PLASTIC BUCKLING OF SSSS THIN RECTANGULAR PLATES … · RECTANGULAR PLATES SUBJECTED TO UNIAXIAL COMPRESSION USING ... buckling of plates may be classified as elastic ... (1948),

168

Int. J. Struct. & Civil Engg. Res. 2013 U G Eziefula et al., 2013

PLASTIC BUCKLING OF SSSS THINRECTANGULAR PLATES SUBJECTED

TO UNIAXIAL COMPRESSION USINGTAYLOR-MACLAURIN SHAPE FUNCTION

D O Onwuka1, O M Ibearugbulem1 and U G Eziefula2*

1 Department of Civil Engineering, Federal University of Technology, Owerri, Nigeria.2 School of Engineering Technology, Imo State Polytechnic, Umuagwo, Nigeria

*Corresponding author:U G Eziefula, [email protected]

ISSN 2319 – 6009 www.ijscer.comVol. 2, No. 4, November 2013

© 2013 IJSCER. All Rights Reserved

Int. J. Struct. & Civil Engg. Res. 2013

Research Paper

INTRODUCTIONBased on the stress-strain relationship,

buckling of plates may be classified as elastic

buckling or plastic buckling. Elastic buckling

is based on Hooke’s law where it is assumed

that the proportional limit of the plate material

In this paper, a solution for the plastic buckling of a thin rectangular isotropic plate with foursimply supported edges under uniform in-plane compression is presented. The plastic bucklingequation was derived using a deformation theory of plasticity and a work principle. The plateanalysis was carried out through a theoretical formulation based on Taylor-Maclaurin series andapplication of energy method. The approximate shape function for the plate boundary conditionsusing the Taylor-Maclaurin series was truncated at the fifth term. The shape function wassubstituted into the plastic buckling equation and the critical plastic buckling load was obtained.The plate buckling coefficient was determined for aspect ratios within the range of 0.1 and 1.0 atincrements of 0.1. The results were compared with solutions from previous studies and theaverage percentage difference was 0.091%. This difference demonstrates that the Taylor-Maclaurin series shape function is a very good approximation of the exact values for thedisplacement function of the deformed SSSS plate.

Keywords: Critical buckling load, Deformation plasticity theory, Displacement function, In-plane compression, Taylor-Maclaurin series, Thin plate

is greater than the buckling stress. In many

practical cases, however, buckling may occur

in the plastic range. The actual buckling load

in the plastic range is always lower than the

buckling load in the elastic range. Hence, it is

necessary to carry out plastic buckling analysis

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using plasticity theories so as to determine theaccurate buckling load when buckling occursin the plastic range.

The two commonly used plasticity theoriesin plate buckling are the deformation theorypioneered by Ilyushin (1947) and the flow (orincremental) theory developed by Handelmanand Prager (1948). The deformation theory ofplasticity is mathematically less consistent incomparison with the flow theory of plasticity.However, most researchers accept that theplastic buckling loads by deformation theoryare always in better agreement withexperimental results and that they have lowernumerical values than those obtained from theflow theory. This is the well-known paradox ofplate plastic buckling, and a universallyaccepted solution of the plastic bucklingparadox has not yet been presented (Prideand Heimerl, 1949; Iskason and Pifko, 1969;Becque, 2010).

In finding solutions to plate bucklingproblems for both the elastic and plasticranges, the use of Fourier series or

trigonometric series in estimating the shapefunction of the deformed plate exists inliterature. Irrespective of the plasticity theoryused, some researchers used the numericalapproach while others used the equilibriumand energy approaches in finding solutions toplastic buckling of plates. Studies byresearchers such as Stowell (1948), Iyengar(1988), Shen (1990) and Wang et al. (2004)involved the use of trigonometric series. Theuse of Taylor’s series in solving plate bucklingproblems has attracted very little attention.

To the best of the researchers’ knowledge,the Taylor’s series has not been used in theenergy approach for analyzing the plasticbuckling of SSSS plates. Therefore, the aimof this study is to use the Taylor-Maclaurinseries to solve the plastic buckling problem ofa thin rectangular isotropic plate with foursimply supported edges subjected to uniaxialin-plane compressive loads. The problemdefinition is illustrated in Figure 1. Thegoverning equation derived in the analysis isbased on the deformation theory of plasticityusing Stowell’s approach.

Figure 1: SSSS Thin Rectangular Plate under Uniaxial In-plane Loading

b Nx

SSSS PLATENx

a

y

x

170


3

9sE t

D ...(4)

w = AH ...(5)

p is the aspect ratio, t is the plate thickness,a and b are the length and width of the platerespectively, H is the plate buckling coefficientand A is amplitude of the shape function.

Ibearugbulem (2012) expanded the shapefunction using Taylor-Maclaurin series andobtained

( ) ( )0 0

0 00 0

,

..

! !

m nm n

m n

w w x y

F x F yx x y y

m n

...(6)

where F(m)(x0) is the mth partial derivative of the

function with respect to x and F(n)(y0) is the nth

partial derivative of the function w and respectto y. m! and n! are the factorials of m and nrespectively while x

0 and y

0 are the points of

origin. He truncated the infinite power seriesat m = n = 4 and got

4

0

4

0

.n

nmnm

m

QRKJw ...(7)

where

0

!

m m

m

F aJ

m

...(8a)

0

!

n n

n

F bK

n

...(8b)

The boundary conditions for an SSSS plateare

0 0; 0 0Rw R w R ...(9)

MATHEMATICAL FORMULATIONStowell (1948) expressed the differentialequation of equilibrium for the plastic bucklingof a thin, flat, rectangular plate under uniformcompression in the x-direction as:

4 4 4 2

4 2 2 4 2

312 0

4 4t x

s

E Nw w w w

E x x y y D x

...(1)

Where Etis the tangent modulus, E

s is the

secant modulus, Nx is the buckling load, D

— is

the plastic flexural rigidity of the plate and w isthe displacement in the z-axis. Transformingthe x – y coordinate system to R – Q coordinatesystem, we have

;x y

R Qa b

It should be noted that R and Q aredimensionless parameters.

Eziefula (2013) applied a technique basedon Ibearugbulem et al. (2013) where Equation(1) was transformed using the principle ofconservation of work in a static continuum.Eziefula (2013) made N

x the subject of formula

and obtained

4 4 41 3 22 1 1

0 0 4 4 2 2 2 44 4

22 1 1

0 0 4

Nx

EH H H H HtDp H R Q

p E R p R Q Qs

Hb H R Q

R

...(2)

where

p = a/b ...(3)

2

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1 0; 1 0Rw R w R ...(10)

0 0; 0 0Rw Q w Q ...(11)

1 0; 1 0Rw Q w Q ...(12)

Substituting Equations (9) and (11) intoEquation (7) gave

J0 = J

2 = 0; K

0 = K

2 = 0

Substituting Equation (10) into Equation (7)and solving the resulting two equationssimultaneously gave

J1 = J

4 ; J

3 = –2J

4

Similarly, substituting Equation (12) intoEquation (7) and solving the resulting twoequations simultaneously gave

K1 = K

4 ; K

3 = –2K

4

Substituting the values of J0, J

1, J

2, J

3, J

4,

K0, K

1, K

2, K

3 and K

4 into Equation (7) gave

3 4 3 44 4 4 4 4 42 2w J R J R J R K Q K Q K Q

3 4 3 44 4 2 2J K R R R Q Q Q ...(13)

From Equations (5), (7) and (13), we have

4 4A J K ...(14)

3 4 3 42 2H R R R Q Q Q ...(15)

Partial derivatives of Equation (15) withrespect to R, Q or both R and Q gave

4

3 4 3 44

24 2 2H

H R R R Q Q QR

...(16)

4

23 4 3 44

24 2 2H

H R R R Q Q QR

...(17)

2

Q

4

2 24 4

144H

H R R Q QR Q

3 4 3 42 2R R R Q Q Q ...(18)

2

2 3 42

12 2H

H R R R R RR

23 42Q Q Q ...(19)

Expanding and integrating Equations (16),(17), (18), and (19) partially with respect to Rand Q in a closed domain respectively resultedin

1 1 4

40 0

0.23619H

H R QR

...(20)

1 1 4

40 0

0.23619H

H R QQ

...(21)

1 1 4

2 20 0

2 0.47183H

H R QR Q

...(22)

1 1 2

20 0

0.02390H

H R QR

...(23)

Substituting the values in Equations (20),(21), (22) and (23) into Equation (2) gave

0.23619 1 3 20.47183 0.236192 2 4 4

0.2390

Nx

ED t pEb p s

...(24)

The plastic buckling load may be expressedas

0.02390

2 2

2 2

3 4

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2

2

b

DHNx

...(25)

where

2

200130.100027.2

4

3

4

100130.1p

E

E

pH

s

t

...(26)

RESULTS AND DISCUSSIONThe results from this study gave the equationof critical plastic buckling load as

2

00130.100027.24

3

4

12

00130.12

2

, psE

tE

pb

DCRxN

...(27)

From Iyengar (1988), the exact solution forthe plastic buckling of an SSSS plate usingStowell’s approach is

242

2

2

2

, 24

3

4

1

m

pnn

p

m

E

E

b

DN

s

tCRx

...(28)

In Equation (28), m and n are the bucklingmodes. For the first mode of buckling, m = 1.Also, since we are interested in finding thelowest value of N

x at which the plate buckles,

n must be equal to one (Iyengar, 1988). Hence,Equation (28) may be simplified to

2

22

2

, 24

3

4

11p

E

E

pb

DN

s

tCRx

...(29)

The factor, Et/E

s is equal to one in elastic

buckling but its value is always less than unityin plastic buckling. In this paper, the numericalvalue of E

t/E

s is taken to be equal to 0.9.

Table 1 shows the values of H from this presentstudy and Iyengar (1988) for different aspectratios using E

t/E

s = 0.9.

Table 1: Values of H for Plastic Bucklingof Uniaxially Compressed SSSS Thin Rectangular Plate

p = a/b H from Present Study H from Iyengar (1988) Percentage Difference

0.1 94.6305 94.5100 0.1275

0.2 25.1954 25.1650 0.1208

0.3 12.3815 12.3678 0.1108

0.4 7.9490 7.9413 0.0970

0.5 5.9554 5.9500 0.0908

0.6 4.9335 4.9294 0.0832

0.7 4.3811 4.3778 0.0754

0.8 4.0883 4.0853 0.0733

0.9 3.9547 3.9520 0.0689

1.0 3.9277 3.9251 0.0662

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From Table 1, the highest percentagedifference is 0.1275% for p = 0.1, while thelowest percentage difference is 0.0662% forp = 1.0. The average percentage differencebetween the solution from this present studyand Iyengar’s solution is 0.091%. Iyengar’ssolution is an exact solution obtained fromtrigonometric series while the solution from thepresent study is an approximate solutionbased on Taylor-Maclaurin series. The solutionfrom the present study is an upper boundsolution. It can be observed that the closenessof the two solutions improves as the aspectratio increases from 0.1 to 1.

CONCLUSIONIn this study, plastic buckling analysis of a thin,flat, rectangular, isotropic SSSS plate wascarried out using Stowell’s plasticity theory andTaylor-Maclaurin series shape function. A worktechnique was applied to determine the platebuckling coefficient for different aspect ratiosof the plate. The results showed that thesolution is a very close approximation of theexact solution. Therefore, the Taylor-Maclaurinseries is adequate for approximating thedeformed shape of the SSSS plate in theplastic buckling analysis.

REFERENCES1. Becque J (2010), “Inelastic Plate

Buckling”, ASCE Journal of EngineeringMechanics, Vol. 136, No. 9, pp. 1123-1130.

2. Eziefula U G (2013), “Plastic BucklingAnalysis of Thin Rectangular IsotropicPlates”, M.Eng Thesis Submitted toPostgraduate School, Federal Universityof Technology, Owerri.

3. Handelman G H and Prager W (1948),“Plastic Buckling of Rectangular Platesunder Edge Thrusts”, NACA TechnicalReport, No. 946.

4. Ibearugbulem O M (2012), “Applicationof a Direct Variational Principle in ElasticStability Analysis of Thin Rectangular FlatPlates”, Ph.D Dissertation, FederalUniversity of Technology, Owerri.

5. Ibearugbulem O M, Ettu L O and Ezeh JC (2013), “Direct Integration and WorkPrinciple as New Approach in BendingAnalyses of Isotropic Rectangular Plates”.The International Journal of Engineeringand Science, Vol. 2, No. 3, pp. 28-36.

6. Ilyushin A A (1947), “The Elasto-plasticStability of Plates”, NACA TechnicalMemorandum, No. 1188.

7. Isakson G and Pifko A (1969), “A FiniteElement Method for the Plastic BucklingAnalysis of Plates”. AIAA Journal, Vol. 7,No. 10, pp. 1950-1957.

8. Iyengar N G R (1988), “Structural Stabilityof Columns and Plates”, Ellis Horwood,Chichester.

9. Pride R A and Heimerl G J (1949), “PlasticBuckling of Simply Supported CompressedPlates”, NACA Technical Note, No.1817.

10. Shen H S (1990), Elasto-plastic Analysisfor the Buckling and Postbuckling ofRectangular Plates under UniaxialCompression. Applied Mathematics andMechanics, English Edition, Vol. 11, No.10, pp. 931-939.

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Plastic Buckling of Columns and Plates”,NACA Technical Report, No. 898.

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“Plastic Bucking of Rectangular PlatesSubjected to Intermediate and End In-planeLoads”, International Journal of Solids andStructures, Vol. 41, pp. 4279-4297.

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PLASTIC BUCKLING OF SSSS THIN RECTANGULAR PLATES … · RECTANGULAR PLATES SUBJECTED TO UNIAXIAL COMPRESSION USING ... buckling of plates may be classified as elastic ... (1948),