thin plate theory and to
the case where the external shear stress is small. Hitherto, there are apparently no study
being done on the plastic buckling of rectangular Mindlin plates under uniaxial
compression and shear loads.
The convergence studies of buckling stress parameters are shown in Tables 4.10
and 4.11 for rectangular plates under uniaxial (in the x-direction) and shear stresses.
The converged results are obtained within reasonable accuracy when the degree of
polynomial p =12 is used. Owing to the lack of plastic buckling solutions, the
comparison study is performed against existing elastic buckling solutions and shown in
Table 4.12. The buckling stress parameters obtained in this study are slightly lower
than the corresponding existing elastic solutions obtained by Batdorf and Stein (1947).
This is due to the fact that the existing solutions were obtained by using 10 terms of
the Fourier series that represent the deflected shape and it is believed that these
solutions have not converged yet.
The interaction curves for the plastic buckling shear and uniaxial stresses are shown
in Fig. 4.30 for simply supported, rectangular plates with a/b = 1 and a/b = 4. It can be
seen that the differences between the plastic buckling stress parameters obtained by IT
and DT are more pronounced when the plate is under pure shear (i.e. 0=xλ ). With the
Rectangular Plates
99
presence of a small value of uniaxial compression, the reductions in shear plastic
buckling stress parameters given by IT are larger than their corresponding DT
counterparts. This is consistent with Tugcu’s (1996) observation that the plastic
buckling shear stress parameters based on IT are more sensitive to the presence of
uniaxial compression than their DT counterparts. Moreover, the interaction curves
based on IT are affected significantly by the aspect ratios whereas the corresponding
DT interaction curves are negligibly affected by the aspect ratios for the thickness-to-
width ratio 05.0=τ .
In order to examine the effect of transverse shear deformation, the interaction
curves based on the classical thin plate theory are presented in Fig. 4.30 as well. As
expected, the classical thin plate theory overpredicts the plastic buckling load
parameters as it normality assumption stiffens the plate by not permitting any
transverse shear deformation. It is found that the reductions due to the effect of shear
deformation in the results obtained by IT are generally larger than those reductions in
their DT counterparts.
Peters (1954) found that the experimentally obtained plastic buckling loads of
simply supported plates are in good agreement with the theoretical plastic buckling
results computed by Stowell (1948). He went on to propose a simple interaction
formula, 122 =+ xxy RR , where xyR is the ratio of shear stress to the critical shear stress
due to pure shear and xR is the ratio of uniaxial stress to the critical buckling stress
due to pure uniaxial compression. In order to investigate the validity of Peters’ formula
for different thickness-to-width ratios τ , the normalized theoretical interaction curves
based on DT for various thickness to width ratios are determined and compared with
the interaction curves of Peters’ formula (see Figs. 4). It can be seen from these figures
Rectangular Plates
100
that the shapes of the interaction buckling curves depend on the thickness-to-width
ratios and that the theoretical curves are bounded by the curves obtained by Peters’
formula 122 =+ xxy RR and the elastic interaction formula, 12 =+ xxy RR (Batdorf and
Stein, 1947). Therefore it can be concluded that Peters’ formula provides an upper
bound solution and it overestimates the buckling capacity of the plates with small to
moderate thickness-to-width ratios.
Table 4.10 Convergence study of plastic buckling shear stress parameters xyλ for simply supported plates under combined loadings (τ = 0.03).
a/b = 1 a/b = 2 a/b = 4 p xλ IT DT IT DT IT DT
8 1 5.0118 4.3807 4.2216 3.9907 4.0267 3.8502 10 5.0067 4.3793 4.2201 3.9898 3.9846 3.8230 12 5.0062 4.3972 4.2199 3.9897 3.9825 3.8216 14 5.0060 4.3792 4.2198 3.9897 3.9824 3.8215 8 2 4.0189 3.8152 3.5760 3.4740 3.4271 3.3440 10 4.0173 3.8144 3.5753 3.4735 3.3953 3.3178 12 4.0170 3.8142 3.5752 3.4735 3.3940 3.3167 14 4.0169 3.8142 3.5752 3.4734 3.3939 3.3166
Table 4.11 Convergence study of plastic buckling shear stress parameters xyλ for clamped square plates under combined loadings. (τ = 0.03)
a/b = 1 a/b = 2 a/b = 4 p xλ IT DT IT DT IT DT
8 1 10.0704 5.1586 7.3376 4.8265 7.8656 4.8904 10 10.0527 5.1578 7.1084 4.7996 6.7100 4.7229 12 10.0518 5.1577 7.0984 4.7985 6.4496 4.6833 14 10.0517 5.1577 7.0981 4.7985 6.4132 4.6780 8 2 8.9635 4.8768 6.4573 4.5381 7.1827 4.6471 10 8.9478 4.8761 6.2306 4.5082 5.8685 4.4233 12 8.9469 4.8760 6.2197 4.5069 5.6013 4.3744 14 8.9468 4.8760 6.2193 4.5069 5.5577 4.3666
Rectangular Plates
101
Table 4.12 Comparison study of elastic buckling shear stress parameters xyλ for simply supported plates under combined loadings. ( ,001.0=τ ν = 0.3)
a/b xλ 1 2 3
Present study 7.92 6.46 4.55 1 Batdorf and Stein (1947) 8.11 6.62 4.68
Present study 5.60 4.56 3.21 2 Batdorf and Stein (1947) 5.71 4.66 3.29
Present study 4.85 4.02 2.85 4 Batdorf and Stein (1947) 5.00 4.10 2.94
0
1
2
3
4
5
6
7
0 0.5 1 1.5 2 2.5 3 3.5 4
IT, a/b = 1
Fig. 4.34 Interaction curves for SSSS rectangular plates under combined uniaxial compression and shear stresses (τ = 0.05)
DT, a/b = 1
λx = σxhb2/(π2D)
λ xy = τ xy
hb2 /(π
2 D)
Mindlin plate theory
Classical thin plate theory
IT, a/b = 2
IT, a/b = 4
DT, a/b = 4
τxy
σx σx a
b
Rectangular Plates
102
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2
Rx
Rxy
τ = 0.001
τ = 0.025 τ = 0.05
τ = 0.1
12 =+ xxy RR
122 =+ xxy RR
Fig. 4.36 Normalized interaction curves for clamped rectangular plates (a/b = 4)
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2
Rx
Rxy
Fig. 4.35 Normalized interaction curves for simply supported rectangular plate (a/b = 4)
τ = 0.001
τ = 0.025
τ = 0.05
τ = 0.1
12 =+ xxy RR
122 =+ xxy RR
Rectangular Plates
103
4.5 Concluding Remarks
The plastic buckling behaviour of rectangular Mindlin plates under uniaxial, biaxial
and shear stresses are examined based on the extensive buckling results generated by
the Ritz computer code.
The effect of transverse shear deformation is taken into account in the analysis by
adopting the Mindlin plate theory. As in the case of elastic buckling, the effect of
transverse shear deformation lowers the plastic buckling stress parameter and the
effect is more pronounced for the results obtained by IT when compared with their DT
counterparts.
The effects of aspect ratios a/b, thickness-to-width ratios τ , loading ratio β on the
plastic buckling stress parameters were investigated. It was found that the buckling
mode switches at particular aspect ratios for SSSS and CCCC plates. The locations of
cusps, at the aspect ratios of coincident buckling modes, are different from those
locations in the case of elastic buckling. Moreover, the cusps locations also depend on
the type of plasticity theory used. This indicates that the plastic buckling mode shape
may be different from its elastic counterpart for a given aspect ratio.
For small thickness-to-width ratios, the plastic buckling solutions are close to their
elastic counterparts while, for large thickness ratios, the plastic buckling solutions are
substantially lower than their elastic counterparts. Generally, the plastic buckling stress
parameters obtained by IT are larger than their DT counterparts. The difference
increases with increasing thickness-to-width ratios.
With the presence of a small value of uniaxial compression, the reductions in shear
plastic buckling stress parameters given by IT are larger than their corresponding DT
counterparts.
104
TRIANGULAR AND ELLIPTICAL PLATES
5.1 Introduction
In the previous chapter, plastic buckling problems of rectangular plates were
investigated by using the Ritz method based on the Cartesian coordinate system. In
fact, the Ritz method can be employed to solve arbitrary plate shapes which are
expressed by polynomial functions. All that is needed is to modify the integration
limits and the basic functions depending on the plate shapes and boundary conditions.
This will be illustrated in this chapter by considering the plastic buckling problems of
isosceles triangular plates, right angled triangular plates and elliptical plates.
5.2 Triangular Plates
Consider isosceles and right-angled triangular plates under uniform compressive
stress as shown in Figs 5.1 and 5.2, respectively. The boundary conditions for the
triangular plates are represented by a three-letter designation which is arranged in the
CHAPTER 5
Triangular and Elliptical Plates
105
order starting from the left end edge (x = 0) followed by the other edges in an
anticlockwise direction. For example, see Fig. 5.1 for CS*F isosceles triangular plate
and Fig. 5.2 for CS*F right-angled triangular plate.
x, ξ
y, η
C
S*
F
a
b
Fig. 5.1 CS*F isosceles triangular plate and coordinate system
σ
σ
σ
x, ξ
y, η
C
S*
F
a
b
Fig. 5.2 CS*F right-angled triangular plate and coordinate system
σ
σ
σ
Triangular and Elliptical Plates
106
In order to illustrate the compositions of the basic functions, we use the CS*F
isosceles and CS*F right angled triangular plates examples. The basic functions for
these plates are given by
for CS*F isosceles triangular plate
)5.05.0)(1()5.05.0()5.05.0()1( 011 +−+=−++−+= ξηξξηξηξφ wb (5.1)
)1()5.05.0()5.05.0()1( 001 +=−++−+= ξξηξηξφ xb (5.2)
)1()5.05.0()5.05.0()1( 001 +=−++−+= ξξηξηξφ yb (5.3)
for CS*F right angled triangular plate
)1)(1()()1()1( 011 ++=+++= ηξηξηξφ wb (5.4)
)1()()1()1( 001 +=+++= ξηξηξφ xb (5.5)
)1()()1()1( 001 +=+++= ξηξηξφ yb (5.6)
Note that a relationship between the plastic buckling solutions of polygonal Mindlin
plates and their corresponding elastic buckling solutions based on classical thin plate
theory was presented by Wang (2004). The relationship is given by
ep
p
GE
λ
κτλπ
να
λ=
⎟⎟⎠
⎞⎜⎜⎝
⎛−− 2
222
121
(5.7)
where the subscript p refers to the plastic buckling solutions, the subscript e refers to
the elastic buckling solutions and the parameters G,α are given by Eqs. (4.1b) and
(4.1e) or Eqs. (4.2b) and (4.2e) depending on the type of plasticity theory used.
Triangular and Elliptical Plates
107
Based on the classical thin plate theory, Han (1960) gave the elastic buckling
solution 3/16=eλ for simply supported equilateral triangular plates ( 2/3/ =ba ).
In view of this result, Eq. (5.7) can be expressed as
222
22
49
)1(48
ατπκ
ακλ
GE
vp
+
−= (5.8)
Note that the above equation is valid only for simply supported edge of hard type
(S). However, the adopted Ritz functions in Ritz method fail to satisfy the geometric
boundary condition of this hard type of simply supported edge when the plate edges
are inclined (as in triangular plates) or curved (as in elliptical plates) edges. In order to
handle the hard type of simply supported edge, a modification in the formulations is
needed and it will be discussed in Chapter 8.
Table 5.1 presents the comparison and convergence studies on the plastic buckling
stress parameters of equilateral triangular plates with various boundary conditions;
S*S*S*, CCC and S*CC. The convergence study for right-angled triangular plate with
a/b = 1 is presented in Table 5.2. From Tables 5.1 and 5.2, it can be seen that the
results associated with the degree of polynomial p = 10 show satisfactory convergence.
Therefore, p = 10 will be adopted for the subsequent calculations of plastic buckling
solutions of triangular plates. For simply supported equilateral triangular plates, it can
be seen that present results are slightly less than the corresponding results obtained by
Eq. (5.8). This is due to the different implementation of simply support boundary
condition as discussed in the previous paragraph. However, the buckling results for the
simply supported edge of soft type become closer to their hard type counterparts when
the plate thickness is thin (for example τ = 0.001). For other boundary conditions, no
Triangular and Elliptical Plates
108
plastic buckling results are available in the literature and thus comparison studies are
made only with existing elastic buckling results. It can be seen that results agree well
with existing elastic solutions when the thickness-to-width ratio 001.0=τ .
Table 5.1. Convergence and comparison studies of plastic buckling stress factors λ for equilateral triangular plates (a/b = 2/3 ) under uniform compression.
τ = 0.001 τ = 0.050 τ = 0.075 BC p
IT DT IT DT IT DT 8 5.3331 5.3331 2.7492 2.6572 1.6106 1.3670 10 5.3331 5.3331 2.7441 2.6542 1.6046 1.3662 12 5.3330 5.3330 2.7428 2.6533 1.6038 1.3661 S*S*S*
Wang et al. (2004) 5.3332 5.3332 2.8033 2.6885 1.8038 1.3912
8 14.8351 14.8351 4.8175 3.2301 4.5489 1.5973 10 14.8340 14.8340 4.8173 3.2301 4.5487 1.5973 CCC
Xiang et al. (1994) 14.8340 (Elastic) __ __ __ __
8 10.6051 10.6051 3.6207 3.0497 3.0713 1.5219 10 10.6050 10.6050 3.6171 3.0491 3.0684 1.5218 S*CC 12 10.6050 10.6050 3.6164 3.0490 3.0681 1.5218
Table 5.2 Convergence and comparison studies of plastic buckling stress factors λ for right angled triangular plates under uniform compression (a/b = 1).
τ = 0.001 τ = 0.050 τ = 0.075 BC p
IT DT IT DT IT DT 8 4.9998 4.9998 2.6971 2.6180 1.5605 1.3530 10 4.9998 4.9998 2.6920 2.6147 1.5548 1.3520 12 4.9997 4.9997 2.6905 2.6138 1.5538 1.3518 S*S*S*
Xiang et al. (1994) 4.9997 (Elastic) __ __ __ __
8 14.1625 14.1625 4.6217 3.2074 4.3551 1.5883 10 14.1417 14.1417 4.6193 3.2072 4.3537 1.5883 12 14.1412 14.1412 4.6193 3.2072 4.3537 1.5883 CCC
Xiang et al. (1994) 14.1413 (Elastic) __ __ __ __
8 10.4139 10.4139 3.5857 3.0405 3.0192 1.5183 S*CC 10 10.4130 10.4130 3.5801 3.0396 3.0133 1.5163
Triangular and Elliptical Plates
109
Effect of transverse shear deformation
Tables 5.3 and 5.4 show the effect of transverse shear deformation on the plastic
buckling stress parameters of simply supported and clamped isosceles triangular
plates. It can be seen that the inclusion of the effect of transverse shear deformation in
the analysis lowers buckling stress parameters. The reductions are shown as the
percentages of the classical thin plate solutions. It can be seen that the reduction
increases with increasing thickness-to-width ratios. Generally, those reductions
associated with IT are larger than their DT counterparts.
Effect of aspect ratio
The variations of plastic buckling stress parameter λ with respect to the aspect ratio
are shown Figs. 5.3 and 5.4 for simply supported, isosceles and right angled triangular
plates. In order to show the effect of thickness-to-width ratios, three thickness-to-width
ratios are considered (i.e. 1.0,05.0,001.0=τ ). It should be noted that the buckling
curve of 001.0=τ may be regarded as an elastic buckling curve. Hence the deviation
of plastic buckling stress parameters from the corresponding elastic parameters can be
seen for various aspect ratios. Generally, the plastic buckling stress parameter
decreases with increasing aspect ratios.
Triangular and Elliptical Plates
110
Table 5.3 Effect of transverse shear deformation on λ for S*S*S* isosceles triangular plates
a/b = 0.5 a/b = 2 τ Plate theory
Elastic IT DT Elastic IT DT Kirchhoff 10.0000 10.0000 10.0000 2.7262 2.7262 2.7262 Mindlin 9.9990 9.9990 9.9990 2.7262 2.7262 2.7262 0.001
Difference 0% 0% 0% 0% 0% 0% Kirchhoff 10.0000 8.6119 8.5474 2.7262 2.7261 2.7261 Mindlin 9.5888 8.4259 8.3714 2.6893 2.6893 2.6893 0.025
Difference 4.11% 2.16% 2.06% 1.53% 1.53% 1.53% Kirchhoff 10.0000 3.3500 0.9343 2.7262 0.9676 0.7760 Mindlin 7.1522 2.0271 0.8929 2.4068 0.8677 0.7577 0.1
Difference 28.48% 39.49% 4.43% 11.72% 10.32% 2.36%
Table 5.4 Effect of transverse shear deformation on λ for CCC
isosceles triangular plates
a/b = 0.5 a/b = 2 τ Plate theory Elastic IT DT Elastic IT DT
Kirchhoff 28.2837 28.2837 28.2837 7.8271 7.8271 7.8271 Mindlin 28.2820 28.2820 28.2820 7.8280 7.8280 7.8280 0.001
Difference 0% 0% 0% 0% 0% 0% Kirchhoff 28.2837 12.4332 11.4777 7.8271 7.4427 7.4190 Mindlin 26.6956 12.3117 11.3982 7.7001 7.3587 7.3361 0.025
Difference 5.61% 0.97% 0.69% 1.62% 1.13% 1.12% Kirchhoff 28.2837 9.4750 1.0584 7.8271 2.6221 0.9054 Mindlin 14.8924 7.2632 1.0166 6.2244 2.4129 0.8917 0.1
Difference 47.34% 23.34% 3.95% 20.48% 7.98% 1.51%
Triangular and Elliptical Plates
111
0
1
2
3
4
5
6
7
8
0.25 0.5 0.75 1 1.25 1.5 1.75 2
τ = 0.001 τ = 0.05 τ = 0.10
IT DT
λ = σh
b2 /(π2 D
)
Fig. 5.3 Variation of λ with respect to a/b ratio for S*S*S* isosceles triangular plates
σ
a
b
Aspect ratio a/b
0
1
2
3
4
5
1 1.5 2 2.5 3 3.5 4
τ = 0.001 τ = 0.05 τ = 0.10
IT DT
λ = σh
b2 /(π2 D
)
Fig. 5.4 Variation of λ with respect to a/b ratio for S*S*S* right angled triangular plates
σ
a
b
Aspect ratio a/b
Triangular and Elliptical Plates
112
5.3 Elliptical Plates
Consider an elliptical plate of constant thickness h and aspect ratio a/b as shown in
Fig. 5.5. The plate is subjected to in-plane uniform compressive stress σ . The problem
is to determine the plastic buckling stress of such loaded elliptical plates.
The basic functions for the Ritz formulations are given by:
for simply supported elliptical plate
)1()1( 22122 −+=−+= ηξηξφ wb (5.9)
1)1( 022 =−+= ηξφ xb (5.10)
1)1( 022 =−+= ηξφ yb (5.11)
for clamped elliptical plate
)1()1( 22122 −+=−+= ηξηξφ wb (5.12)
)1()1( 22122 −+=−+= ηξηξφ xb (5.13)
)1()1( 22122 −+=−+= ηξηξφ yb (5.14)
a
b
x, ξ
y, η
Fig. 5.5 Elliptical plate under uniform compression and coordinate system
σ
Triangular and Elliptical Plates
113
There are hitherto, no studies been done on plastic buckling of elliptical plates
despite a few studies (Voinovsky-Krieger, 1937; Shibaoka, 1956; Wang and Liew,
1993b; Wang et al., 1994a) been conducted on the elastic buckling of such plates.
Therefore, comparison studies can only be carried out against existing elastic buckling
solutions. In order to obtain elastic buckling stress parameters, very small thickness-to-
width ratio ( 001.0=τ ) is adopted in the Ritz plastic buckling analysis. Note that, as a
special case, the elliptical plate becomes a circular plate when a/b = 1 and the
analytical plastic buckling solutions for circular plates are available (Wang et al.,
2001a). The exact expression of plastic buckling parameter for circular plates under
uniform compression can be expressed in the transcendental form of
αβ
−=ΩΩΩ 1)()(
1
0
JJ for simply supported plates (5.15)
2
0
2
22
8225.06722.0
)1(
ατσ
κ
ακλ Ev
+
−= for clamped plates (5.16)
where )(•nJ is Bessel function of the first kind of order n and
( ) 02222
22
/1(1212
στλπκαλκπ
Evb
−−=Ω (5.17)
The convergence and comparison studies are shown in Tables 5.5 and 5.6 for
simply supported and clamped circular and elliptical plates for various aspect ratios
(a/b = 1, 2). It can be seen that satisfactory convergence is achieved when the degree
of polynomial 10=p is used. The converged solutions are found to agree well with
the existing solutions.
Triangular and Elliptical Plates
114
Table 5.5 Convergence and comparison studies of plastic buckling stress factors )/( 22 Dhb πσ for circular plates under uniform compression (a/b=1).
τ = 0.001 τ = 0.050 τ = 0.075 BC p IT DT IT DT IT DT
8 1.7322 1.7322 1.6623 1.6617 1.0690 1.066310 1.7322 1.7322 1.6623 1.6617 1.0690 1.0663Simply
supported Wang et al. (2001a) 1.7322 1.7322 1.6622 1.6617 1.0690 1.0663
8 5.9503 5.9503 2.9035 2.7536 1.9673 1.415210 5.9503 5.9503 2.9035 2.7536 1.9673 1.4152Clamped
Wang et al. (2001a) 5.9503 5.9503 2.9035 2.7536 1.9673 1.4152
Table 5.6 Convergence and comparison studies of plastic buckling stress factors )/( 22 Dhb πσ for elliptical plates under uniform compression (a/b=2).
τ = 0.001 τ = 0.050 τ = 0.075 BC p IT DT IT DT IT DT
8 1.2467 1.2467 1.2281 1.2280 0.9814 0.9762 10 1.2467 1.2467 1.2271 1.2270 0.9811 0.9759 12 1.2467 1.2467 1.2266 1.2265 0.9810 0.9758
Simply supported
Wang et al. (1994a) 1.236 (Elastic) __ __ __ __
8 4.2286 4.2286 2.6054 2.5403 1.5564 1.3379 10 4.2286 4.2286 2.6054 2.5403 1.5564 1.3379 Clamped
Wang et al. (1994a) 4.231 (Elastic) __ __ __ __
Effect of transverse shear deformation
Tables 5.7 and 5.8 show the effect of transverse shear deformation on the plastic
buckling stress parameters of simply supported and clamped circular and elliptical
plates. It can be seen that the effect of transverse shear deformation lower the plastic
buckling stress parameters. The effect on the plastic buckling stress parameters
Triangular and Elliptical Plates
115
obtained by using both plasticity theories is not as significant as that on corresponding
elastic buckling stress parameters. Generally, the reductions are larger for the buckling
stress associated with IT when compared with the reduction in DT counterparts.
Table 5.7 Effect of transverse shear deformation on λ for simply supported
circular (a/b = 1) and elliptical plates (a/b = 3)
a/b = 1 a/b = 3 h/b Plate theory Elastic IT DT Elastic IT DT
Thin plate 1.7322 1.7322 1.7322 1.1834 1.1834 1.1834Mindlin 1.7322 1.7322 1.7322 1.1834 1.1834 1.18340.001
Difference 0% 0% 0% 0% 0% 0% Thin plate 1.7322 1.7322 1.7322 1.1834 1.1834 1.1834Mindlin 1.7267 1.7267 1.7267 1.1787 1.1787 1.17870.025
Difference 0.32% 0.32% 0.32% 0.40% 0.40% 0.40%Thin plate 1.7322 0.6788 0.6761 1.1834 0.6686 0.6522Mindlin 1.6481 0.6762 0.6735 1.1299 0.6611 0.64590.100
Difference 4.86% 0.38% 0.38% 4.52% 1.12% 0.97%
Table 5.8 Effect of transverse shear deformation on λ for clamped
circular (a/b = 1) and elliptical plates (a/b = 3)
a/b = 1 a/b = 3 h/b Plate theory Elastic IT DT Elastic IT DT
Thin plate 5.9504 5.9504 5.9504 4.0392 4.0392 4.0392Mindlin 5.9503 5.9503 5.9503 4.0392 4.0392 4.03920.001
Difference 0% 0% 0% 0% 0% 0% Thin plate 5.9504 5.9037 5.9001 4.0392 4.0378 4.0377Mindlin 5.8859 5.8439 5.8405 4.0063 4.0050 4.00490.025
Difference 1.08% 1.01% 1.01% 0.81% 0.81% 0.81%Thin plate 5.9504 1.9937 0.8728 4.0392 1.3583 0.8258Mindlin 5.0628 1.8833 0.8631 3.5721 1.3030 0.81790.100
Difference 14.91% 5.54% 1.11% 11.56% 4.07% 0.96%
Triangular and Elliptical Plates
116
Effect of aspect ratio
The variations of plastic buckling stress parameter with respect to the aspect ratio
are shown in Figs. 5.6 and 5.7 for simply supported and clamped elliptical plates.
Generally, plastic buckling stress parameters decreases with increasing aspect ratios.
For large thickness-to-width ratios, the variation of plastic buckling stress parameters
with respect to aspect ratio is negligible. Interestingly, the plastic buckling stress
parameters obtained by using IT and DT are close to each other for simply supported
elliptical plates.
0.5
0.75
1
1.25
1.5
1.75
2
1 1.5 2 2.5 3
τ = 0.001 τ = 0.050 τ = 0.100
IT DT
λ = σh
b2 /(π2 D
)
Aspect ratio a/b
Fig. 5.6 Variation of λ with respect to a/b ratio for simply supported elliptical plates
σ
a
b
Triangular and Elliptical Plates
117
5.4 Concluding Remarks
The applicability of the Ritz method for various plate shapes is demonstrated by
solving the plastic buckling problems of isosceles triangular plates, right angled
triangular plates and elliptical plates. The results are verified by comparing with the
limited existing elastic and plastic solutions. Based on the triangular and elliptical plate
examples, it can be concluded that the developed Ritz method can be readily applied
for the plastic buckling analyses of arbitrarily shaped plates with sides are defined by
polynomial functions.
0
1
2
3
4
5
6
7
1 1.5 2 2.5 3
τ = 0.001 τ = 0.050 τ = 0.100
IT DT
λ = σh
b2 /(π2 D
)
Fig. 5.7 Variation of λ with respect to a/b ratio for clamped elliptical plates
σ
a
b
Aspect ratio a/b
118
SKEW PLATES
6.1 Introduction
Buckling behaviour of skew plates is especially important to engineers who are
involved in designing aircraft wings, missile fins and skew bridges. The elastic
buckling of skew plates has been studied extensively by many researchers who
employed various numerical methods for solutions. Anderson (1951), Wittrick (1953),
Mizusawa et al. (1980) and Wang et al. (1992) used the Ritz method, York and
Williams (1995) adopted the Lagrangian multiplier method, Huyton and York (2001)
made used of the commercial finite element software ABAQUS and Wang et al.
(2003) employed the differential quadrature method. When the skew plates are thick,
the effect of transverse shear deformation must be considered. Otherwise the buckling
capacity will be overestimated. The buckling problem of thick skew plates was
CHAPTER 6
Skew Plates
119
investigated by Kitipornchai et al. 1993, Wang et al. 1993a and Xiang et al. 1995.
However, there are hitherto no plastic buckling solutions for thick skew plates
published in the literature. This chapter presents for the first time the plastic buckling
solutions of thick skew plates.
The plastic bifurcation buckling problem of skew Mindlin plates is solved by using
the Ritz method. The skew coordinate system is adopted because it is the most natural
coordinate system for this plate shape. The detailed formulation of the Ritz method
using the skew coordinate system was presented in Section 3.4 and the computer code
called BUCKRITZSKEW was developed (see in Appendix A). In the calculations of
plastic buckling stress, both the incremental theory (IT) and the deformation theory
(DT) of plasticity were adopted. The effects of transverse shear deformation,
thickness-to-width ratios, aspect ratios and skew angles on the plastic buckling stress
parameters are investigated.
For convenience and clarity, the letters, IT and DT, will be stated in the parentheses
to indicate the plasticity theories used. For example, λ (IT) refers to the plastic
buckling stress parameter λ based on IT. In generating the numerical results, a typical
aerospace material Al 7075 T6 is adopted. The material has the following properties:
Poisson’s ratio ν = 0.33, 6105.10 ×=E psi, 707.00 == σσ ksi, 672.0 =σ ksi and the
Ramberg-Osgood parameters k = 3/7, 2.9=c .
6.2 Uniaxial and Biaxial Loading
Consider a skew plate of uniform thickness h, length a, oblique width b and skew
angle ψ as shown in Fig. 6.1. The plate is subjected to in-plane uniaxial or biaxial
compressive stresses and the boundary conditions of the plate edges may take any
Skew Plates
120
combination of simply supported, clamped and free edge conditions. The problem at
hand is to determine the plastic bifurcation stress of such a loaded plate.
The plastic buckling stress parameters yx λλ , used in the program code are defined
by:
λπσ
λ ==D
hbxx 2
2
(6.1)
λβπσ
λ ==D
hbyy 2
2
(6.2)
where λ is the key parameter to be evaluated.
Before generating the buckling results, convergence studies of the plastic buckling
stress parameter )/( 22 Dhbx πσλ = with respect to the degree of polynomial functions
were conducted. Tables 6.1 to 6.4 present typical convergence studies for simply
supported and clamped skew plates. Monotonic convergence is observed with an
increasing degree of polynomials for the Ritz approximate functions. The convergence
rate is found to be slower when the skew angle increases. Generally, a polynomial
Fig. 6.1. Skew plate under biaxial compression stresses
b/2
a/2 ξ
η
xσ
xy σβσ =
xy σβσ =
xσ a/2
ψ
Skew Plates
121
degree set of p = 14 is found to furnish converged solutions. This degree of
polynomial will be used to calculate all subsequent buckling results.
Table 6.1. Convergence study of plastic buckling stress parameters λ
for simply supported skew plate under uniaxial compression ( 0=β , a/b=1)
o0=ψ o30=ψ o45=ψ h/b p
IT DT IT DT IT DT
8 4.0000 4.0000 6.0105 6.0105 10.8428 10.842810 4.0000 4.0000 5.9637 5.9637 10.5315 10.531512 __ __ 5.9361 5.9361 10.3651 10.365114 __ __ 5.9184 5.9184 10.2528 10.2528
0.001
16 __ __ 5.9062 5.9062 10.1710 10.171
8 3.9850 3.9848 5.9621 5.9544 9.6919 9.3467 10 3.9850 3.9848 5.9163 5.9092 9.4476 9.1711 12 __ __ 5.8893 5.8825 9.3321 9.0815 14 __ __ 5.8719 5.8652 9.2549 9.0202
0.025
16 __ __ 5.8599 5.8534 9.1977 8.9736
8 3.4955 2.7954 4.6798 3.0723 7.8187 3.4149 10 3.4955 2.7954 4.6069 3.0632 7.3994 3.3831 12 3.4955 2.7954 4.5547 3.0574 7.2705 3.3696 14 __ __ 4.5192 3.0535 7.1860 3.3614
0.05
16 __ __ 4.4937 3.0508 7.1015 3.3551
In order to check the correctness of the program, comparison studies are shown in
Tables 6.5 and 6.6 against existing elastic buckling solutions because there are no
plastic buckling solutions available in the open literature. The elastic buckling
solutions can readily be obtained from the plastic buckling analysis program by setting
EEE st == . Alternatively, elastic buckling solutions can also be obtained by setting
the thickness-to-width ratio to a small value (for eg. τ = 0.001) in the plastic buckling
analysis. This is because the plastic buckling solutions approach the corresponding
Skew Plates
122
elastic buckling solutions for relatively thin plates. It can be seen from Tables 6.5 and
6.6 that thin plate (τ = 0.001) plastic buckling results agree well with all existing
elastic buckling solutions for clamped skew plates. For simply supported skew plates,
the present results agree with some existing solutions (Fried and Schmitt, 1972;
Kitipornchai et al, 1993; Saadatpour et al., 2002) but are slightly higher than the
existing solutions obtained by Saadatpour et al. (1998), Huyton and York (2001) and
Wang et al. (2003). This is due to the treatment of the corner condition as discussed in
the paper by Wang et al. (1992).
Table 6.2. Convergence study of plastic buckling stress parameters λ for simply supported skew plate under biaxial compression σσσ == yx ( 1=β , a/b=1)
o0=ψ o30=ψ o45=ψ
h/b p IT DT IT DT IT DT
8 2.0000 2.0000 2.5757 2.5757 3.8364 3.8364 10 2.0000 2.0000 2.5597 2.5597 3.7640 3.7640 12 __ __ 2.5500 2.5500 3.7167 3.7167 14 __ __ 2.5437 2.5437 3.6836 3.6836
0.001
16 __ __ 2.5393 2.5393 3.6592 3.6592
8 1.9927 1.9927 2.5634 2.5634 3.8068 3.8068 10 1.9927 1.9927 2.5474 2.5474 3.7348 3.7347 12 __ __ 2.5378 2.5378 3.6874 3.6874 14 __ __ 2.5315 2.5315 3.6541 3.6540
0.025
16 __ __ 2.5271 2.5271 3.6293 3.6293 8 1.8713 1.8649 2.1672 2.1497 2.5255 2.4742 10 1.8713 1.8649 2.1607 2.1436 2.5093 2.4605 12 __ __ 2.1567 2.1398 2.4985 2.4512 14 __ __ 2.1541 2.1373 2.4909 2.4446
0.05
16 __ __ 2.1523 2.1356 2.4852 2.4397
Skew Plates
123
Table 6.3. Convergence study of plastic buckling stress parameters λ for clamped skew plate under uniaxial compression xσ ( 0=β , a/b = 1)
o0=ψ o30=ψ o45=ψ
h/b p IT DT IT DT IT DT
8 10.0764 10.0764 13.6543 13.6543 22.6083 22.608310 10.0739 10.0739 13.5476 13.5476 20.617 20.617 12 10.0737 10.0737 13.5399 13.5399 20.2042 20.204214 __ __ 13.5382 13.5382 20.1346 20.1346
0.001
16 __ __ 13.5378 13.5378 20.1172 20.1172
8 8.9361 8.7840 10.4743 10.0177 15.6945 11.787910 8.9352 8.7833 10.3926 9.9772 13.6173 11.417912 8.9351 8.7832 10.3885 9.9753 12.9930 11.300714 8.9350 8.7831 10.3881 9.9751 12.8328 11.2756
0.025
16 __ __ 10.3880 9.9750 12.8009 11.2716
8 5.3547 3.2398 7.8350 3.4300 13.1786 3.6760 10 5.3451 3.2392 7.3715 3.4146 11.6271 3.6272 12 5.3441 3.2392 7.2784 3.4115 11.0101 3.6098 14 5.3439 3.2392 7.2690 3.4113 10.8042 3.6054
0.05
16 __ __ 7.2680 3.4112 10.7524 3.6046
Skew Plates
124
Table 6.4. Convergence study of plastic buckling stress parameters λ for clamped skew plate under biaxial compression σσσ == yx ( 1=β , a/b = 1)
o0=ψ o30=ψ o45=ψ
h/b p IT DT IT DT IT DT
8 5.3044 5.3044 6.8618 6.8618 10.0546 10.054610 5.3037 5.3037 6.8561 6.8561 9.9203 9.9203 12 5.3036 5.3036 6.8550 6.8550 9.8953 9.8953 14 __ __ 6.8550 6.8550 9.8883 9.8883
0.001
16 __ __ 6.8549 6.8549 9.8854 9.8854
8 5.2298 5.2285 6.6278 6.6178 8.5286 8.4654 10 5.2295 5.2282 6.6254 6.6155 8.4919 8.4308 12 5.2294 5.2281 6.6251 6.6151 8.4866 8.4259 14 5.2293 5.2280 6.6250 6.6150 8.4855 8.4248
0.025
16 __ __ 6.6249 6.6150 8.4851 8.4245
8 2.7957 2.6829 3.0443 2.8326 3.5891 3.0328 10 2.7957 2.6829 3.0439 2.8324 3.5746 3.0297 12 2.7957 2.6828 3.0439 2.8324 3.5726 3.0293 14 __ __ 3.0438 2.8324 3.5722 3.0292
0.05
16 __ __ - - 3.5721 3.0292
Skew Plates
125
Table 6.5. Comparison study against existing elastic solutions for simply supported skew plates (τ = 0.001)
a/b References Method o0=ψ o30=ψ o45=ψ
Uniaxial compression ( 0=β )
Durvasula (1971) Ritz Method1 4.000 6.41 12.3 Fried and Schmitt (1972) F.E.M __ 5.91 10.2
Kitipornchai et al. (1993) Ritz Method2 4.000 5.897 10.103
Saadatpour et. al. (1998) Galerkin Method 4.000 5.868 9.756 Huyton and York. (2001) ABAQUS __ 5.85 9.67
Saadatpour et al. (2002) Ritz Method3 4.000 5.904 10.061
Wang et al. (2003) Differential Quadrature Method 4.000 5.83 9.39
1
Present Study Ritz Method3 4.000 5.918 10.253
Durvasula (1971) Ritz Method1 4.000 6.03 10.3 Kitipornchai et al. (1993) Ritz Method2 4.000 5.621 8.905
Huyton and York (2001) ABAQUS __ 5.59 8.80
2
Present Study Ritz Method3 4.000 5.625 8.927
Biaxial Compression ( 1=β )
Wang et al. (1993) Ritz Method3 2.000 2.544 3.684 1 Present Study Ritz Method3 2.000 2.544 3.684
Wang et al. (1993) Ritz method3 1.250 1.616 2.356 2 Present Study Ritz Method3 1.250 1.616 2.356
1 double Fourier series; 2 orthogonal polynomial functions; 3 simple polynomial functions
Skew Plates
126
Table 6.6. Comparison study against existing elastic solutions for clamped skew plates (τ = 0.001)
a/b Method Ψ = 0˚ Ψ = 30˚ Ψ = 45˚
Uniaxial compression ( 0=β )
Wittrick (1953) Ritz Method1 __ 13.636 21.64 Durvasula (1970) Galerkin Method 10.07 13.58 20.44 Kitipornchai et al. (1993) Ritz Method2 10.074 13.538 20.112 Huyton and York. (2001) ABAQUS __ 13.5 20.1 Saadatpour et al. (2002) Ritz Method3 10.107 13.538 20.106
Wang et al. (2003) Differential Quadrature Method
10.07 13.54 20.10
1
Present Study Ritz Method3 10.074 13.538 20.135
Kitipornchai et al. (1993) Ritz Method2 7.867 10.283 15.197 Huyton and York (2001) ABAQUS __ 10.30 15.1 2
Present Study Ritz Method3 7.867 10.284 15.172
Biaxial compression ( 1=β )
Ashton(1969) Ritz Method4 5.39 6.981 10.092 Wang et al. (1993) Ritz Method3 5.304 6.854 9.888 1
Present Study Ritz Method3 5.304 6.854 9.888
Wang et al. (1993) Ritz Method3 3.923 5.208 7.788 2
Present Study Ritz Method3 3.923 5.208 7.788 1 trigonometric series; 2 double Fourier series; 3 simple polynomial functions; 4 beam characteristic functions
Effect of transverse shear deformation and thickness-to-width ratio
Tables 6.7 and 6.8 show the effect of transverse shear deformation on the plastic
buckling stress parameter λ for simply supported and clamped skew plates under
uniaxial compressive stress. The Kirchhoff plate (classical thin plate) solutions are
obtained by setting the shear correction factor 10002 =κ as discussed in Chapter 4. It
Skew Plates
127
can be seen that the effect of transverse shear deformation lowers the plastic buckling
stress parameter λ . The effect is more pronounced as the thickness-to-width ratio and
the skew angle increases. Moreover, the effect of transverse shear deformation is more
significant in the case of clamped boundary condition than in the case of simply
supported boundary condition. Generally, the reductions due to the effect of transverse
shear deformation for the plastic buckling stress parameter xλ (IT) are larger than
those for xλ (DT).
Table 6.7 Effect of transverse shear deformation on λ for simply supported skew plates under uniaxial compression (a/b = 1)
o30=ψ o45=ψ h/b Plate theory
Elastic IT DT Elastic IT DT Kirchhoff 5.9184 5.9184 5.9184 10.2530 10.2530 10.2530Mindlin 5.9184 5.9184 5.9184 10.2528 10.2528 10.25280.001
Difference 0% 0% 0% 0% 0% 0% Kirchhoff 5.9184 5.9019 5.8950 10.2530 9.3182 9.0724 Mindlin 5.8876 5.8719 5.8652 10.1496 9.2549 9.0202 0.025
Difference 0.52% 0.51% 0.50% 1.01% 0.68% 0.58% Kirchhoff 5.9184 4.6296 3.0700 10.2530 7.5279 3.3895 Mindlin 5.7978 4.5192 3.0536 9.8671 7.1861 3.3614 0.05
Difference 2.04% 2.38% 0.53% 3.76% 4.54% 0.83% Kirchhoff 5.9184 4.6199 0.9638 10.2530 7.5265 1.0275 Mindlin 5.4651 4.1714 0.9513 8.8833 6.2560 1.0034 0.1
Difference 7.66% 9.71% 1.30% 13.35% 16.88% 2.35%
Skew Plates
128
Table 6.8 Effect of transverse shear deformation on λ for clamped skew plates under uniaxial compression (a/b = 1)
o30=ψ o45=ψ h/b Plate theory
Elastic IT DT Elastic IT DT Kirchhoff 13.5387 13.5387 13.5387 20.1366 20.1368 20.1368Mindlin 13.5382 13.5382 13.5382 20.1346 20.1346 20.13460.001
Difference 0% 0% 0% 0% 0% 0% Kirchhoff 13.5387 10.4965 10.0531 20.1366 13.1523 11.3825Mindlin 13.2683 10.3881 9.9751 19.4957 12.8328 11.27560.025
Difference 2.00% 1.03% 0.78% 3.18% 2.43% 0.94% Kirchhoff 13.5387 7.9399 3.4633 20.1366 12.3635 3.6796 Mindlin 12.5310 7.2690 3.4113 17.8833 10.8042 3.6054 0.05
Difference 7.44% 8.45% 1.5% 11.19% 12.61% 2.02% Kirchhoff 13.5387 7.9365 1.0420 20.1366 12.3555 1.0947 Mindlin 10.2673 5.9412 1.0006 13.4987 8.1688 1.0363 0.1
Difference 24.16% 25.14% 3.97% 32.96% 33.89% 5.33%
The variations of plastic buckling stress parameter λ with respect to b/h ratio are
shown in Figs. 6.2 and 6.3. The thin plate buckling curves are also shown as dashed
lines for comparison purposes. It can be seen that the plastic buckling stresses
approach the elastic buckling stresses as the plate thickness decreases. But for larger
plate thicknesses i.e. smaller b/h ratios, the plastic buckling stresses differ appreciably
from the corresponding elastic buckling stresses. Generally, the IT buckling curves are
higher than the corresponding DT buckling curves and the difference increases as b/h
ratio decreases.
Effect of aspect ratio a/b
The variations of plastic buckling stress parameter λ with respect to the aspect
ratio are shown in Figs. 6.4 and 6.5 for, respectively, simply supported and clamped
skew plates under uniaxial compressive stress. As in the rectangular plate solutions,
Skew Plates
129
there are some kinks in the buckling curves. These kinks indicate the mode switching.
However, for relatively thick plates (for example τ = 0.05), these kinks cannot be seen
clearly in the buckling curves and the buckling capacity gained due to mode switching
effect is negligible. It is interesting to note that the plastic buckling stress parameter
λ (DT) is almost constant with respect to the aspect ratio when the thickness-to-width
ratio is large ( 05.0=τ ).
Effect of skew angle ψ
The effect of skew angle on the plastic buckling stress parameter is shown in Figs.
6.6 and 6.7 for, respectively, simply supported and clamped skew plates under uniaxial
compressive stress. It should be noted that the plastic buckling stress parameters for
skew plates with negative skew angles are the same as those for skew plates with
positive skew angles since the loading conditions are symmetric. Therefore, only the
positive skew angles are considered in Figs. 6.6 and 6.7. It can be seen that the plastic
buckling stress parameter increases with increasing skew angles for small thickness-
to-width ratios. However, for relatively large thickness-to-width ratios (for example,
05.0=τ ), the effect of skew angle on the plastic buckling stress parameter is rather
insignificant. Generally, the difference between λ (IT) and λ (DT) increases with
increasing skew angles.
Skew Plates
130
0
0.2
0.4
0.6
0.8
1
1.2
1.4
10 20 30 40 50 60
0=β
1=β
Fig. 6.2 Normalized plastic buckling stress σc/σ0.2 versus b/h ratio for simply supported skew plate (ψ = 30˚)
Elastic IT DT
b/h
σ c/σ
0.2
a b σc σc 30˚
b a σc
σc
0
0.2
0.4
0.6
0.8
1
1.2
1.4
10 20 30 40 50 60
b/h
Fig. 6.3 Normalized plastic buckling stress σc/σ0.2 versus b/h ratio for clamped skew plate (ψ = 30˚)
σ c/σ
0.2
0=β
1=β
Elastic ITDT
a b
σc
σc
σc σc a b 30˚
Skew Plates
131
2
4
6
8
10
0.5 1 1.5 2 2.5 3
ElasticITDT
τ = 0.025 τ = 0.05
λ =
σxh
b2 /(π2 D
)
Aspect ratio a/b
Fig. 6.4 Plastic buckling stress parameter λ versus aspect ratio a/b for simply supported skew plate (ψ = 30˚)
a b σx σx 30˚
0
5
10
15
20
25
30
0.5 1 1.5 2 2.5 3
τ = 0.025 τ = 0.05
ElasticITDT
λ =
σxh
b2 /(π2 D
)
Aspect ratio a/b
Fig. 6.5 Plastic buckling stress parameter λ versus aspect ratio a/b for clamped skew plate (ψ = 30˚)
σx σx a b 30˚
Skew Plates
132
λ =
σxh
b2 /(π2 D
)
Skew angle ψ
Fig. 6.6 Plastic buckling stress parameter λ versus skew angle ψ for simply supported skew plate (a/b = 1)
0
2
4
6
8
10
12
0 15 30 45
τ = 0.001τ = 0.025τ = 0.050
IT DT
σx σx ab
ψ λ
= σ
xhb2 /(π
2 D)
Skew angle ψ
Fig. 6.7 Plastic buckling stress parameter λ versus skew angle ψ for clamped skew plate (a/b = 1)
0
5
10
15
20
25
0 15 30 45
τ = 0.001τ = 0.025τ = 0.050
IT DT
σx σx ab
ψ
Skew Plates
133
6.3 Shear Loading
There are two kinds of in-plane shear loadings when dealing with shear buckling
problem of skew plates. The first kind of shear loading is the shear loading acting
along the two horizontal edges where the traction is a pure shear stress, whereas along
the two other oblique edges, the traction consists of both shear and normal stresses of
such magnitude that every infinitesimal rectangular element is in a state of pure shear.
The first kind of shear loading is depicted in Fig. 6.8a. The second kind is shear
loading being applied uniformly along the plate edges as shown in Fig. 6.8b.
The first kind of shear loading is referred to as R shear loading while the second
kind is referred to as S shear loading. In order to differentiate between the results of R
shear and S shear loads, the subscript R will be used to indicate the former loads while
b a
ψτ tan2 xy
Rxy =τ Rxy =τ
Sxy =τ Sxy =τ
ψτ tan2 xy
a b
(a)
(b)
Fig. 6.8 Two kinds of shear loading conditions on skew plates: (a) R shear loading and (b) S shear loading
ψ
Skew Plates
134
the subscript S will denote the latter kind of load. The following symbols are used to
denote the plastic buckling stress parameters
Rxy λλ = , 0== yx λλ for R shear loading: (6.4)
sxy λλ = , ψλλ tan2 sx −= for S shear loading. (6.5)
Tables 6.9 to 6.12 show the convergence studies that were carried out for the skew
plates with relatively large skew angles (ψ = 45˚ and ψ = -45˚) because it has been
found in the previous section that more polynomial terms are required for highly skew
plates. Generally, it can be seen that the 14th degree of polynomials in the
displacement functions are sufficient to ensure converged solutions. Therefore, the
degree of polynomial 14=p is adopted for all subsequent computations for R shear
and S shear buckling stresses.
Table 6.13 shows the comparison study between the present buckling results and
existing shear buckling solutions. Since no plastic shear buckling solutions are
available, comparisons can only be made with elastic buckling solutions by setting
001.0=τ in the plastic buckling analysis. It can be seen that the results are close
except for some cases. For example, the present result is lower than the existing results
by about 11% for simply supported skew plate of ψ = 30˚ under S shear loading. It is
believed that some of the previously published results may not have converged. The
comparison studies verify somewhat the correctness of the present formulation and the
accuracy of the present Ritz results.
Skew Plates
135
Table 6.9 Convergence study of plastic buckling parameters for simply supported skew plates under shear loading (ψ = 45˚)
001.0=τ 050.0=τ
IT DT IT DT a/b p
Rλ Sλ Rλ Sλ Rλ Sλ Rλ Sλ
8 9.558 61.600 9.558 61.600 7.698 49.983 2.138 1.840 10 9.458 61.482 9.458 61.482 7.619 49.979 2.135 1.840 12 9.393 61.464 9.393 61.464 7.566 49.979 2.133 1.840
1
14 9.347 61.451 9.347 61.451 7.528 49.978 2.132 1.840
8 10.153 39.517 10.153 39.517 7.861 34.201 2.150 1.766 10 10.083 39.392 10.083 39.392 7.775 34.148 2.148 1.766 12 10.039 39.361 10.039 39.361 7.736 34.132 2.147 1.766
2
14 10.007 39.353 10.007 39.353 7.709 34.129 2.146 1.766
Table 6.10 Convergence study of plastic buckling parameters for simply supported skew plates under shear loading (ψ = -45˚)
001.0=τ 050.0=τ
IT DT IT DT a/b p
Rλ Sλ Rλ Sλ Rλ Sλ Rλ Sλ
8 26.205 6.411 26.205 6.411 19.981 5.879 2.414 1.387 10 23.597 5.978 23.597 5.978 17.738 5.495 2.382 1.367 12 23.000 5.853 23.000 5.853 17.150 5.379 2.373 1.361
1
14 22.868 5.794 22.868 5.794 17.009 5.325 2.371 1.358
8 14.508 4.823 14.508 4.283 10.994 4.093 2.248 1.317 10 12.967 4.089 12.967 4.089 9.709 3.913 2.212 1.303 12 12.638 4.053 12.638 4.053 9.336 3.881 2.203 1.300
2
14 12.590 4.043 12.590 4.043 9.265 3.871 2.201 1.300
Skew Plates
136
Table 6.11. Convergence study of plastic buckling parameters for clamped
skew plates under shear loading (ψ = 45˚)
001.0=τ 050.0=τ
IT DT IT DT a/b p
Rλ Sλ Rλ Sλ Rλ Sλ Rλ Sλ
8 24.108 89.471 24.108 89.470 16.459 61.769 2.366 1.897 10 24.051 89.241 24.051 89.241 16.422 61.769 2.365 1.897 12 24.042 89.170 24.042 89.170 16.418 61.769 2.365 1.897
1
14 24.039 89.161 24.039 89.161 16.417 61.769 2.365 1.897
8 19.555 61.626 19.555 61.626 13.823 49.709 2.310 1.836 10 19.198 61.394 19.198 61.394 13.655 49.709 2.308 1.836 12 19.170 61.380 19.170 61.380 13.638 49.685 2.307 1.836
2
14 19.168 61.376 19.168 61.376 13.637 49.685 2.307 1.836
Table 6.12. Convergence study of plastic buckling parameters for clamped skew plates under shear loading (ψ = -45˚)
001.0=τ 050.0=τ
IT DT IT DT a/b p
Rλ Sλ Rλ Sλ Rλ Sλ Rλ Sλ
8 54.049 11.940 54.049 11.940 27.904 8.968 2.500 1.463 10 36.873 9.856 36.873 9.856 23.528 8.258 2.458 1.441 12 32.990 9.376 32.990 9.376 22.176 8.080 2.442 1.433 13 31.742 9.375 31.742 9.375 21.636 8.067 2.437 1.432
1
14 31.738 9.285 31.738 9.285 21.588 8.047 2.436 1.431
8 28.826 20.205 28.826 10.205 18.887 7.768 2.389 1.445 10 22.587 7.628 22.587 7.628 15.836 6.490 2.345 1.404 12 20.725 6.892 20.725 6.892 14.600 6.109 2.324 1.386 13 19.996 6.872 19.996 6.872 14.181 6.081 2.318 1.385
2
14 19.995 6.739 19.995 6.739 14.148 6.032 2.317 1.381
Skew Plates
137
Table 6.13 Comparison study of elastic buckling stress parameter of skew plates
under shear (a/b = 1)
SSSS CCCC Skew angle Researchers
Rλ Sλ Rλ Sλ
Present 22.868 5.794 31.738 9.285 -45˚ Others 27.7a 9.348c 32.56b __ Present 16.080 5.778 21.903 9.432 -30˚ Others 17.1a 6.686c 23.64b 10.76d Present 12.11 6.835 16.963 11.0167 -15˚ Others 12.4a 7.063c 17.24b 11.00d Present 7.539 14.616 14.354 22.292 15˚ Others 7.61a 14.830c 14.39b 22.28d Present 7.150 26.873 16.596 39.886 Others 7.54a 27.674c 16.66b 39.92d 30˚
7.07f __ __ __ Present 9.347 61.451 24.039 89.161 Others 10.7a 64.241c 24.324e __ 45˚
9.04f __ 24.08b __
a Durvasula (1971); b Durvasula (1970); c Yoshimura (1963); d Hamada (1959); e Wittrick (1954); fArgyis (1965) Effect of transverse shear deformation and thickness-to-width ratio
Tables 6.14 to 6.17 show the effect of transverse shear deformation on the plastic
buckling stress parameter Rλ and Sλ for simply supported and clamped skew plates.
In order to examine the effect of transverse shear deformation, the classical thin plate
(Kirchhoff) solutions are presented as well. These thin plate solutions are obtained by
setting the shear correction factor 10002 =κ as discussed in Chapter 4. It can be seen
that the Mindlin plate solutions are smaller than their corresponding classical thin plate
solutions due to the effect of transverse shear deformation. The effect is more
pronounced for the skew plates with larger thickness-to-width ratios and more
Skew Plates
138
restrained boundary conditions. Generally, the buckling stress parameters, Rλ (Elastic
and IT) and Sλ (Elastic and IT) are more influenced by the effect of transverse shear
deformation than their DT counterparts.
Table 6.14 Effect of transverse shear deformation on Rλ for simply supported skew plates (a/b = 1)
o30=ψ o45=ψ h/b Plate
theory Elastic IT DT Elastic IT DT Kirchhoff 7.1496 7.1497 7.1497 9.3471 9.3471 9.3471 Mindlin 7.1496 7.1496 7.1496 9.3470 9.3470 9.3470 0.001
Difference 0% 0% 0% 0% 0% 0% Kirchhoff 7.1496 6.0083 5.7139 9.3471 7.8683 6.4708 Mindlin 7.0953 5.9823 5.6957 9.2590 7.8051 6.4505 0.025
Difference 0.76% 0.43% 0.32% 0.94% 0.80% 0.31% Kirchhoff 7.1496 5.3887 2.0399 9.3471 7.7807 2.1442 Mindlin 6.9371 5.2548 2.0296 9.0132 7.5278 2.1316 0.050
Difference 2.97% 2.48% 0.50% 3.57% 3.25% 0.59%
Table 6.15 Effect of transverse shear deformation on Sλ for simply supported
skew plates (a/b = 1)
o30=ψ o45=ψ h/b Plate theory Elastic IT DT Elastic IT DT
Kirchhoff 26.8742 26.8742 26.8742 61.4549 61.4549 61.4549 Mindlin 26.8731 26.8731 26.8731 61.4509 61.4509 61.4509 0.001
Difference 0% 0% 0% 0.01% 0.01% 0.01% Kirchhoff 26.8742 22.8008 7.0310 61.4549 56.0180 6.4049 Mindlin 26.2551 22.3488 7.0033 59.0707 54.0199 6.3663 0.025
Difference 2.30% 1.98% 0.39% 3.88% 3.57% 0.60% Kirchhoff 26.8742 22.7996 2.1029 61.4549 54.2074 1.8792 Mindlin 24.5714 21.1046 2.0761 53.002 48.6943 1.8400 0.050
Difference 8.57% 7.43% 1.27% 13.75% 10.17% 2.09%
Skew Plates
139
Table 6.16 Effect of transverse shear deformation on Rλ for clamped skew plates (a/b = 1)
o30=ψ o45=ψ h/b Plate
theory Elastic IT DT Elastic IT DT Kirchhoff 16.5968 16.5968 16.5968 24.0404 24.0404 24.0404 Mindlin 16.5960 16.5960 16.5960 24.0389 24.0389 24.0389 0.001
Difference 0% 0% 0% 0.01% 0.01% 0.01% Kirchhoff 16.5968 13.0423 7.4363 24.0404 18.6331 7.9704 Mindlin 16.1519 12.7553 7.3948 23.1649 18.0123 7.9122 0.025
Difference 2.68% 2.20% 0.56% 3.64% 3.33% 0.73% Kirchhoff 16.5968 13.0382 2.3052 24.0404 18.6309 2.4160 Mindlin 14.9718 11.9687 2.2715 20.9019 16.4171 2.3654 0.050
Difference 9.79% 8.20% 1.46% 13.06% 11.88% 2.09%
Table 6.17 Effect of transverse shear deformation on Sλ for clamped skew plates (a/b = 1)
o30=ψ o45=ψ h/b Plate
theory Elastic IT DT Elastic IT DT Kirchhoff 39.8888 39.8888 16.5968 89.1713 89.1713 89.1713 Mindlin 39.8860 39.8860 16.5960 89.1609 89.1609 89.1609 0.001
Difference 0.01% 0.01% 0% 0.01% 0.01% 0.01% Kirchhoff 39.8888 35.4582 7.4363 89.1713 82.8029 6.7018 Mindlin 38.2375 34.1627 7.3948 83.3027 77.7362 6.6344 0.025
Difference 4.14% 3.65% 0.56% 6.58% 6.12% 1.01% Kirchhoff 39.8888 35.4578 2.3052 89.1713 71.2645 1.9607 Mindlin 34.1733 30.8872 2.2715 70.2337 61.8897 1.8966 0.050
Difference 14.33% 12.89% 1.46% 21.24% 13.15% 3.27%
The variations of plastic buckling stresses, R shear and S shear, with respect to b/h
ratios are shown in Figs. 6.9 and 6.10, respectively, for simply supported and clamped
skew plates with o30=ψ . The buckling curves based on the classical thin plate theory
are presented as dashed lines for comparison purposes. Owing to the fact that the S
shear loading is equivalent to the R shear loading plus a uniaxial tensile loading for
0>ψ , the plastic buckling strength of plate in S shear case should be higher than that
Skew Plates
140
in R shear case. This fact is always correct for elastic and IT buckling curves.
However, it can be seen from Fig. 6.10 that, Sλ (DT) may be smaller than Rλ (DT) for
small b/h ratios. This is due to the presence of high tensile stress.
Effect of aspect ratio a/b
The variations of shear plastic buckling stress parameters, Rλ and Sλ , with respect
to aspect ratios are shown in Figs. 6.11 and 6.12. For R shear case, the kinks (that
indicate buckling mode switching) along the curves are obvious for both IT and DT
buckling curves. However, these kinks for DT buckling curves fade away as the
thickness-to-width ratio increases. On the other hand the locations of these kinks for S
shear case are not as obvious as in R shear case.
It can be seen that Rλ and Sλ based on DT are more sensitive to the thickness-to-
width ratios than their IT counterparts. Generally, as discussed before, the shear plastic
buckling stress parameters based on IT are found to be higher than those based on DT.
The difference is more pronounced for the skew plates with large thickness-to-width
ratios and small aspect ratios (i.e. 1/ <ba )
Effect of skew angle ψ
Figures 6.13 to 6.16 show the variations of Rλ and Sλ with respect to the skew
angle ψ for simply supported and clamped skew plates. It can be seen that Rλ (IT)
decreases first and then increases when the skew angle ψ increases from -45˚ to +45˚.
Note that the plastic buckling results for positive shear loading of skew plates with
negative skew angles will be same as those for negative shear loading of skew plates
with positive skew angles. For a given aspect ratio and thickness ratio, Rλ (IT) for a
Skew Plates
141
skew plate with negative skew angle are higher than those for a skew plate with a
corresponding positive skew angle. In another word, for a skew plate with positive
skew angle, Rλ (IT) in the case of positive shear loading is lower that its negative
shear loading counterpart. Therefore, the direction of the shear stresses that cause a
lower plastic buckling stress parameter Rλ (IT) is the one that tends to increase the
skewness of the plate. On the other hand, the values of Sλ (IT) increase with
increasing skew angles ψ from -45˚ to +45˚.
The variation of Rλ (DT) and Sλ (DT) with respect to the skew angle is the same as
their IT counterparts for small thickness-to-width ratios. However, for larger
thickness-to-width ratios, the variations of Rλ (DT) and Sλ (DT) with respect to the
skew angle are almost constant.
0
0.25
0.5
0.75
1
20 40 60 80 100 120
b/h
Fig. 6.9 Normalized buckling stress τxy/σ0.2 versus b/h ratio for simply supported skew plate (a/b =1)
τ xy/σ
0.2
S shear
R shear
ab
a b
30˚
30˚
Elastic IT DT
Skew Plates
142
0
0.25
0.5
0.75
1
25 50 75 100 125 150
S shear
R shear
b/h
Fig. 6.10 Normalized buckling stress τxy/σ0.2 versus b/h ratio for clamped skew plate (a/b = 1)
τ xy/σ
0.2
a b
a b
30˚
30˚
ElasticIT DT
Aspect ratio a/b
λxy
Fig. 6.11. Variation of plastic buckling stress parameters with respect to aspect ratio for SSSS skew plate with ψ = 30˚
ITDT
Aspect ratio a/b
0
2
4
6
8
10
12
14
0.5 1 1.5 2 2.5 3 3.5 40
10
20
30
40
0.5 1 1.5 2 2.5 3 3.5 4
τ = 0.015 τ = 0.025 τ = 0.050
ab
a b
30˚
30˚
Skew Plates
143
Skew angle ψ
Fig. 6.13. Variation of plastic (R shear) buckling stress parameters with respect to skew angle of SSSS skew plate
Rλ
τ = 0.001τ = 0.025τ = 0.050
IT DT +ψ
a b
0
5
10
15
20
25
-45 -30 -15 0 15 30 45
0
5
10
15
20
0.5 1 1.5 2 2.5 3 3.5 40
10
20
30
40
50
60
0.5 1 1.5 2 2.5 3 3.5 4
Aspect ratio a/b
Fig. 6.12. Variation of plastic shear buckling stress parameters with respect to aspect ratio for CCCC skew plate with ψ = 30˚
ITDT
Aspect ratio a/b
τ = 0.01τ = 0.02τ = 0.03
a b
a b
Rλ Sλ
30˚
30˚
Skew Plates
144
0
10
20
30
-45 -30 -15 0 15 30 45
Skew angle ψ
Fig. 6.14. Variation of plastic (S shear) buckling stress parameters with respect to skew angle of SSSS skew plate
Sλ
τ = 0.001 τ = 0.025 τ = 0.050
IT DT
+ψ
a b
0
5
10
15
20
25
30
35
-45 -30 -15 0 15 30 45
Rλ
Skew angle ψ
τ = 0.001τ = 0.02τ = 0.03
IT DT
Fig. 6.15. Variation of plastic (R shear) buckling stress parameters with respect to skew angle of CCCC skew plate
a b
+ψ
Skew Plates
145
6.4 Concluding Remarks
In this chapter, plastic buckling analysis of skew plates was carried out using the Ritz
method based on the skew coordinate system. In order to include the effect of
transverse shear deformation, the Mindlin plate theory was adopted. The plastic
buckling behaviour of skew plates under unaxial, biaxial and shear stresses are
examined. Generally, the effect of transverse shear deformation lowers the buckling
stress parameters and the effect is more pronounced in the case of the plastic buckling
stress parameters obtained by using IT than in the case of their DT counterparts. The
effects of aspect ratios, thickness to width ratios, and skew angles on the plastic
buckling stress parameters were discussed.
0
10
20
30
-45 -30 -15 0 15 30 45
Sλ
Skew angle ψ
Fig. 6.16. Variation of plastic (S shear) buckling stress parameters with respect to skew angle of CCCC skew plate
τ = 0.001τ = 0.02τ = 0.03
IT DT
a b
+ψ
146
CIRCULAR AND ANNULAR PLATES
7.1 Introduction
Circular and annular plates are used in turbine disks, submarines, instrument mounting
bases for space vehicles and microelectromechanical systems (MEMS) devices (Su
and Spearing, 2004). Their buckling capacities need to be calculated when applied in
situations where in-plane compressive of forces act on the plates. The elastic
axisymmetric buckling problems (where the deflection surface is assumed to be
axisymmetric) of circular and annular plates have been studied by many researchers.
(Raju and Rao, 1989; Wang et al., 1993c; Liew et al., 1994a; Laura et al., 2000). The
assumption of axisymmetric buckling, however, does not necessarily lead to the
desired lowest critical load. It had been found that the buckling mode shape for an
CHAPTER 7
Circular and Annular Plates
147
annular plate, which corresponds to the lowest critical load, may be axisymmetric or
asymmetric depending on the inner radius to outer radius ratio and edge boundary
conditions (Yamaki, 1958; Majumdar, 1971). Therefore, the asymmetric buckling
problems of circular and annular plates have also been studied (Gerard, 1978;
Thevendran and Wang, 1996; Wang, 2003).
All the aforementioned studies, however, are confined to the elastic buckling
problems. The elastoplastic buckling of annular plate under pure shear was studied by
Ore and Durban (1989). Recently, Kosel and Bremec (2004) investigated the
elastoplastic buckling of annular plates under uniform loading by using the finite
different method. These studies are based on the classical thin plate (Kirchhoff) theory
which neglects the effect of transverse shear deformation. When dealing with
relatively thick plates, it is necessary to use thick plate theories such as the Mindlin
plate theory for accurate buckling load. Otherwise the buckling load will be
overestimated because the neglect of the effect of transverse shear deformation
stiffens the plates. In this chapter, the plastic buckling of circular and annular Mindlin
plates is studied. The Ritz method using the polar coordinate system is employed for
the analysis and the program is coded in MATHEMATICA language (see Appendix A
for the program called BUCKRITZPOLAR).
In order to verify the numerical plastic buckling results, an analytical approach is
presented for the asymmetric plastic buckling problems of Mindlin circular and
annular plates There are hitherto no analytical solution is available for the asymmetric
buckling problems of Mindlin circular and annular plates, although an analytical
method for elastic buckling was presented by Yamaki (1958) based on the classical
thin plate theory. Readers who are interested in the analytical approach for elastic
buckling problems of such Mindlin plates may refer to the paper by Wang and Tun
Circular and Annular Plates
148
Myint Aung (2005). In the next section, the analytical approach for solving plastic
buckling of circular and annular plates under uniform stress is featured.
7.2 Analytical Method
Consider a circular or annular Mindlin plates subjected to a uniform radial
compressive stress i.e. σσσ θθ ==rr . The governing equations for the buckling of
Mindlin plate under uniform compressive stresses shown in Eqs. (2.36a-c) can be
expressed in the polar coordinate system as
( ) 01222 =⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛∂∂
++∂∂
+∇−θφφφ
κσκ θ
rrrGhwhGh rr (7.1a)
011112
112
1112
2
2
2
2
2
32
23
2
2
22
3
=⎭⎬⎫
⎩⎨⎧
⎟⎠⎞
⎜⎝⎛∂∂
−⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
∂+⎟⎟
⎠
⎞⎜⎜⎝
⎛∂∂
−⎭⎬⎫
⎩⎨⎧
∂∂
++
⎭⎬⎫
⎩⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
∂−
⎭⎬⎫
⎩⎨⎧
∂∂
−⎟⎠⎞
⎜⎝⎛∂∂
−⎟⎠⎞
⎜⎝⎛∂∂
+
θφ
θφ
θφφκ
θφβφφ
θφφα
θθ
θθ
rrrrGh
rwGh
rrEh
rrrrrEh
rr
rrr
(7.1b)
011112
1112
1112
22
2
2
23
223
22
2
2
3
=⎭⎬⎫
⎩⎨⎧
−⎟⎠⎞
⎜⎝⎛∂∂
+∂∂
+⎟⎠⎞
⎜⎝⎛∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
∂+
⎭⎬⎫
⎩⎨⎧
∂∂
+−⎭⎬⎫
⎩⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
∂+
⎭⎬⎫
⎩⎨⎧
⎟⎠⎞
⎜⎝⎛∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
rrrrrrrGh
wr
Ghrr
Ehrr
Eh
rr
rr
θθθ
θθ
φφφθφ
θφ
θφκ
θφβ
θφ
θφα
(7.1c)
where ( ) ( ) ( ) ( ) ( ) ( ) 222222 //1//1/ θ∂•∂+∂•∂+∂•∂=•∇ rrrr .
Following the work of Mindlin (1951), the rotations ( θφφ ,r ) can be defined in
terms of potentials ( H,ϕ ) as
θϕφ
∂∂
+∂∂
=H
rrr1 (7.2a)
Circular and Annular Plates
149
rH
r ∂∂
−∂∂
=θϕφθ
1 (7.2b)
In view of Eqs. (7.2a-b) and after some simplifications, Eqs. (7.1a-c) become
01 22 =⎥
⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ −+∇ w
Ghh
κσϕ (7.3a)
( )⎭⎬⎫
⎩⎨⎧
∂∂
⎟⎠⎞
⎜⎝⎛ −−
∂∂
−⎭⎬⎫
⎩⎨⎧
+−∇∂∂
2
2
2
22 12112
θϕβαϕκϕα
rEG
rrw
EhG
r
021212
2
2
22 =
⎭⎬⎫
⎩⎨⎧
∂∂
⎟⎠⎞
⎜⎝⎛ −−+−∇
∂∂
+rH
EGH
EhGH
EG
rβακ
θ (7.3b)
( )⎭⎬⎫
⎩⎨⎧
∂∂
⎟⎠⎞
⎜⎝⎛ −−++−∇
∂∂
2
2
2
22 2121
rEGw
EhG
rϕβαϕκϕα
θ
0121122
2
2
22 =
⎭⎬⎫
⎩⎨⎧
∂∂
⎟⎠⎞
⎜⎝⎛ −−
∂∂
−⎟⎟⎠
⎞⎜⎜⎝
⎛−∇
∂∂
−θ
βακ HrE
Grr
HEhGH
EG
r (7.3c)
For plates under uniform radial compressive stresses, i.e. σσσ θθ ==rr , the value
of the expression 0/2 =−− EGβα for both IT and DT. In view of this, Eqs. (7.3b)
and (7.3c) become, respectively,
( ) 012122
22
2
22 =⎟⎟
⎠
⎞⎜⎜⎝
⎛−∇
∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛+−∇
∂∂ H
hrEGw
EhG
rκ
θϕκϕα (7.4a)
and
( ) 0121212
22
2
22 =⎟⎟
⎠
⎞⎜⎜⎝
⎛−∇
∂∂
−⎥⎦
⎤⎢⎣
⎡+−∇
∂∂ H
hrEGw
EhG
rκϕκϕα
θ (7.4b)
The potential H may be separated from ϕ and w by differentiation, addition, and
subtraction of Eqs. (7.4a) and (7.4b). These equations become
Circular and Annular Plates
150
0122
222 =⎟⎟
⎠
⎞⎜⎜⎝
⎛−∇∇ H
hκ (7.5a)
( ) 0122
222 =⎥
⎦
⎤⎢⎣
⎡+−∇∇ w
EhG ϕκϕ (7.5b)
It now remains to separate ϕ and w between Eq. (7.3a) and Eq. (7.5b). It is
observed that these equations are satisfied by
w)1( −Λ=ϕ (7.6)
where Λ is a constant. The constant can be obtained by substituting Eq. (7.6) into
Eqs. (7.3a) and (7.5b)
01 =Λ (7.7a)
Ghh
22 κσ
=Λ (7.7b)
Therefore, Eq. (7.5) can be rewritten as
2211 )1()1( ww −Λ+−Λ=ϕ (7.8)
In view of Eqs. (7.7a-b) and (7.8), Eq. (7.5b) can be written as
012 =∇ w (7.9a)
01
12
2
2
32 =
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎟⎠⎞
⎜⎝⎛ −
−∇ w
GhhEh
h
κσα
σ (7.9b)
For clarity, 21 , ww and H will be denoted as 21 ,ΘΘ and 3Θ , respectively. For
generality and convenience, the following nondimensional terms are introduced
Circular and Annular Plates
151
Rww = (7.10a)
Rrr = (7.10b)
Rh
=τ (7.10c)
DhR
2
2
πσλ = (7.10d)
where w is the normalized deflection, r the normalized radial coordinate and τ the
thickness-to-radius ratio.
Therefore, in view of Eqs. (7.2a-b), the transverse deflection w and the rotations
( θφφ ,r ) can be defined in terms of three potential functions Q1, Q2, Q3 as
11)1(12
3222
21
θκνλτφ
∂Θ∂
+∂Θ∂
⎟⎟⎠
⎞⎜⎜⎝
⎛−
−+
∂Θ∂
−=rrG
Err (7.11a)
rrGE
r ∂Θ∂
−∂Θ∂
⎟⎟⎠
⎞⎜⎜⎝
⎛−
−+
∂Θ∂
−= 3222
21 11
)1(121
θκνλτ
θφθ (7.11b)
21 Θ+Θ=w (7.11c)
The governing equations, ie. Eqs. (7.5a), (7.9a-b) can be rewritten in terms of
nondimendional terms 321 and , ΘΘΘ :
012 =Θ∇ (7.12a)
( ) 0222
2 =Θ+∇ δ (7.12b)
( ) 0323
2 =Θ+∇ δ (7.12c)
where
Circular and Annular Plates
152
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛−
−−=
GE
22
22
22
)1(1211
κνλτνα
λδ (7.13a)
2
223
12τκδ −= (7.13b)
It can be seen that Eqs. (7.12a-c) are Bessel equations and the general solutions of
these equations are given by (Chiang, 1995)
)cos(log
)cos( 111 θθ nr
rBnrA n
n
⎭⎬⎫
⎩⎨⎧
+=Θ − (7.14a)
)cos()()cos()( 22222 θθ nrSBnrRA nn ∆+∆=Θ (7.14b)
)sin()()sin()( 33333 θθ nrSBnrRA nn ∆+∆=Θ (7.14c)
where the upper form of Eq. (7.14a) is used for n = 0 (axisymmetric buckling) and the
lower form for n ∫ 0 (asymmetric buckling), n is the number of nodal diameters. The
functions )(),( χχ jnjn SR ∆∆ , and the variables 21 ,∆∆ are given by
3,2 0 if )Im(0 if
2
2
=⎪⎩
⎪⎨⎧
<≥
=∆ jjj
jjj δδ
δδ (7.15)
3,2 0 if )(0 if )(
)( 2j
2
=⎪⎩
⎪⎨⎧
<∆≥∆
=∆ jrIrJ
rRjn
jjnjn δ
δ (7.16)
3,2 0 if )(
0 if )()( 2
j
2
=⎪⎩
⎪⎨⎧
<∆≥∆
=∆ jrKrY
rSjn
jjnjn δ
δ (7.17)
The parameters 321 ,, AAA , and 321 ,, BBB are unknown constants, )(•nJ and )(•nI are
Bessel functions of the first kind and the modified first kind of order n, respectively,
and )(•nY and )(•nK are Bessel functions of the second kind and the modified second
Circular and Annular Plates
153
kind of order n, respectively. In the sequel, plastic buckling problems of circular and
annular plates with and without internal ring supports will be solved by using the Ritz
method and the aforementioned analytical method.
7.3 Circular plates
Consider a moderately thick, isotropic, circular plate of radius R, uniform thickness h,
Young’s modulus E, shear modulus G, and Poisson’s ratio ν , as shown in Figure 7.1.
The plate is subjected to a uniform compressive in-plane stress σ along its circular
edge.
σ
h
R
Fig. 7.1 Clamped circular plate under in-plane compressive stress
σ σ
R
Circular and Annular Plates
154
For numerical results, an aluminum alloy Al 2024 T6 is adopted with the following
properties: Poisson’s ratio 32.0=ν , 6107.10 ×=E psi, 4.612.00 == σσ ksi, and the
Ramberg-Osgood parameters k = 0.3485, c = 20.
The following typical boundary conditions at the edges are considered:
Simply supported edge
0=w ; 0=rrM ; 0=θφ (7.18a-c)
Clamped edge
0=w ; 0=rφ ; 0=θφ (7.19a-c)
Free edge
0=rrM ; 0=θrM ; 0=∂∂
+χ
σ whQr (7.20a-c)
where the bending moment rrM , twisting moment θrM and shear force rQ are given
by
⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛
∂∂
++∂∂
=θφ
φβφα θ
rr
rr rrEhM12
3
(7.21a)
⎥⎦
⎤⎢⎣
⎡∂∂
+⎟⎠⎞
⎜⎝⎛ −∂∂
=rr
GhM rr
θθθ
φφ
θφ1
12
3
(7.21b)
⎟⎠⎞
⎜⎝⎛ +∂∂
= rr rwGhQ φκ 2 (7.21c)
Circular and Annular Plates
155
The Ritz method and the developed analytical method will be used to determine the
plastic buckling stress parameter so as to provide an independent check on the
solutions.
Ritz Method
In order to avoid singularity problems in the case of circular plate, ζ needs to be set
to a very small number (e.g. 10-7) and the inner edge is made free. The inputs needed
in the program code are the material properties such as c, k, 0/σE , geometric
properties such as h/R and the basic functions (boundary conditions). The basic
functions θφφφ brb
wb and ,, , which are the product of boundary equations with
appropriate power, are given by
platescircular supportedsimply for )1()1()1()1()1()1(
01
00
01
⎪⎭
⎪⎬
⎫
+−=+−=+−=
ξξφξξφξξφ
θb
rb
wb
(7.22)
platescircular clampedfor )1()1()1()1()1()1(
01
01
01
⎪⎭
⎪⎬
⎫
+−=+−=+−=
ξξφξξφξξφ
θb
rb
wb
(7.23)
Analytical Method
For the circular plate, the arbitrary constants 321 ,, BBB in Eq. (7.14a-c) must vanish in
order to avoid singularity for the displacement fields θφφ ,, rw at the center of the
plate, since log )(r , )( 2rYn δ and )( 2rK n δ are infinite as r approaches zero.
Circular and Annular Plates
156
In view of Eqs. (7.14a-c), a set of homogeneous equations can be derived by
implementing the foregoing boundary conditions and these equations can be expressed
in the following matrix form
[ ] 131333 0 xK =Φ ×× (7.24)
where [ ]TAAA 32113 =Φ × . It should be noted that, in the case of axisymmetric
buckling, θφ and the derivative with respect to θ are zero everywhere and Eq. (7.24)
becomes:
[ ] 121222 0 xK =Φ ×× (7.25)
where [ ]TAA 2113 =Φ × . The plastic buckling stress parameter λ is evaluated by
setting the determinant of [ ]K in Eq. (7.24) or Eq. (7.25) to be zero. The elements of
the [ ]K matrix are given in the Appendix B. Therefore one can easily write the
algorithm in other programming languages by using these matrix elements.
Results
The convergence and comparison studies are shown in Table 7.1 for simply supported
and clamped circular plates. It can be seen that the degree of polynomial 8=p is
found to furnish the converged Ritz results and these results are in total agreement
with the analytical solutions.
Circular and Annular Plates
157
Table 7.1. Convergence and comparison study of plastic buckling stress parameter λ for circular plates under uniform radial compression
001.0=τ 05.0=τ 1.0=τ BC p IT DT IT DT IT DT
4 0.4306 0.4306 0.4292 0.4292 0.4240 0.4240 6 0.4305 0.4305 0.4291 0.4291 0.4240 0.4240 8 0.4305 0.4305 0.4291 0.4291 0.4240 0.4240
Simply supported
Analytical Method 0.4305 0.4305 0.4291 0.4291 0.4240 0.4240
4 1.5031 1.5031 1.4837 1.4837 0.6331 0.6130 6 1.4876 1.4876 1.4715 1.4715 0.6321 0.6125 8 1.4876 1.4876 1.4715 1.4715 0.6321 0.6125 Clamped
Analytical Method 1.4876 1.4876 1.4715 1.4715 0.6321 0.6125
7.4 Annular plates
Consider a moderately thick, isotropic annular plate of outer radius R and inner radius
b, uniform thickness h. The plate is subjected to a uniform compressive in-plane stress
σ along its edges as shown in Fig. 7.2. The problem at hand is to determine the
plastic buckling stress of such plates.
Owing to the practical importance thereof, an extensive study on the plastic
buckling behaviour of annular Mindlin plates is conducted for all possible
combinations of clamped (C), simply supported (S), free (F) boundary conditions at
the outer and inner edges. For brevity, the boundary conditions of annular plates are
denoted by two letter symbols, eg. C-S denotes an annular plate with clamped (C)
outer edge and simply supported (S) inner edge.
Circular and Annular Plates
158
Ritz Method
The computer program called BUCKRITZPOLAR is used to determine the plastic
buckling stress parameter of the aforementioned buckling problem of annular plate. In
order to demonstrate the input expressions for the basic functions θφφφ brb
wb and ,, , the
sample expressions of basic functions for C-S annular plates are given by
)1)(1()1()1( 11 +−=+−= ξξξξφ wb (7.24a)
)1()1()1( 01 −=+−= ξξξφ rb (7.24b)
)1)(1()1()1( 11 +−=+−= ξξξξφθb (7.24c)
Fig. 7.2 C-S annular plate under in-plane loading
Rb
h
σ σ
σ σ σ σ
Circular and Annular Plates
159
Analytical Method
In view of Eqs. (17.14), a set of homogeneous equations can be derived by
implementing the boundary conditions (7.18a-c, 7.19a-c, 7.20a-c) at the both edges of
annular plate and the system of equations can be expressed in the following matrix
form
[ ] 161666 0 x=×× ΦK (7.25)
where [ ]TBBBAAA 32132116 =Φ × . The elements of the [ ]K matrix are
given in the Appendix B. The plastic buckling stress parameter λ is evaluated by
setting the determinant of [ ]K in Eq. (7.25) to be zero.
Results and Discussions
Convergence studies of the Ritz results are shown in Table 7.2 for annular plate with
4.0/ == Rbζ . It can be seen that converged results are obtained when the degree of
polynomial p = 10.
Table 7.3 compares the results obtained by using the Ritz method and the analytical
method with the existing elastic buckling solutions. It can be seen that the analytical
results agree well with the corresponding Ritz buckling solutions. Moreover, it is
found that these plastic buckling solutions agree well with the existing elastic
solutions when the thickness to outer radius ratio τ = 0.001.
Circular and Annular Plates
160
Table 7.2 Convergence study of plastic buckling stress parameter λ for annular plates ( 4.0/ == Rbζ )
001.0=τ 05.0=τ 1.0=τ BC p
IT DT IT DT IT DT 4 2.9700 2.9700 2.3549 2.3241 1.0255 0.7995 6 2.9652 2.9652 2.3534 2.3227 1.0251 0.7993 8 2.9652 2.9652 2.3534 2.3227 1.0251 0.7993 10 2.9652 2.9652 2.3534 2.3227 1.0251 0.7993
SS
12 2.9652 2.9652 2.3534 2.3227 1.0251 0.7993 4 0.2138 0.2138 0.2135 0.2135 0.2125 0.2125 6 0.2137 0.2137 0.2133 0.2133 0.2124 0.2124 8 0.2137 0.2137 0.2133 0.2133 0.2124 0.2124 10 0.2137 0.2137 0.2133 0.2133 0.2124 0.2124
SF
12 0.2137 0.2137 0.2133 0.2133 0.2124 0.2124 4 11.8197 10.3372 3.8111 3.1168 3.2344 0.9390 6 10.3370 10.3356 3.7826 3.1166 3.2123 0.9390 8 10.3345 10.3343 3.7821 3.1166 3.2119 0.9390 10 10.3343 10.3343 3.7821 3.1166 3.2119 0.9390
CC
12 10.3343 10.3343 3.7821 3.1166 3.2119 0.9390 4 0.7688 0.7688 0.6677 0.6677 0.5758 0.5622 6 0.6829 0.6829 0.6632 0.6632 0.5742 0.5609 8 0.6803 0.6803 0.6623 0.6623 0.5741 0.5608 10 0.6802 0.6802 0.6622 0.6622 0.5741 0.5608
FC
12 0.6802 0.6802 0.6622 0.6621 0.5741 0.5608
Circular and Annular Plates
161
Table 7.3 Comparison study of plastic buckling stress parameter )/( 22 DhR πσλ =
for annular plates ( 4.0/ == Rbζ ).
Thickness-to-outer radius ratio, τ 0.001 0.05 0.1 BC Reference Elastic IT DT IT DT
Analytical Method 10.335 (3) 3.407 (3) 2.586 (3) 3.120 (3) 0.708 (3)
Ritz Method 10.334 (3) 3.407 (3) 2.586 (3) 3.127 (3) 0.708 (3)Wang et al. (1994b) 10.315 (3) __ __ __ __
C-C
Yamaki (1958) 10.336 (3) Analytical Method 6.465 (0) 2.700 (1) 2.497 (1) 2.386 (1) 0.696 (1)
Ritz Method 6.465 (0) 2.700 (1) 2.498 (1) 2.387 (1) 0.696 (1)Wang et al. (1994b) 6.486 (0) __ __ __ __
C-S
Yamaki (1958) 6.485 (0) __ __ __ __ Analytical Method 1.089 (0) 1.080 (0) 1.080 (0) 0.648 (0) 0.610 (0)
Ritz Method 1.089 (0) 1.080 (0) 1.080 (0) 0.648 (0) 0.611 (0)Wang et al. (1994b) 1.103 (0) __ __ __ __
C-F
Yamaki (1958) 1.103 (0) __ __ __ __ Analytical Method 2.961 (0) 2.232 (0) 2.224 (0) 0.948 (0) 0.654 (0)
Ritz Method 2.961 (0) 2.233 (0) 2.224 (0) 0.949 (0) 0.654 (0)Wang et al. (1994b) 2.964 (0) __ __ __ __
S-S
Yamaki (1958) 2.965 (0) __ __ __ __ Analytical Method 5.302 (0) 2.432 (0) 2.399 (0) 1.427 (0) 0.676 (0)
Ritz Method 5.302 (0) 2.432 (0) 2.399 (0) 1.428 (0) 0.676 (0)Wang et al. (1994b) 5.289 (0) __ __ __ __
S-C
Yamaki (1958) 5.296 (0) __ __ __ __
Circular and Annular Plates
162
Table 7.3 Continued
Thickness-to-outer radius ratio, τ
0.001 0.05 0.1 BC Reference Elastic IT DT IT DT
Analytical Method 0.212 (0) 0.212 (0) 0.212 (0) 0.211 (0) 0.211 (0)
Ritz Method 0.212 (0) 0.212 (0) 0.212 (0) 0.211 (0) 0.211 (0)Wang et al. (1994b) 0.214 (0) __ __ __ __
S-F
Yamaki (1958) 0.213 (0) __ __ __ __ Analytical Method 0.671 (2) 0.653 (2) 0.653 (2) 0.543 (1) 0.542 (1)
Ritz Method 0.672 (2) 0.653 (2) 0.653 (2) 0.543 (1) 0.542 (1)Wang et al. (1994b) 0.680 (2) __ __ __ __
F-C
Yamaki (1958) 0.685 (2) __ __ __ __ Analytical Method 0.196 (1) 0.195 (1) 0.195 (1) 0.193 (1) 0.193 (1)
Ritz Method 0.196 (1) 0.195 (1) 0.195 (1) 0.193 (1) 0.193 (1)Wang et al. (1994b) 0.199 (1) __ __ __ __
F-S
Yamaki (1958) 0.199 (1) __ __ __ __
The variations of plastic buckling stress parameter λ with respect to aspect ratios
ζ are shown in Figs. 7.3 to 7.10 for various boundary conditions. It can be seen that
plastic buckling stress parameters increase with increasing value of ζ except for S-F
and F-S annular plates. In the S-F annular plate, the plastic buckling stress parameters
decreases monotonically while in F-S annular plates, the buckling stress increases first
and later decreases as ζ increases.
It can be seen that the number of nodal diameters of the buckled surface of the plate
increases in C-C annular plate as the aspect ratio ζ increases, while in C-S, S-S, and
F-S annular plates, the plates first buckle in asymmetric modes with one nodal
Circular and Annular Plates
163
diameter (i.e. n = 1) and after a particular value of ζ ratio, the buckle mode takes on
an axisymmetric forms (i.e. n = 0). The sample critical buckling mode shapes are
shown in Fig. 7.11 for C-C annular plates of various aspect ratios ζ . Apart from F-C
annular plate, the remaining types of annular plates always buckle in an axisymmetric
mode. In F-C annular plate, the number of nodal diameter n changes from one to two
and then changes back to one and followed by zero as the ratio ζ increases. It is
interesting to note that, as far as axisymmetric buckling (n = 0) is concerned, the
buckling stress parameters are identical for S-F and F-S annular plates for a particular
thickness ratio, irrespective of the hole size i.e. ζ .
The differences between the results from DT and IT plasticity theories are also
examined. It has been founded that the discrepancies increase with increasing aspect
ratio ζ and thickness-to-outer radius ratio τ . This is because the buckling stress
parameter λ increases with increasing aforementioned ratios and consequently the
effective modulus used in IT and DT progressively differ each other. However, in S-F
and F-S annular plates, the buckling stress parameters from both theories are identical
for the considered thickness-to-outer radius ratios. This is because the buckling
stresses for these boundary conditions are much lower than nominal yield stress for
the considered thickness-to-outer radius ratios due to the less restrained boundary
conditions. However, for greater thickness-to-outer radius ratios, the results obtained
by IT and DT may differ each other since the plastic buckling stress increases with
increasing thickness-to-outer radius ratio. It is interesting to note that the buckling
stress parameters obtained by DT do not much vary with respect to ζ ratio when the
boundary conditions are either simply supported or clamped and thickness-to-outer
radius ratio is large.
Circular and Annular Plates
164
0
2
4
6
8
10
12
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
DT & IT( τ = 0.001)
1=n
2=n
3=
n
1=n 2=n
2=n1=n
3=n
3=n
4=n
4=n
4=n
4=n
5=n
5=n
5=n
5=n
6=
n6
=nIT( τ = 0.05)
IT( τ = 0.1)
DT( τ = 0.05)
DT( τ = 0.1)
ζ
λ =
σhR
2 /π2 D
Fig. 7.3 Variation of buckling stress parameter λ with respect to ζ ratio for C-C annular plate
0
2
4
6
8
10
12
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0=n 0=n
0=n
0=n
1=n
1=n
1=n
1=n 0=n1=n
ζ
λ =
σhR
2 /π2 D
IT & DT (τ = 0.001)
IT (τ = 0.05)
IT (τ = 0.1)
DT (τ = 0.05)
DT (τ = 0.1)
Fig. 7.4 Variation of buckling stress parameter λ with respect to ζ ratio for C-S annular plate
Circular and Annular Plates
165
0
1
2
3
4
5
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0=n0=n
Fig. 7.5 Variation of buckling stress parameter λ with respect to ζ ratio for C-F annular plate
ζ
λ =
σhR
2 /π2 D
IT & DT
(τ = 0.001)
IT (τ = 0.05)
IT (τ = 0.1)
DT (τ = 0.1)
DT (τ = 0.05)
0
2
4
6
8
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0=n
0=n
1=n
1=n
ζ
λ =
σhR
2 /π2 D
IT & DT (τ = 0.001)
IT (τ = 0.05)
IT (τ = 0.1)
DT (τ = 0.05)
DT (τ = 0.1)
Fig. 7.6 Variation of buckling stress parameter λ with respect to ζ ratio for S-S annular plate
Circular and Annular Plates
166
0
2
4
6
8
10
12
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0=n
0=n
0=n
1=n
1=n
Fig. 7.7 Variation of buckling stress parameter λ with respect to ζ ratio for S-C annular plate
ζ
λ =
σhR
2 /π2 D
IT & DT
(τ = 0.001) IT (τ = 0.05)
IT (τ = 0.1)
DT (τ = 0.05)
DT (τ = 0.1)
0.0
0.1
0.2
0.3
0.4
0.5
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0=n
Fig. 7.8 Variation of buckling stress parameter λ with respect to ζ ratio for S-F annular plate
ζ
λ =
σhR
2 /π2 D
IT & DT (τ = 0.001, 0.05)
IT & DT (τ = 0.1)
Circular and Annular Plates
167
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0=n
0=n
0=
n
1=n
1=
n
1=n
2=n2=n
2=n1=n
Fig. 7.9 Variation of buckling stress parameter λ with respect to ζ ratio for F-C annular plate
λ =
σhR
2 /π2 D
IT & DT (τ = 0.001)
IT (τ = 0.05)
DT (τ = 0.05)
IT (τ = 0.1)
DT (τ = 0.1)
ζ
0.00
0.05
0.10
0.15
0.20
0.25
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0=n1=n
Fig. 7.10 Variation of buckling stress parameter λ with respect to ζ ratio for F-S annular plate
ζ
λ =
σhR
2 /π2 D
IT & DT (τ = 0.001)
IT & DT (τ = 0.05)
IT & DT (τ = 0.1)
Circular and Annular Plates
168
7.5 Circular and annular plates with intermediate ring supports
Consider a circular of radius R and thickness h or an annular plate of outer radius R,
inner radius b and thickness h under uniform compressive stress σ . The plate may
have simply supported or clamped edge support conditions and m concentric ring
supports with radii b1, b2,…, bm, as shown in Fig. 7.12. The rings are assumed to be
rigid so that zero deflection is imposed along the ring supports. The problem at hand is
to determine the plastic buckling stress of such ring supported circular or annular
plates. The analysis will be carried out by using both Ritz method and developed
analytical method.
05.0=ζ 2.0=ζ 4.0=ζ
5.0=ζ 6.0=ζ 65.0=ζ
Fig. 7.11. Critical buckling mode shapes for C-C annular plates of various ζ ratios (DT, τ = 0.05)
Circular and Annular Plates
169
Ritz Method
The plastic buckling stress parameter for the circular and annular plates with internal
ring supports can be easily obtained by modifying the basic function for transverse
deflection in the Ritz program called BUCKRITZPOLAR. The modification for
internal supports in the basic functions of the Ritz method for general shapes will be
discussed in Chapter 9. The sample modified basic functions for the C-F annular plate
with one internal ring support at Rb /11 =ζ are given by:
[ ])1/()12()1()1( 101 ζζζξξξφ −−−−+−=w
b (7.26)
Fig. 7.12 Simply supported (a) annular plate (b) circular plate with multiple concentric ring supports
(a) (b)
σ σ
R
σ
h
b1 b2
bm
bm
b1 b R
Circular and Annular Plates
170
Analytical Method
The circular or annular plate with m ring support can be divided into m+1 segments.
The governing equations of each segment are given by Eq. (7.1a-c) and the equations
for each segments will be denoted by a superscript i, where i is the integer number
which counts the number of segments starting from the outer edge. Not that for
circular plates with internal ring supports, the arbitrary constants 13
12
11 ,, +++ mmm BBB
must vanish in order to avoid singularity for the displacement fields 111 ,, +++ mmr
mw θφφ at
the center of the m+1th circular segment.
The boundary conditions for the periphery edge of the circular plates or the inner
and outer edges of the annular plates are given in Eqs. (7.18 to 7.20). In order to
satisfy the ring support conditions, and to ensure the continuity requirements of
deflections, rotations and moments along the interface of the ith segment and the
(i+1)th segment, the following essential and natural boundary conditions must be
satisfied between the two segments.
0=iw (7.27a)
01 =+iw (7.27b)
01 =− +iiθθ φφ (7.27c)
0 1 =− +ir
ir φφ (7.27d)
01 =− +irr
irr MM (7.27e)
0 1 =− +ir
ir MM θθ (7.27f)
In views of Eqs. (7.14a-c), a homogeneous system of equations can be derived by
implementing the boundary conditions at the edges and the interface conditions
between the segments as follow:
Circular and Annular Plates
171
for annular plates with internal ring supports
[ ] 1)1(61)1(6)1(6)1(6 ×+×++×+ = mmmm 0ΦK (7.28a)
for circular plates with internal ring supports
[ ] 13)1(613)1(63)1(63)1(6 ×−+×−+−+×−+ = mmmm 0ΦK (7.28b)
The plastic buckling stress parameter λ can be determined by setting the determinant
of matrix [ ]K in Eq. (7.28a) or (7.28b) to be zero.
Results and Discussion
Tables 7.4 and 7.5 show the convergence studies of plastic buckling stress parameter
obtained by using the Ritz method for annular and circular plates with an internal ring
support and the results are compared against the analytical solutions. Note that the
bracketed values shown in the tables indicate the number of nodal diameter for the
critical buckling mode shape. It can be seen that the converged Ritz results agree well
with the analytical solutions. Although the formulations were presented to cater for
multiple supports, only the plastic buckling behaviour of circular and annular plates
with a single ring support will be studied in this section.
Circular and Annular Plates
172
Table 7.4 Convergence study of plastic buckling stress parameter λ for annular plates with one concentric ring support ( 6.0,2.0 1 == ζζ )
001.0=τ 05.0=τ 1.0=τ BC p
IT DT IT DT IT DT 4 8.6983 8.6983 3.2727 2.5727 3.0124 0.7067 6 6.5984 6.5984 2.5880 2.4789 2.0489 0.6911 8 6.5603 6.5603 2.5821 2.4768 2.0307 0.6907 10 6.5592 6.5592 2.5818 2.4766 2.0292 0.6907 12 6.5589 6.5589 2.5817 2.4766 2.0287 0.6907
S-S
Analytical Method
6.5586 (0)
6.5586 (0)
2.5815 (0)
2.4765 (0) 2.0275
(0) 0.6907
(0) 4 1.8805 1.8805 1.8410 1.8398 0.8849 0.6453 6 1.8695 1.8695 1.8312 1.8301 0.8823 0.6451 8 1.8681 1.8681 1.8299 1.8288 0.8822 0.6451 10 1.8679 1.8679 1.8296 1.8285 0.8821 0.6451
S-F
12 1.8678 1.8678 1.8295 1.8284 0.8821 0.6451
Analytical Method
1.8677 (0)
1.8677 (0)
1.8293 (0)
1.8282 (0) 0.8821
(0) 0.6451
(0) 4 13.4819 13.4819 4.4140 2.6430 __ 0.7182 6 12.7939 12.7939 4.2005 2.6326 3.6363 0.7165 8 12.7793 12.7793 4.1960 2.6323 3.6361 0.7165 10 12.7792 12.7792 4.1960 2.6323 3.6361 0.7165
C-C
12 12.7792 12.7792 4.1960 2.6323 3.6361 0.7165
Analytical Method
12.7792 (0)
12.7792 (0)
4.1960 (0)
2.6323 (0) 3.8040
(0) 0.7165
(1) 4 0.9893 0.9893 0.9675 0.9675 0.5693 0.5681 6 0.9886 0.9886 0.9666 0.9666 0.5692 0.5680 8 0.9874 0.9874 0.9655 0.9655 0.5691 0.5679 10 0.9872 0.9872 0.9652 0.9652 0.5691 0.5679
F-C
12 0.9871 0.9871 0.9651 0.9651 0.5691 0.5679
Analytical Method
0.9870 (0)
0.9870 (0)
0.9647 (0)
0.9647 (0) 0.5690
(0) 0.5678
(0)
Effect of the location of ring support
The variations of plastic buckling stress parameters with respect to ring support
radius are shown in Figs. 7.13 to 7.15 for circular plates and annular plates. It can be
seen that the critical buckling mode shape of circular and annular plates with an
internal ring support may not be the same as that of corresponding plates without
Circular and Annular Plates
173
internal ring supports. For example, in the presence of an internal ring support of
small radius, the critical buckling mode shape of a clamped circular plate change from
an axisymmetric buckling mode shape (without ring support) to an asymmetric mode
shape with one nodal diameter (with ring support). The hollow circles along the
buckling curves represent the ring support radius where the two buckling mode shapes
give the same buckling load. Therefore, these hollow circles indicate the change in the
critical buckling mode shape with respect to the ring support radius. Sample 3D
pictures of the critical buckling mode shapes are shown in Fig. 7.16 for C-C annular
plate of 2.0=ζ . Note that the simply supported edge becomes clamped edge when
the internal ring support is close to that edge. For example, when the internal ring
support is close to the inner edge of S-S annular plate, the buckling curve approaches
that of S-C annular plate.
Table 7.5 Convergence study of plastic buckling stress parameter λ for circular plates with one concentric ring support ( 6.01 =ζ )
001.0=τ 05.0=τ 1.0=τ BC p IT DT IT DT IT DT
4 2.7354 2.7354 2.1863 2.1801 0.8726 0.6488 6 2.6664 2.6664 2.1725 2.1669 0.8525 0.6474 8 2.6644 2.6644 2.1721 2.1664 0.8521 0.6474 10 2.6640 2.6640 2.1720 2.1664 0.8520 0.6474 12 2.6638 2.6638 2.1720 2.1663 0.8520 0.6474
S-S
Analytical Method
2.6635 (0)
2.6635 (0)
2.1719 (0)
2.1662 (0) 0.8519
(0) 0.6473
(0) 4 3.1804 3.1804 2.2572 2.2469 1.0358 0.6573 6 3.0483 3.0483 2.2385 2.2295 0.9949 0.6551 8 3.0406 3.0406 2.2373 2.2283 0.9922 0.6550 10 3.0390 3.0390 2.2370 2.2281 0.9916 0.6549
S-F
12 3.0385 3.0385 2.2369 2.2280 0.9914 0.6549
Analytical Method
3.0377 (0)
3.0377 (0)
2.2367 (0)
2.2278 (0) 0.9907
(0) 0.6549
(0)
Circular and Annular Plates
174
Optimal location of ring support
The presence of an internal ring support increases the value of plastic buckling
stress parameters. For example, the buckling stress parameter for a simply supported
circular plate increases from 0.4291 to 2.2192 by introducing a ring support. Of
interest in the design of supported circular plate is the optimal location of the internal
ring support for the maximum buckling load. The locations of optimal ring support are
denoted by black dots in the Figs. 7.13 to 7.15. It can be seen that these optimal
locations are influenced by the boundary conditions,ζ , τ and the type of plasticity
theory used.
However, the nature of calculation of optimal location of a ring support is iterative
and thus the calculation is computationally expensive. Chou at el. (1999) found that
the optimal locations of internal ring supports for vibration of circular plates are found
at the nodal circles of the corresponding internally unsupported plates. This optimality
condition is examined for plastic buckling of circular and annular Mindlin plates in
Table 7.6. It can be seen that the radii of nodal circles of the second buckling modes of
internally unsupported circular and annular plates are not exactly the same as but are
close to the corresponding iteratively computed optimal ring support radii. Therefore,
it can be concluded that the nodal circles of the second buckling modes of internally
unsupported circular and annular plates can be used as optimal location of internal
ring support in order to achieve the maximum buckling load in design. It is interesting
to note that all plates, except S-C annular plate, buckle in an axisymmetric mode
shape (i.e. n = 0) when the plates are supported by a ring support at the optimal
locations.
Circular and Annular Plates
175
Ring support radius Rb /11 =ζ
λ =
σhR
2 /π2 D
S-S
S-C
C-S
C-C
Fig. 7.13 Variation of buckling stress parameter λ with respect to ring support location 1ζ for annular plate ( ,2.0=ζ IT)
2
2.5
3
3.5
4
4.5
0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95
n = 2
n = 1
n = 0n = 1
n = 2
n = 2n =
1
n = 0 n = 1
n = 0n = 1 n = 2
n = 0 n = 1
Ring support locations Rb /11 =ζ
Fig. 7.14 Variation of buckling stress parameter λ with respect to ring support location 1ζ for annular plates ( ,2.0=ζ DT)
λ =
σhR
2 /π2 D
2
2.2
2.4
2.6
2.8
0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95
n = 2 n = 1 n = 0
n = 0
n = 1 n = 2
C-C
n = 2n = 1 n = 1n = 0
n = 2
n = 0 n = 1
C-S
S-C S-S
Circular and Annular Plates
176
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
0 0.2 0.4 0.6 0.8 1
λ =
σhR
2 /π2 D
Ring support locations Rb /11 =ζ
n =
1
n = 1
n = 0
n = 0Clamped
Simply supported
Fig. 7.15 Variation of buckling stress parameter λ with respect to ring support location 1ζ for circular plates
IT DT
Circular and Annular Plates
177
3.0/11 == Rbζ 5.0/11 == Rbζ
65.0/11 == Rbζ 7.0/11 == Rbζ
Fig.7.16 Critical Buckling mode shapes for C-C annular plates with various ring support locations (b/R = 0.2, DT, h/R = 0.05)
Circular and Annular Plates
178
Table 7.6 Comparison study between the optimal ring support radius and radius of nodal circle of second mode shape
Elastic IT DT
BC Optimal 1ζ
Nodal circle radius
Optimal 1ζ
Nodal circle radius
Optimal 1ζ
Nodal circle radius
Annular plates 2.0=ζ
S-S 0.6563 0.6547 0.7101 0.7064 0.6922 0.6902 C-C 0.6031 0.6011 0.6022 0.6016 0.6031 0.6015 S-C 0.7086 0.7238 0.7258 0.7406 0.7208 0.7340 C-S 0.5180 0.5174 0.5547 0.5544 0.5504 0.5445
Annular plates 5.0=ζ
S-S 0.7641 0.7625 0.7859 0.7841 0.7797 0.7782 C-C 0.7500 0.7496 0.7504 0.7499 0.7508 0.7499 S-C 0.8215 0.8230 0.8285 0.8298 0.8269 0.8280 C-S 0.6855 0.6853 0.6953 0.6949 0.6934 0.6926
Simply supported circular plates 0.4625 0.4618 0.4692 0.4699 0.4681 0.4687
Clamped circular plates 0.2672 0.2663 0.2688 0.2663 0.2703 0.2663
7.6 Concluding Remarks
Plastic buckling analyses of Mindlin circular and annular plates with and without
an internal ring support under a uniform stress were carried out by using the Ritz
method. The program code for the Ritz methods using the polar coordinate system are
written in MATHEMATICA and it is given in Appendix A. Moreover, the buckling
problems were also solved analytically. The matrix elements for the analytical method
are also provided in Appendix B for readers who want to use other programming
Circular and Annular Plates
179
languages so that programs can be easily obtained. The analytical solutions will be
useful to researchers who wish to verify their numerical methods and solutions.
The Ritz results and analytical results are found to agree well. Moreover, these
plastic buckling solutions are in good agreement with the existing elastic buckling
solution when the plate is thin. Generally, the plastic buckling stress parameters
obtained by using DT were consistently lower than their IT counterparts.
It was found that the locations of the nodal circles of the second axisymmetric
buckling modes of internally unsupported circular and annular plates are useful in
guiding designers for maximum buckling capacity in designing such plates with an
internal ring support.
180
PLATES WITH COMPLICATING EFFECTS
8.1 Introduction
This chapter considers Mindlin plates with complicating effects such as the presence
of internal line/curved supports, point supports, mixed boundary conditions, elastic
foundations, internal line hinges and intermediate in-plane compressive stress. So far,
the application of the Ritz method for plastic buckling analysis has been presented for
Mindlin plates with various edge supports conditions. In this chapter, we show how
the Ritz method may be used to accommodate the aforementioned complicating effects
in plastic buckling analysis. Although the treatments are presented in the rectangular
coordinate system only, one can readily recast the treatments in the polar coordinates
or the skew coordinates by using suitable coordinate transformation relations. For
numerical results, we adopt the aerospace material Al-7075-T6 with the properties:
CHAPTER 8
Plates with Complicating Effects
181
Poisson’s ratio 33.0=ν , 6105.10 ×=E psi, 707.0 =σ ksi and the Ramberg-Osgood
parameters 7/3=k , 2.9=c .
8.2 Internal line/curved/loop supports
Consider a plate with internal line/curved/loop supports as shown in Fig. 8.1. These
internal supports are assumed to impose a zero deflection (w = 0) along its length. This
addition geometrical constraint can be taken care of in the Ritz method by the
modification of the basic function for the transverse deflection as follows
[ ] ⎟⎟⎠
⎞⎜⎜⎝
⎛Ξ⎟⎟
⎠
⎞⎜⎜⎝
⎛Γ= ∏∏
=
Ωns
jj
ne
ii
wb
wj ),(),(
1
ηξηξφ (8.1)
where the underlined term is contributed by the presence of internal supports, in which
jΞ being the equation of the jth support and ns the number of these internal supports.
The rest of the formulations remain unchanged as presented in Chapter 4.
This method can handle any number of arbitrarily oriented/shaped internal supports
as long as the locations of these supports can be represented by analytical expressions
except for line/curved supports whose ends do not terminate at the plate boundary. The
line/curved supports whose ends do not terminate at the plate edges can be simulated
by a series of point constraints along the supports as shown in Fig. 8.2. The treatment
of these point supports in Ritz method will be discussed in Section 8.3.
Plates with Complicating Effects
182
For sample numerical results, consider a rectangular plate which is supported
internally by a rigid internal line support as shown in Fig. 8.4. The plate is subjected to
a uniaxial in-plane compressive stress xσ . For the considered internal line support, the
underlined term in Eq. 8.1 is given by
Fig. 8.1 Plates with internal (a) line, (b) curved and (c) loop supports
(a) (b) (c)
Line support Curve support Loop support
Fig. 8.2 Simulation of internal supports using point supports
Line support
Curve support Point supports to simulate internal support
Plates with Complicating Effects
183
∏∏ −=Ξns
jj
ns
jj )(),( αξηξ (8.2)
where jα is the coordinate of jth line support in ξ direction as shown in Fig. 8.3.
The convergence and comparison studies are shown in Table 8.1 and the some
sample critical buckling mode shapes are shown in Fig. 8.4 for simply supported
square plates with internal line supports. It can be seen that the satisfactory
convergence is achieved when the degree of polynomial p = 12 is used. It is found that
the converged results agree well with the analytical elastic buckling solutions derived
by Xiang (2003) when the plate is thin ( 001.0=τ ).
Internal line support
b
a xσ
xσ
ajα
Fig. 8.3 Rectangular plate with an internal line support
η
ξ
Plates with Complicating Effects
184
Table 8.1 Convergence and comparison studies of plastic buckling stress parameter λ for rectangular plate with internal line supports
001.0=τ 025.0=τ 05.0=τ
jα p IT DT IT DT IT DT
6 5.3217 5.3217 5.2703 5.2684 3.5287 2.8770 8 5.3177 5.3177 5.2664 5.2645 3.5287 2.8769 10 5.3174 5.3174 5.2661 5.2642 3.5287 2.8769 12 5.3168 5.3168 5.2655 5.2637 3.5287 2.8769
-0.6
Xiang (2003) 5.3165 5.3165 __ __ __ __
6 6.2501 6.2501 6.1340 6.1281 3.5470 2.9192 8 6.2499 6.2499 6.1338 6.1279 3.5469 2.9192 10 6.2499 6.2499 6.1338 6.1279 3.5469 2.9192 12 6.2499 6.2499 6.1338 6.1279 3.5469 2.9192
0
Xiang (2003) 6.2500 6.2500 __ __ __ __
6 8.6996 8.6996 7.9021 7.8658 4.1644 3.0765 8 8.6235 8.6235 7.8599 7.8249 4.1406 3.0721 10 8.6154 8.6154 7.8555 7.8206 4.1401 3.0718 12 8.6154 8.6154 7.8554 7.8205 4.1401 3.0718
-0.7,0.7
Xiang (2003) 8.6119 8.6119 __ __ __ __
6 10.7289 10.7289 8.7659 8.7033 4.4337 3.1495 8 10.3974 10.3974 8.6478 8.5901 4.3686 3.1366 10 10.3883 10.3883 8.6445 8.5869 4.3669 3.1363 12 10.3878 10.3878 8.6443 8.5867 4.3669 3.1363
-0.5,0.5
Xiang (2003) 10.386 10.386 __ __ __ __
Plates with Complicating Effects
185
8.3 Point supports
Consider a Mindlin plate with point supports which are located at the edges or inside
the plate. The transverse deflections are zero at the location of point supports, i.e.
pii niw ,....2,1 0),( ==ηξ (8.2)
where pn is the number of point supports and iξ and iη are coordinates of the ith
point support. These geometric constraints due to point supports can be taken into
Fig. 8.4 Critical buckling mode shapes of square plates with internal line support
6.01 −=α 01 =α
5.0,5.0 21 =−= αα 7.0,7.0 21 =−= αα
line support line support
line supports
Plates with Complicating Effects
186
account by several ways. The following are some common methods that are used to
simulate the point supports in the Ritz method.
(1) Lagrangian multiplier method
The point constraint may be incorporated into the total energy functional by using
Lagrangian multipliers as
∑=
++=Πpn
iiii wLVU
1),( ηξ (8.3)
where iL is the ith Lagrangian multiplier. This yields the eigenvalue equation as
0
000
=
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
Ledc
KKK
HKKK
ee
dedd
cecdcc
(8.4)
where the elements of the H submatrix are given by
pjjwiij njNih ,.....,2,1 ,,.......,2,1 ),( === ηξφ (8.5)
where N is the number of polynomial terms given by Eq. (3.14).
(2) Using constraint functions
This approach was first used by Liew et al. (1994) in order to solve the vibration
problems of Mindlin plate on point supports. In this approach, the constraint
conditions (Eq. 8.2) is used to reduce the dimensions of the eigenfunction while the
matrix remains positive definite. By substituting the coordinates of internal point
Plates with Complicating Effects
187
supports in to the approximate function for transverse deflection (Eq. 3.12a), the
constraint functions (Eq. 8.2) can be expressed as
[ ] 01 =×× pp nNn cC (8.6)
Assuming that pn is less than N, then the Eq. (8.6) can be rearranged as
[ ] [ ] cRRS =
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
=
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
+
+
N
n
n
n c
cc
c
cc
p
p
pˆ:
ˆˆ
ˆ:ˆˆ
2
1
2
1
(8.7)
where S is the positive definite matrix of pp nn × , R the matrix of )( pp nNn −× ,
ic ( i = 1 to N) the elements of c with the order rearranged by a position matrix P of
NN × with elements value 0 or 1 satisfying the relation cPc ˆ= . From Eq. (8.7), we
have
cIRS
c ⎥⎦
⎤⎢⎣
⎡=
−1
ˆ (8.8)
where I is the identity matrix of )()( pp nNnN −×− . The unknown coefficient c can
now be expressed in terms of modified unknown coefficient c as
cBcIRS
PcPc =⎥⎦
⎤⎢⎣
⎡==
−1
ˆ (8.9)
The substitution of Eq. (8.9) into the usual eigenvalue equation (Eq. 3.20) yields
Plates with Complicating Effects
188
[ ][ ] 0=⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
edc
BK (8.10)
where
[ ]⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
II
BB
000000
(8.11)
(3) Using elastic spring supports
The rigid point supports can be simulated by elastic springs with a sufficiently large
spring constant. The strain energy due to these springs is given by
[ ]∑=
=pn
kkkk wSQ
1
2),(21 ηξ (8.12)
where kS is the spring constant of the kth point support, which needs to be sufficiently
large number in order to simulate rigid point support.
The augmented total energy functional becomes
VQU ++=Π (8.13)
The minimization with respect to the Ritz coefficients yields
[ ] [ ]( ) 0=⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧+
edc
GK (8.14)
where the matrix [ ]G is contributed by the strain energy due to the springs and is
given by
Plates with Complicating Effects
189
[ ]⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
=I
IG
G00
0000cc
(8.15)
where
[ ] NjNiSpn
kkk
wjkk
wik
cc ,..,2,1 ,..,2,1 ),(),(1
=== ∑=
ηξφηξφG (8.16)
where I is the identity matrix of order N.
Note that the first approach increases the dimensions of the eigen matrix that results
in the increase of computational time whereas the second approach reduces the
dimensions of the matrix but it needs more effort in forming the matrix [ ]B . The last
approach is the simplest and easy to implement.
So far the treatment for point supports in the direction of transverse deflection is
presented and the treatments for the point supports of normal and tangential rotation
can also be done in the same way. It is to be remarked that the difficulty of
implementing the simple support (hard type) for oblique line and curved edges in
rectangular coordinates system may be overcome by using the combination of soft
type simple support (S*) and point constraints on the tangential rotation tφ on the
concerned edges as in Wang et al. (1995). Note that the mixed boundary conditions
can also be simulated by using suitable point restraints such as simple point supports
and clamped point supports along the boundary as in Liew and Wang (1992).
For sample buckling results, the uniaxially loaded square plate supported by four
symmetrically placed internal point supports as shown in Fig. 8.5 is considered. In
order to compare with the elastic buckling results of Venugopal et al. (1989),
Poisson’s ratio 3.0=ν is adopted. The convergence and comparison studies are
Plates with Complicating Effects
190
shown in Table 8.2 for squares plate resting on four internal point supports. It is found
that the converged results agree well with existing solutions. It can be seen that the
plastic buckling stress parameters obtained by IT agree well with their DT
counterparts for 05.0=τ . This is due to the less restrained boundary condtion.
Table 8.2 Convergence and comparison study of plastic buckling stress parameter λ for four points supported square plate
001.0=τ 05.0=τ
Uniaxial loading Biaxial loading α p Uniaxial loading
Biaxial loading IT DT IT DT
6 0.9221 0.7391 0.9157 0.9157 0.7351 0.7351 8 0.9217 0.7390 0.9152 0.9152 0.7350 0.7350 10 0.9217 0.7390 0.9151 0.9151 0.7349 0.7349 0.5
Venugopal (1989) 0.9217 0.7390 __ __ __ __
ξ
η
a point supports
point supports
α
Fig. 8.5 Four point supported square plate
Plates with Complicating Effects
191
8.4 Elastically restrained edges and mixed boundary conditions
The elastically restrained edge support can be modeled by combining three
different spring models whose stiffness are given by
wkS : spring stiffness of kth plate edge for the transverse elastic support,
xkS : spring stiffness kth plate edge for the rotational elastic support about y-axis and
ykS : spring stiffness kth plate edge for the rotational elastic support about x-axis.
Note that the arbitrary boundary conditions can be expressed by the combination of
free edges and aforementioned spring supports. For example, the clamped boundary
can be expressed by the combination of the free edge and the springs supports with the
very large values of yk
xk
wk SSS and , .
The potential energy stored in the elastic supports can be derived as
( ) ( ) k
ne
ky
ykx
xk
wk dSSwSQ
k
Γ++= ∑ ∫= Γ1
222
21 φφ (8.17)
where ne is the number of edge and kΓ refers to kth the plate boundary.
The augmented total potential energy becomes
VQU ++=Π (8.18)
The minimization of the energy functional yields the following augmented
eigenvalue equation
[ ] [ ]( ) 0=⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧+
edc
QK (8.19)
Plates with Complicating Effects
192
where
[ ]⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
=ee
dd
cc
000000
(8.20)
in which
[ ] NjNidS k
ne
k
wj
wi
wk
cc
k
,..,2,1 and ,..,2,1 ),(),(1
==Γ=∑ ∫= Γ
ηξφηξφQ (8.21a)
[ ] NjNidS k
ne
k
xj
xi
xk
dd
k
,..,2,1 and ,..,2,1 ),(),(1
==Γ=∑ ∫= Γ
ηξφηξφQ (8.21b)
[ ] NjNidS k
ne
k
yj
yi
yk
ee
k
,..,2,1 and ,..,2,1 ),(),(1
==Γ=∑ ∫= Γ
ηξφηξφQ (8.21c)
Note that the elements of the matrix [ ]K are the same as those given by Eqs.(3.232a-f)
in which the basic functions are chosen by assuming that all edges are free edges.
The aforementioned procedure can also handle mixed boundary conditions such as
those shown in Fig. 8.6 by separating the boundary integral Eq. (8.17) into several
parts according to the numbers of boundary types along the edge of the plate. For
example, the energy stored in the elastic support for the plate edge 1−=η (denoted as
edge no. 1) of case 1 in Fig. 8.6 is given by
[ ] ξηξφηξφη
dS wj
wi
wcc ∫−
−==
1
11111 ),(),(Q (8.22a)
[ ] ξηξφηξφη
dS xj
xi
xdd ∫−
−==
1
11111 ),(),(Q (8.22b)
[ ] ξηξφηξφα
ηdS y
jy
iyee ∫
−−=
=1
11121 ),(),(Q (8.22c)
Plates with Complicating Effects
193
Note that the first subscript of spring constant S refers to the edge number while the
second subscript refers to the segment number along the edge, for example, yS12 refers
to the rotational spring constant of second segment of the edge no. 1. It is to be
remarked that, since most of the edges are simply supported for cases shown in Fig.
(8.6), the mixed boundary condition can also be achieved by combining simply
supported edges and rotational spring about x-axis along the portion of the edge that
are clamped.
In order to verify the correctness of the method, comparison studies are shown in
Table 8.3 for three cases shown in Fig. 8.6. Note that 001.0=τ and 3.0=ν are
adopted in the analysis in order to compare with the existing elastic buckling results.
The present results are slightly higher than those of Mizusawa and Leonard (1990)
when the value of α is somewhere between 0 and 2. This may be due to the fact that
the boundary conditions are imposed by using the boundary integral in the present
Case 1 Case 2 Case 3
Fig.8.6 Square plates with mixed boundary conditions under uniaxial compression
a
α α α
a a
ξ
η
Plates with Complicating Effects
194
treatment while in spline element method (Mizusawa and Leonard, 1990) the
boundary condition are imposed only at the nodes.
Table 8.3 Comparison studies of buckling stress parameter λ for square plates with mixed boundary conditions
α Present (Ritz method)
Mizusawa and Leonard (1990) (Spline element method)
Case 1 0 4.0000 4.000 1/3 4.3573 4.209 1 5.4654 5.198 4/3 5.7059 5.628 2 5.7401 5.740 Case 2 0 4.0000 4.000 1/3 5.5240 5.265 1 5.7252 5.684 4/3 5.7388 5.732 2 5.7401 5.740 Case 3 0 4.0000 4.000 1/3 6.9300 6.524 1 7.6217 7.430 4/3 7.6837 7.640 2 7.6911 7.690
8.5 Elastic foundations
The elastic buckling of plates resting on elastic foundations has been studied by
many researchers (Naidu et al., 1990; Shen, 1999; Wang, 2005). This elastic
foundation effect can be taken into account in the proposed Ritz plastic buckling
analysis by augmenting strain energy given in Eq. (3.9) with FU
Plates with Complicating Effects
195
∫∫=A
kF dAwSU 2
21 (8.23)
where kS is the modulus of subgrade reaction for the foundation. The resulting
eigenvalue equation is given by:
0=⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡ +
edc
KKKKKKK
ee
dedd
cecdccF
cc
symmetric (8.24)
where the elements of ccFK are given by
[ ]∫∫=A
wj
wik
ccFij ddSk ηξφφ (8.25)
Note that, in above formulations, it is assumed that foundation reacts in
compression as well as in tension. However, in many practical applications, the
foundation cannot provide tensile reaction. For example, in the case of plates resting
on soils that lacking both adhesive and cohesive properties. Other applications of the
tensionless foundation are found in the delamination of laminated plates and
retrofitting metal plates which are attached to concrete or steel plates in composite
structures. In these cases, the foundations can be assumed as rigid foundation and the
buckling problem of plates in such cases is also known as unilateral buckling. The
buckling problem of these unilateral constrained plates is complicated because the
location and extent of the contact region are not known at the outset. In order to
demonstrate the ability of the Ritz plastic buckling analysis, unilateral buckling
problems of rectangular plates are considered in this study.
One of the ways of incorporating a unilateral constraint condition in the buckling
analysis is to model the foundation as a bed of springs that exhibits a deformation
Plates with Complicating Effects
196
sign-dependent relationship. Such behaviour of foundation can be modeled by
introducing a contact function X that is defined by
0=X (separation) (8.26a)
1=X (contact) (8.26b)
In applying contact condition, the plate area is discretized into a series of grids, and
the contact condition is assessed at the grid locations. Sufficiently large spring
stiffness is adopted to simulate a rigid surface. In this study, a grid spacing of 3030×
and the foundation stiffness of 104 are adopted. The matrix elements of ccFK given in
Eq. (8.25) become
∑=
=pN
k
wj
wik
ccFij XSk
1φφ (8.27)
where N is the number of grid points.
The nature of solution procedure is iterative since the final contact region between
the plate and the foundation surface is unknown before the solution is sought. In order
to define the contact region and sole the problem, the following iterative procedure is
adopted;
Step (1) Solve the eigenvalue equation by assuming that there is no contact between
plate and foundation.
Step (2) Check the contact status at the grid points. The elastic foundation is activated
or deactivated by using the contact function X and then solve the eigenvalue equation.
Step (3) Compare the eigenvalue with the previous iteration. If the convergence is
satisfactory, then stop and adopt the last eigenvalue as solution. If not, go to Step 2
and repeat.
Plates with Complicating Effects
197
Table 8.4 shows the comparison study between the present results ( 001.0=τ ) and
the existing elastic unilateral buckling solutions. It can be seen that the results agree
well.
Table 8.4 Comparison study of unilateral buckling stress parameter λ
for rectangular plates under shear Unilateral Buckling
BC Aspect ratio a/b
Bilateral Buckling Present
Method Smith et
al. (1999)
Influence of unilateral buckling
(%)
1 9.325 9.355 9.355 0.32 2 6.546 6.719 6.720 2.64 SSSS
3 5.840 6.683 6.689 14.43
1 14.642 15.242 15.26 4.10 2 10.248 12.025 12.03 17.34 CCCC
3 9.535 12.269 12.21 28.67
Tables 8.5 and 8.6 show the plastic buckling stress parameters for simply supported
and clamped rectangular plates under pure shear. Both bilateral and unilateral buckling
stress parameters are presented and denoted as bilλ and uniλ respectively. The increase
in the buckling stress parameter due to the presence of rigid restraint is also shown in
the percentage of bilλ . It is found that the difference between bilλ and uniλ increases as
the aspect ratio increases. This is because, at larger aspect ratios, the plate buckles in
multiple half-wavelengths in bilateral buckling case and as a result the contact area
between the plate and rigid foundation increases. Typical buckling mode shapes for
bilateral and unilateral buckling cases are presented in Fig. 8.7 for the clamped
rectangular plate.
Plates with Complicating Effects
198
Table 8.5 Unilateral plastic buckling stress parameters for rectangular plates
under shear (IT)
025.0=τ 05.0=τ BC a/b
bilλ uniλ %diff bilλ uniλ %diff
1 6.782 6.787 0.07 6.139 6.145 0.10 2 5.595 5.792 3.52 4.594 5.043 9.77 SSSS
3 5.151 5.563 8.00 4.275 5.591 30.78
1 11.177 11.582 3.62 10.554 10.967 3.91 2 8.059 9.595 19.06 7.618 9.134 19.90 CCCC
3 7.604 9.796 28.83 7.151 9.344 30.67
Table 8.6 Unilateral plastic buckling stress parameters for rectangular plates under shear (DT)
025.0=τ 05.0=τ
BC a/b bilλ uniλ %diff bilλ uniλ %diff
1 6.201 6.205 0.06 2.090 2.091 0.05 2 5.432 5.555 2.26 1.991 2.016 1.26 SSSS
3 5.151 5.563 8.00 1.964 2.056 4.68
1 7.189 7.251 0.86 2.236 2.248 0.54 2 6.562 6.903 5.20 2.138 2.192 2.53 CCCC
3 6.421 6.943 8.13 2.119 2.199 3.78
Plates with Complicating Effects
199
8.6 Internal line hinges
The internal hinges may be used in the modeling of folding gates and hatches. In
order to cater the slope discontinuity along the hinge line, the domain decomposition
method can be employed. In the domain decomposition method, the plate is divided
Bilateral (a/b = 2) Unilateral (a/b = 2)
Bilateral (a/b = 3) Unilateral (a/b = 3)
Fig. 8.7 Typical buckling mode shapes for bilateral and unilateral buckling of clamped rectangular plates
Bilateral (a/b = 1) Unilateral (a/b = 1)
Plates with Complicating Effects
200
into several subdomains along the hinges depending on the number of hinges. In order
to illustrate the treatment of internal line hinge in the Ritz method, a rectangular plate
with one internal hinge is considered as shown in Fig. 8.8. The adopted admissible p-
Ritz functions for the deflection and rotations of the subdomain 1 are given by
( ) ),(, 110 0
1111 ηξφηξ w
m
p
q
q
imcw ∑∑
= =
= (8.28a)
( ) ),(, 110 0
1111 ηξφηξφ x
m
p
q
q
imx d∑∑
= =
= (8.28b)
( ) ),(, 110 0
1111 ηξφηξφ y
m
p
q
q
imy e∑∑
= =
= (8.28c)
and the similar functions for subdomain 2 are given by
( ) ),(, 220 0
2222 ηξφηξ w
m
p
q
q
imcw ∑∑
= =
= (8.29a)
( ) ),(, 220 0
2222 ηξφηξφ x
m
p
q
q
imx d∑∑
= =
= (8.29b)
( ) ),(, 220 0
2222 ηξφηξφ y
m
p
q
q
imy e∑∑
= =
= (8.29c)
in which 11,ηξ and 22 ,ηξ are the normalized coordinate systems, which have the
integration limits from -1 to 1, for each subdomain.
Along the internal hinge, the compatibility conditions for the transverse deflections
are given by
),1(),1( 2211 ηη −= ww (8.30)
Plates with Complicating Effects
201
In view of Eq. (8.30), the Ritz approximate function for transverse deflection for
subdomain 2 can be redefined as follows (Su and Xiang, 1999):
),()1(),1(),(
),1(),( 22222
221222 ηξξ
ηφηξφ
ηηξ www wb
wb −+−
= (8.31)
The energy functional for each domain can be expressed as in Eq. (3.11) by
substituting corresponding displacement fields, i.e. 111 ,, yxw φφ and 222 ,, yxw φφ . The
corresponding eigenvalue equation is given by
0=⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
edc
KKKKKK
ee
dedd
cecdcc
symmetric (8.32)
in which
aα a)1( α−
b xσ
xy σβσ =
1ξ
1η 2η
2ξ
Fig. 8.8 A rectangular plate with an internal hinge under in-plane compressive stress
internal line hinge
Plates with Complicating Effects
202
[ ] NNcc
NN
cccc
22222
1
000
×
×
+⎥⎦
⎤⎢⎣
⎡= K
KK (8.33)
[ ] NNcd
NN
cdcd
22222
1 000
0 ×
×
+⎥⎦
⎤⎢⎣
⎡= K
KK (8.34)
[ ] NNce
NN
cece
22222
1 000
0 ×
×
+⎥⎦
⎤⎢⎣
⎡= K
KK (8.35)
NNdd
dddd
222
1
00
×
⎥⎦
⎤⎢⎣
⎡=
KK
K (8.36)
NNde
dede
222
1
00
×
⎥⎦
⎤⎢⎣
⎡=
KK
K (8.37)
NNee
eeee
222
1
00
×
⎥⎦
⎤⎢⎣
⎡=
KK
K (8.38)
Note that the expressions for eei
ddi
cei
cdi
cci KKKKK and ,,, (i = 1, 2) are the same as
those in the usual formulation for rectangular plates which are given by Eqs. (3.23a-f),
except the fact that the corresponding Ritz functions need to be adopted accordingly,
i.e. Eqs.(8.28a-c) for subdomain 1, Eqs.(8.29b-c) and Eq.(8.31) for subdomain 2.
For sample results, a square plate with an internal hinge under uniaxial loading is
considered. Table 8.6 shows a comparison study against existing elastic buckling
results. Note that, in order to compare with existing elastic buckling results, the
thickness-to-width ratio τ and Poisson’s ratio ν are set to be 0.001 and 0.3,
respectively. It can be seen that the results agree well with the existing solutions.
Plates with Complicating Effects
203
Table 8.7 Buckling stress parameter λ for square plate with an internal hinge under uniaxial loading
α
p 0.1 0.2 0.3 0.4 0.5
4 1.5773 1.7481 1.8976 2.0042 2.0436 6 1.5764 1.7473 1.8969 2.0035 2.0429 8 1.5761 1.7472 1.8968 2.0035 2.0428 SSSS
Xiang et al. (2001) 1.5770 1.7474 1.8969 2.0035 2.0429
4 8.3647 6.1853 4.6843 4.3519 4.3432 6 7.9073 6.0684 4.6281 4.2919 4.2486 8 7.8868 6.0582 4.6166 4.2790 4.2366 10 7.8858 6.0556 4.6125 4.2749 4.2330
CCCC
Wang et al. (2001b) 7.8857 6.0545 4.6108 4.2727 4.2306
8.7 Intermediate in-plane loads
Consider a Mindlin plate which is subjected to an end compressive stress 1σ and an
intermediate compressive stress 2σ as shown in Fig. 8.8. Note that a positive value of
σ implies a compressive load while negative value implies a tensile uniaxial load.
Since all the parameters G,,, γβα depend on the stress level, the plate needs to be
divided into two domains along the location of intermediate load in order to handle the
different levels of stresses. The first domain is to the left of the vertical line defined by
ax α= (see Fig. 8.8) and the second domain is to the right of this line. The Ritz
approximate functions for deflection and rotations can be adopted as usual for the
whole plate. The resulting eigenvalue equation is given by
Plates with Complicating Effects
204
0
2
22
222
1
11
111
=⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
+⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
edc
KKKKKK
KKKKKK
ee
dedd
cecdcc
ee
dedd
cecdcc
symmetricsymmetric (8.39)
where subscript 1 and 2 refer to the domain 1 and 2 respectively.
The sample results of )/( 2211 Dhb πσλ = are shown in Table 8.7 for a rectangular
plate of 04.0=τ and for a given value of 50)/()( 22212 −=+= Dhb πσσλ . The same
material properties 20 ,3485.0 ksi, 4.61 ksi, 10700 0 ==== ckE σ are adopted as in
Wang et al. (2004) for comparison purpose. It can be seen that the present results are
found to be slightly lower than existing results. This is due to the effect of transverse
shear deformation which is neglected in Wang et al. (2004).
1σ 2σ
21 σσ +
x
y
b
aα a)1( α−
Fig. 8.9 Rectangular plate under edge and intermediate compressive stress
Plates with Complicating Effects
205
Table 8.8 Convergence and comparison of plastic buckling stress parameters 1λ of rectangular plates under edge and intermediate loads. ( 04.0,502 =−= τλ )
a/b = 1 a/b = 2 a/b = 3 BC p
IT DT IT DT IT DT 8 4.4568 3.9141 3.8084 3.6628 3.7939 3.6577 10 4.4010 3.9021 3.7869 3.6566 3.7528 3.6389 12 4.3769 __ 3.7809 __ 3.7361 __ 14 4.3769 __ 3.7809 __ 3.7361 __
SSSS
Wang et al. (2004) 4.470 4.002 3.799 3.712 3.738 3.674
8 6.3389 4.1166 4.2094 3.8374 4.3579 3.7954 10 6.0809 4.0946 4.1114 3.8150 4.0434 3.7744 12 5.9898 4.0847 4.0871 3.8074 3.9486 3.7609 14 5.9458 __ 4.0777 __ 3.9195 __
CSCS
Wang et al. (2004) 6.214 4.198 4.108 3.878 3.917 3.802
8 2.3641 2.2225 2.2454 2.1438 2.2467 2.2146 10 2.3322 2.1872 2.2333 2.1299 2.2237 2.1995 12 2.3182 __ 2.2194 __ 2.2133 __ 14 2.3092 __ 2.2105 __ 2.2028 __
FSFS
Wang et al. (2004) 2.451 2.451 2.289 2.289 2.266 2.266
8.8 Concluding Remarks
This chapter demonstrates the treatments for the complicating effects such as the
presence of internal line/point/curved/loop supports, elastically restrained edges,
mixed boundary conditions, elastic foundations, internal line hinges, intermediate in-
plane loads when using the Ritz method for plastic buckling analysis. The various
treatments are demonstrated by solving sample problems and comparing the results
with existing elastic buckling solutions. It can be seen that the Ritz method is very
Plates with Complicating Effects
206
versatile and powerful to handle all the aforementioned complicating effects for plastic
buckling analysis of Mindlin plates.
207
CONCLUSIONS AND RECOMMENDATIONS
9.1 Conclusions
An automated Ritz method is presented for the plastic buckling analysis of Mindlin
plates of various shapes, boundary and loading conditions. The effect of transverse
shear deformation was taken in to account by adopting the Mindlin plate theory. In the
Ritz method, the assumed displacement functions were defined by the product of basic
functions and mathematically complete two-dimensional polynomial functions whose
degree may be increased until the desired accuracy is achieved. The basic functions
are defined by the product of boundary equations raised to the appropriate power
(either 0 or 1) that ensure automatic satisfaction of the geometric boundary conditions.
As it is more convenient to handle certain plates in their natural coordinate systems,
three versions of the Ritz software codes are given. The software code BUCKRITZRE
CHAPTER 9
Conclusions and Recommendations
208
is based on the Cartesian coordinate system and it can be used for determining the
plastic buckling stress of plates of any shape whose edges are defined by polynomial
functions. Another software code, called BUCKRITZSKEW, is written based on the
skew coordinate system and it is to compute the plastic buckling stress of skew plates.
The last software code, called BUCKRITZPOLAR, is to solve the plastic buckling
problems of circular and annular plates and it is based on the polar coordinate system.
It is worth noting that the presented Ritz program codes can be used to study not
only the plastic buckling of plates but also the elastic buckling of plates. One can
calculate the elastic buckling stress parameter based on the classical thin plate theory
by setting the thickness-to-width ratio to a small value. If one wish to calculate the
elastic buckling stress parameter based on Mindlin plate theory, it can be easily done
by setting the tangent modulus and secant modulus equal to Young’s modulus.
The capability of the Rayleigh-Ritz method was demonstrated by solving the plastic
buckling problems of rectangular, skew, triangular, elliptical, circular and annular
plates. In addition to conventional loading and boundary conditions, various
treatments were presented for the presence of complicating effects such as internal
line/curved/point supports, internal line hinges, intermediate in-plane loads and elastic
foundation. In all these situations, it has been shown that Ritz method is simple and
easy to formulate since it does not require mesh discretization of the plate. Being an
approximate continuum method, it eliminates the need for mesh generation and thus
large numbers of degrees of freedom. Therefore it requires less memory and is easy to
implement when compared with the finite element method. Since the eigen problem
needs to be solved in an incremental and iterative fashion in plastic buckling problems,
the computational superiority of the Ritz method is even more convincing.
Conclusions and Recommendations
209
The plastic buckling stress parameters for various shapes such as rectangular, skew,
triangular, elliptical, circular and annular Mindlin plates were calculated for various
aspect ratios, thickness-to-width ratios and boundary conditions. So far, most of the
plastic buckling solutions available focus mainly on the rectangular plates. Therefore,
the extensive plastic buckling data presented for various plate shapes will be useful as
a useful reference source for designers and researchers who wish to verify their plastic
buckling results.
It was found that the plastic buckling stress parameters approach their elastic
counterparts when the plate is thin or the plate is supported by less restrained boundary
conditions. But for plates with larger thickness-to-width ratios and more restrained
boundary conditions, the plastic buckling parameters differ appreciably from their
elastic counterparts. Generally, the plastic buckling results obtained by IT are larger
than their DT counterparts and the difference increases with increasing thickness-to-
width ratios.
As expected, the effect of transverse deformation lowers the plastic buckling stress
parameters. The effect is more pronounced in the cases of thicker plates and plates
with more constrained boundary conditions. Generally, the reductions in the plastic
buckling stress parameters based on IT are larger than their DT counterparts.
Owing to the lack of analytical buckling solutions for asymmetric buckling
problems of circular and annular Mindlin plates, an analytical method was featured in
this thesis for the first time. By following the procedure used by Mindlin (1951), the
governing equations of the plates were transformed into the standard forms of partial
differential equations in terms of three potential functions. These transformed
governing equations were solved analytically and the exact plastic buckling solutions
were obtained. The validity of the results was confirmed by comparing with existing
Conclusions and Recommendations
210
analytical elastic buckling thin plate results (by setting the plate thickness to very
small value) and with plastic buckling results obtained by the Ritz method. It was
found that the results agree well. However, this analytical method is not applicable for
the cases of non-uniform stress distributions. If the variations of stresses are
considered, the general solutions of governing equations are not possible due to the
complexity arising from the variations of stresses. Nonetheless, since the buckling
problems of circular and annular plate under uniform compression are fundamental in
structural analysis, the developed analytical method will be useful for researchers who
wish to check the validity and convergence of their numerical methods.
9.2 Recommendations
There are some interesting directions for future works in the area of research presented
in this thesis.
One of the interesting areas is to combine the finite element analysis with the Ritz
method. The major set back in the Ritz method is that it requires prebuckling stress
distribution. In the case of plastic buckling analysis, it is hard to obtain the analytical
expression for stress distribution. The assumption of uniform stress distribution, i.e. no
stress dissipation in the x-y plane, was adopted in the present study. Although the
effect of stress dissipation is very marginal in the cases of uniform loadings, the effect
is more pronounced in the cases of plates under patch loading i.e. locally distributed
loading and point loading (Liu and Pavlovic, 2005) and plates with holes (Uenoya and
Redwood, 1978). Uenoya and Redwood (1978) combined the Ritz method and the
finite element method in their elasto-plastic bifurcation analysis of perforated plates,
with the Ritz method being used for the bifurcation analysis and the finite element
Conclusions and Recommendations
211
being used for determining in-plane stress. However, their study is confined to the
classical thin plate theory (Kirchhoff) and the rectangular plates with simply supported
or clamped boundary conditions. In future, it is desirable to extend the Ritz plastic
buckling analysis of Mindlin plate of various shapes and boundary conditions.
Another possible area for future work is the extension of presented Ritz method to
postbuckling analysis and allowing for the effect of imperfections. This study mainly
focused on the plastic bifurcation analysis i.e. small deflection theory was adopted.
Therefore, the effect of initial imperfections cannot be taken into consideration in the
present study. In order to include the effect of imperfections, a postbuckling analysis
(with large deflection theory) needs to be carried out. The applicability of the Ritz
method to the postbuckling analysis was shown in Jane et al. (2003). Therefore, the
application of the Ritz method to the plastic postbuckling analysis of Mindlin plates,
with the consideration of effect of imperfections, should be made in future.
212
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Appendix A: Ritz Program Codes
This Appendix is prepared in MATHEMATICA since it is more convenient to
print the program code directly from MATHEMATICA than copying to word file.
In MATHEMATICA, comments on the code can be written in (*.....*) or seperate
cells. In oder to explain the program code properly, comments are written in seper-
ate cells as a text.
A.1 Ritz Program Code with Catesian Coordinate System
à BUCKRITZRE
This program is based on the Catesian coordinate formulations of Ritz method
and is to solve the plastic buckling problems of Mindlin plates under any combi-
nations of in-plane and shear stresses with various boundary conditions.
† Firstly basic functions need to be defined and the sample basic functions for
a SSSS rectangular plate are given by
fw1 = Hx + 1L Hh + 1L Hx - 1L Hh - 1L;fx1 = Hx + 1L0 Hh + 1L Hx - 1L0 Hh - 1L;fy1 = Hx + 1L Hh + 1L0 Hx - 1L Hh - 1L0;
222
† The mathematically complete polynomial functions are formed as a list.
Hence the displacement functions w, fx, fy are obtained by multiplying
the corresponding basic function with polynomial terms.
p = 6;H∗ p is the degree of polynomial equation ∗LNmax =
H# + 1L H# + 2L2
&@pD;
polyfunction = Flatten@Reverse ê@Table@ξi ηq−i , 8q, 0, p<, 8i, 0, q<DD;
w = polyfunction@@#DD φw1 &;φx = polyfunction@@#DD φx1 &;φy = polyfunction@@#DD φy1 &;
† The integration of the matrix ·A
i
k
jjjjjjjjjjjjjjj
1 hx xh
∫ hN-1
∫ xhN-1
ª ª
xN-1 xN-1 h
∏ ª
∫ xN-1 hN-1
y
zzzzzzzzzzzzzzz „ A are
carried out over the plate domain for one time only and store it in a variable
named inte as an array. Then a user defined integration function is written
in such a way that the exponents of x and h in all the integrands are tested
and use the coorsponding integrated values in inte. It is more computation-
ally efficient since the integration of polynomial are repeatedly involved in
the calculation.
Appendix A
223
Table@ξi ηq , 8i, 0, 2 p + 5<, 8q, 0, 2 p + 5<D;
inte = ‡−1
1
‡−1
1
# ξ η & ê@ %;
co =Coefficient@Coefficient@#, ξ, Exponent@#, ξDD,
η, Exponent@#, ηDD &;H∗co is the function to get coefficient
of the polynomial function∗Lpreintefunct = If@TrueQ@# 0D, 0, co@#D Part@inte,
Exponent@#, ξD + 1, Exponent@#, ηD + 1DD &;intefunction = If@TrueQ@Head@#D PlusD,
Map@preintefunct, #D, preintefunct@#DD &;
† The matrix elements are formed according to the Eqs. (3.23a-f)
Table@∂ξ w@iD ∂ξ w@jD, 8i, 1, Nmax<, 8j, 1, Nmax<D;Expand ê@ %;kcc1 = Map@intefunction, %, 82<D;
Table@∂η w@iD ∂η w@jD, 8i, 1, Nmax<, 8j, 1, Nmax<D;Expand ê@ %;kcc2 = Map@intefunction, %, 82<D;
Table@∂ξ w@iD ∂η w@jD, 8i, 1, Nmax<, 8j, 1, Nmax<D;Expand ê@ %;kcc3 = Map@intefunction, %, 82<D;
Table@∂η w@iD ∂ξ w@jD, 8i, 1, Nmax<, 8j, 1, Nmax<D;Expand ê@ %;kcc4 = Map@intefunction, %, 82<D;
Table@∂ξ w@iD φx@jD, 8i, 1, Nmax<, 8j, 1, Nmax<D;Expand ê@ %;kcd1 = Map@intefunction, %, 82<D;
Table@∂η w@iD φy@jD, 8i, 1, Nmax<, 8j, 1, Nmax<D;Expand ê@ %;kce1 = Map@intefunction, %, 82<D;
Table@∂ξ φx@iD ∂ξ φx@jD,8i, 1, Nmax<, 8j, 1, Nmax<D;
Expand ê@ %;kdd1 = Map@intefunction, %, 82<D;
Appendix A
224
Table@ ∂η φx@iD ∂η φx@jD,8i, 1, Nmax<, 8j, 1, Nmax<D;
Expand ê@ %;kdd2 = Map@intefunction, %, 82<D;
Table@φx@iD φx@jD, 8i, 1, Nmax<, 8j, 1, Nmax<D;Expand ê@ %;kdd3 = Map@intefunction, %, 82<D;
Table@∂ξ φx@iD ∂η φx@jD,8i, 1, Nmax<, 8j, 1, Nmax<D;
Expand ê@ %;kdd4 = Map@intefunction, %, 82<D;
Table@∂η φx@iD ∂ξ φx@jD,8i, 1, Nmax<, 8j, 1, Nmax<D;
Expand ê@ %;kdd5 = Map@intefunction, %, 82<D;
Table@ ∂ξ φx@iD ∂η φy@jD,8i, 1, Nmax<, 8j, 1, Nmax<D;
Expand ê@ %;kde1 = Map@intefunction, %, 82<D;
Table@ ∂η φx@iD ∂ξ φy@jD,8i, 1, Nmax<, 8j, 1, Nmax<D;
Expand ê@ %;kde2 = Map@intefunction, %, 82<D;
Table@ ∂η φx@iD ∂η φy@jD,8i, 1, Nmax<, 8j, 1, Nmax<D;
Expand ê@ %;kde3 = Map@intefunction, %, 82<D;
Table@ ∂ξ φx@iD ∂ξ φy@jD,8i, 1, Nmax<, 8j, 1, Nmax<D;
Expand ê@ %;kde4 = Map@intefunction, %, 82<D;
Table@∂η φy@iD ∂η φy@jD,8i, 1, Nmax<, 8j, 1, Nmax<D;
Expand ê@ %;kee1 = Map@intefunction, %, 82<D;
Table@∂ξ φy@iD ∂ξ φy@jD,8i, 1, Nmax<, 8j, 1, Nmax<D;
Expand ê@ %;kee2 = Map@intefunction, %, 82<D;
Appendix A
225
Table@∂ξ φy@iD ∂η φy@jD,8i, 1, Nmax<, 8j, 1, Nmax<D;
Expand ê@ %;kee3 = Map@intefunction, %, 82<D;
Table@∂η φy@iD ∂ξ φy@jD,8i, 1, Nmax<, 8j, 1, Nmax<D;
Expand ê@ %;kee4 = Map@intefunction, %, 82<D;
Table@φy@iD φy@jD, 8i, 1, Nmax<, 8j, 1, Nmax<D;Expand ê@ %;kee5 = Map@intefunction, %, 82<D;
† The stiffness matrix can be obtained as a function of lx, ly and lxy. The
algorithm shown here is for the deformation theory of plasticity.
<< LinearAlgebra`MatrixManipulation`
stiffness@λx_, λy_, λxy_D := ModuleA8t, s, c11, c12, c13, c22, c23, c33, c44<,
λe ="#####################################################
λx2 − λx λy + λy2 + 3 λxy2 ;ly = 1;ζ = lxêly;τ = hêly;
t = 1 ì ikjjjj1 + c k i
kjj 1
12 H1 − ν2L Hλe e τ2 Lyzz
c−1yzzzz;
s = 1 ìi
kjjjjj1 + k
ikjjjj
λe e τ2
12 H1 − ν2Lyzzzz
c−1y
zzzzz;
c11 = 1 − 3 H1 − têsL ikjjjj
λy2
4 λe2+
λxy2
λe2
yzzzz;
c12 = −1
2ikjjjj1 − H1 − 2 νL t − 3 H1 − têsL
ikjjjj
λx λy
2 λe2+
λxy2
λe2
yzzzzyzzzz;
c13 =3
2H1 − têsL i
kjj 2 λx − λy
λeyzz
λxy
λe;
c22 = 1 − 3 H1 − têsL ikjjjj
λx2
4 λe2+
λxy2
λe2
yzzzz;
c23 =3
2H1 − têsL i
kjj 2 λy − λx
λeyzz
λxy
λe;
Appendix A
226
k c33 = 3 t ês − H1 − 2 νL t + 9 H1 − têsL
λxy2
λe2;
c44 =1
κ H3 t ês − H1 − 2 νL tL;
α = c22 c33 − c232;β = c13 c23 − c12 c33;χ = c12 c23 − c13 c22;γ = c11 c33 − c132;µ = c12 c13 − c11 c23;δ = c11 c22 − c122;ρ = 1êt Det@88c11, c12, c13<,
8c12, c22, c23<, 8c13, c23, c33<<D;
kc =t
c44 h2ikjj 1
ζkcc1 + ζ kcc2y
zz;
gcc =1
12 H1 − ν2L ly2 ikjj 1
ζ
λx
λkcc1 +
λy
λζ kcc2 +
λxy
λHkcc3 + kcc4Ly
zz;
kcc = kc;
kcd =ly t
2 c44 h2kcd1;
kce =ly t
2 c44 h2ζ kce1;
kdd =α
12 ρ
1
ζkdd1 +
δ
12 ρζ kdd2 +
t
4 c44 τ2ζ kdd3 +
χ
12 ρkdd4 +
χ
12 ρkdd5;
kde =β
12 ρkde1 +
δ
12 ρkde2 +
µ
12 ρζ kde3 +
χ
12 ρ
1
ζkde4;
kee =γ
12 ρζ kee1 +
δ
12 ρ
1
ζkee2 +
µ
12 ρkee3 +
µ
12 ρkee4 +
t
4 c44 τ2ζ kee5;
rowc = AppendRows@kcc, kcd, kceD;rowd = AppendRows@Transpose@kcdD, kdd, kdeD;rowe = AppendRows@
Transpose@kceD, Transpose@kdeD, keeD;mat = Map@Chop, AppendColumns@
rowc, rowd, roweD, 82<D;zeromat = ZeroMatrix@NmaxD;matkm = BlockMatrix@88gcc, zeromat, zeromat<,
8zeromat, zeromat, zeromat<,8zeromat, zeromat, zeromat<<D;
Return@mat − λ matkmD;E;
Appendix A
227
† The input data for material properties, geometrical properties and loading
conditions need to be defined here (sample input data are for AL 7075 T6
aluminum alloy). Then an angorithm (the detailed procedure was discussed
in Chapter 3) is written in oder to determine the plastic buckling stress
parameter l. Since, the stiffness matrix was formed without having input
data, the analysis can be perform for another material properties or aspect
ratio without having to recalculate stiffness matrix.
λx := λ;λy := 0;λxy := 0;
κ = 5ê6;ν = 0.33;e = 150; H∗e means Eêσ0∗Lk = 3ê7;c = 9.2;h = 0.05;lx = 1;
BucklingLoadFinder@λ0_D :=Module@8n, num, W, W1, Wmid, d<,
λ = λ0; d = 5;Do@
Do@n = Eigenvalues@
Chop@stiffness@λx, λy, λxyD êê NDD;num = Length@Select@n, Sign@#D −1 ⅅIf@num > 0, Print@num,
" buckling load is less than ",λ êê ND; Break@D, λ = λ + d
D, 8i, 50<D;If@num > 1, λ = λ − d;
d = d ê2; λ = λ + d; Print@λD, Break@DD, 8a, 10<D;If@num < 1,
Print@"Initial Value of λ is too small"D;Abort@D
Appendix A
228
D;
W2 = Det@stiffness@λx, λy, λxyDD;λ = λ − d;
W1 = Det@stiffness@λx, λy, λxyDD;
If@Sign@W1D Sign@W2D,Print@"Limit is not OK"D;Abort@D, Print@"Limit is OK"D
D;Do@d = d ê2; λmid = λ + d;
If@d < 0.00005, Break@DD;λ = λmid;Wmid = Det@stiffness@λx, λy, λxyDD;Print@"λ=", " ", SetPrecision@λmid, 8D,
"W1=", " ", W1, "Wmid=", " ", WmidD;If@Sign@W1D ≠ Sign@WmidD, λ = λmid − dD,8i, 40<D;
Print@"h=", h, " ", "c=",c, " ", "λ=", " ", λmid êPi2D
D;
H∗λ0 is the initial value of λ∗LH∗The following command will
load the algorithm and determine λ∗L
Appendix A
229
† The buckling mode shape can be ploted as follow:
Clear@λD;λ = λmid;eigen = Eigenvalues@
8Chop@mat êê ND, Chop@matkm êê ND<D;eigenew = If@# ≤ 0, ∞, #D & ê@ eigen;po = Ordering@eigenew, 1D;Print@"λ=", eigenew@@po@@1DDDDD;v = Eigenvectors@
8Chop@mat êê ND, Chop@matkm êê ND<D;Co = v@@po@@1DDDD;defl = HTable@Co@@iDD polyfunction@@iDD φw1,
8i, Nmax<DL ê. List → Plus;Plot3D@defl, 8ξ, −1, 1<, 8η, −1, 1<, Axes → False,
BoxRatios −> 8lx, 1, 0.3<, PlotRange → AllD
Clear@"Global`∗"D
A.2 Ritz Program Code with Skew Coordinate System
à BUCKRITZSKEW
This program is based on the skew coordinate formulations of Ritz method and
is to solve the plastic buckling problems of skew Mindlin plates under any
combinations of in-plane and shear stresses with various boundary conditions.
The choice of basic functions and the use of integration function are similar to
those in BUCKRITZRE.
† Firstly basic functions need to be defined and the sample basic functions for
a SSSS skew plate are given by
Appendix A
230
fw1 = Hx + 1L Hh + 1L Hx - 1L Hh - 1L;fx1 = Hx + 1L0 Hh + 1L Hx - 1L0 Hh - 1L;fy1 = Hx + 1L Hh + 1L0 Hx - 1L Hh - 1L0;
† The mathematically complete polynomial functions are formed as a list.
Hence the displacement functions w, fx, fy are obtained by multiplying the
corresponding basic function with polynomial terms.
p = 6;H∗ p is the degree of polynomial equation ∗LNmax =
H# + 1L H# + 2L2
&@pD;
polyfunction = Flatten@Reverse ê@Table@ξi ηq−i , 8q, 0, p<, 8i, 0, q<DD;
w = polyfunction@@#DD φw1 &;φx = polyfunction@@#DD φx1 &;φy = polyfunction@@#DD φy1 &;
† The integration of the matrix ·A
i
k
jjjjjjjjjjjjjjj
1 hx xh
∫ hN-1
∫ xhN-1
ª ª
xN-1 xN-1 h
∏ ª
∫ xN-1 hN-1
y
zzzzzzzzzzzzzzz „ A are
carried out over the plate domain for one time only. Then an integration
function is written in such a way that the exponents of x and h in all the
integrands are tested and use the coorsponding integrated values. It is more
computationally efficient since the integration of polynomial are repeatedly
involved in the calculation.
Appendix A
231
Table@ξi ηq , 8i, 0, 2 p + 5<, 8q, 0, 2 p + 5<D;
inte = ‡−1
1
‡−1
1
# ξ η & ê@ %;
co =Coefficient@Coefficient@#, ξ, Exponent@#, ξDD,
η, Exponent@#, ηDD &;H∗co is the function to get coefficient
of the polynomial function∗Lpreintefunct = If@TrueQ@# 0D, 0, co@#D Part@inte,
Exponent@#, ξD + 1, Exponent@#, ηD + 1DD &;intefunction = If@TrueQ@Head@#D PlusD,
Map@preintefunct, #D, preintefunct@#DD &;H∗intefunction is the function to integrate
the polynomial by using known value inte∗L
† The matrix elements are formed according to the Eqs.(3.34a-f)
H∗Elements for kcc∗L
Table@ ∂ξ w@iD ∂ξ w@jD, 8i, 1, Nmax<, 8j, 1, Nmax<D;Expand ê@ %;k1 = Map@intefunction, %, 82<D;
Table@ ∂η w@iD ∂η w@jD, 8i, 1, Nmax<, 8j, 1, Nmax<D;Expand ê@ %;k2 = Map@intefunction, %, 82<D;
Table@∂ξ w@iD ∂η w@jD, 8i, 1, Nmax<, 8j, 1, Nmax<D;Expand ê@ %;k3 = Map@intefunction, %, 82<D;
Table@∂η w@iD ∂ξ w@jD, 8i, 1, Nmax<, 8j, 1, Nmax<D;Expand ê@ %;k4 = Map@intefunction, %, 82<D;
H∗Elements for kcd∗L
Table@∂ξ w@iD φx@jD, 8i, 1, Nmax<, 8j, 1, Nmax<D;Expand ê@ %;k5 = Map@intefunction, %, 82<D;
Table@ ∂η w@iD φx@jD, 8i, 1, Nmax<, 8j, 1, Nmax<D;Expand ê@ %;k6 = Map@intefunction, %, 82<D;
Appendix A
232
H∗Elements for kce∗L
Table@ ∂η w@iD φy@jD, 8i, 1, Nmax<, 8j, 1, Nmax<D;Expand ê@ %;k7 = Map@intefunction, %, 82<D;
Table@∂ξ w@iD φy@jD, 8i, 1, Nmax<, 8j, 1, Nmax<D;Expand ê@ %;k8 = Map@intefunction, %, 82<D;
H∗Elements for kdd∗L
Table@ ∂ξ φx@iD ∂ξ φx@jD,8i, 1, Nmax<, 8j, 1, Nmax<D;
Expand ê@ %;k9 = Map@intefunction, %, 82<D;
Table@∂η φx@iD ∂η φx@jD,8i, 1, Nmax<, 8j, 1, Nmax<D;
Expand ê@ %;k10 = Map@intefunction, %, 82<D;
Table@∂η φx@iD ∂ξ φx@jD,8i, 1, Nmax<, 8j, 1, Nmax<D;
Expand ê@ %;k11 = Map@intefunction, %, 82<D;
Table@∂ξ φx@iD ∂η φx@jD,8i, 1, Nmax<, 8j, 1, Nmax<D;
Expand ê@ %;k12 = Map@intefunction, %, 82<D;
Table@ φx@iD φx@jD, 8i, 1, Nmax<, 8j, 1, Nmax<D;Expand ê@ %;k13 = Map@intefunction, %, 82<D;
H∗Elements for Kde∗L
Table@∂ξ φx@iD ∂ξ φy@jD,8i, 1, Nmax<, 8j, 1, Nmax<D;
Expand ê@ %;k14 = Map@intefunction, %, 82<D;
Table@ ∂ξ φx@iD ∂η φy@jD,8i, 1, Nmax<, 8j, 1, Nmax<D;
Expand ê@ %;k15 = Map@intefunction, %, 82<D;
Appendix A
233
Table@ ∂η φx@iD ∂ξ φy@jD,8i, 1, Nmax<, 8j, 1, Nmax<D;
Expand ê@ %;k16 = Map@intefunction, %, 82<D;
Table@∂η φx@iD ∂η φy@jD,8i, 1, Nmax<, 8j, 1, Nmax<D;
Expand ê@ %;k17 = Map@intefunction, %, 82<D;
Table@ φx@iD φy@jD, 8i, 1, Nmax<, 8j, 1, Nmax<D;Expand ê@ %;k18 = Map@intefunction, %, 82<D;
H∗Elements for kee∗L
Table@ ∂η φy@iD ∂η φy@jD,8i, 1, Nmax<, 8j, 1, Nmax<D;
Expand ê@ %;k19 = Map@intefunction, %, 82<D;
Table@ ∂ξ φy@iD ∂ξ φy@jD,8i, 1, Nmax<, 8j, 1, Nmax<D;
Expand ê@ %;k20 = Map@intefunction, %, 82<D;
Table@∂η φy@iD ∂ξ φy@jD,8i, 1, Nmax<, 8j, 1, Nmax<D;
Expand ê@ %;k21 = Map@intefunction, %, 82<D;
Table@∂ξ φy@iD ∂η φy@jD,8i, 1, Nmax<, 8j, 1, Nmax<D;
Expand ê@ %;k22 = Map@intefunction, %, 82<D;
Table@ φy@iD φy@jD, 8i, 1, Nmax<, 8j, 1, Nmax<D;Expand ê@ %;k23 = Map@intefunction, %, 82<D;
† The stiffness matrix can be obtained as a function of lx,ly and lxy.The
algorithm shown here is for the deformation theory of plasticity.
<< LinearAlgebra`MatrixManipulation`stiffness@λx_, λy_, λxy_D :=
Appendix A
234
ModuleA8t, s, c11, c12, c13,c22, c23, c33, α, β, γ, δ, ρ, ζ<,
λe ="#####################################################
λx2 − λx λy + λy2 + 3 λxy2 ;ly = 1;ζ = lyêlx;τ = hêly;
t = 1 ìi
kjjjjj1 + c k
ikjjjj
λe e τ2
12 H1 − ν2Lyzzzz
c−1y
zzzzz;
s = 1 ìi
kjjjjj1 + k
ikjjjj
λe e τ2
12 H1 − ν2Lyzzzz
c−1y
zzzzz;
c11 = 1 − 3 H1 − têsL ikjjjj
λy2
4 λe2+
λxy2
λe2
yzzzz;
c12 = −1
2ikjjjj1 − H1 − 2 νL t − 3 H1 − têsL
ikjjjj
λx λy
2 λe2+
λxy2
λe2
yzzzzyzzzz;
c13 =3
2H1 − têsL i
kjj 2 λx − λy
λeyzz
λxy
λe;
c22 = 1 − 3 H1 − têsL ikjjjj
λx2
4 λe2+
λxy2
λe2
yzzzz;
c23 =3
2H1 − têsL i
kjj 2 λy − λx
λeyzz
λxy
λe;
c33 = 3 tês − H1 − 2 νL t + 9 H1 − têsL λxy2
λe2;
c44 =1
κ H3 t ês − H1 − 2 νL tL;
α = c22 c33 − c232;β = c13 c23 − c12 c33;χ = c12 c23 − c13 c22;γ = c11 c33 − c132;µ = c12 c13 − c11 c23;δ = c11 c22 − c122;ρ = 1êt Det@88c11, c12, c13<,
8c12, c22, c23<, 8c13, c23, c33<<D;
kcc1 =t
c44 h2Sec@ωD
ikjj ζ k1 +
1
ζk2 − Sin@ωD k3 − Sin@ωD k4y
zz;
gcc =1
12 H1 − ν2L ly2
ikjjζ Cos@ωD λx
λk1 + Sec@ωD λy
λikjj 1
ζk2 −
Sin@ωD k3 − Sin@ωD k4 + Sin@ωD2 ζ k1yzz −
Appendix A
235
λxy
λHk3 + k4 − 2 Sin@ωD ζ k1Ly
zz;
kcc = kcc1;
kcd =t ly
2 c44 h2 ikjj k5 − Sin@ωD k6
1
ζyzz;
kce =t ly
2 c44 h2 ikjj 1
ζk7 − Sin@ωD k8y
zz;
kdd1 =α
12 ρζ Cos@ωD3 k9;
kdd2 =γ
12 ρTan@ωD Sin@ωD i
kjjSin@ωD2 k9 ζ +
1
ζk10 − Sin@ωD k11 − Sin@ωD k12y
zz;
kdd3 =β
12 ρSin@ωD Cos@ωD
H2 Sin@ωD k9 ζ − k11 − k12L;
kdd4 =δ
12 ρCos@ωD i
kjj4 Sin@ωD2 k9 ζ −
2 Sin@ωD k11 − 2 Sin@ωD k12 + k101
ζyzz;
kdd5 =χ
12 ρCos@ωD2 Hk11 + k12 − 4 Sin@ωD ζ k9L;
kdd6 =µ
12 ρSin@ωD i
kjj−
2
ζk10 +
3 Sin@ωD Hk11 + k12L − 4 Sin@ωD2 ζ k9yzz;
kdd7 =t
4 c44 τ2Cos@ωD 1
ζk13;
kdd =kdd1 + kdd2 + kdd3 + kdd4 + kdd5 + kdd6 + kdd7;
kde1 =γ
12 ρTan@ωD i
kjj− Sin@ωD2 k14 ζ +
Sin@ωD k15 + Sin@ωD k16 − k171
ζyzz;
kde2 =β
12 ρCos@ωD H− Sin@ωD k14 ζ + k15L;
kde3 =δ
12 ρCos@ωD H− 2 Sin@ωD k14 ζ + k16L;
kde4 =χ
12 ρCos@ωD2 ζ k14;
kde5 =µ
12 ρikjj3 Sin@ωD2 ζ k14 +
1
ζk17 − 2 Sin@ωD Hk15 + k16Ly
zz;
kde6 = −t
4 c44 τ2Sin@ωD Cos@ωD 1
ζk18;
Appendix A
236
ζkde = kde1 + kde2 + kde3 + kde4 + kde5 + kde6;
kee1 =γ
12 ρSec@ωD i
kjjk19
1
ζ+
Sin@ωD2 k20 ζ − Sin@ωD k21 − Sin@ωD k22yzz;
kee2 =δ
12 ρCos@ωD k20 ζ;
kee3 =µ
12 ρH−2 Sin@ωD ζ k20 + k21 + k22L;
kee4 =t
4 c44 τ2Cos@ωD 1
ζk23;
kee = kee1 + kee2 + kee3 + kee4;
rowc = AppendRows@kcc, kcd, kceD;rowd = AppendRows@Transpose@kcdD, kdd, kdeD;rowe = AppendRows@
Transpose@kceD, Transpose@kdeD, keeD;mat = Map@Chop, AppendColumns@
rowc, rowd, roweD, 82<D;zeromat = ZeroMatrix@NmaxD;matkm = BlockMatrix@88gcc, zeromat, zeromat<,
8zeromat, zeromat, zeromat<,8zeromat, zeromat, zeromat<<D;
Return@mat − λ matkmD;E;
† The input data or material properties and geometrical properties need to be
defined here (sample input data are for AL 7075 T6 aluminum alloy and S
shear loading).The angorithm for the calculation of buckling stress parame-
ter is the same as in BUCKRITZRE.
ω = −30 ∗ Piê180; H∗skew angle in radian∗Lλx = −2 λ Tan@ωD;λy = 0;λxy = λ ;ν = 33ê100;κ = 5ê6;e = 150; H∗e means Eêσ0∗Lk = 3ê7;BucklingLoadFinder@λ0_D :=
Module@8n, num, W, W1, Wmid, d<,λ = λ0; d = 5;
Appendix A
237
Do@Do@
n = Eigenvalues@Chop@stiffness@λx, λy, λxyD êê NDD;
num = Length@Select@n, Sign@#D −1 ⅅIf@num > 0, Print@num,
" buckling load is less than ",λ êê ND; Break@D, λ = λ + d
D, 8i, 50<D;If@num > 1, λ = λ − d;
d = d ê2; λ = λ + d; Print@λD, Break@DD, 8a, 10<D;If@num < 1, Print@
"Initial Value of λ is too small"D; Abort@DD;
W2 = Det@stiffness@λx, λy, λxyDD;λ = λ − d;
W1 = Det@stiffness@λx, λy, λxyDD;
If@Sign@W1D Sign@W2D,Print@"Limit is not OK"D;Abort@D, Print@"Limit is OK"D
D;Do@d = d ê2; λmid = λ + d;
If@d < 0.00005, Break@DD;λ = λmid;Wmid = Det@stiffness@λx, λy, λxyDD;Print@"λ=", " ", SetPrecision@λmid, 8D,
"W1=", " ", W1, "Wmid=", " ", WmidD;If@Sign@W1D ≠ Sign@WmidD, λ = λmid − dD,8i, 40<D;
Print@"h=", h, " ", "c=",c, " ", "λ=", " ", λmidêPi2D
D;
H∗λ0 is the initial value of λ∗LH∗The following command will
load the algorithm and determine λ∗L
Appendix A
238
† The buckling mode shape can be ploted as follow:
Clear@λD;λ = λmid;stiffness@λx, λy, λxyD;eigen = Eigenvalues@
8Chop@mat êê ND, Chop@matkm êê ND<D;eigenew = If@# ≤ 0, ∞, #D & ê@ eigen;po = Ordering@eigenew, 1D;Print@"λ=", eigenew@@po@@1DDDDD;v = Eigenvectors@
8Chop@mat êê ND, Chop@matkm êê ND<D;Co = v@@po@@1DDDD;defl = HTable@Co@@iDD polyfunction@@iDD φw1,
8i, Nmax<DL ê. List → Plus;Needs@"DrawGraphics`DrawingMaster`"DIteratorSubstitution@8ξ, η, defl<,
8ξ, −1 + η Tan@ωD, 1 + η Tan@ωD<, xD êê Simplify;function = %@@1DD;ParametricPlot3D@function, 8x, 0, 1<,8η, −Cos@ωD, Cos@ωD<, PlotPoints → 825, 25<,BoxRatios → 81.5773, 1, 0.3<,ViewPoint → 80, −3, 2<, PlotRange → All,Boxed → False, Axes → FalseD
Clear@"Global`∗"D
A.3 Ritz Program Code with Polar Coordinate System
à BUCKRITZPOLAR
This program is based on the polar coordinate formulations of Ritz method and
is to solve the plastic buckling problems of circular and annular Mindlin plates
under in-plane radial stress with various boundary conditions.
Appendix A
239
† Firstly basic functions need to be defined and the sample basic functions
for a simply supported annular plate are given by
φw1 = Hξ + 1L Hξ − 1L ; H∗SSSS∗Lφx1 = Hξ + 1L0 Hξ − 1L0;φy1 = Hξ + 1L Hξ − 1L;
† The mathematically complete one dimensional polynomial functions are
formed as a list. Hence the displacement functions w, fx, fy are obtained
by multiplying the corresponding basic function with polynomial terms.
p = 10;H∗ p is the degree of polynomial equation ∗Lζ = 0.4; H∗ζ=bêR∗LΩ = 2 Pi;polyfunction = Table@ξi−1, 8i, p<D;Nmax = p;
w = polyfunction@@#DD φw1 &;φx = polyfunction@@#DD φx1 &;φy = polyfunction@@#DD φy1 &;
† The integration of the matrix ·A
i
k
jjjjjjjjjjjjjj
1x
ª
x2 p+5
y
zzzzzzzzzzzzzz „ A, ·
A
H1+zL+H1-zL xÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ2
i
k
jjjjjjjjjjjjjj
1x
ª
x2 p+5
y
zzzzzzzzzzzzzz „ A,
·A
2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅH1+zL+H1-zL x
i
k
jjjjjjjjjjjjjj
1x
ª
x2 p+5
y
zzzzzzzzzzzzzz „ A are carried out over the plate domain for one
time only. Then an integration function is written in such a way that the
exponents of x and h in all the integrands are tested and use the coorspond-
ing integrated values. It is more computationally efficient since the integra-
tion of polynomial are repeatedly involved in the calculations.
Appendix A
240
function = Table@ξi, 8i, −1, 2 p + 5<D;
inte = ‡−1
1
# ξ & ê@ function;
co = Coefficient@#, ξ, Exponent@#, ξDD &;H∗co is the function to get
coefficient of the polynomial function∗L
preintefunct = If@TrueQ@# 0D, 0,co@#D Part@inte, Exponent@#, ξD + 1DD &;
intefunction = If@TrueQ@Head@#D PlusD,Map@preintefunct, #D, preintefunct@#DD &;
function1 =
TableA H1 + ζL + H1 − ζL ξ
2 ξi, 8i, 0, 2 p + 5<E;
inte1 = ‡−1
1
# ξ & ê@ function1;
function2 =
TableA 2
H1 + ζL + H1 − ζL ξξi, 8i, 0, 2 p + 5<E;
inte2 = ‡−1
1
# ξ & ê@ function2;
preintefunct1 = If@TrueQ@# 0D, 0,co@#D Part@inte1, Exponent@#, ξD + 1DD &;
intefunction1 = If@TrueQ@Head@#D PlusD,Map@preintefunct1, #D, preintefunct1@#DD &;
preintefunct2 = If@TrueQ@# 0D, 0,co@#D Part@inte2, Exponent@#, ξD + 1DD &;
intefunction2 = If@TrueQ@Head@#D PlusD,Map@preintefunct2, #D, preintefunct2@#DD &;
Appendix A
241
† The matrix elements are formed according to the Eqs.(3.47a-f)
Table@∂ξ w@iD ∂ξ w@jD, 8i, 1, Nmax<, 8j, 1, Nmax<D;Expand ê@ %;kcc1 = Map@intefunction1, %, 82<D;
Table@w@iD w@jD, 8i, 1, Nmax<, 8j, 1, Nmax<D;Expand ê@ %;kcc2 = Map@intefunction2, %, 82<D;
Table@∂ξ w@iD φx@jD, 8i, 1, Nmax<, 8j, 1, Nmax<D;Expand ê@ %;kcd1 = Map@intefunction1, %, 82<D;
Table@w@iD φy@jD, 8i, 1, Nmax<, 8j, 1, Nmax<D;Expand ê@ %;kce1 = Map@intefunction, %, 82<D;
Table@∂ξ φx@iD ∂ξ φx@jD,8i, 1, Nmax<, 8j, 1, Nmax<D;
Expand ê@ %;kdd1 = Map@intefunction1, %, 82<D;
Table@ φx@iD φx@jD, 8i, 1, Nmax<, 8j, 1, Nmax<D;Expand ê@ %;kdd2 = Map@intefunction2, %, 82<D;
Table@φx@iD φx@jD, 8i, 1, Nmax<, 8j, 1, Nmax<D;Expand ê@ %;kdd3 = Map@intefunction1, %, 82<D;
Table@φx@iD φx@jD, 8i, 1, Nmax<, 8j, 1, Nmax<D;Expand ê@ %;kdd4 = Map@intefunction2, %, 82<D;
Table@∂ξ φx@iD φx@jD, 8i, 1, Nmax<, 8j, 1, Nmax<D;Expand ê@ %;kdd5 = Map@intefunction, %, 82<D;
Table@φx@iD ∂ξ φx@jD, 8i, 1, Nmax<, 8j, 1, Nmax<D;Expand ê@ %;kdd6 = Map@intefunction, %, 82<D;
Table@ ∂ξ φx@iD φy@jD, 8i, 1, Nmax<, 8j, 1, Nmax<D;Expand ê@ %;kde1 = Map@intefunction, %, 82<D;
Appendix A
242
Table@ φx@iD ∂ξ φy@jD, 8i, 1, Nmax<, 8j, 1, Nmax<D;Expand ê@ %;kde2 = Map@intefunction, %, 82<D;
Table@ φx@iD φy@jD, 8i, 1, Nmax<, 8j, 1, Nmax<D;Expand ê@ %;kde3 = Map@intefunction2, %, 82<D;
Table@ φx@iD φy@jD, 8i, 1, Nmax<, 8j, 1, Nmax<D;Expand ê@ %;kde4 = Map@intefunction2, %, 82<D;
Table@φy@iD φy@jD, 8i, 1, Nmax<, 8j, 1, Nmax<D;Expand ê@ %;kee1 = Map@intefunction2, %, 82<D;
Table@∂ξ φy@iD ∂ξ φy@jD,8i, 1, Nmax<, 8j, 1, Nmax<D;
Expand ê@ %;kee2 = Map@intefunction1, %, 82<D;
Table@ φy@iD φy@jD, 8i, 1, Nmax<, 8j, 1, Nmax<D;Expand ê@ %;kee3 = Map@intefunction2, %, 82<D;
Table@∂ξ φy@iD φy@jD, 8i, 1, Nmax<, 8j, 1, Nmax<D;Expand ê@ %;kee4 = Map@intefunction, %, 82<D;
Table@ φy@iD ∂ξ φy@jD , 8i, 1, Nmax<, 8j, 1, Nmax<D;Expand ê@ %;kee5 = Map@intefunction, %, 82<D;
Table@φy@iD φy@jD, 8i, 1, Nmax<, 8j, 1, Nmax<D;Expand ê@ %;kee6 = Map@intefunction1, %, 82<D;
Appendix A
243
† The stiffness matrix can be obtained as a function of lx,ly and lxy.The
algorithm shown here is for the deformation theory of plasticity. drop-
Zeros[m_List?MatrixQ] is to drop the matrix comlumns and rows whose
elements are zeros. This can be happen when n = 0 because
all the elements of Kce, Kde, Kee are zeros for n = 0.
<< LinearAlgebra`MatrixManipulation`
S1 := 2 ê; n 0S1 := 1 ê; n > 0S2 := 0 ê; n 0S2 := 1 ê; n > 0
notZero@ro_ListD := Or @@ H# ≠ 0 &L ê@ rodropZeroes@m_List? MatrixQD :=
Transpose@Select@Transpose@Select@m, notZeroDD, notZeroDD
stiffness@λx_, λy_, λxy_D := ModuleA8t, s, c11, c12, c13, c22, c23, c33, c44<,
λe ="#####################################################
λx2 − λx λy + λy2 + 3 λxy2 ;ly = 1;τ = hêly;
t = 1 ì ikjjjj1 + c k i
kjj 1
12 H1 − ν2L Hλe e τ2 Lyzz
c−1yzzzz;
s = 1 ìi
kjjjjj1 + k
ikjjjj
λe e τ2
12 H1 − ν2Lyzzzz
c−1y
zzzzz;
c11 = 1 − 3 H1 − têsL ikjjjj
λy2
4 λe2+
λxy2
λe2
yzzzz;
c12 = −1
2ikjjjj1 − H1 − 2 νL t − 3 H1 − têsL
ikjjjj
λx λy
2 λe2+
λxy2
λe2
yzzzzyzzzz;
c13 =3
2H1 − têsL i
kjj 2 λx − λy
λeyzz
λxy
λe;
c22 = 1 − 3 H1 − têsL ikjjjj
λx2
4 λe2+
λxy2
λe2
yzzzz;
c23 =3
2H1 − têsL i
kjj 2 λy − λx
λeyzz
λxy
λe;
Appendix A
244
k c33 = 3 t ês − H1 − 2 νL t + 9 H1 − têsL
λxy2
λe2;
c44 =1
κ H3 t ês − H1 − 2 νL tL;
α = c22 c33 − c232;β = c13 c23 − c12 c33;χ = c12 c23 − c13 c22;γ = c11 c33 − c132;µ = c12 c13 − c11 c23;δ = c11 c22 − c122;ρ = 1êt Det@88c11, c12, c13<,
8c12, c22, c23<, 8c13, c23, c33<<D;H∗This is for Deformation theory
of plasticity∗Lkc =
t
c44 τ2
ikjj Ω
H1 − ζL S1 kcc1 +
H1 − ζL Ω
4n2 S2 kcc2y
zz;
gcc =1
12 H1 − ν2L ikjj Ω
H1 − ζLλx
λS1 kcc1 +
λy
λn2 S2
H1 − ζL Ω
4kcc2y
zz;
kcc = kc;
kcd =t
c44 τ2
Ω
2S1 kcd1;
kce = −t
c44 τ2
H1 − ζL Ω
4n S2 kce1;
kdd =α
12 ρ
Ω
H1 − ζL S1 kdd1 +δ
12 ρ
H1 − ζL Ω
4n2 S2 kdd2 +
t
c44 τ2
H1 − ζL Ω
4S1 kdd3 +
γ
12 ρ
Ω H1 − ζL4
S1 kdd4 +β Ω
24 ρS1 kdd5 +
β Ω
24 ρS1 kdd6;
kde =β Ω
2 12 ρn S1 kde1 −
δ Ω
2 12 ρn S2 kde2 +
γ
12 ρH1 − ζL Ω
4n S1 kde3 +
δ
12 ρ
H1 − ζL Ω
4n S2 kde4;
kee =γ
12 ρ
H1 − ζL Ω
4n2 S1 kee1 +
δ
12 ρ
Ω
H1 − ζL S2 kee2 +δ
12 ρ
H1 − ζL Ω
4 S2 kee3 −
δ
12 ρ
Ω
2 S2 kee4 −
δ
12 ρ
Ω
2 S2 kee5 +
t
c44 τ2
H1 − ζL Ω
4S2 kee6;
rowc = AppendRows@kcc, kcd, kceD;
Appendix A
245
rowd = AppendRows@Transpose@kcdD, kdd, kdeD;rowe = AppendRows@
Transpose@kceD, Transpose@kdeD, keeD;mat = AppendColumns@rowc, rowd, roweD;matkm = PadRight@dropZeroes@gccD,8Length@matD, Length@matD<D;
Return@mat − λ matkmD;E;
† The input data or material properties and geometrical properties need to be
defined here (sample input data are for AL 2024 T6 aluminum alloy).The
angorithm for the calculation of buckling stress parameter is the same as in
BUCKRITZRE. Note that asymetric buckling mode shape (n ∫0) is possi-
ble and the numbers of nodal diameter n is unknown. So, the lowest buck-
ling stress parameter and corresponding nodal number n are determined by
an iteration fashion.
Clear@n, λD;λx := λ;λy := λ;λxy = 0;
κ =5
6;
ν = 32ê100;e = 10700ê61.4; H∗e means Eêσ0∗Lk = 0.3485;c = 20;h = 0.05;BucklingLoadFinder@λ0_D :=
Module@8n, num, W, W1, Wmid, d<,λ = λ0; d = 5;Do@
Do@n = Eigenvalues@
Chop@stiffness@λx, λy, λxyD êê NDD;num = Length@Select@n, Sign@#D −1 ⅅIf@num > 0, Print@num,
" buckling load is less than ",λ êê ND; Break@D, λ = λ + d
D, 8i, 50<
Appendix A
246
D;If@num > 1, λ = λ − d;
d = d ê2; λ = λ + d; Print@λD, Break@DD, 8a, 10<D;If@num < 1, Print@
"Initial Value of λ is too small"D; Abort@DD;
W2 = Det@stiffness@λx, λy, λxyDD;λ = λ − d;
W1 = Det@stiffness@λx, λy, λxyDD;
If@Sign@W1D Sign@W2D,Print@"Limit is not OK"D;Abort@D, Print@"Limit is OK"D
D;Do@d = d ê2; λmid = λ + d;
If@d < 0.00005, Break@DD;λ = λmid;Wmid = Det@stiffness@λx, λy, λxyDD;Print@"λ=", " ", SetPrecision@λmid, 8D,
"W1=", " ", W1, "Wmid=", " ", WmidD;If@Sign@W1D ≠ Sign@WmidD, λ = λmid − dD,8i, 40<D;
Print@"h=", h, " ", "c=", c, " ", "λ="," ", λmidêPi2 êê ND; Return@λmidD;
D;
n = 0; λt = BucklingLoadFinder@1D;n = 1; λn = BucklingLoadFinder@1D;While@λt ≥ λn, λt = λn; n = n + 1;
λn = BucklingLoadFinder@1D;D; Print@"n = ", n − 1, " ",
"λ = ", SetPrecision@λtêPi2, 10DD;
† The buckling mode shape can be ploted as follow:
Appendix A
247
stiffness@λt, λt, 0D;Eigenvalues@8mat, matkm<DIf@# ≤ 0, ∞, #D & ê@ %po = Ordering@%, 1Dv = Eigenvectors@8mat, matkm<D;H∗Eigenvectors@%D;∗LCo = %@@po@@1DDDDTable@Co@@iDD polyfunction@@iDD φw1, 8i, Nmax<D;defl = % ê. List → Plus;pic = ParametricPlot3D@8H1ê2 HH1 + ζL + H1 − ζL ξLL ∗ Cos@1ê2 Ω ηD,H1ê2 HH1 + ζL + H1 − ζL ξLL ∗ Sin@1ê2 Ω ηD,defl Cos@n 1 ê2 Ω ηD<, 8ξ, −1, 1<, 8η, −1,
1<, BoxRatios −> 81, 1, 0.25<, Axes −> False,Boxed −> False, PlotPoints → 820, 60<D
Clear@"Global`∗"D
Appendix A
248
249
Appendix B: Elements of Matrices
B.1: Plastic buckling of circular plate under uniform in-plane compression
The non-zero elements of the matrix [ ] 33×K in Eq. (7.22) are given by
(i) for simply supported edge
111 =k (B.1)
)( 212 δnJk = (B.2)
)()1(21 βα −−= nnk (B.3)
[ ( ) )(22)(2)(1)1(124
12
22221222
2222
2
22 δβαδδβδδαδκν
λτnnn JnJJ
GEk +−+⎟⎟
⎠
⎞⎜⎜⎝
⎛−
−= −−
])()( 2222212 δαδδβδ ++ ++ nn JJ (B.4)
( ) )()(1)( 313323 δδδβα ++−−= nn IInnk (B.5)
nk =31 (B.6)
)()1(12
1 222
2
32 δκν
λτnnJ
GEk ⎟⎟
⎠
⎞⎜⎜⎝
⎛−
−= (B.7)
⎩⎨⎧
≠−−=
=+ 0for )()(
0for 0
313333 nInI
nk
nn δδδ (B.8)
(ii) for clamped edge
The first row and third row of matrix elements of [ ] 33×K in Eq. (7.22) are same as the
elements in corresponding rows for simply supported edge. The non-zero elements in
second row are
⎩⎨⎧
≠−=
=0for 0for 0
21 nnn
k (B.9)
Appendix B
250
)()(1)1(12 22222
2
22 δδδκν
λτ+−⎟⎟
⎠
⎞⎜⎜⎝
⎛−
−= nn JnJ
GEk (B.10)
)( 323 δnnIk = (B.11)
B.2: Plastic buckling of annular plate under uniform in-plane compression
In this appendix, the non-zero elements of the matrix [ ] 66×K in Eq. (7.25) are
presented. The elements in the first three rows depend on the boundary condition of the
outer edge while the elements in the next three rows depend on the inner edge
condition.
If the outer edge is
(i). clamped edge
111 =k (B.12)
)( 212 δnJk = (B.13)
⎩⎨⎧
≠=
=0for 10for 0
14 nn
k (B.14)
)( 215 δnYk = (B.15)
⎩⎨⎧
≠−=
=0for 0for 0
21 nnn
k (B.16)
)()(1 222222 δδδκσ
+−⎟⎠⎞
⎜⎝⎛ −= nn JnJ
Gk (B.17)
)( 323 δnnIk = (B.18)
⎩⎨⎧
≠=−
=0for 0for 1
24 nnn
k (B.19)
Appendix B
251
)()(1 222225 δδδκσ
+−⎟⎠⎞
⎜⎝⎛ −= nn YnY
Gk (B.20)
)( 326 δnnKk = (B.21)
nkk == 3431 (B.22)
)(1 2232 δκσ
nnJG
k ⎟⎠⎞
⎜⎝⎛ −= (B.23)
⎩⎨⎧
≠−−=
=+ 0for )()(
0for 0
313333 nInI
nk
nn δδδ (B.24)
)(1 2235 δκσ
nnYG
k ⎟⎠⎞
⎜⎝⎛ −= (B.25)
⎩⎨⎧
≠+−=
=+ 0for )()(
0for 0
313336 nKnK
nk
nn δδδ (B.26)
(ii). simply supported edge
The first row and third row of matrix elements are same as the elements in
corresponding rows for clamped edge (i.e. Eqs. B.12 to B.26). The non-zero elements
in second row are
)()1(21 βα −−= nnk (B.27)
[ ( ) )(22)(2)(141
222
22122222222 δβαδδβδδαδ
κσ
nnn JnJJG
k +−+⎟⎠⎞
⎜⎝⎛ −= −−
])()( 2222212 δαδδβδ ++ ++ nn JJ (B.28)
( ) )()(1)( 313323 δδδβα ++−−= nn IInnk (B.29)
⎩⎨⎧
≠−+−=−
=0for ))(1(0for
24 nnnn
kβα
βα (B.30)
[ ( ) )(22)(2)(141
222
22122222225 δβαδδβδδαδ
κσ
nnn YnYYG
k +−+⎟⎠⎞
⎜⎝⎛ −= −−
])()( 2222212 δαδδβδ ++ + nn YY (B.31)
Appendix B
252
( ) )()(1)( 313326 δδδβα +−−−= nn KKnnk (B.32)
iii. free edge
The elements in the second row of the matrix are same as simply supported case (Eqs.
B.27-B.32). The remaining elements in the first and thirds rows are given by
Gnk 211 κσ
−= (B.33)
)( 313 δnnIk = (B.34)
⎪⎩
⎪⎨
⎧
≠
=−=
0for
0for
2
2
14
nG
n
nGk
κσκσ
(B.35)
)( 316 δnnKk = (B.36)
)1(231 −= nnk (B.37)
)()()1(12 2122232 δδδκσ
++−⎟⎠⎞
⎜⎝⎛ −= nn JJn
Gnk (B.38)
( ) ( )⎪⎩
⎪⎨⎧
≠+−−−+−
==
++ 0for )()1(2)(2220for 0
322331
223
3
33 nInnInnn
knn δδδδ
δ (B.39)
)1(234 +−= nnk (B.40)
)()()1(12 2122235 δδδκσ
++−⎟⎠⎞
⎜⎝⎛ −= nn YYn
Gnk (B.41)
( ) ( )⎪⎩
⎪⎨⎧
≠+−−−+
==
++ 0for )()1(2)(2220for 0
322331
223
3
36 nKnnKnnn
knn δδδδ
δ (B.42)
Appendix B
253
If the inner edge is
i. clamped edge
nbk =41 (B.43)
)( 242 δnJk = (B.44)
⎩⎨⎧
≠=
= − 0for 0for )log(
44 nbnb
k n (B.45)
)( 245 δnYk = (B.46)
151
−−= nnbk (B.47)
)()(12 21212
252 δδ
κσδ bJbJG
k nn +− −⎟⎠⎞
⎜⎝⎛ −= (B.48)
)( 353 δbIbnk n= (B.49)
⎩⎨⎧
≠=−
=+−
−
0for 0for
)1(
1
54 nnbnb
k n (B.50)
)()(12 21212
255 δδ
κσδ bYbYG
k nn +− −⎟⎠⎞
⎜⎝⎛ −= (B.51)
)( 356 δbKbnk n= (B.52)
161
−= nnbk (B.53)
)(1 2262 δκσ bJ
bn
Gk n⎟
⎠⎞
⎜⎝⎛ −= (B.54)
⎪⎩
⎪⎨⎧
≠+−
==
−+ 0for )( )(2
0for 0
3131363 nbIbI
nk
nn δδδ (B.55)
)1(64
+−= nnbk (B.56)
)(1 2265 δκσ bY
bn
Gk n⎟
⎠⎞
⎜⎝⎛ −= (B.57)
Appendix B
254
⎪⎩
⎪⎨⎧
≠+
==
−+ 0for )( )(2
0for 0
3131366 nbKbK
nk
nn δδδ (B.58)
ii. simply supported edge
The fourth row and sixth row of matrix elements are same as the elements in
corresponding rows for clamped edge (i.e. Eqs. B.43-B.58). The non-zero elements in
fifth row are
)()1(251 βα −−= − nnbk n (B.59)
[ ( ) )(22)(2)(141
222
22
2122222
22252 δβδαδδβδδα
κσ bJnbbJbbJbGb
k nnn +−+⎟⎠⎞
⎜⎝⎛ −= −−
])()(2 2222
2212 δδαδδβ bJbbJb nn ++ ++ (B.60)
( ) )()(1)( 3133253 δδδβα bIbbInbnk nn ++−−= (B.61)
( )( )⎩
⎨⎧
≠−+−=−
=+−
−
0for )1(0for
)2(
2
54 nnnbnb
k n βαβα
(B.62)
[ ( ) )(22)(2)(141
222
22
2122222
22255 δβδαδδβδδα
κσ bYnbbYbbYbGb
k nnn +−+⎟⎠⎞
⎜⎝⎛ −= −−
])()( 2222
2212 δδαδδβ bYbbYb nn ++ ++ (B.63)
( ) )()(1)( 3133256 δδδβα bKbbKnbnk nn +−−−= (B.64)
iii. free edge
The elements in the fifth row of the matrix are same as simply supported case (Eqs.
(B.59-B.64). The remaining elements in the fourth and sixth rows are given by
Appendix B
255
Gnbk
n
2
1
41 κσ−
−= (B.65)
)( 343 δbIbnk n= (B.66)
⎪⎩
⎪⎨
⎧
≠
=−=
+ 0for
0for
21
2
44
nGb
n
nGbk
n κσκσ
(B.67)
)( 346 δbKbnk n= (B.68)
nnbk n )1(2 261 −= − (B.69)
)()()1(1221222262 δδδ
κσ bJbbJnGb
nk nn ++−⎟⎠⎞
⎜⎝⎛ −= (B.70)
( )⎪⎩
⎪⎨⎧
≠⎟⎠⎞
⎜⎝⎛ +−−−+−
==
++ 0for )()1(2)(2220for 0
3223231
223
2
33
63 nbInbnbInb
bn
nk
nn δδδδδ
(B.71)
)1(2 )2(64 +−= +− nnbk n (B.72)
)()()1(1221222265 δδδ
κσ bYbbYnGb
nk nn ++−⎟⎠⎞
⎜⎝⎛ −= (B.73)
( )⎪⎩
⎪⎨⎧
≠⎟⎠⎞
⎜⎝⎛ +−−−+
==
++ 0for )()1(2)(2220for 0
3223231
223
2
33
66 nbKnbnbKnb
bn
nk
nn δδδδδ
(B.74)
256
List of Author’s Publications
Journal Papers
1. Wang, C.M. and Tun Myint Aung, “Plastic buckling analysis of thick plates
using p-Ritz method.” International Journal of Solids and Structures,
submitted for possible publication.
2. Tun Myint Aung, Wang, C. M. and Chakrabarty, J., (2005). “Plastic buckling
of moderately thick annular plates.” International Journal of Structural
Stability and Dynamics, 5 (3), 337-357.
3. Tun Myint Aung and Wang, C.M, (2005). “Buckling of circular plates under
intermediate and edge radial loads.” Thin-Walled Structures, Vol. 43, 1926-
1933.
4. Wang, C.M. and Tun Myint Aung, (2004). “Buckling of circular Mindlin plates
with an internal ring support and elastically restrained edge.” Journal of
Engineering Mechanics, ASCE, Vol. 131 (4), 359-366.
5. Wang, C.Y., Wang, C.M. and Tun Myint Aung, (2004). “Buckling of a
weakened column.” Journal of Engineering Mechanics, ASCE, Vol. 130 (11),
1373-1376.
Conferences Papers
1. Tun Myint Aung and Wang, C. M. (2005). “Plastic buckling skew plates via
Ritz method.” Proceedings of the Eighteenth KKCNN Symposium on Civil
Engineering, Kaohsiung, Taiwan, 471-476.
List of Author’s Publications
257
2. Wang, C. M. and Tun Myint Aung, (2005). “Ritz method for plastic buckling
analysis of thick plates.” Third International Conference of Structural Stability
and Dynamics, Kissimmee, Florida, USA.
3. Wang, C. M., Tun Myint Aung and Vo, K.K. (2005). “Plastic buckling analysis
of Mindlin plates using the Ritz method.” Proceedings of the Tenth
International Conference on Civil, Structural and Environmental Engineering
Computing, Rome, Italy.
4. Tun Myint Aung and Wang, C. M. (2004). “Plastic buckling of annular plates.”
Proceedings of the 17th KKCNN Symposium on Civil Engineering, Ayutthaya,
Thailand, 367-372.
5. Wang, C.M. and Tun Myint Aung, (2003). “Buckling Of Circular Plates With
An Internal Ring Support: Asymmetric Mode And Optimal Ring Support
Radius.” Proceedings of the 16th KKCNN Symposium on Civil Engineering,
Kyungju, Korea.