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BUCKLING OF RECTANGULAR PLATES UNDER
INTERMEDIATE AND END LOADS
Chen Yu
NATIONAL UNIVERSITY OF SINGAPORE
2003
BUCKLING OF RECTANGULAR PLATES UNDER
INTERMEDIATE AND END LOADS
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.
Chapter 1 Introduction 3
• 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)
Chapter 1 Introduction 4
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
Chapter 1 Introduction 5
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
Chapter 1 Introduction 6
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
Chapter 1 Introduction 7
(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.
Chapter 1 Introduction 8
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)
Chapter 1 Introduction 9
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,
Chapter 1 Introduction 10
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
Chapter 1 Introduction 11
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
Chapter 1 Introduction 12
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,
Chapter 1 Introduction 13
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
Chapter 1 Introduction 14
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)
Chapter 2 Buckling of Plates under End Loads 17
( ) 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)
Chapter 2 Buckling of Plates under End Loads 18
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)
Chapter 2 Buckling of Plates under End Loads 19
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
Chapter 2 Buckling of Plates under End Loads 20
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
Chapter 2 Buckling of Plates under End Loads 21
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
Chapter 2 Buckling of Plates under End Loads 22
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)
Chapter 2 Buckling of Plates under End Loads 23
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
Chapter 2 Buckling of Plates under End Loads 24
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
Chapter 2 Buckling of Plates under End Loads 25
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)
Chapter 2 Buckling of Plates under End Loads 26
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)
Chapter 2 Buckling of Plates under End Loads 27
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
Chapter 2 Buckling of Plates under End Loads 28
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)
Chapter 2 Buckling of Plates under End Loads 29
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.
Chapter 2 Buckling of Plates under End Loads 30
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
INTERMEDIATE AND END LOADS
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
Chapter 3 Elastic Buckling of Plates under Intermediate and End Loads 33
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)
Chapter 3 Elastic Buckling of Plates under Intermediate and End Loads 34
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)
Chapter 3 Elastic Buckling of Plates under Intermediate and End Loads 35
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.
Chapter 3 Elastic Buckling of Plates under Intermediate and End Loads 36
• 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
Chapter 3 Elastic Buckling of Plates under Intermediate and End Loads 37
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.
Chapter 3 Elastic Buckling of Plates under Intermediate and End Loads 38
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
Chapter 3 Elastic Buckling of Plates under Intermediate and End Loads 39
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).
Chapter 3 Elastic Buckling of Plates under Intermediate and End Loads 40
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 ).
Chapter 3 Elastic Buckling of Plates under Intermediate and End Loads 41
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.
Chapter 3 Elastic Buckling of Plates under Intermediate and End Loads 42
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)
Chapter 3 Elastic Buckling of Plates under Intermediate and End Loads 43
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
Chapter 3 Elastic Buckling of Plates under Intermediate and End Loads 44
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.
Chapter 3 Elastic Buckling of Plates under Intermediate and End Loads 45
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.
Chapter 3 Elastic Buckling of Plates under Intermediate and End Loads 46
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
Compressive Stress StateZero Stress State Tensile Stress State
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
Chapter 3 Elastic Buckling of Plates under Intermediate and End Loads 47
-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
Chapter 3 Elastic Buckling of Plates under Intermediate and End Loads 48
-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.
Chapter 3 Elastic Buckling of Plates under Intermediate and End Loads 49
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
Chapter 3 Elastic Buckling of Plates under Intermediate and End Loads 50
-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
Chapter 3 Elastic Buckling of Plates under Intermediate and End Loads 51
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.
Chapter 3 Elastic Buckling of Plates under Intermediate and End Loads 52
-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
Chapter 3 Elastic Buckling of Plates under Intermediate and End Loads 53
-5
0
5
10
15
-15 -10 -5 0 5 10 15
Λ1
Λ2
(c) Rectangular plate with a/b = 2.0
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.
Chapter 3 Elastic Buckling of Plates under Intermediate and End Loads 54
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
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 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).
Chapter 3 Elastic Buckling of Plates under Intermediate and End Loads 55
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.
Chapter 3 Elastic Buckling of Plates under Intermediate and End Loads 56
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.
Compressive Stress StateZero Stress State Tensile Stress State
Q
P
Chapter 3 Elastic Buckling of Plates under Intermediate and End Loads 57
-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
Chapter 3 Elastic Buckling of Plates under Intermediate and End Loads 58
-4
-2
0
2
4
-8 -4 0 4 8
Λ1
Λ2
(c) Rectangular plate with a/b = 2.0
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.
Chapter 3 Elastic Buckling of Plates under Intermediate and End Loads 59
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
Intermediate Load Location χ
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).
Chapter 3 Elastic Buckling of Plates under Intermediate and End Loads 60
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
INTERMEDIATE AND END LOADS
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)
Chapter 4 Plastic Buckling of Plates under Intermediate and End Loads 63
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)
Chapter 4 Plastic Buckling of Plates under Intermediate and End Loads 64
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)
Chapter 4 Plastic Buckling of Plates under Intermediate and End Loads 65
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
Chapter 4 Plastic Buckling of Plates under Intermediate and End Loads 66
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)
Chapter 4 Plastic Buckling of Plates under Intermediate and End Loads 67
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)
Chapter 4 Plastic Buckling of Plates under Intermediate and End Loads 68
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
Chapter 4 Plastic Buckling of Plates under Intermediate and End Loads 69
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.
Chapter 4 Plastic Buckling of Plates under Intermediate and End Loads 70
• 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.
Chapter 4 Plastic Buckling of Plates under Intermediate and End Loads 71
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.
Chapter 4 Plastic Buckling of Plates under Intermediate and End Loads 72
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
Chapter 4 Plastic Buckling of Plates under Intermediate and End Loads 73
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
Chapter 4 Plastic Buckling of Plates under Intermediate and End Loads 74
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.
Chapter 4 Plastic Buckling of Plates under Intermediate and End Loads 75
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
Chapter 4 Plastic Buckling of Plates under Intermediate and End Loads 76
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.
Chapter 4 Plastic Buckling of Plates under Intermediate and End Loads 77
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
Chapter 4 Plastic Buckling of Plates under Intermediate and End Loads 78
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
Chapter 4 Plastic Buckling of Plates under Intermediate and End Loads 79
-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
Chapter 4 Plastic Buckling of Plates under Intermediate and End Loads 80
-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
Chapter 4 Plastic Buckling of Plates under Intermediate and End Loads 81
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
Chapter 4 Plastic Buckling of Plates under Intermediate and End Loads 82
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
Chapter 4 Plastic Buckling of Plates under Intermediate and End Loads 83
-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
Chapter 4 Plastic Buckling of Plates under Intermediate and End Loads 84
-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
400/ 0 =σE Results based on IT
Chapter 4 Plastic Buckling of Plates under Intermediate and End Loads 85
-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
800/ 0 =σE Results based on IT
200/ 0 =σE Results based on DT
Chapter 4 Plastic Buckling of Plates under Intermediate and End Loads 86
-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
400/ 0 =σE Results based on DT
800/ 0 =σE Results based on DT
Chapter 4 Plastic Buckling of Plates under Intermediate and End Loads 87
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.
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