buckling of rectangular plates under intermediate and end loads

BUCKLING OF RECTANGULAR PLATES UNDER

INTERMEDIATE AND END LOADS

Chen Yu

NATIONAL UNIVERSITY OF SINGAPORE

2003

BUCKLING OF RECTANGULAR PLATES UNDER


Chen Yu

(B. Eng.)

A THESIS SUBMITTED

FOR THE DEGREE OF MASTER OF ENGINEERING

DEPARTMENT OF CIVIL ENGINEERING

NATIONAL UNIVERSITY OF SINGAPORE

2003

i

ACKNOWLEDGEMENTS

The author wishes to express her 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 Associate Professor Xiang Yang of University of Western

Sydney, Australia for his valuable discussions.

The author is grateful to the National University of Singapore for providing a

handsome research scholarship during the two-year study.

Finally, the author wishes to express her deep gratitude to her family, for their love and

continuous support during the course of this research.

ii

TABLE OF CONTENTS

ACKNOWLEDGEMENTS i

TABLE OF CONTENTS ii

SUMMARY iv

NOMENCLATURE v

LIST OF TABLES vii

LIST OF FIGURES viii

CHAPTER 1: INTRODUCTION 1

1.1 Background 1

1.2 Literature Review 2

1.2.1 Elastic buckling of rectangular plates 2

1.2.2 Plastic buckling of rectangular plates 8

1.3 Objectives and Scope of Study 12

1.4 Outline of Thesis 13

CHAPTER 2: BUCKLING OF PLATES UNDER END LOADS 15

2.1 Elastic Buckling Theory 15

2.1.1 Derivation of differential equation for elastic buckling 15

2.1.2 Boundary conditions 19

2.2 Plastic Buckling Theory 20

2.2.1 Derivation of constitutive relations based on Hencky’s deformation theory

20

2.2.2 Derivation of constitutive relations based on Prandtl-Reuss material 24

2.2.3 Derivation of differential equation for plastic buckling 25

2.2.4 Boundary conditions 27

iii

CHAPTER 3: ELASTIC BUCKLING OF PLATES UNDER

INTERMEDIATE AND END LOADS 31

3.1 Mathematical Modeling 32

3.1.1 Problem definition 32

3.1.2 Method of solution 32

3.2 Results and Discussions 38

3.2.1 SSSS plates 38

3.2.2 CSCS plate 41

3.2.3 FSFS plate 42

3.3 Concluding Remarks 43

CHAPTER 4: PLASTIC BUCKLING OF PLATES UNDER

INTERMEDIATE AND END LOADS 61

4.1 Mathematical Modeling 62

4.1.1 Problem definition 62

4.1.2 Method of solution 62

4.2 Results and Discussions 70

4.2.1 Effect of various aspect ratios a/b 72

4.2.2 Effect of various loading positions χ 74

4.2.3 Effect of various boundary conditions 74

4.2.4 Effect of various material properties 75

4.2.5 Effect of using two different theories 75

4.3 Concluding Remarks 75

CHAPTER 5: CONCLUSIONS AND RECOMMENDATIONS 88

5.1 Conclusions 88

5.2 Recommendations for Future Studies 89

REFERENCES 90

AUTHOR’S LIST OF PUBLICATIONS 95

iv

SUMMARY This thesis is concerned with the new buckling problem of rectangular plates subjected

to intermediate and end uniaxial loads. The considered plate has two opposite simply

supported edges that are parallel to the load direction and the other remaining edges may

take any combination of free, simply supported or clamped condition. The

aforementioned buckling problem is solved by decomposing the plate into two sub-plates

at the location where the intermediate uniaxial load acts. Each sub-plate buckling

problem is solved exactly using the Levy approach and the two solutions brought

together by matching the continuity equations at the separated edge.

Both elastic and plastic theories have been used to formulate the problem. For the

elastic theory, there exists five possible solutions for each sub-plate. Thus, when we

combine the two sub-plate problems, we need to consider twenty-five possible different

solution combinations. It is found that the stability curves consist of a number of these

combinations depending on the boundary conditions, aspect ratios, and intermediate load

positions. For the plastic buckling part, two competing theories, namely incremental

theory and deformation theory have been adopted to bound the plastic buckling solutions.

Unlike its elastic counterpart, there are eight possible solutions for each sub-plate when

considering plastic buckling. Thus sixty-four possible solution combinations are

considered for the whole plate. The final solution combination depends on various ratios

of the intermediate load to the end load, the intermediate load locations, aspect ratios,

boundary conditions and material properties.

Extensive stability criteria curves were presented to elucidate the buckling behavior of

such loaded rectangular plates. The results will be useful for engineers designing walls or

plates that have to support intermediate floors/loads.

Keywords: Elastic buckling; Plastic buckling; Thin plate theory; Incremental Theory of

Plasticity; Deformation Theory of Plasticity; Rectangular plates; Intermediate load; Levy

method; Stability criteria.

v

NOMENCLATURE

a length of plates

b width of plates

c dimensionless constant describing the shape of the Ramberg-Osgood stress-strain relation

D flexural rigidity

E Young’s modulus

G shear modulus

H plastic modulus

h thickness of plates

k horizontal distance between the knee of ∞=c curve and the intersection of the c curve with the 1/ 0 =σσ line in the Ramberg-Osgood stress-strain relation

yyxx MM , bending moments per unit length on x and y planes

xyM twisting moment per unit length on x plane

m number of half waves of the buckling mode along y direction

n number of half waves of the buckling mode along x direction

1N end load on sub-plate 1 per unit length

2N intermediate load on sub-plate 2 per unit length

xN uniaxial load on x plane

xQ shear force per unit length on x plane

S secant modulus

T tangent modulus

U strain energy

xV effective shear force per unit length

vi

W work done due to uniaxial loads

w transverse deflection of a point on the mid-plane

ργβα ,,, parameters in incremental theory of plasticity and deformation theory of plasticity

xzyzxy γγγ ,, shear strain in the xy, yz and xz plane

yyxx εε , normal strain in x and y directions

η contraction rate at current stress state

1Λ end buckling load factor

2Λ intermediate buckling load factor

ν Poisson’s ratio

∏ potential energy

0σ nominal yield stress

1σ end buckling load stress

2σ intermediate buckling load stress

yyxx σσ , normal stress on the x and y planes

xyσ shear stress on the x plane and parallel to the y direction

σ effective stress

χ intermediate load position

ei

ei ψφ , parameters in elastic solutions

pi

pi ψφ , parameters in plastic solutions

vii

LIST OF TABLES

Table 3.1 Twenty-five combinations of solutions 35

Table 3.2 Buckling factors 2Λ for simply supported rectangular plates

subjected to inplane load in sub-plate 2 only ( 01 =N )

41

Table 4.1 Types of solutions depending on values of 321 ,, ∆∆∆ 65

Table 4.2 Buckling stresses 1σ for a simply supported, square plate under

uniaxial end load (i.e. no intermediate load)

71

Table 4.3 Comparison of buckling factors of full plates with uniaxial

intermediate and end loads and their corresponding end loaded

sub-plates with different interfacial edge conditions

73

viii

LIST OF FIGURES

Fig. 1.1 Fig. 1.1 Buckling of plates under(a) point loads; (b) partially distributed loads; (c) patch loads at edge center; (d) patch loads near corners.

8

Fig. 2.1 Thin rectangular plate under end uniaxial load 29

Fig. 2.2 Stress resultants on a plate element. 29

Fig. 2.3 Ramberg-Osgood stress-strain relation 30

Fig. 3.1 Geometry and coordinate systems for rectangular plate subjected to intermediate and end uniaxial inplane loads. (a) Original plate; (b) Sub-plate 1; and (c) Sub-plate 2

45

Fig. 3.2 Typical stability criterion curves for SSSS plates subjected to end and intermediate inplane loads: (a) plate with integer aspect ratio a/b, and (b) plate with non-integer aspect ratio a/b.

46

Fig. 3.3 Stability criteria for SSSS rectangular plates with (a) a/b = 1.0, (b) a/b = 1.5, and (c) a/b = 2.0.

48

Fig. 3.4 Variations of buckling intermediate load factor 2Λ with respect to location χ for SSSS square plate (Note that

0000.4=Λ cr is the buckling load factor for square SSSS plate under end load only).

49

Fig. 3.5 Normalized modal shapes and modal moment distributions in the x-direction for SSSS square plate subjected to intermediate load N2 (N1 = 0): (a) modal shapes; and (b) modal moment distributions

50

Fig. 3.6 Variation of buckling factors Λ2 versus plate aspect ratio a/b for SSSS plates subjected to inplane load in sub-plate 2 only.

51

Fig. 3.7 Stability criteria for CSCS rectangular plates with plate aspect ratios (a) a/b = 1.0, (b) a/b = 1.5, and (c) a/b = 2.0.

53

Fig. 3.8 Variations of buckling intermediate load factor 2Λ with respect to location χ for CSCS square plate

54

Fig. 3.9 Variations of buckling factors Λ2 versus plate aspect ratio a/b for CSCS rectangular plates subjected to inplane load in sub-plate 2 only.

55

Fig. 3.10 Typical stability criterion curve for FSFS rectangular plates subjected to end and intermediate inplane loads.

56

Fig. 3.11 Stability criteria for FSFS rectangular plates with plate aspect ratios (a) a/b = 1.0, (b) a/b = 1.5, and (c) a/b = 2.0.

58

ix

Fig. 3.12 Variations of buckling intermediate load factor Λ2 with respect to location χ for FSFS square plate

59

Fig. 3.13 Variations of buckling factors Λ2 versus plate aspect ratio a/b for FSFS rectangular plates subjected to inplane load in sub-plate 2 only.

60

Fig. 4.1 Rectangular plate under intermediate and end uniaxial loads 77

Fig. 4.2 Typical stability criterion curve 77

Fig. 4.3 Stability criteria for SSSS rectangular plates with h/b = 0.04 and different aspect ratios a/b = 1, 2, 3 by (a) IT and (b) DT. The intermediate load is placed at χ = 0.5

78

Fig. 4.4 Stability criteria for CSCS rectangular plates with h/b = 0.04 and different aspect ratios a/b = 1, 2, 3 by (a) IT and (b) DT. The intermediate load is placed at χ = 0.5

79

Fig. 4.5 Stability criteria for FSFS rectangular plates with h/b = 0.04 and different aspect ratios a/b = 1, 2, 3 by (a) IT and (b) DT. The intermediate load is placed at χ = 0.5

80

Fig. 4.6 Rectangular plate YSZS and the corresponding sub-plate YSXS under uniaxial load

81

Fig. 4.7 Variation of buckling factors 2Λ with respect to χ for rectangular plates with 1Λ =0 and 2 by (a) IT (b) DT.

82

Fig. 4.8 Stability criteria for rectangular plates with h/b = 0.04, aspect ratio a/b = 2, intermediate load position 5.0=χ and different boundary conditions by (a) IT and (b) DT

83

Fig. 4.9 Stability criteria for SSSS square plates with h/b = 0.04, χ =

0.5, for different 0σ

E = (a) 200, (b) 400, (c) 800 by IT

85

Fig. 4.10 Stability criteria for SSSS square plates with h/b = 0.04, χ =

0.5, for different 0σ

E = (a) 200, (b) 400, (c) 800 by DT

86

Fig. 4.11 Buckling load factors 2Λ for SSSS square plates with 01 =Λ 87

INTRODUCTION

1.1 Background

Plates are widely used in many engineering structures such as aircraft wings, ships,

buildings, and offshore structures. Most plated structures, although quite capable of

carrying tensile loadings, are poor in resisting compressive forces. Usually, the buckling

phenomena observed in compressed plates take place rather suddenly and may lead to

catastrophic structural failure. Therefore it is important to know the buckling capacities

of the plates in order to avoid premature failure.

The first significant treatment of plate buckling occurred in the 1800s. Based on

Kirchhoff assumptions, the stability equation of rectangular plates was derived by Navier

(1822). Since then, investigations on the buckling of plates with all sorts of shapes,

boundary and loading conditions have been reported in standard texts (e.g. Timoshenko

and Gere 1961, Bulson 1970), research reports (e.g. Batdorf and Houbolt 1946) and

technical papers (e.g. Wang et al. 2001; Xiang et al. 2001). Research on the buckling of

plates may be categorized under elastic buckling and plastic buckling. In the elastic

buckling research, it is assumed that the critical load remains below the elastic limit of

the plate material. However, in practical problems the plate may be stressed beyond the

elastic limit before buckling occurs. Therefore, buckling theories of plasticity are

Chapter 1

Chapter 1 Introduction 2

introduced for practical uses. Generally there are two competing plastic theories, namely,

the deformation theory (DT) and the incremental theory (IT) of plasticity.

The buckling of rectangular plates under intermediate and end loads has hitherto not

been treated. The present study tackles such a problem by considering both elastic

buckling and the plastic buckling behavior of these loaded problems.

1.2 Literature Review

In the following, a literature review on the bucking of rectangular plates is presented to

provide the background information for the present investigation. The review focuses on

homogenous, isotropic, thin plates. Studies on sandwich, composite and orthotropic

plates are not covered.

1.2.1 Elastic buckling of rectangular plates

This part is concerned with the research done for the elastic buckling of rectangular

plates under various in-plane loads and boundary conditions for the plate edges.

Navier (1822) derived the basic stability equation for rectangular plates under lateral

load by including the twisting action. The inclusion of the ‘twisting’ term is very

important because the resistance of the plate to twisting can considerably reduce

deflections under lateral load. Saint-Venant (1883) modified the equation by including

axial edge forces and shearing forces. The modified equation formed the basis for much

of the work on plate stability of plates with various loads and boundary conditions.


• Buckling of plates under uniaxial compression

The most basic form of plate buckling problem is a simply supported plate under

uniaxial compression. Bryan (1891) gave the first solution for the problem by using the

energy method to obtain the values of the critical loads. He assumed that the deflection

surface of the buckled plate could be represented by a double Fourier series. Timoshenko

(1925) used another method to solve the problem. He assumed that the plate buckled into

several sinusoidal half waves in the direction of compression. When satisfying the

boundary conditions, the equations formed a matrix problem which upon solving yields

the critical load. The problem was discussed in many standard textbooks such as

Timoshenko and Gere (1961) and Bulson (1970).

Apart from simply supported plates, Timoshenko (1925) explored the buckling of

uniformly compressed rectangular plates that are simply supported along two opposite

sides perpendicular to the direction of compression and having various edges along the

other two sides. The various boundary conditions considered include SSSS, CSCS, FSSS,

FSCS, CSES (S - simply supported edge, F - free edge, C - clamped or built-in edge and

E - elastically restrained edge). The theoretical results were in good agreement with

experimental results obtained by Bridget et al. (1934). Lundquist and Stowell (1942) used

the integration method to solve ESES plates by assuming that the surface deflection was

the sum of a circular arc and a sine curve. They also discussed the critical load of ESFS

plates by both integration method and the energy method by assuming that transverse

deflection was the sum of a straight line and the cantilever deflection curve. Schleicher

(1931) gave the theoretical solution by using the integration method for CSCS plate with

the loaded edges clamped. The earliest accurate solution available is due to Levy (1942)


for the case of CCCC plate with one direction uniaxial compression. He regarded the

plate as simply supported, and then made the edge slopes equal to zero by a suitable

distribution of edge-bending moments. Bleich (1952) obtained the critical load for the

ESES plates with loaded edges elastically restrained. The results are for the symmetric

mode only and values of aspect ratio are less than 1.0.

For the elastic buckling of rectangular plates with linearly varying axial compression

there is no exact analytical solution. For these cases, recourse is made by considering the

energy or similar method, based on an assumed deflected form. The best-known analysis

for simply supported plates is due to Timoshenko and Gere (1961), who employed the

principle of conservation of energy and assumed the buckled form of the plate consisted

of several half-waves in the loading direction. Kollbrunner and Hermann (1948)

examined the CSSS plates. They found when the clamped edge is on the tension side of

the plate, the critical load factors do not differ greatly from those with both edges simply

supported. Schuette and Mcculloch (1947) employed the Lagrangian multiplier to solve

the buckling problem of ESSS plates. Walker (1967) used the Galerkin’s method to give

accurate values of critical load for a number of the edge conditions as mentioned before.

He also studied the case of ESFS plates. Xiang et al. (2001) considered the elastic

buckling of a uniaxially loaded rectangular plate with an internal line hinge. Using the

Levy’s method, they succeeded in presenting the exact solution for many different

boundary conditions such as SSSS, FSFS, CSCS, FSSS and SSCS plates.

• Buckling of plates under biaxial compression

Bryan (1891) first considered the SSSS plates under biaxial compressions by assuming

that the deflection could be written as a double Fourier series. Wang (1953) solved the


same problem by finite-difference method. Timoshenko and Gere (1961) solved the

CCCC plates under two-direction loads by the energy method. Bulson (1970) cited many

research works on the buckling problem of plates under biaxial compressions. One

example is a rigorous analysis for ESFS plates by using the exact solution of the

differential equation of equilibrium. An extra term in the equation of equilibrium was

added to allow for the transverse force. It is found that the effect of a restraint along one

side ranged between simply supported and clamped boundary condition. Another

example is for examining the FSFS plates by using two buckling forms, i.e. symmetric

and anti-symmetric forms. It is worth noting that the buckling loads associated with the

symmetric buckling form were much lower than those of anti-symmetric form.

Xiang et al. (2003) used the Ritz method to solve the buckling problem of rectangular

plates with an internal line hinge under both uniaxial and biaxial loads. The buckling

factors are generated for rectangular plates of various aspect ratios, hinge locations and

support conditions.

• Buckling of plates under in-plane shear forces

Wang (1953) and Timoshenko and Gere (1961) applied the energy method to solve the

buckling problem of SSSS plates under in-plain shear forces. Since it is not possible to

make assumptions about the number of half-waves, Timoshenko assumed that the

deflection surface was taken in the form of infinite series. Timoshenko and Gere (1961)

studied further to consider SSCC plates and also the behavior of an infinitely long plate

subjected to shear forces. Lundquist and Stowell (1942) examined the ESES plates by

employing the energy method, and also the exact analysis to solve the differential


equation of equilibrium. More recently, Reddy (1999) applied the Rayleigh-Ritz

approximation to solve the CCCC plates under shear forces.

• Buckling of plates under combined loads

Batdorf and Stein (1947) evaluated the buckling problem under combined shear and

compression combinations for simply supported plates by adopting the deflection

function in the form of infinite series. Batdorf and Houbolt (1946) gave a solution to the

equation of equilibrium for infinitely long plates with restrained edges under shear and

uniform transverse compression. Johnson and Buchert (1951) used the energy method to

explore the buckling behavior of rectangular plates with compression edge simply

supported or elastically restrained, tension edge simply supported. Researchers who are

interested in this field of research may refer to Bulson (1970), in which many research

papers were cited. More recently, Kang and Leissa (2001) presented exact solutions for

the buckling of rectangular plates having two opposite, simply supported edges subjected

to linearly varying normal stresses causing pure in-plane moments, the other two edges

being free.

• Buckling of plates under body forces

Farvre (1948) is probably the first researcher to work out approximate buckling

solutions of rectangular plates under selfweight and uniform in-plane compressive forces.

However, he treated only plates with all four edges simply supported. Wang and Sussman

(1967) solved the same problem using the Rayleigh-Ritz method and concluded that the

average stress in the plate at buckling is less than that for a plate with uniform

compression at buckling. Both Favre (1948) and Wang and Sussman (1967) did not give

numerical values in their papers. Using the conjugate load-displacement method, Brown


(1991) investigated the buckling of rectangular plates under (a) a uniformly distributed

load, (b) a linearly increasing distributed load and (c) a varying sinusoidal load across the

plate width. The second type of load is equivalent to the plate’s selfweight. In his study,

Brown treated a number of combinations of boundary conditions. More recently, Wang et

al. (2002) considered the buckling problem of vertical plates under body

forces/selfweight. The vertical plate is either clamped or simply supported at its bottom

edge while its top edge is free. The two sides of the plate may either be free, simply

supported or clamped. Xiang et al. (2003) treated yet another new elastic buckling

problem where the buckling capacities of cantilevered, vertical, rectangular plates under

body forces are computed.

• Buckling of plates under other forms of loads

Bulson cited Yamaki’s buckling studies on SSSS, CSCS and CCCC plates under equal

and opposite point loads as shown in Fig. 1.1a. Bulson (1970) also cited Yamaki’s

research on buckling problems of CSCS and SSSS plates under partially distributed

loads which are acted upon the simply supported edges as shown in Fig. 1.1b. Lee et al.

(2001) considered the elastic buckling problem of square EEEE and ESES plates

subjected to in-plane loads of different configurations acting on opposite sides of plates

as shown in Figs 1.1c and 1.1d. The effects of Kinney’s fixity factor (introduced to

describe the support conditions at the edges covering the boundary conditions of simply

supported and fixed edges) and the width factor on critical load factors were treated.


1.2.2 Plastic buckling of rectangular plates

This part is concerned with the development of the plastic stability theories.

Incremental theory of plasticity (IT) and the deformation theory of plasticity (DT) are

considered in detail. As an alternative method, the strip method is also briefly reviewed.

The earliest development of DT is due to Engesser (1895) and Von Karman (1910).

They developed a theory based on the fact that for a fiber which is compressed beyond

the elastic limit, the tangent modulus (i.e. the ratio of the variation of strain to the

corresponding variation of stress) assumes different values depending on whether the

variation of stress constitutes an increase or a relief of the existing compressive stress.

q q

(a)

Fig. 1.1 Buckling of plates under (a) point loads; (b) partially distributed loads; (c) patch loads at edge center; (d) patch loads near corners.

q

Lχ

L

χ/q

(d)

(b)

q q

simply supported

q

Lχ5.0

Lχ5.0

L

L

χ/q

(c)


Bleich (1924) and Timoshenko (1936) applied Engesser-Von Karman theory to the

plastic buckling of plates by introducing the “reduced modulus” into the formulas for the

elastic buckling of plates. The results of their theory were obtained in the case of a

narrow rectangular strip with its compressed short edge simply supported and the long

edges free.

Kaufmann (1936) and Ilyushin (1944) developed the basis of deformation theory of

plasticity by presenting another route for application of Engesser-Von Karman theory.

They went back to the considerations by which the reduced modulus was derived and

applied to the case of buckled a plate. Ilyushin (1946) reduced the problem to the solution

of two simultaneous nonlinear partial differential equations of the fourth order in the

deflection and stress function, and in the approximate analysis to a single linear equation.

Solutions were given for the special cases of a rectangular plate buckling into a

cylindrical form, and of an arbitrarily shaped plate under uniform compression. Stowell

(1948) assumed that the plate remained in the purely plastic state during buckling. He

used Ilyushin’s general relations to derive the differential equation of equilibrium of

plates under combined loads. The corresponding energy expressions were also found.

Bijlaard (1949) also used the assumption of “plastic deformation”. 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. The

differential equation for plate buckling was derived and results of its application to

several kinds of loading and boundary conditions were given. El-Ghazaly and Sherbourne

(1986) employed the deformation theory for the elastic-plastic buckling analysis of plates

under non-proportional external loading and non-proportional stresses. Loading,


unloading, and reloading situations were considered. Comparison between experiments

and analysis results showed that the deformation theory of plasticity was applicable in

situations involving plastic buckling under non-proportional loading and non-uniform

stress fields.

The incremental theory of plasticity was first developed in the early work by

Handelman and Prager (1948). They assumed that for a given state of stress there existed

a one-to-one correspondence between the rates of change of stress and strain in such a

manner that the resulting relation between stress and strain cannot be integrated so as to

yield a relation between stress and strain along. Pearson (1950) modified Handelman and

Prager’s assumption of initial loading. His analytical results showed that the incremental

was improved by incorporating Shanley’s concept of continuous loading.

Deformation theory and incremental theory of plasticity are two competing plastic

theories. Consequently much work and comparison studies have been done by using both

of them. Shrivastava (1979) analyzed the inelastic buckling by including the effects of

transverse shear by both theories. Three cases were discussed: (1) for infinitely long

simply supported plates, (2) for square simply supported plates, and (3) for infinitely long

ones simply supported on three sides and free on one unloaded edge. Ore and Durban

(1989) presented a linear buckling analysis for annular elastoplastic plates under shear

loads. They found that deformation theory predicts critical loads which were considerably

below the predictions obtained with the flow theory. Furthermore, comparison with

experimental data for different metals showed a good agreement with the deformation

theory. Tugcu (1991) employed both theories for simply supported plates under biaxial

loads. It was shown that the incremental theory predictions for the critical buckling stress


were susceptible to significant reductions due to a number of factors pertinent to testing,

while the deformation theory analysis was shown to be more or less insensitive to all of

these factors. Durban and Zuckerman (1999) examined the elastoplastic buckling of a

rectangular plate with three sets of boundary conditions (four simply supported

boundaries and the symmetric combinations of clamped/simply supported sides). It was

found that for thicker plates, the deformation theory gives lower critical stresses than

those obtained from the incremental theory.

There is a general agreement among engineers and researchers that (a) deformation

theory is physically less correct than incremental theory, but (b) deformation theory

predicts buckling loads that are smaller than those obtained with incremental theory, and

(c) experimental evidence points in favor of deformation theory results. Onat and

Drucker (1953) through an approximate analysis showed that incremental theory

predictions for the maximum support load of long plates supported on three sides will

come down to the deformation theory bifurcation load if small but unavoidable

imperfections were taken into account. Later, the plate buckling paradox was examined

by Sewell (1963) who obtained somewhat lower flow theory buckling loads by allowing

a variation in the direction of the unit normal. Sewell (1973) in a subsequent study

illustrated that use of Tresca yield surface brings about significant reductions in the

buckling loads obtained using incremental theory. Neale (1975) examined the sensitivity

of maximum support load predictions to initial geometric imperfections, using

incremental theory. A similar study was performed by Needleman and Tvergaard (1976)

which also included the effect of in-plane boundary conditions for square plates under

uniaxial compression. An exhaustive discussion of the buckling paradox in general is


given by Hutchinson (1974). While imperfection sensitivity provided a widely accepted

explanation for the buckling paradox in general, reservations concerning the mode and

amplitude of the imposed imperfections for some buckling problems are not uncommon.

Readers who are interested in plastic buckling of plates may obtain further information

from these published papers: Shrivastava (1995), Betten and Shin (2000), Soh et al.

(2000), Chakrabarty (2000), Wang et al. (2002) and Wang (2003).

From the literature review above, we can see that although much work has been done,

the buckling of rectangular plates subjected to end and intermediate loads remain hitherto

untouched. This has prompted the author to work on this project.

1.3 Objectives and Scope of Study

The buckling of rectangular plates with various plate boundary and load conditions has

been studied extensively and there is an abundance of buckling results in the open

literature. However, a new plate buckling problem where a rectangular plate is subjected

to not only end loads, but also an intermediate uniaxial load remains to be studied.

The aim of the study is to determine the buckling factors of rectangular plates under

intermediate and end loads. The considered plates have two opposite simply supported

edges that are parallel in direction to the applied uniaxial loads while the other two

remaining edges may take any other combinations of clamped, simply supported and free

edge. Both elastic theory and plastic theories including incremental theory (IT) and

deformation theory (DT) are used to explore the problem. Further the study investigates

the effects of various plate aspect ratios, intermediate load positions, boundary conditions,


and material properties on the buckling factors. In the plastic buckling of plates, the

differences between results by IT and DT are examined.

1.4 Outline of Thesis

In this Chapter 1, the background information, literature review, objectives and scope

of the study are presented.

In Chapter 2, the governing equations are derived for both elastic and plastic buckling

of rectangular plates under uniaxial end loads. Equations for various boundary conditions

are also presented.

Chapter 3 is concerned with the elastic buckling of rectangular plates subjected to

intermediate and end uniaxial in-plane loads. The plate has two opposite simply

supported edges that are parallel to the load direction and the other remaining edges may

take any combination of free, simply supported or clamped condition. The buckling

problem is solved by decomposing the plate into two sub-plates at the location where the

intermediate uniaxial load acts. Each sub-plate buckling problem is solved exactly using

the Levy approach and the two solutions brought together by matching the continuity

equations at the interfacial edge. There are five possible solutions for each sub-plate and

consequently there are twenty-five combinations of solutions to be considered. The

effects of various aspect ratios, intermediate load positions and boundary conditions are

investigated.

In Chapter 4 we extend the elastic buckling problem to the more practical plastic

buckling of plates. Both the Incremental Theory of Plasticity and the Deformation Theory

of Plasticity are considered in bounding the plastic behavior of the plate. In contrast to the


five possible solutions for the elastic problem, there exist eight possible solutions for each

sub-plate. Consequently, there are sixty-four combinations of solutions to be considered

for the entire plate. The solution combination depends on the aspect ratios, the

intermediate load positions, the intermediate to end load ratios, the material properties

and the boundary conditions. The effects of the aforementioned parameters and the

adoption of DT and IT on the buckling factors are also investigated.

Finally, Chapter 5 summarizes the main research findings in conclusions. Suggestions

for future investigations are also provided.

BUCKLING OF PLATES UNDER END LOADS

This chapter presents the governing equations for the elastic buckling and plastic

buckling of thin rectangular plates under uniaxial end loads. For plastic buckling of plates,

we consider two competing theories of plasticity, namely the deformation theory of

plasticity (DT) and the incremental theory of plasticity (IT).

2.1 Elastic Buckling Theory

2.1.1 Derivation of differential equation for elastic buckling

Consider a rectangular thin plate of length a, width b, and thickness h, subjected to

uniaxial compressive loads xN as shown in Fig. 2.1. Adopting the rectangular Cartesian

coordinates x, y, z, where x and y lie in the middle plane of the plate and z is pointing

downward from the middle plane, the uniaxial load xN is parallel to x axis.

The simplest plate theory is that proposed by Kirchhoff (1850). The assumptions for

the Kirchhofff plate theory are:

(a) deflections are small (i.e. less than the thickness of the plate),

(b) the middle plane of the plate does not stretch during bending, and remains a

neutral surface, analogous to the neutral axis of a beam,

Chapter 2

Chapter 2 Buckling of Plates under End Loads 16

(c) plane sections rotate during bending to remain normal to the neutral surface, and

do not distort, so that stresses and strains are proportional to their distance from

the neutral surface,

(d) the loads are entirely resisted by bending moments induced in the elements of the

plate and the effect of shearing forces is neglected,

(e) the thickness of the plate is small compared with other dimensions.

Based on the foregoing assumptions, the displacement field could be expressed as

xwzzyxu∂∂

−=),,( , (2.1a)

ywzzyxv∂∂

−=),,( , (2.1b)

),(),,( yxwzyxw = , (2.1c)

where ),,( wvu are the displacement components along the (x, y, z) coordinate directions,

respectively, and w is the transverse deflection of a point on the mid-plane (i.e., z = 0).

The non-zero linear strains associated with the displacement field are

2

2

xwz

xu

xx ∂∂

−=∂∂

=ε (2.2a)

2

2

ywz

yv

yy ∂∂

−=∂∂

=ε (2.2b)

yxwz

xv

yu

xy ∂∂∂

−=∂∂

+∂∂

=2

2γ (2.2c)

where ),( yyxx εε are the normal strains and xyγ is the shear strain.

The virtual strain energy U of the Kirchhoff plate theory is given by (see Ugural, 1999)


( ) dxdydzUh

h xyxyyyyyxxxx∫ ∫Ω −

++=

0

2/

2/δγσδεσδεσδ

dxdyyxwM

ywM

xwM xyyyxx∫Ω

∂∂

∂+

∂∂

+∂∂

−=0

2

2

2

2

2

2 δδδ (2.3)

where 0Ω denotes the domain occupied by the mid-plane of the plate, ( )yyxx σσ , the

normal stresses, xyσ the shear stress, and ( )xyyyxx MMM ,, the moments per unit length, as

shown in Fig. 2.2. Note that the virtual strain energy associated with the transverse shear

strains is zero as 0== xzyz γγ in the Kirchhoff plate theory.

The relationship between the moments and stresses are given by

zdzMh

h xxxx ∫−=2/

2/σ (2.4a)

zdzMh

h yyyy ∫−=2/

2/σ (2.4b)

zdzMh

h xyxy ∫−=2/

2/σ . (2.4c)

The work W done by the uniaxial load xN , due to displacement w only, equals (see

Ugural, 1999)

dxdyxwNW x

2

021

∂∂

−= ∫Ω . (2.5)

The virtual work Wδ due to the uniaxial load xN is given by

dxdyxw

xwNW x ∂∂

∂∂

= ∫Ωδδ

0

. (2.6)

The principle of virtual displacements requires that 0=−=∏ WU δδδ , i.e.

020

2

2

2

2

2

=

∂∂

∂∂

+∂∂

∂+

∂∂

+∂∂

−=∏ ∫Ω dxdyxw

xwN

yxwM

ywM

xwM xxyyyxx

δδδδδ (2.7)


By using the divergence theorem, one obtains

wdxdyxwNMMM xyyyyxyxyxxxx δδ ∫Ω

∂∂

−++−=∏0

2

2

,,, 2

( ) ( ) dsywnMnM

xwnMnM yyyxxyyxyxxx∫Γ

∂∂

++∂∂

+−δδ

( ) 0,,,, =

++

∂∂

−++ ∫Γ wdsnMMnxwNMM yxxyyyyxxyxyxxx δ (2.8)

where a comma followed by subscripts denotes differentiation with respect to the

subscripts, i.e., x

MM xxxxx ∂

∂=, , and so on, ( )yx nn , denote the direction cosines of the unit

normal n) on the boundary Γ , and ds denotes the incremental length along boundary. If

the unit normal vector n) is oriented at an angle θ from the positive x-axis, then

θcos=xn and θsin=yn . Since wδ is arbitrary in 0Ω , and it is independent of xw ∂∂ /δ ,

and yw ∂∂ /δ on the boundary Γ , it follows that

02 2

2

2

22

2

2

=∂∂

−∂

∂+

∂∂∂

+∂

∂xwN

yM

yxM

xM

xyyxyxx in 0Ω . (2.9)

Eq. (2.9) represents the equilibrium equation of the Kirchhoff plate theory for rectangular

plates under uniaxial load xN .

Assuming the material of the plate to be isotropic and obeys Hooke’s law, then the

stress-strain relations are given by

( )yyxxxxE νεεν

σ +−

= 21 (2.10a)

( )xxyyyyE νεεν

σ +−

= 21 (2.10b)


xyxyxyEG γν

γσ)1(2 +

== (2.10c)

where E denote the Young’s modulus, G the shear modulus, and ν the Poisson’s ratio. By

substituting Eqs. (2-10) into Eqs. (2.4) and carrying out the integration over the plate

thickness, one obtains

( )

∂∂

+∂∂

−=+−

== ∫∫ −− 2

2

2

22/

2/2

2/

2/ 1 yw

xwDzdzEzdzM

h

h yyxx

h

h xxxx ννεεν

σ (2.11a)

( )

∂∂

+∂∂

−=+−

== ∫∫ −− 2

2

2

22/

2/2

2/

2/ 1 xw

ywDzdzEzdzM

h

h xxyy

h

h yyyy ννεεν

σ (2.11b)

yxwDzdzGzdzM

h

h xy

h

h xyxy ∂∂∂

−−=== ∫∫ −−

22/

2/

2/

2/)1( νγσ (2.11c)

where D is the flexural rigidity )1(12 2

3

ν−=

EhD . (2.12)

By substituting Eq. (2.11-13) into Eq. (2.7), the governing equation for buckling of

plate subjected to a uniaxial load is obtained:

02 2

2

4

4

22

4

4

4

=∂∂

+

∂∂

+∂∂

∂+

∂∂

xwN

yw

yxw

xwD x (2.13)

2.1.2 Boundary conditions

We take the boundary conditions that apply along the edge x = a of a rectangular plate

with edges parallel to the x and y axes as examples to explain the boundary conditions for

rectangular plates.

Clamped Edge (C)

In this case both the deflection and slope must vanish along the edge x=a, that is


0=w and 0=∂∂

xw (2.14a,b)

Simply Supported Edge (S)

Along the simply supported edge x = a, the deflection and the bending moment are both

zero. Hence

0=w and 02

2

2

2

=

∂∂

+∂∂

−=yw

xwDM xx ν (2.15a,b)

Free Edge (F)

Such an edge is free of moment and vertical shear force along the edge x=a. That is

02

2

2

2

=

∂∂

+∂∂

−=yw

xwDM xx ν (2.16a)

Because the plate is under axial load xN which is parallel to the x axis, and we assume

that compressive force as positive, the effective vertical shear force along the edge x=a is

xwN

yM

QV xxy

xx ∂∂

−∂

∂+=

0)2( 2

3

3

3

=∂∂

−

∂∂∂

−+∂∂

−=xwN

yxw

xwD xν (2.16b)

2.2 Plastic Buckling Theory

2.2.1 Derivation of constitutive relations based on Hencky’s deformation theory

Consider a thin rectangular plate in which the material is bounded between the planes

2hz ±= . The bounding planes are unstressed, while uniform compressive stresses of

magnitudes hN 11 σ= act in the x- directions, to represent the plastic state. If the


transverse shear rates on the incipient deformation mode at bifurcation are disregarded,

the admissible velocity field may be written as

xwzu∂∂

−= , (2.17a)

ywzv∂∂

−= , (2.17b)

ww = , (2.17c)

where ),,( wvu are the displacement components along the (x, y, z) coordinate directions,

respectively, and w is the transverse deflection of a point on the mid-plane (i.e., z=0).

It is assumed that the relationship between the stress rate and the rate of deformation at

the point of bifurcation is that corresponding to the incremental form of the DT suggested

by Hencky. Since the strain rate vector in that 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 parameter σ in this modified

theory is simply a measure of the length of the current deviatoric stress vector, rather than

that of the radius of an isotropically expanding Mises cylinder. The incremental form of

the Hencky equation ijpp

ij s)2/3( σεε = is easily found as:

ij

p

ij

ppp

ij dssdddd

σε

σε

σε

σσε

23

23

+

−= (2.18)

where ijs is the deviatoric stress vector, and ijds is its time incremental, which must be

considered in the Jaumann sense, so that it vanishes in the event of an instantaneous rigid

body rotation. The elastic strain increment, given by the generalized Hooke’s Law, is


kkijijeij d

Eds

Ed σδννε

3211 −

+

+

= (2.19)

where ijδ is the Kronecker delta.

Combining Eq. (2.18) and Eq. (2.19), the rate form of the complete stress-strain

relation is obtained as:

ijkkijijij sSE

TEs

SEE

−+

−+

−

−=σσσδννε

23

321

221

23 &

&&& . (2.20)

during the continued loading of a plastically stressed element.. In the above, T is the

tangent modulus equal to εσ dd / , and S is the secant modulus equal to εσ / , where ε is

the total effective strain which is equal to pe εε + .

Let 1σ− denote the non-zero principal stresses whose directions coincide with the x

axis, at the point of bifurcation. Since the effective σ is given by

2222 3 xyyyyyxxxx σσσσσσ ++−= , (2.21)

a straightforward differentiation gives

211

22

σσσσσ

σσ yyxx ddd −

−= (2.22)

on setting 1σσ −=x , 0=xyτ , 0=yσ at bifurcation. The constitutive Eq. (2.20) therefore

furnishes

( ) ( )

−++

−+−

−

−= yyxxyyxxyyxxxxvv

SEE σσσσσσε &&&&&&&

21

3212

321

21 (2.23a)

( ) ( )

−++

−+−

−

−= yyxxyyxxxxyyyyvv

SEE σσσσσσε &&&&&&&

21

3212

321

21 (2.23b)

( ) xyxy vSEE σε &&

−−= 213

21 . (2.23c)


After some algebraic manipulations, the first two results are reduced to

yyxxxxT σησε &&& −= (2.24a)

xxyyyy STT σησ

σσε &&& −

−−= 2

211

431 (2.24b)

whereη is the contraction ratio at the current state of stress. On using the expression

221 σσ = , the above relations can be inverted to give the constitutive relations in the form

( )yyxxxx E εβεασ &&& += , (2.25a)

( )yyxxyy E εγεβσ &&& += , (2.25b)

( )[ ]1/322

−+=

SEvE xy

xy

εσ

&& (2.25c)

where

( ) ( )

−−−+=

ETv

SE νρ 212213 . (2.26a)

−−=

ST1341

ρα (2.26b)

( )

−−=

ETν

ρβ 21221 (2.26c)

ργ 4= (2.26d)

Assuming that the plate material obeys the Ramberg-Osgood constitutive law,

+=

−1

0

1c

kE σ

σσε (2.27)

where 0σ is a nominal yield stress, c is a dimensionless constant that describes the shape

of the stress-strain relationship with c = ∞ for elastic-perfectly plastic response, k 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. 2.3.

By differentiating both sides of Eq. (2.27), and considering that T is the tangent

modulus equal to εσ dd / , and S is the secant modulus equal to εσ / , one obtains

1

0

1−

+=

c

kSE

σσ , (2.28a)

1

0

1−

+=

c

ckTE

σσ . (2.28b)

2.2.2 Derivation of constitutive relations based on Prandtl-Reuss material

For a Prandtl-Reuss material, the plastic strain rate vector, in a nine dimensional space,

is directed along the deviator stress vector. Stated mathematically, the flow rule is

ijijij

pijp

ij sET

sH

s

−===

1123

23

23

σσ

σσ

σε

ε&&&

& (2.29)

since ETd

ddd

ddd

dd

H

eep 111−=−=

−==

σε

σε

σεε

σε . (2.30)

The complete Prandtl-Reuss equation relating the stress rate to the strain rate is given

by

ijijkkijij sTEsE

−+

−

++= 123

321)1(

σσδσννε&

&&& (2.31)

This equation may be compared with Eq. (2.20), which evidently reduces Eq. (2.31) on

setting S = E in the first and last terms on the right hand side. By using similar method of


derivation of the biaxial constitutive relations that employed in section 2.2.2, we can get

the constitutive relations based on the Prandtl-Reuss material in the form

( )yyxxxx E εβεασ &&& += , (2.32a)

( )yyxxyy E εγεβσ &&& += , (2.32b)

νε

σ+

=1

xyxy

E && (2.32c)

where

( ) ( )

−−−+=

ETvv 212213ρ . (2.33a)

−−=

ST1341

ρα (2.33b)

( )

−−=

ETv21221

ρβ (2.33c)

ργ 4= (2.33d)

2.2.3 Derivation of differential equation for plastic buckling

The expression for the stress rates is

( ) 222 42 xyyyyyxxxxijij GE εεγεεβεαετ &&&&&&& +++= . (2.34)

To obtain the condition for bifurcation of the plate in the elastic/plastic range, consider

the uniqueness criterion in the form

02 >

∂∂

∂∂

−−∫ dVxv

xv

j

k

i

kjkijijijij εεσεσ &&&& (2.35)


Since the only nonzero components of the stress tensor are 1σσ −=xx only, which are

small compared to the modulus of elasticity E, the condition for uniqueness becomes

∫ >

∂∂

+

∂∂

− 022

1 dVxw

xv

ijij σεσ && (2.36)

In view of Eqs. (2.17a-c), the strain rates and the velocity gradients are given by

2

2

xwzxx ∂

∂−=ε& , (2.37a)

2

2

ywzyy ∂

∂−=ε& , (2.37b)

yxwzxy ∂∂

∂−=

2

22ε& , (2.37c)

yxwz

xv

∂∂∂

−=∂∂ 2

, (2.37d)

Inserting Eq. (2.37a-d) into the inequality Eq. (2.36), and integrating through the

thickness of the plate, the condition for uniqueness is given by

∫∫

∂∂

∂+

∂∂

+∂∂

∂∂

+

∂∂ dxdy

yxw

EG

yw

yw

xw

xwh

22

2

2

2

2

2

22

2

22 412

γβα

02

1 >

∂∂

− ∫∫ dxdyxw

Eσ . (2.38)

All the integrals appearing in the above expression extend over the middle surface of the

plate. The Euler-Langrange differential equation, associated with the minimization with

respect to arbitrary variations of w, is easily shown to be

( ) 2

21

24

4

22

4

4

4 122xw

Ehyw

yxw

xw

∂∂

−=∂∂

+∂∂

∂++

∂∂ σγµβα , (2.39)


where )1/(1 νµ += . The solution to the bifurcation problem is therefore reduced to the

solution of the differential equation (2.39) under appropriate boundary conditions. It is

worth noting that Eq. (2.39) is applicable for both DT and IT, with different definition

forα , β , γ , and ρ . When the bifurcation occurs in the elastic range (S = T = E), we

have )1/(1 2νγµβα −==+= , and Eq. (2.39) reduces to the well-known governing

equation for elastic buckling as given in Eq. (2.13).

2.2.4 Boundary conditions

In order to establish the static boundary conditions in terms of w, it is convenient to

take the components of the nominal stress rate ijs& as approximately equal to those of ijτ& .

This is justified by the fact that the stresses at bifurcation will be small compared to the

elastic and plastic moduli. In view of the relation Eqs. (2.25) and Eqs. (2.37), the rates of

change of the resultant bending and twisting moments per unit length are given by

∂∂

+∂∂

−== ∫− 2

2

2

232/

2/ 12 yw

xwEhzdzM

h

h xxxx βασ&& , (2.40a)

∂∂

+∂∂

−== ∫− 2

2

2

22/

2/

3

12 yw

xwEhzdzM

h

h yyyy γβσ&& , (2.40b)

∫− ∂∂∂

+−==

2/

2/

23

)1(12h

h xyxy yxwEhzdzM

νσ&& . (2.40c)

We take the boundary conditions that apply along the edge x=a of a rectangular plate

with edges parallel to the x and y axes as examples to explain the boundary conditions for

rectangular plates.

Clamped Edge (C)

In this case both the deflection and slope must vanish along the edge x=a, that is


0=w and 0=∂∂

xw (2.41a,b)

Simply Supported Edge (S)

Along the simply supported edge x=a, the deflection and the bending moment rate must

vanish. Hence

0=w and 012 2

2

2

23

=

∂∂

+∂∂

−=yw

xwEhM xx βα& (2.42a,b)

Free Edge (F)

Such an edge is free of moment and vertical shear force along the edge x=a. That is

012 2

2

2

23

=

∂∂

+∂∂

−=yw

xwEhM xx βα& (2.43a)

Because the plate under axial stress xσ which is parallel to the x axis, and we assume that

compressive stress as positive, the effective vertical shear force along the edge x=a is

xwN

yM

QV xxy

xx ∂∂

−∂

∂−=

( ) 0212 2

3

3

33

=∂∂

−

∂∂∂

++∂∂

−=xwh

yxw

xwEh

xσµβα . (2.43b)


a

z

xN

y

x

xN

b

h

Figure 2.1 Thin rectangular plate under end uniaxial load

x

y z

yyNyQ yyM

xyMyxNxQ

xyMxxM

xxN

xyN

nnNnQ

nsN

nnMnsM

nQnnN nnM

nsM

nsN

xxNxxM

xyM

xyN

xQ

yxM

yyM

yQ yyN

yxN

Figure 2.2 Stress resultants on a plate element. The in-plane resultants yyxx NN , and xyN do not enter the equations in the pure bending case, and they are the specified forces in a buckling problem.


Fig. 2.3 Ramberg-Osgood stress-strain relation

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 1 2 3 4

C

o

o

EE

+=

σσσ

ασ

ε

o

Eσε

oσσ

∞=c

2 0c =

1 0c =

5c =3c =2c =

α+1

c

Ek

E

+=

0

0

σσσσε

k+1

ELASTIC BUCKLING OF PLATES UNDER


This chapter is concerned with the elastic buckling of rectangular plates subjected to

intermediate and end uniaxial inplane loads, whose direction is parallel to two simply

supported edges. The aforementioned buckling problem is solved by decomposing the

plate into two sub-plates at the location where the intermediate uniaxial load acts. Each

sub-plate buckling problem is solved exactly using the Levy approach and the two

solutions brought together by matching the continuity equations at the interfacial edge. It

is worth noting that there are five possible solutions for each sub-plate and consequently

there are twenty-five combinations of solutions to be considered. For different boundary

conditions, the buckling solutions comprise of different combinations. For each boundary

condition, the correct solution combination depends on the ratio of the intermediate load

to the end load. The exact stability criteria, presented both in tabulated and in graphical

forms, should be useful for engineers designing walls or plates that have to support

intermediate floors/loads.

Chapter 3

Chapter 3 Elastic Buckling of Plates under Intermediate and End Loads 32

3.1 Mathematical Modeling

3.1.1 Problem definition

Consider an isotropic, rectangular thin plate with two simply supported edges that are

parallel to the uniaxial inplane load direction as shown in Fig. 3.1. The other two sides of

the plate may take any combination of free, simply supported and clamped edges. The

plate is of length a, width b, thickness h, modulus of elasticity E, and Poisson’s ratio ν .

The plate is subjected to an end uniaxial inplane load 1N at the edge x = 0 and an

intermediate uniaxial inplane load 2N at the location ax χ= . The problem at hand is to

determine the buckling load for such a loaded plate.

3.1.2 Method of solution

The plate is first divided into two sub-plates. The first sub-plate is to the left of the

vertical line defined by ax χ= and the second sub-plate is to the right of this line.

Adopting the coordinate systems as shown in Figs. 3.1b and 3.1c, the governing buckling

equation (see Eq. (2.13)) for each sub-plate based on the thin plate theory may be

canonically written as Eq. (3.1)

2,1,02 2

2

4

44

22

42

4

4

==∂∂

+∂∂

+∂∂

∂+

∂∂ i

xw

yw

xyw

xw

i

ii

i

ii

ii

ii

i

i λξξ (3.1)

in which

( )

( )

( ) ( )D

aNNDaN

ba

ba

byy

axx

axx

bww i

ii

i

22

212

22

11

212

21

1

1,

,1,,,1

,,

χλχλ

χξχξχχ

−+==

−===

−===

(3.2a-h)

w is the transverse displacement at the midplane of the plate, and )]1(12/[ 23 ν−= EhD


the flexural rigidity of the plate.

The essential and natural boundary conditions for the two simply supported edges at

0=iy and 1=iy associated with the i-th sub-plate are given by

0=iw (3.3)

02

2

2

2

2 =∂∂

+∂∂

i

i

i

i

i yw

xw

ξν (3.4)

By using the Levy approach, the transverse displacement of the i-th sub-plate may be

expressed as

( ) ( ) 2,1,sin, == iymxAyxw iiimiii π (3.5)

where m (= 1, 2, …, ∞ ) is the number of half waves of the buckling mode in the y

direction. The transverse displacement given in Eq. (3.5) satisfies the boundary

conditions of the two parallel simply supported edges as given by Eqs. (3.3) and (3.4).

In view of Eq. (3.5), the partial differential equations in Eq. (3.1) may be reduced to a

fourth-order ordinary differential equations as

2,1,02 2

2444

2

2222

4

4

==++− ixdAdAm

xdAdm

xdAd

i

imiimi

i

imi

i

im λπξπξ (3.6)

Depending on the roots of the characteristics equation of the differential equation,

there are five general solutions to the above fourth order differential equation as given

below.

Solution A (for 2,1,0 =< iiλ )

iiieiii

eiii

eiii

eii ymCxCxCxCxw πψψφφ sin)coshsinhcosh(sinh 4321 +++= (3.7a)


in which

iiiiiei mm λπξλπξλφ 2222222 4

21)2(

21

−+−−= (3.7b)

iiiiiei mm λπξλπξλψ 2222222 4

21)2(

21

−−−−= (3.7c)

Solution B (for 2,1,0 == iiλ )

iiieiiii

eiii

eiiii

eii ymCxxCxCxxCxw πφφφφ sin)coshcoshsinh(sinh 4321 +++= (3.8a)

in which

πξφ miei = (3.8b)

Solution C (for 2,1,40 222 =<< imii πξλ )

321 sinsinhcoscoshcos(sinh iieii

eiii

eii

eiii

eii

eii CxxCxxCxxw ψφψφψφ ++=

iiieii

ei ymCxx πψφ sin)sincosh 4+ (3.9a)

in which

4

222 ii

ei m λπξφ −= , 2/i

ei λψ = (3.9b,c)

Solution D (for 2,1,4 222 == imii πξλ )

iiieiiii

eiii

eiiii

eii ymCxxCxCxxCxw πφφφφ sin)sinsincos(cos 4321 +++= (3.10a)

in which

πξφ miei = (3.10b)


Solution E (for 2,1,4 222 => imii πξλ )

iiieiii

eiii

eiii

eii ymCxCxCxCxw πψψφφ sin)sincossin(cos 4321 +++= (3.11a)

in which

iiiiiei mm λπξλπξλφ 2222222 4

21)2(

21

−−−= (3.11b)

iiiiiei mm λπξλπξλψ 2222222 4

21)2(

21

−+−= (3.11c)

To solve the buckling problem of the rectangular plate that consists of two sub-plates,

twenty-five combinations of the solutions must be considered. The designated

combinations of solutions are given in Table 3.1.

Table 3.1 Twenty-five combinations of solutions

Solutions for Sub-plate 1 Solution Combinations A B C D E

A Combination 1

Combination 6

Combination 11

Combination 16

Combination 21

B Combination 2

Combination 7

Combination 12

Combination 17

Combination 22

C Combination 3

Combination 8

Combination 13

Combination 18

Combination 23

D Combination 4

Combination 9

Combination 14

Combination 19

Combination 24

Solu

tions

for S

ub-p

late

2

E Combination 5

Combination 10

Combination 15

Combination 20

Combination 25

The eight arbitrary constants )2,1(,,, 4321 =iCCCC iiii in Eqs. (3.7-3.11) are to be

determined by the boundary and interfacial conditions. The essential and natural

boundary conditions of the plate at the edge 01 =x and edge 12 =x are defined as

follows.


• For simply supported edges:

⇒= 0iw 0=imA , and (3.12a)

⇒=∂∂

+∂∂ 01

2

2

2

2

2i

i

i

i

i yw

xw ν

ξ 01 22

2

2

2 =−∂

∂im

i

im

i

AmxA πν

ξ, i = 1, 2 (3.12b)

• For clamped edges:

⇒= 0iw 0=imA , and (3.13a)

⇒=∂∂ 0

i

i

xw 0=

i

im

xddA i = 1, 2 (3.13b)

• For free edges:

⇒=∂∂

+∂∂ 01

2

2

2

2

2i

i

i

i

i yw

xw ν

ξ01 22

2

2

2 =− imi

im

i

AmxdAd πν

ξ, and (3.14a)

( ) 0213

2

2

3

3

3

3 =∂∂

+∂∂

∂−+

∂∂

i

i

i

i

ii

i

ii

i

i xw

yxw

xw

ξλ

ξν

ξ

( ) 0213

222

3

3

3 =+−

−⇒i

im

i

i

i

im

ii

im

i xddA

xddAm

xdAd

ξλ

ξπν

ξ (3.14b)

To ensure displacement continuities and equilibrium conditions at the interface of the

two sub-plates, the following essential and natural conditions must be satisfied

⇒=−==

00211

21 xxww 0

021121

=−== xmxm AA , (3.15)

⇒=∂∂

−∂∂

==

011

02

2

211

1

1 21 xx xw

xw

ξξ011

022111 21

=−== x

im

x

im

xddA

xddA

ξξ (3.16)

011

022

22

22

22

221

21

12

21

12

21

21

=

∂

∂+

∂∂

−

∂∂

+∂∂

== xxyw

xw

yw

xw ν

ξν

ξ


011

0

2222

2

221

222

1

2

21

21

=

−−

−⇒

== x

imim

x

imim Am

xdAdAm

xdAd νπ

ξνπ

ξ (3.17)

( )

( ) 021

21

02

232

22

222

23

232

23

32

11

13

1

21

211

13

131

13

31

2

1

=

∂∂

+∂∂

∂−+

∂∂

−

∂∂

+∂∂

∂−+

∂∂

=

=

x

x

xw

yxw

xw

xw

yxw

xw

ξλ

ξν

ξ

ξλ

ξν

ξ

⇒

( )

( ) 021

21

0232

22

22

22

32

3

32

113

1

21

11

22

31

3

31

2

1

=

+

−−

−

+

−−

=

=

x

imimim

x

imimim

xddA

xddAm

xdAd

xddA

xddAm

xdAd

ξλ

ξπν

ξ

ξλ

ξπν

ξ (3.18)

When assembling the sub-plates to form the whole plate via the implementation of the

boundary conditions of the plate along the two edges parallel to the y-axis Eqs. (3.12-14)

and the interface conditions between two sub-plates as given by Eqs. (3.15-18), a system

of homogenous equations is obtained:

[ ] 0=CK (3.19)

in which TCCCCCCCCC 2423222114131211= . For a nontrivial solution, the

determinant of [ ]K must vanish. Each solution combination for the determinant of [ ]K is

examined. The valid solution combinations should satisfy the following requirements:

• The buckling loads satisfy the limits of validity for the solution combinations

which they belong to;

• The buckling load factor is the lowest value among possible solutions; and

• The stability curves are continuous.


3.2 Results and Discussions

The proposed solution procedure is applied to study the buckling behaviour of

rectangular plates subjected to intermediate and end inplane loads. Rectangular plates

with various combinations of edge support conditions and aspect ratios are considered.

The Poisson’s ratio is taken to be 3.0=ν for all calculations. The buckling factors for the

end and intermediate loads are expressed as Λ1 = N1b2/(π2D) and Λ2 = N2b2/(π2D),

respectively.

3.2.1 SSSS plates

A simply supported rectangular plate (or simply referred to as an SSSS plate) subjected

to intermediate and end inplane loads is first considered. Fig. 3.2 presents the typical

stability criterion curves for SSSS plates with integer and non-integer aspect ratios. It is

found that when the plate aspect ratio a/b is an integer, the typical stability criterion curve

consists of four regimes as shown in Fig. 3.2(a). Regimes I, II, III and IV are defined by

solution combinations 5, 15, 23 and 21, respectively. The critical points P, Q and R that

connect the regimes are defined by the solution combinations 10, 19 and 22, respectively.

Point P represents the loading case in which the inplane load is applied to sub-plate 2

only (N1 = 0). Point R is for the loading case where only sub-plate 1 is loaded (N1 + N2 =

0). Point Q shows the buckling load condition that the plate is subjected to end load only

(N2 = 0). However, when the plate aspect ratio a/b is not an integer (for example a/b =

1.5), the typical stability criterion curve consists of five regimes as shown in Fig. 3.2(b).

Regimes I, II, III, IV and V are defined by solution combinations 5, 15, 25, 23 and 21.

The regimes are connected by critical points P, Q, R and S defined by solution


combinations 10, 20, 24 and 22, respectively. Points P and S are for the loading cases

where only sub-plate 2 or sub-plate 1 is loaded, respectively.

The exact stability criteria for SSSS plates with various aspect ratios (a/b = 1, 1.5, 2)

and intermediate load locations ( )7.0and5.0,3.0=χ are presented in Figs. 3.3a to 3.3c.

The critical points P, Q and R for plates with a/b = 1 and 2, and P, Q, R and S for plates

with a/b = 1.5 are marked on the stability curves. We observe that when the intermediate

inplane load is positive (N2 > 0), the buckling factor Λ1 decreases almost linearly as the

buckling factor Λ2 increases for all cases in Fig. 3.3. On the other hand, if the

intermediate inplane load is negative (N2 < 0), the buckling factor Λ1 increases almost

linearly as the value of the buckling factor Λ2 increases. The increase of Λ1 is more

pronounced when the location factor of the intermediate load χ is small. It is evident that

the stability curves for all cases in Fig. 3.3 have a highly nonlinear portion when the

buckling factor Λ2 is close to zero.

The effect of the location χ of the intermediate load on the buckling loads of square

SSSS plates can be observed more clearly in Fig. 3.4. As expected, the buckling factor Λ2

increases with increasing χ values. What are unexpected, however, are the kinks in these

buckling load variations with respect to the intermediate load location χ . These kinks

imply that there are buckling mode switchings. Take the example of the square SSSS

plate that is subjected to only an intermediate inplane load (i.e. end inplane load N1 = 0).

A mode switch is observed when the location of the intermediate load χ is in the vicinity

of 0.5. This can be confirmed by plotting the buckling mode shapes and the modal

bending moment distributions at χ = 0.4, 0.5, and 0.6, as shown in Figs. 3.5(a) and 5(b).


It is evident from the figures that the mode shapes and modal bending moment

distribution for χ = 0.4 are not similar to those for χ = 0.6. There is a portion of

bending moment distribution with a negative sign for the case of χ = 0.4. No negative

bending moment distribution portion is observed for the case of χ = 0.6. The double

curvature mode shape for the case of χ = 0.4 reinforces the fact that the mode shape is

different from the single curvature associated with the case of χ = 0.6.

Fig. 3.6 presents the variations of the buckling factor Λ2 with respect to the aspect ratio

a/b for SSSS plates subjected to inplane load in sub-plate 2 only (i.e. N1 = 0 and χ > 0).

The buckling results in Fig. 3.6 are obtained by using solution combination 10 for both

integer and non-integer aspect ratios a/b. For comparison purposes, the buckling factor

for SSSS plates subjected to end loads only (i.e. 0=χ ) is also plotted in Fig. 3.6, and the

values in brackets indicate the locations and buckling factors at the kinks in the curve. As

expected, the buckling factors Λ2 for plates subjected to inplane load in sub-plate 2 only

(i.e. 0>χ ) are always higher than the ones subjected to end inplane load (i.e. 0=χ ),

especially when the location factor χ of the intermediate load is large. As the aspect

ratio a/b increases, the buckling factors for all cases approach the value of 4 as shown in

Fig. 3.6. For benchmark purposes, Table 3.2 presents the exact buckling factors Λ2 for

SSSS plates subjected to inplane load in sub-plate 2 only (i.e. N1 = 0 and 0>χ ).


Table 3.2 Buckling factors 2Λ for simply supported rectangular plates subjected to

inplane load in sub-plate 2 only ( 01 =N )

χ a/b = 1 a/b = 2 a/b = 3 a/b = 4

0.3 5.31343 4.35397 4.15794 4.11248

0.5 6.37793 4.54296 4.32523 4.18006

0.7 6.64427 5.81516 4.71776 4.36124

3.2.2 CSCS plates

We consider a rectangular plate with the two edges parallel to the x-axis simply

supported while the two edges parallel to the y-axis are clamped (this plate is referred to

as a CSCS plate). The typical stability criterion curve for a CSCS plate with an integer or

non-integer aspect ratio a/b subjected to end and intermediate loads is similar to that of

an SSSS plate with a non-integer aspect ratio a/b as shown in Fig. 3.2(b). Regimes I, II,

III, IV and V are defined by solution combinations 5, 15, 25, 23 and 21, respectively. The

critical points P, Q, R and S that connect the regimes can be obtained from the solution

combinations 10, 20, 24 and 22, respectively.

Exact stability criteria for CSCS plates with various aspect ratios (a/b = 1, 1.5 and 2)

and intermediate load locations ( )7.0and5.0,3.0=χ are presented in Figs. 3.7(a) to

3.7(c). The stability criterion curves for the CSCS plates show very similar trends as for

SSSS plates. The variations of the buckling factor Λ2 with respect to the intermediate load

location χ for square CSCS plates are presented in Fig. 3.8.


By using the solution combination 25, we can obtain the variations of buckling load

factors versus the aspect ratios a/b for CSCS rectangular plates under end inplane load

only (i.e. N1 = 0 and 0=χ ), as shown in Fig 3.9. The values in brackets indicate the

locations and buckling factors on the kinks where the number of half waves n of the

buckling mode along the x direction switches. For example, if the plate aspect ratio a/b is

less than 1.732, the number of half waves n = 1. If 1.732 ≤ a/b < 2.828, the number of

half waves n = 2. An interesting relationship is obtained between the number of half

waves n and the coordinates of the points where the mode shape switching occurs. For

the point before which the number of half waves is n and after which the number of half

wave is (n+1), the aspect ratio a/b = )2( +nn and the buckling factor Λ2 = (2n +

2)2/[n(n + 2)].

Fig. 3.9 also presents the variation of the buckling factor Λ2 against the plate aspect

ratio a/b for CSCS rectangular plates subjected to inplane load in sub-plate 2 only (i.e. N1

= 0 and 0>χ ). As expected, the buckling factors for plates under such loading a case

are always higher than the ones subjected to end load only, especially when the location

factor χ of the intermediate load is large. Kinks, present in the curves in Fig. 3.9,

indicate mode shape switching at the particular aspect ratio a/b.

3.2.3 FSFS plates

A rectangular plate with the two edges parallel to the x-axis simply supported and the

two edges parallel to the y-axis free is considered (referred from hereon as a FSFS plate).

The typical stability criterion curve of FSFS plates with the location of the intermediate

load 5.0,3.0=χ and 0.7 is shown in Fig. 3.10. There are only three regimes (I, II, and III)


on the stability criterion curve that are determined by the solution combinations 3, 13 and

11, respectively. The critical points P and Q that connect the regimes are defined by the

solution combinations 8 and 12, respectively.

Exact stability criteria for FSFS plates with various aspect ratios (a/b = 1, 1.5 and 2)

and intermediate load locations ( χ = 0.3, 0.5 and 0.7) are presented in Fig. 3.11. For

rectangular FSFS plates, the stability criterion curves are very close to each other while

the intermediate load location χ varies from 0.3 to 0.5 to 0.7. The variations of the

buckling factor Λ2 versus the intermediate load location χ for square FSFS plates are

presented in Fig. 3.12.

The relationship between the buckling factor Λ2 and the aspect ratio a/b is presented in

Fig. 3.13 for FSFS rectangular plates subjected to inplane load in sub-plate 2 only (N1 =

0). For FSFS plates with the intermediate load acting at χ = 0.1 and 0.3, the buckling

factor increases as the plate aspect ratio increases. For χ = 0.5 to 0.9, the buckling factor

decreases as the plate aspect ratio increases. When the plate aspect ratio is large, the

buckling factor approaches the value 2.437 for all cases as shown in Fig. 3.13.

3.3 Concluding Remarks

This chapter presents an analytical method to investigate the elastic buckling behaviour

of Levy-type plates subjected to the end and intermediate inplane loads. A rectangular

plate is divided into two sub-plates at the location of the intermediate load and the five

feasible exact solutions of the governing differential equation for each sub-plate are

derived. The critical buckling load is determined from one of the twenty-five possible

solution combinations for the two sub-plates. Exact stability criterion curves are

presented for several selected Levy-type plates subjected to the end and intermediate


inplane loads. The influence of the intermediate load locations on the stability criterion

curves of the plates is discussed. The exact buckling solutions are valuable as benchmark

values and for engineers designing walls or plates that have to support intermediate

floors/loads.


simply supported edges b

aχ a)1( χ−

1N

2N

21 NN +

(a) Original Plate

1y

1x

1N

aχ

any B.C.

2y

2x

21 NN +interface

a)1( χ−

any B.C.

(b) Sub-plate 1 (c) Sub-plate 2

Figure 3.1 Geometry and coordinate systems for a rectangular plate subjected to intermediate and end uniaxial inplane loads. (a) Original plate; (b) Sub-plate 1; and (c) Sub-plate 2.


Regime III

Regime II

Regime I

2Λ

Regime IV

1Λ

Compressive Stress StateZero Stress State Tensile Stress State

Figure 3.2 (a) Plate with integer aspect ratio a/b

R

Q

P

1Λ

2Λ

Regime I

Regime II

Regime V


Regime IIIRegime IV

Figure 3.2 (b) Plate with non-integer aspect ratio a/b

Figure 3.2 Typical stability criterion curves for SSSS plates subjected to end and intermediate inplane loads: (a) plate with integer aspect ratio a/b, and (b) plate with non-integer aspect ratio a/b.

S

P

QR


-2

0

2

4

6

8

-8 -6 -4 -2 0 2 4 6 8

Λ1

Λ2

3.0=χ5.0=χ

7.0=χ

(b) Rectangular plate with a/b=1.5

-2

0

2

4

6

8

-8 -6 -4 -2 0 2 4 6 8

Λ1

Λ2

(a) Square plate (a/b = 1)

3.0=χ 5.0=χ

3.0=χ


-4

-2

0

2

4

6

8

-8 -6 -4 -2 0 2 4 6 8

Λ1

Λ2

(c) Rectangular plate with a/b = 2.0

3.0=χ5.0=χ

7.0=χ

Fig. 3.3 Stability criteria for SSSS rectangular plates with (a) a/b = 1.0, (b) a/b = 1.5, and (c) a/b = 2.0.


Figure 3.4 Variations of buckling intermediate load factor 2Λ with respect to location χ for SSSS square plate (Note that 0000.4=Λ cr is the buckling load factor for square SSSS plate under end load only).

0

1

2

3

4

5

6

7

8

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Intermediate Load Location χ

Buc

klin

g In

term

edia

te lo

ad F

acto

r Λ2

Λ1 = 0

Λ1 = 0.5Λ cr

Λ1 = 0.8 Λcr


-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0 0.2 0.4 0.6 0.8 1

k = 0.4k = 0.5k = 0.6

x/a

Nor

mal

ized

Mod

al S

hape

(a) Modal shapes

4.0=χ5.0=χ6.0=χ

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0 0.2 0.4 0.6 0.8 1

k = 0.4k = 0.5k = 0.6

x/a

Nor

mal

ized

Mod

alB

endi

ngM

omen

t

(b) Modal moment distributionsFigure 3.5 Normalized modal shapes and modal moment distributions in the x-direction for SSSS square plate subjected to intermediate load N2 (N1 = 0): (a) modal shapes; and (b) modal moment distributions

4.0=χ5.0=χ6.0=χ


0

2

4

6

8

10

12

0 1 2 3 4 5 6

(1.414,4.5) (2.449,4.167)

2Λ

a/b

0=χ

3.0=χ

5.0=χ 7.0=χ

2N 2N

aχ a)1( χ−

b (3.464,4.083)

Figure 3.6 Variation of buckling factors Λ2 versus plate aspect ratio a/b for SSSS plates subjected to inplane load in sub-plate 2 only.


-5

0

5

10

15

-15 -10 -5 0 5 10 15

Λ1

Λ2

3.0=χ5.0=χ

7.0=χ

(a) Square plate (a/b = 1.0)

-5

0

5

10

15

-15 -10 -5 0 5 10 15

Λ1

Λ2

(b) Rectangular plate with a/b = 1.5

3.0=χ5.0=χ

7.0=χ


-5

0

5

10

15

-15 -10 -5 0 5 10 15

Λ1

Λ2


3.0=χ

5.0=χ

7.0=χ

Fig. 3.7 Stability criteria for CSCS rectangular plates with plate aspect ratios (a) a/b = 1.0, (b) a/b = 1.5, and (c) a/b = 2.0.


0

2

4

6

8

10

12

14

16

18

20

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8


Buc

klin

g In

term

edia

te lo

ad F

acto

r Λ2

Λ1 = 0

Λ1 = 0.5Λ cr

Λ1 = 0.8 Λ cr

Figure 3.8 Variations of buckling intermediate load factor 2Λ with respect to location χ for CSCS square plate (Note that Λcr = 6.7432 is the buckling load factor for square CSCS plate under end load only).


3

4

5

6

7

8

9

10

11

0 1 2 3 4 5 6 7

2Λ

a/b

0=χ

3.0=χ

5.0=χ 7.0=χ

(1.732,5.333)

(5.916,4.114)

2N 2N

aχ a)1( χ−

b

(2.828,4.5) (3.873,4.267)(4.899,4.167)

Figure 3.9 Variations of buckling factors Λ2 versus plate aspect ratio a/b for CSCSrectangular plates subjected to inplane load in sub-plate 2 only.


Regime I

Regime II

Regime III

1Λ

2Λ

Figure 3.10 Typical stability criterion curve for FSFS rectangular plates subjected to end and intermediate inplane loads.


Q

P


-4

-2

0

2

4

-8 -6 -4 -2 0 2 4 6 8

Λ1

Λ2

(b) Rectangular plate with a/b = 1.5

3.0=χ

7.0,5.0=χ

-4

-2

0

2

4

-8 -6 -4 -2 0 2 4 6 8

Λ1

Λ2

(a) Square plate

3.0=χ5.0=χ

7.0=χ


-4

-2

0

2

4

-8 -4 0 4 8

Λ1

Λ2


7.0,5.0,3.0=χ

Fig. 3.11 Stability criteria for FSFS rectangular plates with plate aspect ratios (a) a/b = 1.0, (b) a/b = 1.5, and (c) a/b = 2.0.


0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8


Buc

klin

g In

term

edia

te lo

ad F

acto

r Λ2

Λ1 = 0

Λ1 = 0.5Λcr

Λ1 = 0.8Λ cr

Figure 3.12 Variations of buckling intermediate load factor Λ2 with respect to location χ for FSFS square plate (Note that Λcr = 2.0429 is the buckling load factor for square FSFS plate under end load only).


0

2

4

6

8

10

12

14

16

0 1 2 3 4 5 6

2Λ

aχ a)1( χ−

2N b 2NF F

S

S

0=χ 1.0=χ

3.0=χ 5.0=χ

7.0=χ

a/b

9.0=χ

Figure 3.13 Variations of buckling factors Λ2 versus plate aspect ratio a/b for FSFSrectangular plates subjected to inplane load in sub-plate 2 only.

PLASTIC BUCKLING OF PLATES UNDER


This chapter is concerned with the plastic buckling of rectangular plates subjected to

both intermediate and end uniaxial loads. The plate has two opposite simply supported

edges that are parallel to the load direction and the other remaining edges may take any

combination of free, simply supported or clamped conditions. Both the Incremental

Theory of Plasticity and the Deformation Theory of Plasticity are considered in bounding

the plastic behavior of the plate. The buckling problem is solved by decomposing the

plate into two sub-plates at the boundary where the intermediate load acts. Each sub-plate

buckling problem is solved exactly using the Levy approach and the two solutions

brought together by the continuity equations at the separated interface. There are eight

possible solutions for each sub-plate and consequently there are sixty-four combinations

of solutions to be considered for the entire plate. The final solution combination depends

on the nature of the ratio of the intermediate load to the end load, the intermediate load

location, aspect ratio, and material properties. Typical plastic stability criteria are

presented in graphical forms which should be useful for engineers designing plated walls

that have to support intermediate floors/loads.

Chapter 4

Chapter 4 Plastic Buckling of Plates under Intermediate and End Loads 62

4.1 Mathematical Modeling

4.1.1 Problem definition

Consider an isotropic, rectangular thin plate as shown in Fig. 1a. The plate has length a,

width b, and thickness h and is simply supported along the edges y = 0 and y = b. The

other two edges of the plate may take any combination of free (F), simply supported (S)

and clamped (C) conditions. For convenience, a four-letter symbol is used to denote the

support conditions of the plate. For example, an FSCS plate has a free left edge, a simply

supported bottom edge, a clamped right edge and a simply supported top edge.

The plate is subjected to an end load hN 11 σ= (per unit length) at the edge x = 0 and

an intermediate uniaxial load hN 22 σ= (per unit length) at the location ax χ= . Thus the

end reaction force at the right edge ax = is ( )hNN 2121 σσ +=+ as shown in Fig. 4.1.

Note that a positive value of σ implies a compressive load while a negative value implies

a tensile load. The material of the plate is assumed to obey the Ramberg-Osgood

constitutive law. The problem at hand is to determine the plastic buckling load for such a

loaded plate.

4.1.2 Method of solution

The plate is first divided into two sub-plates. The first sub-plate is to the left of the

vertical line defined by ax χ= (see Fig. 4.1b) and the second sub-plate is to the right of

this line (see Fig. 4.1c). Adopting the x-y coordinates system as shown in Figs. 4.1b and

4.1c, the governing plastic buckling equation (see Eq. (2.39)) for each sub-plate may be

canonically written as Eq. (4.1)


2

2

2

2

4

4

4

4

22

4

2

2

4

4 12)(2

i

iii

i

ii

ii

iii

i

ii x

wEh

ayw

ba

yxw

ba

xw

∂∂

−=∂∂

+∂∂

∂++

∂∂ σγµβα , (4.1)

in which

,,,byy

axx

bww i

ii

ii

ii === (4.2a-c)

where i = 1,2 respectively denotes the sub-plates 1 and 2; aa χ=1 , and aa )1(2 χ−= .

The parameters µγβα ,,, are defined as follows:

• Based on Incremental Theory of Plasticity (DT):

−−−+=

ETvv

SE )21(2)21(3ρ , (2.26a)

−−=

ST1341

ρα (2.26b)

−−=

ETv)21(221

ρβ , (2.26c)

ργ 4= , (2.26d)

• Based on Deformation Theory of Plasticity (IT):

ETvv 2)21()45( −−−=ρ , (2.33a)

−−=

ST1341

ρα (2.33b)

−−=

ETv)21(221

ρβ , (2.33c)


ργ 4= , (2.33d)

where v is the Poisson ratio, and the ratios of the elastic modulus E to the tangential

modulus T and the secant modulus S at the onset of buckling are expressed as

1

0

1−

+=

c

kSE

σσ ; 1>c (2.28a)

　;11

0

−

+=

c

ckTE

σσ .1>c (2.28b)

where 0σ is a nominal yield stress, c a dimensionless constant that describes the shape of

the stress-strain relationship with ∞=c for elastic-perfectly plastic response, and k the

horizontal distance between the knee of ∞=c and the intersection of the c curve with the

1/ 0 =σσ line as shown in Fig. 2.3.

The essential and natural boundary conditions for the two simply supported edges

at 0=iy and 1=iy associated with the i-th sub-plate are given by

0=iw (4.3)

02

2

22

2

2 =∂∂

+∂∂

i

i

i

i

i

ii

xw

ayw

bβα (4.4)

Based on the Levy approach (Timoshenko and Woinowsky-Krieger 1959), the solution

to the partial differential equation may take the form of

( ) ( ) 2,1,sin, == iymxAyxw iiimiii π (4.5)


In view of Eq. (4.5), the partial differential equation (4.1) may be reduced into an

ordinary differential equation given by

0)(2124

444

2

2

2

222

2

2

4

4

=+

+−+

i

ii

i

im

i

ii

i

ii

i

im

bma

xdAd

bam

Eha

xdAd

αγπ

απµβ

ασ . (4.6)

Three parameters ,, 21 ∆∆ and 3∆ are defined as follows:

i

ii

i

ii

i

ii

bma

bam

Eha

αγπ

απµβ

ασ

4

4442

2

222

2

2

14)(212

−

+−=∆ , (4.7)

i

ii

bmaα

γπ4

444

2 =∆ , (4.8)

i

ii

i

ii

bam

Eha

απµβ

ασ

2

222

2

2

3)(212 +

−=∆ . (4.9)

Depending on the values of ,, 21 ∆∆ and 3∆ , there are eight possible solutions for the

fourth-order differential equation (4.6). These solutions, designated as Solution A-H (see

Table 4.1), are given below.

Table 4.1 Types of solutions depending on values of 321 ,, ∆∆∆

1∆ 2∆ 3∆ Solution > 0 A > 0 < 0 B > 0 C = 0 < 0 D

> 0 < 0 Any value E

> 0 F = 0 =

4

23∆ < 0 G

< 0 Any value Any value H


Solution A:

ip

iiip

iiip

iiip

iiim xCxCxCxCA ψψφφ sincossincos 4321 +++= , (4.10a)

in which 2

,2

1313 ∆−∆=

∆+∆= p

ip

i ψφ . (4.10b,c)

Solution B:

ip

iiip

iiip

iiip

iiim xCxCxCxCA ψψφφ sinhcoshsinhcosh 4321 +++= , (4.11a)

in which 2

,2

1313 ∆−∆−=

∆+∆−= p

ip

i ψφ . (4.11b,c)

Solution C:

4321 sincos iiiip

iiip

iiim CxCxCxCA +++= φφ , (4.12a)

in which 3∆=piφ . (4.12b)

Solution D:

4321 sinhcosh iiiip

iiip

iiim CxCxCxCA +++= φφ , (4.13a)

in which, 3∆−=piφ . (4.13b)

Solution E:

ip

iiip

iiip

iiip

iiim xCxCxCxCA ψψφφ sincossinhcosh 4321 +++= , (4.14a)

in which .2

,2

1313 ∆+∆=

∆+∆−= p

ip

i ψφ (4.14b,c)


Solution F:

ip

iiiip

iiip

iiiip

iiim xxCxCxxCxCA φφφφ sinsincoscos 4321 +++= , (4.15a)

in which 2

3∆=piφ . (4.15b)

Solution G:

ip

iiiip

iiip

iiiip

iiim xxCxCxxCxCA φφφφ sinhsinhcoshcosh 4321 +++= , (4.16a)

in which 2

3∆−=piφ . (4.16b)

Solution H:

ip

ip

iiip

ip

iiip

iip

iiim xCxCxxCA ψφψφψφ cossinhsincoshcoscosh 321 ++=

ip

iip

ii xxC ψφ sinsinh4+ (4.17a)

in which 12

3312

33 21,

21

∆−∆+∆=∆−∆+∆−= pi

pi ψφ (4.17b,c)

The eight constants 4,3,2,1,, 21 =iCC ii resulting from the solution combination can be

evaluated using the boundary equations at the edges 01 =x and 12 =x ,

• For simply supported edges:

⇒= 0iw 0=imA , and (4.18a)

⇒=∂∂

+∂∂ 02

2

22

2

2i

ii

i

i

i

i

yw

bxw

aβα

02

22

2

2

2 =− imi

i

im

i

i Abm

xdAd

aπβα , i = 1,2 (4.18b)


where )1/(1 νµ += .

• For clamped edges:

⇒= 0iw 0=imA and (4.19a)

⇒=∂∂

0i

i

xw

0=i

im

xddA i = 1, 2 (4.19b)

• For free edges:

⇒=∂∂−

+∂

∂02

2

22

2

2i

ii

i

i

i

i

yw

bxw

aµβα

02

22

2

2

2 =− imi

i

im

i

i Abm

xdAd

aπβα and (4.20a)

⇒=∂∂

+∂∂

∂++

∂∂ 0122

22

3

23

3

3i

i

i

i

ii

i

i

i

i

i

i

i

xw

aEhb

yxw

baxw

aσµβα

( ) 02122

22

23

3

3 =

+−+

∂ i

im

i

i

i

i

i

im

i

i

xddA

bam

aEhb

xAd

aπµβσα , i = 1, 2 (4.20b)

and the continuity equations at the separation interface

⇒=−==

00211

21 xxww 0

021121

=−== xmxm AA , (4.21a)

⇒=∂∂

−∂∂

==

011

02

2

211

1

121 xx x

wax

wa

011

02

2

211

1

1 21

=−== x

m

x

m

xddA

axddA

a, (4.21b)

00

22

22

22

22

22

22

2

121

12

21

21

12

21

1

21

=

∂∂

+∂∂

−

∂∂

+∂∂

== xxyw

bxw

ayw

bxw

aβαβα


00

222

22

22

22

22

2

1

121

22

21

12

21

1

2

1

=

−−

−⇒

=

=

x

mm

x

mm

Ab

mxdAd

a

Ab

mxdAd

a

βπα

βπα

, (4.21c)

012)2(

12)2(

02

2

22

2222

23

22

232

23

32

2

11

1

12

1211

13

21

131

13

31

1

2

1

=

∂∂

+∂∂

∂++

∂∂

−

∂∂

+∂∂

∂++

∂∂

=

=

x

x

xw

aEhb

yxw

baxw

a

xw

aEhb

yxw

baxw

a

σµβα

σµβα

0)2(12

)2(12

02

22

2

222

22

232

23

32

2

11

12

1

122

12

131

13

31

1

2

1

=

+−+−

+−+⇒

=

=

x

mm

x

mm

xddA

bam

aEhb

xdAd

a

xddA

bam

aEhb

xdAd

a

µβπσα

µβπσα

(4.21d)

By substituting the appropriate solutions expressed in Eqs. (4.10) to (4.17) into Eqs.

(4.18) to (4.21), one obtains a set of homogeneous equations which may be expressed in

the following matrix form

[ ] 0=CK , (4.22)

in which TCCCCCCCCC 2423222114131211= . For a nontrivial buckling solution, the

determinant of [ ]K must vanish. The characteristic equation furnishes the stability

criterion. The valid solution combinations should satisfy the following requirements:

• The buckling load satisfies the limits of validity for the solution combinations

which it belongs to;

• The buckling load is the lowest value among possible solutions; and it was found

that m=1 gives the lowest load for all possible m values for the considered cases.


• The stability curves are continuous.

4.2 Results and Discussions

In order to examine the plastic buckling criteria of rectangular plates, we have adopted

the following material properties: ,10700 ksiE = ,4.610 ksi=σ 32.0=v and the

Ramberg-Osgood parameters 20=c and 3485.0=k . The influences of material

properties including 0/σE and c will also be investigated. The buckling factors for the

end and intermediate loads are expressed as )/( 2211 Dhb πσ=Λ and ),/( 22

22 Dhb πσ=Λ

respectively where the symbol ( )[ ]23 112/ ν−= EhD denotes the flexural rigidity of the

plate.

Table 4.2 presents the buckling stress for a simply supported, square plate under a

uniaxial compressive end load. The results are compared with those obtained by Wang et

al. (2001). We can see that both results are in close agreement. Note that the results from

Wang et al. (2001) are based on the Mindlin shear deformable plate theory. As exapected,

the results from Wang et al. (2001) are slightly smaller than the present results which are

based on the classical thin plate theory.


Table 4.2 Buckling stresses 1σ for a simply supported, square plate under uniaxial end

load (i.e. no intermediate load)

Buckling Stresses 1σ (in ksi)

IT DT

b/h

Present study Wang et al. (2001) Present study Wang et al. (2001)

22 71.61 70.84 61.90 60.08

24 61.15 60.71 58.54 57.40

26 54.88 54.60 54.39 53.81

28 49.41 49.11 49.32 48.96

Figure 4.2 presents a typical stability criterion curve for plates subjected to

intermediate and end loads. Note that negative values of 2,1, =Λ ii denote tensile in-

plane loads. The stability criterion curve consists of various regimes. For example, in

Figure 4.2 we consider the case of a CSCS square plate with thickness to width ratio h/b

= 0.04, intermediate load location 5.0=χ , there are five Regimes I, II, III, IV and V

which are defined by solution combinations B+A, H+A, A+A, A+H and A+B,

respectively. The critical points P, Q, R and S that connect the regimes are defined by the

solution combinations G+A, F+A, A+F and A+G, respectively. Point P represents the

loading case in which the inplane load is applied to sub-plate 2 only (i.e. N1 = 0) while

Point S for the loading case where only sub-plate 1 is loaded (i.e. N1 + N2 = 0). However,

for different material properties, aspect ratios, intermediate load positions, and boundary

conditions the solution combinations may change somewhat.


4.2.1 Effect of aspect ratios a/b

Figures 4.3 to 4.5 show the stability criterion curves of SSSS, CSCS and FSFS

rectangular plates with aspect ratios a/b = 1.0, 2.0 and 3.0, loading position χ = 0.5,

thickness to width ratio h/b = 0.04 and various end load to intermediate load ratios.

Figures 4.3a, 4.4a and 4.5a show the buckling results using IT while Figs. 4.3b, 4.4b and

4.5b show results using DT. The two plasticity theories furnish somewhat similar results

except for the case of CSCS square plates.

Referring to Figs. 4.3 and 4.4, we can see that for SSSS and CSCS plates, the buckling

factors decrease with the increasing aspect ratios a/b. However, for FSFS plates, the

buckling factors with a/b = 1.0 could be smaller than that of a/b = 2.0 and a/b = 3.0 as

shown in Fig. 4.5. For SSSS and CSCS plates, the differences between the buckling

factors for a/b = 2.0 and a/b = 3.0 are however much smaller than the differences

between those for a/b = 1.0 and a/b = 2.0.

For negative values of the intermediate load factor 2Λ (i.e. 2Λ is tensile in nature), the

end buckling factor 1Λ takes on almost a constant value. This is because sub-plate 2 does

not buckle as it is under a very low compressive stress state or in a tensile stress state. So

the buckling deformation of the plate mainly occurs in sub-plate 1. In order to reinforce

this point, consider a rectangular plate with two opposite edges simply supported,

intermediate load position 5.0=χ , and is under end and intermediate loads as shown in

Fig. 4.6a. Suppose we let sub-plate 2 under a large tensile stress state, for

example 502 −=Λ . Depending on different boundary conditions of the other two edges,

the buckling factors 1Λ are given in Table 4.3. We now compare the buckling stresses of


rectangular plates whose aspect ratio 5.0/ =ba and same boundary conditions as sub-

plate 1 along three edges, as shown in Fig. 4.6b, but the X edge may be clamped, simply-

supported or free edge. The corresponding buckling factors are given in Table 4.3. Based

on the results presented in Table 4.3 for SSSS, CSCS and FSFS plates, we can conclude

that when sub-plate 2 is under a large tensile stress state, the interface edge may be

regarded as approaching the condition of a clamped edge, except for the FSFS plate with

a/b=2.0, where the interface edge imposes a constraint to the plate similar to the one from

a simply supported edge.

Table 4.3 Comparison of buckling factors of full plates with uniaxial intermediate and

end loads and their corresponding end loaded sub-plates with different interfacial edge

conditions

IT DT IT DT IT DT a/b=1.0 a/b=2.0 a/b=3.0

Buckling factor 1Λ for SSSS with

502 −=Λ 4.470 4.002 3.799 3.712 3.738 3.674

Buckling factors for SSXS Sub-plate a/b = 0.5 a/b = 1.0 a/b = 1.5 X edge takes on a clamped edge 4.543 4.015 3.806 3.717 3.742 3.679

X edge takes on a simply-supported edge 3.893 3.797 3.678 3.605 3.631 3.609X edge takes on a free edge 1.990 1.990 2.311 2.311 2.253 2.253

a/b=1.0 a/b=2.0 a/b=3.0 Buckling factor 1Λ for CSCS with 502 −=Λ 6.214 4.198 4.108 3.878 3.917 3.802

Buckling factors for CSXS Sub-plate a/b = 0.5 a/b = 1.0 a/b = 1.5 X edge takes on a clamped edge 6.367 4.205 4.123 3.884 3.924 3.806

X edge takes on a simply-supported edge 4.543 4.015 3.806 3.717 3.742 3.679 X edge takes on a free edge 2.568 2.568 2.333 2.333 2.270 2.270

a/b=1.0 a/b=2.0 a/b=3.0 Buckling factor 1Λ for FSFS with 502 −=Λ 2.451 2.451 2.289 2.289 2.266 2.266

Buckling factors for FSXS Sub-plate a/b = 0.5 a/b = 1.0 a/b = 1.5 X edge takes on a clamped edge 2.568 2.568 2.333 2.333 2.270 2.270

X edge takes on a simply-supported edge 1.990 1.990 2.311 2.311 2.253 2.253 X edge takes on a free edge 1.539 1.539 1.990 1.990 2.210 2.210


4.2.2 Effect of loading positions χ

Based on IT and DT, Figures 4.7a and 4.7b show the variations of the buckling factors

2Λ of SSSS rectangular plates with respect to loading positions χ when ,01 =Λ 2, a/b =

1, 2, and thickness to width ratio h/b = 0.04 respectively. It can be seen that the buckling

factor 2Λ increases with the increasing value of χ , albeit gradually for 7.0<χ and

more steeply for 7.0≥χ . It can be seen that the two theories of plasticity furnish

significantly different buckling values when 7.0≥χ . Generally, buckling factors

associated with IT are higher than their DT counterparts. It is worth noting that the

buckling factors 2Λ for 01 =Λ differ from their counterparts associated with 21 =Λ by

about 2.

4.2.3 Effect of various boundary conditions

The influence of boundary conditions (such as SSSS, CSCS, FSFS, CSSS, FSSS and

CSFS) on the buckling factors is highlighted in Fig. 4.8. In computing these buckling

factors, we have used the following parameters: aspect ratio a/b = 2.0, thickness to width

ratio h/b = 0.04 and loading position 5.0=χ . It can be seen from Fig. 4.8 that the

buckling criterion curves merge at some portions even though the rectangular plates have

one of its edge conditions different from the other provided the sub-plate with this edge is

under a high tensile stress state. For example CSFS plate with a high tensile stress state in

sub-plate 2 will behave in a similar manner to a CSCS plate with a high tensile stress state

or a low compressive stress state in sub-plate 2 because the tensile stressed sub-plate does

not buckle at all and thus its edge condition plays no part in the buckling phenomenon.


4.2.4 Effect of various material properties

The buckling factors of SSSS square plates with various material properties 0/σE =

200, 400, 800 and c = 2, 3, 20 are given in Figs. 4.9 and 4.10 by using IT and DT,

respectively. A thickness to width ratio h/b = 0.04, and loading position 5.0=χ were

used in the calculations. It can be seen from Figs. 4.9 and 4.10 that the buckling factors

decrease with increasing values of parameter c, i.e. as the plate material approaches the

perfectly elastic-plastic constitutive relation. Furthermore, the shape of the stability

criterion curve for large c values approach a bilinear curve with a horizontal portion when

2Λ is negative and a linear curve when 2Λ is positive. The slope of the linear portion is

given by *2

*1 /ΛΛ where *

1Λ is 1Λ value when 2Λ = 0 (i.e. the case when the plate is

subjected to only end load) and *2Λ is 2Λ value when 1Λ = 0 (i.e the case when the plate

is subjected to only intermediate load).

4.2.5 Effect of using two different theories

The buckling factors of SSSS square plates with various material properties 0/σE =

200, 400, 800 and under intermediate load are given in Fig. 4.11 by using IT and DT,

respectively. From Fig. 4.11 we can see the difference between IT and DT increases with

increasing 0/σE values. For the same value of 0/σE , the differences also increases with

increasing values of parameter c.

4.3 Concluding Remarks

This chapter presents an analytical method to determine the exact plastic buckling

factors of rectangular plates subjected to end and intermediate uniaxial loads and where


two opposite edges (parallel to the loads) are simply supported. In this method, the

rectangular plate is divided into two sub-plates at the intermediate load location. Each

sub-plate buckling problem is then solved using the Levy approach. There are eight

feasible solutions for each sub-plate. The critical buckling load is determined from one of

the sixty-four possible solution combinations for the two sub-plates. The solution

combination depends on the aspect ratio, the intermediate load position, the intermediate

to end load ratio, the material properties and the boundary conditions. The effects of the

aforementioned parameters and the adoption of DT and IT on the buckling factors are

also investigated.

The presented exact buckling solutions should be very useful for engineers designing

plated walls that have to support intermediate floors/loads and they should serve as

benchmark values for checking the convergence, validity and accuracy of numerical

methods for plate buckling analysis.


1Λ

-2

-1

0

1

2

3

4

5

-6 -4 -2 0 2 4 6

Regime I

Regime II

Regime V

2Λ

Compressive Stress State Zero Stress State Tensile Stress State

Regime IV Regime IIIS R

P

Fig. 4.2 Typical stability criterion curve

Q

Fig. 4.1 Rectangular plate under intermediate and end uniaxial loads

simply supported edges b

aχ a)1( χ−

1N

2N

21 NN +

(a)

x

y

any B.C.any B.C.

1y

1x

1N

aχ

(b)

any B.C. b

(c)

2y

2x

21 NN +interface

a)1( χ−

any B.C.b


Fig. 4.3 Stability criteria for SSSS rectangular plates with h/b= 0.04 and different aspect ratios a/b = 1,2,3 by (a) IT and (b) DT. The intermediate load is placed at χ = 0.5

-2

-1

0

1

2

3

4

5

-6 -4 -2 0 2 4 6

1Λ

2Λ

(a)

a/b=1

a/b=3 a/b=1

a/b=3

Results based on IT

a/b=2

a/b=2

-2

-1

0

1

2

3

4

5

-6 -4 -2 0 2 4 6

1Λ

2Λ

(b)

a/b=1

a/b=3

a/b=1

a/b=3

Results based on DT

a/b=2

a/b=2


-2

0

2

4

6

-6 -4 -2 0 2 4 6

1Λ

2Λ

a/b=1

a/b=3

a/b=2

Results based on IT

(a)

Fig. 4.4 Stability criteria for CSCS rectangular plates with h/b = 0.04 and different aspect ratios a/b = 1, 2, 3 by (a) IT and (b) DT. The intermediate load is placed at χ = 0.5

-2

-1

0

1

2

3

4

5

-6 -4 -2 0 2 4 6

1Λ

2Λ

(b)

a/b=1

a/b=3

a/b=2

Results based on DT


-2

-1

0

1

2

3

-4 -3 -2 -1 0 1 2 3 4

1Λ

2Λ

(a)

a/b=2

a/b=1

Results based on IT

a/b=3

-2

-1

0

1

2

3

-4 -3 -2 -1 0 1 2 3 4

1Λ

2Λ

(b)

a/b=1

Results based on DT

Fig. 4.5 Stability criteria for FSFS rectangular plates with h/b = 0.04 and different aspect ratios a/b = 1, 2, 3 by (a) IT and (b) DT. The intermediate load is placed at η = 0.5

a/b=2

a/b=3


Fig. 4.6 Rectangular plate YSZS and the corresponding sub-plate YSXS under uniaxial load

(b) Sub-plate YSXS

N

a5.0

N

bY edge

b

a5.0 a5.0

1N

2N

21 NN −

(a) Full plate YSZS

S

S

Y edge Z edge


0

1

2

3

4

5

6

7

8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

01 =Λ , a/b=1

21 =Λ , a/b=1

2Λ

01 =Λ , a/b=2

21 =Λ , a/b=2

η

(a)

2N 21 NN +

η a (1-η )a

b

1N )/( 22

11 DbN π=Λ)/( 22

22 DbN π=Λ

Results based on IT

0

1

2

3

4

5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

01 =Λ , a/b=1

21 =Λ , a/b=1

01 =Λ , a/b=2

21 =Λ , a/b=2

η

Fig. 4.7 Variation of buckling factors 2Λ with respect to χ for rectangular plates with 1Λ =0 and 2 by (a) IT (b) DT.

Results based on DT

(b)

2Λ


-2

-1

0

1

2

3

4

5

-6 -4 -2 0 2 4 6

CSCS, CSFS, CSSS

SSSS

FSFS, FSSS

FSFS, CSFS

CSCS

CSFS FSSS

CSSS, FSSS

1Λ

2Λ

(a)

Results based on IT

Fig. 4.8 Stability criteria for rectangular plates with h/b=0.04, aspect ratio a/b = 2, intermediate load position 5.0=χ and different boundary conditions by (a) IT and (b) DT

-2

-1

0

1

2

3

4

5

-6 -4 -2 0 2 4 6

CSCS, CSFS, CSSS

SSSS

FSFS, FSSS

FSFS, CSFS

CSCS

CSFS FSSS

CSSS, FSSS

1Λ

2Λ

(b)

Results based on DT


-2

0

2

4

6

-6 -4 -2 0 2 4 6 8

1Λ

2Λ

c=2

c=20

c=3

(b)

-2

0

2

4

6

-6 -4 -2 0 2 4 6 8

1Λ

2Λ

c=2

c=20

c=3

(a)

200/ 0 =σE Results based on IT



-2

0

2

4

6

-6 -4 -2 0 2 4 6 8

1Λ

2Λ

c=2

c=20

c=3

(c)

Fig. 4.9 Stability criteria for SSSS square plates with h/b = 0.04, χ = 0.5, for

different 0σ

E = (a) 200, (b) 400, (c) 800 by IT

-4

-2

0

2

4

6

-8 -6 -4 -2 0 2 4 6 8

1Λ

2Λ

(a)

c=3

c=2

c=20


200/ 0 =σE Results based on DT


-4

-2

0

2

4

6

-8 -6 -4 -2 0 2 4 6 8

1Λ

2Λ

(c)

c=3

c=20 c=2

Fig. 4.10 Stability criteria for SSSS square plates with h/b = 0.04, χ = 0.5, for

different 0σ

E = (a) 200, (b) 400, (c) 800 by DT

-4

-2

0

2

4

6

-8 -6 -4 -2 0 2 4 6 8

1Λ

2Λ

c=3

c=20

(b)

c=2




0

1

2

3

4

5

6

0 5 10 15 20 25c

2Λ

200/ 0 =σE , IT

400/ 0 =σE , DT

800/ 0 =σE , DT

200/ 0 =σE , DT

400/ 0 =σE , IT

800/ 0 =σE , IT

Fig. 4.11 Buckling load factors 2Λ for SSSS square plates with 01 =Λ

IT

DT

0.5a 0.5a

b=a

2N 2N

CONCLUSIONS AND RECOMMENDATIONS

5.1 Conclusions

The thesis is concerned with the buckling of rectangular plates subjected to

intermediate and end uniaxial inplane loads. The plate has two opposite simply supported

edges that are parallel to the load direction while the other remaining two edges may take

any combination of free, simply supported or clamped condition. The buckling problem

is solved by decomposing the plate into two sub-plates at the location where the

intermediate uniaxial load acts. Each sub-plate buckling problem is solved exactly using

the Levy approach and the two solutions brought together by matching the continuity

equations at the interfacial edge.

We use both elastic and plastic theories to solve this problem. For the elastic buckling

problem, there exist five possible solutions for each sub-plate. Consequently twenty-five

possible solution combinations have to be considered for the whole plate. The buckling

solutions comprise of different solution combinations depending on various boundary

conditions, aspect ratios, and intermediate load positions. The influences of

aforementioned factors are discussed and the stability criterion curves are presented in

graphs. In contrast to its elastic counterpart, the plastic buckling problem has eight

possible solutions for each sub-plate. Consequently sixty-four possible solution

Chapter 5

Chapter 5 Conclusions and Recommendations 89

combinations have to be considered for the whole plate. The effect of various aspect

ratios, intermediate load positions, boundary conditions and material properties are

discussed. It is found that deformation theory of plasticity always yields lower buckling

factors than incremental theory of plasticity. Our research should be useful for engineers

designing plated walls that have to support intermediate floors/loads.

5.2 Recommendations for Future Studies

Further studies along this line of investigation could include:

1. Treating the buckling problem of rectangular plates subjected to end and several

intermediate uniaxial loads. This problem has application in plated walls

supporting several intermediate floor loads or shelf loads.

2. Treating plates with various boundary conditions including elastically restricted

edges that simulate practical edge restraints.

3. This study shows that the two competing plastic theories, IT and DT yield

different buckling results. Experiments may be conducted to establish which of

these two plasticity theories predict the buckling factors more accurately.

4. Considering tapered (or varying thickness) plates under end and intermediate

loads which find practical applications in airplane wings.

90

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Wang, C.M., Chen, Y. and Xiang, Y. (2002). “Exact buckling solutions for simply supported, rectangular plates under intermediate and end uniaxial loads.” Proceedings of the 2nd International Conference on Structural Stability and Dynamics, edited by C.M. Wang, G.R. Liu and K.K. Ang, World Scientific, 16-18 December 2002, Singapore, 424-429.

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95

AUTHOR’S LIST OF PUBLICATIONS

Wang, C. M., Chen, Y. and Xiang, Y., (2002). “Exact buckling solutions for simply supported, rectangular plates under intermediate and end uniaxial loads.” Proceedings of the 2nd International Conference on Structural Stability and Dynamics, 424-429, edited by C.M. Wang, G.R. Liu and K.K. Ang, World Scientific, 16-18 December 2002, Singapore.

Wang, C. M. and Chen, Y. (2002). “Elastic/plastic buckling of simply supported, rectangular plates under intermediate and end uniaxial loads.” KKCNN Symposium on Civil Engineering, December 19-20, 2002, Singapore

Wang, C. M., Chen, Y. and Xiang, Y. “Stability criteria for rectangular plates subjected to intermediate and end inplane loads.” Thin-Walled Structures, in press

Wang, C. M., Chen, Y. and Xiang, Y. “Plastic buckling of rectangular plates subjected to intermediate and end inplane loads.” International Journal of Solids and Structures, submitted.

buckling of rectangular plates under intermediate and end loads

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