kim buckling analysis of plates using two variable refined plate theory

8/20/2019 KIM Buckling Analysis of Plates Using Two Variable Refined Plate Theory

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Buckling analysis of plates using the two variable refined plate theory

Seung-Eock Kim a, Huu-Tai Thai a, Jaehong Lee b,

a Department of Civil and Environmental Engineering, Sejong University, 98 Kunja-dong Kwangjin-ku, Seoul 143-747, Republic of Koreab Department of Architectural Engineering, Sejong University, 98 Kunja Dong, Kwangjin Ku, Seoul 143-747, Republic of Korea

a r t i c l e i n f o

Article history:

Received 16 January 2008

Received in revised form

17 July 2008

Accepted 5 August 2008Available online 19 September 2008

Keywords:

Refined plate theory

Buckling analysis

Isotropic plate

Orthotropic plate

Navier method

a b s t r a c t

Buckling analysis of isotropic and orthotropic plates using the two variable refined plate theory is

presented in this paper. The theory takes account of transverse shear effects and parabolic distribution

of the transverse shear strains through the thickness of the plate, hence it is unnecessary to use shear

correction factors. Governing equations are derived from the principle of virtual displacements. The

closed-form solution of a simply supported rectangular plate subjected to in-plane loading has been

obtained by using the Navier method. Numerical results obtained by the present theory are compared

with classical plate theory solutions, first-order shear deformable theory solutions, and available exact

solutions in the literature. It can be concluded that the present theory, which does not require shear

correction factor, is not only simple but also comparable to the first-order shear deformable theory.

& 2008 Elsevier Ltd. All rights reserved.

1. Introduction

The buckling of rectangular plates has been a subject of study

in solid mechanics for more than a century. Many exact solutions

for isotropic and orthotropic plates have been developed, most of

them can be found in Timoshenko and Woinowsky-Krieger [1],

Timoshenko and Gere [2], Bank and Jin [3], Kang and Leissa [4],

Aydogdu and Ece [5], and Hwang and Lee [6]. In company with

studies of buckling behavior of plate, many plate theories have

been developed. The simplest one is the classical plate theory

(CPT) which neglects the transverse normal and shear stresses.

This theory is not appropriate for the thick and orthotropic plate

with high degree of modulus ratio. In order to overcome this

limitation, the shear deformable theory which takes account of

transverse shear effects is recommended. The Reissner [7] and

Mindlin [8] theories are known as the first-order shear deform-

able theory (FSDT), and account for the transverse shear effects bythe way of linear variation of in-plane displacements through the

thickness. However, these models do not satisfy the zero traction

boundary conditions on the top and bottom faces of the plate, and

need to use the shear correction factor to satisfy the constitutive

relations for transverse shear stresses and shear strains. For these

reasons, many higher-order theories have been developed to

improve in FSDT such as Levinson [9] and Reddy [10]. Shimpi and

Patel [11] presented a two variable refined plate theory (RPT) for

orthotropic plates. This theory which looks like higher-order

theory uses only two unknown functions in order to derive two

governing equations for orthotropic plates. The most interesting

feature of this theory is that it does not require shear correction

factor, and has strong similarities with the CPT in some aspects

such as governing equation, boundary conditions and moment

expressions. The accuracy of this theory has been demonstrated

for static bending and free vibration behaviors of plates by Shimpi

and Patel [11], therefore, it seems to be important to extend this

theory to the static buckling behavior.

In this paper, the two variable RPT developed by Shimpi and

Patel [11] has been extended to the buckling behavior of

orthotropic plate subjected to the in-plane loading. Using the

Navier method, the closed-form solutions have been obtained.

Numerical examples involving side-to-thickness ratio and mod-

ulus ratio are presented to illustrate the accuracy of the present

theory in predicting the critical buckling load of isotropic and

orthotropic plates. Numerical results obtained by the presenttheory are compared with CPT solutions, FSDT solutions with

different value of shear correction factor.

2. RPT for orthotropic plates

2.1. Basic assumptions of RPT

Assumptions of the RPT are as follows:

i. The displacements are small in comparison with the plate

thickness h and, therefore, strains involved are infinitesimal.

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0263-8231/$ - see front matter & 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.tws.2008.08.002

Corresponding author. Tel.: +822 3408 3287; fax: +82 2 3408 4331.

E-mail address: [email protected] (J. Lee).

Thin-Walled Structures 47 (2009) 455–462

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ii. The transverse displacement w includes two components of

bending wb and shear ws. Both these components are functions

of coordinates x , y and time t only.

wð x; y; t Þ ¼ wbð x; y; t Þ þ wsð x; y; t Þ (1)

iii. The transverse normal stress s z is negligible in comparisonwith in-plane stresses s x and s y.

iv. The displacements u in x-direction and n in y-direction consist

of extension, bending, and shear components

u ¼ u0 þ ub þ us; v ¼ v0 þ vb þ vs (2)

The bending components ub and vb are assumed to be similarto the displacements given by the classical plate theory.

Therefore, the expression for ub and vb can be given as

ub ¼ z qwbq x

; vb ¼ z qwbq y

(3a)

The shear components us and vs give rise, in conjunctionwith ws, to the parabolic variations of shear strains g xz , g yz and hence to shear stresses s xz , s yz through the thickness of the plate, h, in such a way that shear stresses s xz , s yz arezero at the top and bottom faces of the plate. Consequently,

the expression for us and vs can be given as

us ¼ 1

4 z

5

3 z

z

h

2 qwsq x

; vs ¼ 1

4 z

5

3 z

z

h

2 qwsq y

(3b)

2.2. Kinematics

Based on the assumptions made in the preceding section, the

displacement field can be obtained using Eqs. (1)–(3b) as

uð x; y; z ; t Þ ¼ u0ð x; y; t Þ z qwbq x

þ z 1

4

5

3

z

h

2 qwsq x

vð x; y; z ; t Þ ¼ v0ð x; y; t Þ z

qwb

q y þ z

1

4

5

3

z

h 2 qws

q y

wð x; y; t Þ ¼ wbð x; y; t Þ þ wsð x; y; t Þ (4)

This displacement field accounts for zero traction on boundary

conditions on the top and bottom faces of the plate, and the

quadratic variation of transverse shear strains (and hence

stresses) through the thickness. Thus, there is no need to use

shear correction factors. The strain field obtained by using strain-

displacement relations can be given as

x ¼ 0 x þ z kb

x þ f ks

x

y ¼ 0 y þ z kb

y þ f ks

y

z ¼ 0

g xy ¼ g0

xy

þ z kb

xy

þ f ks

xy

g yz ¼ 5

4 5

z

h

2 qwsq y

g xz ¼ 5

4 5

z

h

2 qwsq x

(5)

where

0 x ¼ qu0q x

; kb x ¼ q

2wb

q x2 ; ks x ¼

q2

wsq x2

0 y ¼ qv0q y

; kb y ¼ q

2wb

q y2 ; ks y ¼

q2

wsq y2

g0 xy ¼ qu0q y

þ qv0q x

; kb xy ¼ 2q

2wb

q x@ y ,

ks xy ¼ 2 q2

wsq xq y

; f ¼ 14

z þ 53

z z h

2(6)

2.3. Constitutive equations

The constitutive equations of an orthotropic plate can be

written as

s xs y

s xys yz s xz

8>>>>>>>>>>>:

9>>>>>>=>>>>>>;

¼

Q 11 Q 12 0 0 0

Q 12 Q 22 0 0 0

0 0 Q 66 0 00 0 0 Q 44 0

0 0 0 0 Q 55

26666664

37777775

x y

g xyg yz g xz

8>>>>>>>>>>>:

9>>>>>>=>>>>>>;

(7)

where Q ij are the plane stress reduced elastic constants in the

material axes of the plate, and are defined as

Q 11 ¼ E 1

1 n12n21; Q 12 ¼

n12E 21 n12n21

; Q 22 ¼ E 2

1 n12n21,

Q 66 ¼ G12; Q 44 ¼ G23; Q 55 ¼ G13 (8)

in which E 1, E 2 are Young’s modulus, G12, G23, G13 are shear

modulus, and n12, n21 are Poisson’s ratios. For the isotropic plate,these above material properties reduce to E 1 ¼ E 2 ¼ E ,

G12 ¼ G23 ¼ G13 ¼ G, n12 ¼ n21 ¼ n. The subscripts 1, 2, 3 corre-spond to x, y, z directions of Cartesian coordinate system,

respectively.

2.4. Equation of motions

The strain energy of the plate can be written as

U ¼ 1

2

Z V

sijijdV ¼ 1

2

Z V

ðs x x þ s y y þ s xyg xy þ s yz g yz þ s xz g xz Þ dV

(9)

Substituting Eqs. (5) and (7) into Eq. (9) and integrating through

the thickness of the plate, the strain energy of the plate can be

rewritten as

U ¼ 1

2

Z A

ðN x0 x þ N y0

y þ N xyg0

xy þ M b

xkb

x þ M b

ykb

y þ M b

xykb

xyÞ d x d y

þ1

2

Z A

ðQ yz g yz þ Q xz g xz þ M s

xks

x þ M s

yks

y þ M s

xyks

xyÞ d x d y (10)

where the stress resultants N , M and Q are defined by

ðN x; N y; N xyÞ ¼

Z h=2h=2

ðs x;s y;s xyÞ d z

ðM b x ; M b

y; M b

xyÞ ¼

Z h=2h=2

ðs x;s y;s xyÞ z d z

ðM s x; M s

y; M s

xyÞ ¼ Z h=2

h=2

ðs x;s y;s xyÞ f d z

ðQ xz ; Q yz Þ ¼

Z h=2h=2

ðs xz ;s yz Þ d z (11)

Substituting Eqs. (5) and (7) into Eq. (11) and integrating

through the thickness of the plate, the stress resultants are related

to the generalized displacements (u0, v0, wb, ws) by the relations

N x

N y

N xy

8>><>>:

9>>=>>; ¼

A11 A12 0

A12 A22 0

0 0 A66

2664

3775

qu0=q x

qv0=q y

qu0=q y þ qv0=q x

8>><>>:

9>>=>>;

M b x

M b y

M b xy

8>>>>>>>:

9>>>>=>>>>;

¼

D11 D12 0

D12 D22 0

0 0 D66

2664

3775

q2

wb=q x2

q2

wb=q y2

2q2

wb=q xq y

8

>>>>>:

9

>>>=>>>;

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M s x

M s y

M s xy

8>>><>>>:

9>>>=>>>; ¼

1

84

D11 D12 0

D12 D22 0

0 0 D66

2664

3775

q2

ws=q x2

q2

ws=q y2

2q2

ws=q xq y

8>>><>>>:

9>>>=>>>;

Q yz

Q xz

( )¼

A44 0

0 A55

" # qws=q y

qws=q x

( ) (12)

where Aij and Dij are called extensional and bending stiffness,respectively, and are defined in terms of the stiffness Q ij as

ð Aij; DijÞ ¼

Z h=2h=2

Q ijð1; z 2Þ d z ði; j ¼ 1; 2; 6Þ

Aij ¼ 5

6Q ijh ði; j ¼ 4; 5Þ (13)

The work done of the plate by applied forces can be written as

V ¼ 1

2

Z A

N o xq

2ðwb þ wsÞ

q x2 þ N o y

q2

ðwb þ wsÞ

q y2

"

þ2N o xyq

2ðwb þ wsÞ

q xq y

#d xd y

Z A

qðwb þ wsÞd xd y (14)

where q and N x0

, N y0

, N xy0

are transverse and in-plane distributedforces, respectively.

The kinetic energy of the plate can be written as

T ¼ 1

2

Z V r €uiidV ¼

1

2

Z A

I 0ð €u20 þ €v

20 þ €w

2b þ €w

2s þ 2 €wb €wsÞd xd y

þ1

2

Z A

I 2q €wbq x

2þ

q €wbq y

2" #þ

I 284

q €wsq x

2þ

q €wsq y

2" #( )d xd y

(15)

where r is mass of density of the plate and I i (i ¼ 0, 2) are theinertias defined by

ðI 0; I 2Þ ¼

Z h=2h=2

ð1; z 2Þrd z (16)

Hamilton’s principle is used herein to derive the equations of motion appropriate to the displacement field and the constitutive

equation. The principle can be stated in analytical form as

0 ¼

Z t 0dðU þ V T Þdt (17)

where d indicates a variation with respect to x and y.

Substituting Eqs. (10), (14) and (15) into Eq. (17) and

integrating the equation by parts, collecting the coefficients of

du0, dv0, dwb, and dws, the equations of motion for the orthotropic

plate are obtained as follows:

du0 :qN xq x

þ qN xyq y

¼ I 0 €u0

dv0 :qN

xyq x þ

qN y

q y ¼ I 0 €v0

dwb : q

2M b x

q x2 þ 2

q2

M b xyq xq y

þq

2M b y

q y2

" #þ q

þ N 0 xq

2w

q x2 þ 2N 0 xy

q2

w

q xq y þ N 0 y

q2

w

q y2

" #¼ I 0 €w I 2r

2 €wb

dws : q

2M s x

q x2 þ 2

q2

M s xyq xq y

þq

2M s y

q y2 þ

qQ xz q x

þqQ yz q y

" #þ q

þ N 0 xq

2w

q x2 þ 2N 0 xy

q2

w

q xq y þ N 0 y

q2

w

q y2

" #¼ I 0 €w

I 284

r 2 €ws (18)

where

r 2 ¼ q2

q x2 þ q

2

q y2 (19)

The boundary conditions of a plate (of length a and width b) are

given as follows:

Clamped–clamped boundaries:On edges x ¼ 0 and a

wb ¼ ws ¼ qwbq x

¼ qwsq x

¼ 0 (20a)

On edges y ¼ 0 and b

wb ¼ ws ¼ qwbq y

¼ qwsq y

¼ 0 (20b)

Simply supported boundaries:On edges x ¼ 0 and a

wb ¼ ws ¼ D11q

2wb

q x2 þ D12

q2

wbq y2

!¼

1

84 D11

q2

wsq x2

þ D12q

2ws

q y2

!¼ 0

(21a)


wb ¼ ws ¼ D12q

2wb

q x2 þ D22

q2

wbq y2

!¼

1

84 D12

q2

wsq x2

þ D22q

2ws

q y2

!¼ 0

(21b) Free–free boundaries:

On edges x ¼ 0 and a

D11q

2wb

q x2 þ D12

q2

wbq y2

!¼

1

84 D11

q2

wsq x2

þ D12q

2ws

q y2

!¼ 0

D11q

3wb

q x3 þ ðD12 þ 4D66Þ

q3

wbq xq y2

" #

¼ A55qwsq x

1

84 D11

q3

wsq x3

þ ðD12 þ 4D66Þ q

3ws

q xq y2

" #¼ 0 (22a)


D12q

2wbq x2

þ D22q

2wbq y2

!¼

1

84 D12

q2wsq x2

þ D22q

2wsq y2

!¼ 0

ðD12 þ 4D66Þ q

3wbq x2q y

þ D22q

3wbq y3

" #

¼ A44qwsq y

1

84 ðD12 þ 4D66Þ

q3

wsq x2q y

þ D22q

3ws

q y3

" #¼ 0 (22b)

3. Buckling of a simply supported rectangular plate under

compressive loads

When a plate is subjected to in-plane compressive forces

(Fig. 1b), and if the forces are small enough, the equilibrium of the

plate is stable and the plate remains flat until a certain load is

reached. At that load, called the buckling load, the stable state of

the plate is disturbed and plate seeks an alternative equilibrium

configuration accompanied by a change in the load-deflection

behavior. The critical buckling loads of simply supported,

orthotropic, rectangular plate will be determined in this paper

by using the Navier solution. The governing equations of plate in

case of static buckling are given by

D11q

4wb

q x4 þ 2ðD12 þ 2D66Þ

q4

wbq x2q y2

þ D22q

4wb

q y4

¼ N 0q

2w

q x2 þ gq

2w

q y2 ! 1

84 D11

q4

ws

q x4 þ 2ðD12 þ 2D66Þ

q4

ws

q x2q y2"

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S.-E. Kim et al. / Thin-Walled Structures 47 (2009) 455–462 457


4/8

þD22q

4ws

q y4

# A44

q2

wsq y2

þ A55q

2ws

q x2

" #

¼ N 0q

2w

q x2 þ g

q2

w

q y2

! (23)

The Navier method is only applied for simply supported

boundary conditions on all four edges of the rectangular plate,as shown in Fig. 1a. The simply supported boundary conditions on

all four edge of the rectangular plate can be expressed as

wð x; 0Þ ¼ wð x; bÞ ¼ wð0; yÞ ¼ wða; yÞ ¼ 0 (24a)

M xð0; yÞ ¼ M xða; yÞ ¼ M yð x; 0Þ ¼ M yð x; bÞ ¼ 0 (24b)

The following displacement functions wb and ws are chosen to

automatically satisfy the boundary conditions in Eqs. (24a) and

(24b)

wb ¼X1m¼1

X1n¼1

W bmn sina x sinb y

ws ¼X1m¼1

X1n¼1

W smn sina x sinb y (25)

where a ¼ mp/a, b ¼ np/b and W bmn, W smn are coefficients.Substituting Eq. (25) into Eq. (23), the following system is

obtained:

k11 k12

k12 k22

" # W bmn

W smn

( )¼

0

0

(26)

where

k11 ¼ ½D11a4 þ 2ðD12 þ 2D66Þa

2b2

þ D22b4

N 0ða2 þ gb2Þ

k12 ¼ N 0ða2 þ gb2Þ

k22 ¼ 1

84½D11a

4 þ 2ðD12 þ 2D66Þa2b

2

þ D22b4

þ A55a2 þ A44b

2 N 0ða

2 þ gb2Þ (27)

For nontrivial solution, the determinant of the coefficient

matrix in Eq. (26) must be zero. This gives the following

expression for buckling load:

N 0 ¼ D

a2 þ gb2ðD=84 þ A55a2 þ A44b

2Þ

D þ ðD=84 þ A55a2 þ A44b2

Þ(28)

where

D ¼ D11a4 þ 2ðD12 þ 2D66Þa

2b2

þ D22b4

(29)

Clearly, when the effect of transverse shear deformation is

neglected, the Eq. (28) yields the result obtained using the

classical plate theory. It indicates that transverse shear deforma-

tion has the effect of reducing the buckling load. For each choice of

m and n, there is a corresponsive unique value of N 0. The criticalbuckling load is the smallest value of N 0(m, n).

4. Numerical results and discussion

For verification purposes, a simply supported rectangular plate

subjected to the loading conditions, as shown in Fig. 2, is

considered to illustrate the accuracy of the present theory in

predicting the buckling behavior of the plate. In order to

ARTICLE IN PRESS

N0 N0

x

y y

x

N0N0

N0

N0

N0

x

y

N0

N0

N0

Fig. 2. The loading conditions of square plate for (a) uniaxial compression, (b) biaxial compression and (c) tension in the x direction and compression in the y direction.

a

b

y

x

at y=b

w = M y = 0

at y = 0

w = M y = 0

at x = 0

w = M x = 0

at x = a

w = M x = 0

a

y

x b

N0xx

0N

0yN

Ny0

Fig. 1. Rectangular plate: (a) boundary condition and (b) in-plane forces.



5/8

investigate the effects of side-to-thickness ratio and modulus

ratio, the first example is applied for isotropic and orthotropic

square plates. Many shear correction factors (k ¼ 2/3, k ¼ 5/6 and

k ¼ 1) are also used for the FSDT in comparison with the present

theory. The following engineering constants are used [12]

E 1=E 2 varied; G12=E 2 ¼ G13=E 2 ¼ 0:5; G23=E 2 ¼ 0:2; n12 ¼ 0:25

(30)For convenience, the following nondimensional buckling load is

used:

¯ N ¼ N cr a

2

E 2h3

(31)

where a is the length of the square plate and h is the thickness of

the plate.

The results of critical buckling load of simply supported square

plate are presented in Tables 1–3 and Figs. 3–6. In the case of

isotropic plate (Fig. 3a), the results obtained by RPT and FSDT are

in excellent agreement event though the plate is very thick. In

case of square plate (a ¼ b ¼ 5h), the maximum difference of RPT

and FSDT with the shear correction factor 5/6 is 0.24%, as shownin Table 3. When the orthotropic plate is used, the difference

between RPT and FSDT will increase with respect to the increase

of side-to-thickness ratio (Fig. 3) and modulus ratio (Figs. 4–6). As

presented in Table 1, the differences between RPT and FSDT

(k ¼ 5/6), and RPT and FSDT (k ¼ 1) are 16.14% and 2.24%,

respectively, for the same case of square plate (a ¼ b ¼ 5h and

E 1/E 2 ¼ 40). It can be seen from Tables 1–3 that the difference of

critical buckling load between RPT and FSDT depends on not only

the side-to-thickness and modulus ratios, but also the in-plane

loading conditions (Fig. 2). In case of square plate (a ¼ b ¼ 10h),

the difference between RPT and FSDT (k ¼ 5/6) is 9.62% for

uniaxial compression (Fig. 4 and Table 1), 9.36% for biaxial

compression (Fig. 5 and Table 2), and 2.92% for tension in the x-

direction and compression in the y-direction (Fig. 6 and Table 3).

ARTICLE IN PRESS

Table 3

Comparison of nondimensional critical buckling load of square plates subjected to

tension in the x direction and compression in the y direction

a/h Theories Isotropic

n ¼ 0.3Orthotropic

E 1/E 2 ¼ 10 E 1/E 2 ¼ 25 E 1/E 2 ¼ 40

5 RPT 4.8274a 4.0258b 4.1044c 4.1525c

FSDT (k ¼ 2/3) 4.4175a 3.2849d 3.3001e 3.3053e

FSDT (k ¼ 5/6) 4.8158a 3.9241c 3.9794c 4.0075d

FSDT (k ¼ 1) 5.1237a 4.4488b 4.5691c 4.6073c

10 RPT 6.6024a 7.7863a 8.5471b 9.1638b

FSDT (k ¼ 2/3) 6.4032a 7.2656a 7.7820b 8.1208b

FSDT (k ¼ 5/6) 6.6010a 7.7748a 8.4774b 8.9039b

FSDT (k ¼ 1) 6.7398a 8.0651a 9.0153b 9.5197b

20 RPT 7.2754a 9.2811a 11.6347b 12.8031b

FSDT (k ¼ 2/3) 7.2139a 9.1310a 11.2544b 12.1990b

FSDT (k ¼ 5/6) 7.2753a

9.2782a

11.6015b

12.6339b

FSDT (k ¼ 1) 7.3168a 9.3790a 11.8453b 12.9428b

50 RPT 7.4895a 9.8101a 12.9531b 14.4177b

FSDT (k ¼ 2/3) 7.4790a 9.7830a 12.8751b 14.2839b

FSDT (k ¼ 5/6) 7.4895a 9.8097a 12.9463b 14.3789b

FSDT (k ¼ 1) 7.4965a 9.8275a 12.9942b 14.4430b

100 RPT 7.5211a 9.8907a 13.1666b 14.6827b

FSDT (k ¼ 2/3) 7.5185a 9.8838a 13.1463b 14.6474b

FSDT (k ¼ 5/6) 7.5211a 9.8906a 13.1648b 14.6724b

FSDT (k ¼ 1) 7.5229a 9.8951a 13.1772b 14.6891b

CPT 7.5317a 9.9179a 13.2393b 14.7732b

a Mode for plate is (m, n) ¼ (1,2).b Mode for plate is (m, n) ¼ (1,3).c Mode for plate is (m, n) ¼ (1,4).d Mode for plate is (m, n) ¼ (1,5).e Mode for plate is (m, n) ¼ (1,6).

Table 2


biaxial compressive load


n ¼ 0.3Orthotropic

E 1/E 2 ¼ 10 E 1/E 2 ¼ 25 E 1/E 2 ¼ 40

5 RPT 1.4756 2.8549a 3.3309a 3.4800a

FSDT (k ¼ 2/3) 1.4100 2.5042a 2.7332a 2.8303a

FSDT (k ¼ 5/6) 1.4749 2.8319a 3.1422a 3.2822a

FSDT (k ¼ 1) 1.5216 3.1027a 3.4933a 3.6793a

10 RPT 1.7112 4.6718a 6.0646a 7.2536a

FSDT (k ¼ 2/3) 1.6886 4.4259 5.4351a 6.0797a

FSDT (k ¼ 5/6) 1.7111 4.6367 5.8370a 6.6325a

FSDT (k ¼ 1) 1.7265 4.7708 6.1425a 7.0690a

20 RPT 1.7825 5.3267 7.6643a 9.6614a

FSDT (k ¼ 2/3) 1.7763 5.2463 7.3701a 8.9895a

FSDT (k ¼ 5/6) 1.7825 5.3100 7.5546a 9.3049a

FSDT (k ¼ 1) 1.7866 5.3533 7.6834a 9.5297a

50 RPT 1.8036 5.5390 8.2784a 10.6576a

FSDT (k ¼ 2/3) 1.8025 5.5249 8.2199a 10.5111a

FSDT (k ¼ 5/6) 1.8036 5.5361 8.2566a

10.5810a

FSDT (k ¼ 1) 1.8042 5.5436 8.2812a 10.6282a

100 RPT 1.8066 5.5707 8.3744a 10.8172a

FSDT (k ¼ 2/3) 1.8063 5.5672 8.3593a 10.7788a

FSDT (k ¼ 5/6) 1.8066 5.5700 8.3687a 10.7972a

FSDT (k ¼ 1) 1.8068 5.5719 8.3751a 10.8095a

CPT 1.8076 5.5814 8.4069 10.8715a

a Mode for plate is (m, n) ¼ (1,2).

Table 1


uniaxial compression


n ¼ 0.3Orthotropic

E 1/E 2 ¼ 10 E 1/E 2 ¼ 25 E 1/E 2 ¼ 40

5 RPT 2.9512 6.3478 9.1039 10.5785

FSDT (k ¼ 2/3) 2.8200 5.5679 7.1122 7.7411

FSDT (k ¼ 5/6) 2.9498 6.1804 8.2199 9.1085

FSDT (k ¼ 1) 3.0432 6.6715 9.1841 10.3463

10 RPT 3.4224 9.3732 16.7719 22.2581

FSDT (k ¼ 2/3) 3.3772 8.8988 14.7011 18.3575

FSDT (k ¼ 5/6) 3.4222 9.2733 15.8736 20.3044FSDT (k ¼ 1) 3.4530 9.5415 16.7699 21.8602

20 RPT 3.5650 10.6534 21.3479 31.0685

FSDT (k ¼ 2/3) 3.5526 10.4926 20.4034 28.85

FSDT (k ¼ 5/6) 3.5650 10.6199 20.9528 30.0139

FSDT (k ¼ 1) 3.5733 10.7066 21.3363 30.8451

50 RPT 3.6071 11.0780 23.1225 34.9717

FSDT (k ¼ 2/3) 3.6051 11.0497 22.9366 34.4886

FSDT (k ¼ 5/6) 3.6071 11.0721 23.0461 34.7487

FSDT (k ¼ 1) 3.6085 11.0871 23.1197 34.9244

100 RPT 3.6132 11.1415 23.4007 35.6120

FSDT (k ¼ 2/3) 3.6127 11.1343 23.3527 35.4852

FSDT (k ¼ 5/6) 3.6132 11.1400 23.3810 35.5538

FSDT (k ¼ 1) 3.6135 11.1438 23.3999 35.5996

CPT 3.6152 11.1628 23.4949 35.8307



6/8

The difference between RPT and FSDT is also due to the shear

correction factors using in FSDT. In case of tension in the x

direction and compression in the y direction (Fig. 2c), the buckling

mode shape (Fig. 7) switches from asymmetric to symmetric or,

conversely, from symmetric to asymmetric, depending on the

shear correction factors.

The next comparison is carried out for the orthotropic

rectangular plates subjected to uniaxial compression with thevariation of aspect ratio and side-to-thickness ratio. The following

material properties are used [13]:

E 2=E 1 ¼ 0:52; G12=E 1 ¼ G23=E 1 ¼ 0:26,

G13=E 1 ¼ 0:16; n12 ¼ 0:44; n21 ¼ 0:23 (32)

The results of buckling stress parameter, k x ¼ (P cr/E 1)(12/p2)

(b/h)2, are shown in Table 4 and Fig. 8. It is observed that the

present theory overpredicts the asymptotic of buckling stress

parameter and is more accurate than the Mindlin theory. Even thequite thick plate (b ¼ 5h), the error is only 0.19%.

ARTICLE IN PRESS

0

10

20

30

40

0

CPTFSDP (k = 5/6)RPT

N

0

10

20

30

40

0


N

E1 /E2 E1 /E2

10 20 30 40 10 20 30 40

Fig. 4. The effect of modulus ratio on the critical buckling load of square plate subjected to uniaxial compression: (a) a ¼ 10h and (b) a ¼ 20h.

00

3

6

9

12 CPTFSDP (k = 5/6)RPT

N

0

0

3

6

9

12 CPTFSDP (k = 5/6)RPT

N

E1 /E2 E1 /E2

10 20 30 40 10 20 30 40

Fig. 5. The effect of modulus ratio on the critical buckling load of square plate subjected to biaxial compression: (a) a ¼ 10h and (b) a ¼ 20h.

.

2.7

2.9

3.1

3.3

3.5

3.7

CPT

FSDT (k = 5/6)RPT

.

N

N

4

6

8

10

12

CPTFSDT (k = 5/6)RPT

N

N

0

6

10

14

18

22

26

a/h


0

5

12

19

26

33

40

a/h


25 50 75 10025 50 75 100

0

a/h

0

a/h

25 50 75 10025 50 75 100

Fig. 3. The effect of side-to-thickness and modulus ratios on the critical buckling load of square plate subjected to uniaxial compression: (a) isotropic, (b) E 1/E 2 ¼ 10,

(c) E 1/E 2 ¼ 25 and (d) E 1/E 2 ¼ 40.



7/8

5. Concluding remarks

An efficient two variable refined plate theory proposed by

Shimpi and Patel [11] has been applied in this paper for bucklingbehavior of isotropic and orthotropic plates. The theory takes

account of transverse shear effects and parabolic distribution of

the transverse shear strains through the thickness of the plate,

hence it is unnecessary to use shear correction factors. The

governing equations are strong similarity with the classical platetheory in many aspects. It can be concluded that the two variable

ARTICLE IN PRESS

Fig. 7. Buckling mode shapes of orthotropic square plate: (a) ( m, n) ¼ (1,1); (b) (m, n) ¼ (1,2); (c) (m, n) ¼ (1,3); (d) (m, n) ¼ (1,4); (e) (m, n) ¼ (1,5); (f) (m, n) ¼ (1,6).

6

9

12

15

18

0


N

6

9

12

15

18

0


N

E1 /E2 E1 /E2

10 20 30 40 10 20 30 40

Fig. 6. The effect of modulus ratio on the critical buckling load of square plate subjected to tension in the x direction and compression in the y direction: (a) a ¼ 10h and

(b) a ¼ 20h.



8/8

refined plate theory can accurately predict the critical buckling

loads of the isotropic plates.

Acknowledgements

The support of the research reported here by Ministry of Commerce, Industry and Energy through Grant 10020379-2006-

12 and Korea Ministry of Construction and Transportation

through Grant 2006-C106A103001-06A050300220 are gratefully

acknowledged.

References

[1] Timoshenko SP, Woinowsky-Krieger S. Theory of plates and shells. New York:McGraw-Hill; 1959.

[2] Timoshenko SP, Gere JM. Theory of elastic stability. New York: McGraw-Hill;1961.

[3] Bank L, Yin J. Buckling of orthotropic plates with free and rotationallyrestrained unloaded edges. Thin Wall Struct 1996;24(1):83–96.

[4] Kang JH, Leissa AW. Exact solutions for the buckling of rectangular plates

having linearly varying in-plane loading on two opposite simply supportededges. Int J Solids Struct 2005;42(14):4220–38.

[5] Aydogdu M, Ece MC. Buckling and vibration of non-ideal simply supportedrectangular isotropic plates. Mech Res Commun 2006;33(4):532–40.

[6] Hwang I, Lee JS. Buckling of orthotropic plates under various inplane loads.KSCE J Civ Eng 2006;10(5):349–56.

[7] Reissner E. The effect of transverse shear deformation on the bending of elastic plates. J Appl Mech-T ASME 1945;12(2):69–77.

[8] Mindlin RD. Influence of rotary inertia and shear on flexural motions of isotropic, elastic plates. J Appl Mech-T ASME 1951;18(1):31–8.

[9] Levinson M. An accurate simple theory of the statics and dynamics of elasticplates. Mech Res Commun 1980;7(6):343–50.

[10] Reddy JN. A refined nonlinear theory of plates with transverse sheardeformation. Int J Solids Struct 1984;20(9):881–96.

[11] Shimpi RP, Patel HG. A two variable refined plate theory for orthotropic plateanalysis. Int J Solids Struct 2006;43(22–23):6783–99.

[12] Reddy JN. Mechanics of laminated composite plate: theory and analysis. New York: CRC Press; 1997.

[13] Srinivas S, Rao AK. Bending, vibration and buckling of simply supported thickorthotropic rectangular plates and laminates. Int J Solids Struct 1970;6(11):1463–81.

ARTICLE IN PRESS

Table 4

Asymptotic k x ¼ (P cr/E 1)(12/p2)(b/h)2 for buckling of orthotropic plate under normal stress P on edges x ¼ 0 and x ¼ a

b/h Exacta CPT Mindlina RPT Error-CPT (%) Error-Mindlin (%) Error-RPT(%)

20 2.966 3.039 2.965 2.9665 2.46 0.03 0.02

10 2.770 3.039 2.768 2.7709 9.71 0.07 0.03

5 2.210 3.039 2.204 2.2143 37.51 0.27 0.19

a Take from Ref. [13].

0

a/b

a

P

x

P

2

3

4

5

6

B u c k l i n g s t r e s s p a r a m e t e r , k

CPTRPT (b = 20h)RPT (b = 10h)RPT (b = 5h)

b

y

1 2 3 4

Fig. 8. Variation of buckling stress parameter k x ¼ (P cr/E 1)(12/p2)(b/h)2 with

respect to b/h and a/b for orthotropic plates under load on x ¼ 0 and x ¼ a.


kim buckling analysis of plates using two variable refined plate theory

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