ARTICLE IN PRESS. Thin-Walled Structures

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1 RTICLE I PRESS Thin-Walled Structures 47 () 4 4 Contents lists availale at ScienceDirect Thin-Walled Structures journal homepage: Buckling analsis of plates using the two variale refined plate theor Seung-Eock Kim a, Huu-Tai Thai a, Jaehong Lee, a Department of Civil and Environmental Engineering, Sejong Universit, Kunja-dong Kwangjin-ku, Seoul , Repulic of Korea Department of rchitectural Engineering, Sejong Universit, Kunja Dong, Kwangjin Ku, Seoul , Repulic of Korea article info rticle histor: Received 1 Januar Received in revised form 17 Jul ccepted ugust vailale online 1 Septemer Kewords: Refined plate theor Buckling analsis Isotropic plate Orthotropic plate avier method astract Buckling analsis of isotropic and orthotropic plates using the two variale refined plate theor is presented in this paper. The theor takes account of transverse shear effects and paraolic distriution of the transverse shear strains through the thickness of the plate, hence it is unnecessar to use shear correction factors. Governing equations are derived from the principle of virtual displacements. The closed-form solution of a simpl supported rectangular plate sujected to in-plane loading has een otained using the avier method. umerical results otained the present theor are compared with classical plate theor solutions, first-order shear deformale theor solutions, and availale eact solutions in the literature. It can e concluded that the present theor, which does not require shear correction factor, is not onl simple ut also comparale to the first-order shear deformale theor. & Elsevier Ltd. ll rights reserved. 1. Introduction The uckling of rectangular plates has een a suject of stud in solid mechanics for more than a centur. Man eact solutions for isotropic and orthotropic plates have een developed, most of them can e found in Timoshenko and Woinowsk-Krieger [1], Timoshenko and Gere [], Bank and Jin [3], Kang and Leissa [4], dogdu and Ece [], and Hwang and Lee []. In compan with studies of uckling ehavior of plate, man plate theories have een developed. The simplest one is the classical plate theor () which neglects the transverse normal and shear stresses. This theor is not appropriate for the thick and orthotropic plate with high degree of modulus ratio. In order to overcome this limitation, the shear deformale theor which takes account of transverse shear effects is recommended. The Reissner [7] and Mindlin [] theories are known as the first-order shear deformale theor (FSDT), and account for the transverse shear effects the wa of linear variation of in-plane displacements through the thickness. However, these models do not satisf the zero traction oundar conditions on the top and ottom faces of the plate, and need to use the shear correction factor to satisf the constitutive relations for transverse shear stresses and shear strains. For these reasons, man higher-order theories have een developed to improve in FSDT such as Levinson [] and Redd [1]. Shimpi and Patel [] presented a two variale refined plate theor () for orthotropic plates. This theor which looks like higher-order Corresponding author. Tel.: ; fa: address: jhlee@sejong.ac.kr (J. Lee). theor uses onl two unknown functions in order to derive two governing equations for orthotropic plates. The most interesting feature of this theor is that it does not require shear correction factor, and has strong similarities with the in some aspects such as governing equation, oundar conditions and moment epressions. The accurac of this theor has een demonstrated for static ending and free viration ehaviors of plates Shimpi and Patel [], therefore, it seems to e important to etend this theor to the static uckling ehavior. In this paper, the two variale developed Shimpi and Patel [] has een etended to the uckling ehavior of orthotropic plate sujected to the in-plane loading. Using the avier method, the closed-form solutions have een otained. umerical eamples involving side-to-thickness ratio and modulus ratio are presented to illustrate the accurac of the present theor in predicting the critical uckling load of isotropic and orthotropic plates. umerical results otained the present theor are compared with solutions, FSDT solutions with different value of shear correction factor.. for orthotropic plates.1. Basic assumptions of ssumptions of the are as follows: i. The displacements are small in comparison with the plate thickness h and, therefore, strains involved are infinitesimal. 3-31/$ - see front matter & Elsevier Ltd. ll rights reserved. doi:1./j.tws...

2 RTICLE I PRESS 4 S.-E. Kim et al. / Thin-Walled Structures 47 () 4 4 ii. The transverse displacement w includes two components of ending w and shear w s. Both these components are functions of coordinates, and time t onl. wð; ; tþ ¼w ð; ; tþþw s ð; ; tþ (1) iii. The transverse normal stress s z is negligile in comparison with in-plane stresses s and s. iv. The displacements u in -direction and n in -direction consist of etension, ending, and shear components u ¼ u þ u þ u s ; v ¼ v þ v þ v s () The ending components u and v are assumed to e similar to the displacements given the classical plate theor. Therefore, the epression for u and v can e given as u ¼ z qw q ; v ¼ z qw q (3a) The shear components u s and v s give rise, in conjunction with w s, to the paraolic variations of shear strains g z, g z and hence to shear stresses s z, s z through the thickness of the plate, h, in such a wa that shear stresses s z, s z are zero at the top and ottom faces of the plate. Consequentl, the epression for u s and v s can e given as u s ¼ 1 4 z 3 z z h q ; v s ¼ 1 4 z 3 z z (3) h q.3. Constitutive equations The constitutive equations of an orthotropic plate can e written as s s s s z s z 3 Q Q Q Q ¼ Q 4 Q 44 7 Q g g z g z where Q ij are the plane stress reduced elastic constants in the material aes of the plate, and are defined as E 1 Q ¼ ; Q 1 n n ¼ n E ; Q 1 1 n n ¼, 1 1 n n 1 Q ¼ G ; Q 44 ¼ G 3 ; Q ¼ G 13 () in which E 1, E are Young s modulus, G, G 3, G 13 are shear modulus, and n, n 1 are Poisson s ratios. For the isotropic plate, these aove material properties reduce to E 1 ¼ E ¼ E, G ¼ G 3 ¼ G 13 ¼ G, n ¼ n 1 ¼ n. The suscripts 1,, 3 correspond to,, z directions of Cartesian coordinate sstem, respectivel. E (7).4. Equation of motions.. Kinematics Based on the assumptions made in the preceding section, the displacement field can e otained using Eqs. (1) (3) as uð; ; z; tþ ¼u ð; ; tþ z qw z q þ z h z h vð; ; z; tþ ¼v ð; ; tþ z qw q þ z q q wð; ; tþ ¼w ð; ; tþþw s ð; ; tþ (4) This displacement field accounts for zero traction on oundar conditions on the top and ottom 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 otained using straindisplacement relations can e given as ¼ þ zk þ f ks ¼ þ zk þ f ks z ¼ g ¼ g þ zk þ fks g z ¼ 4 z h q g z ¼ 4 z h q where ¼ qu q ; ¼ qv q ; g ¼ qu q þ qv q ; k s ¼ q w s qq ; w k ¼ q q ; w s ks ¼ q q w k ¼ q q ; w s ks ¼ q q k ¼ q w q@, f ¼ 1 4 z þ 3 z z h () () The strain energ of the plate can e written as U ¼ 1 Z s ij ij dv ¼ 1 Z ðs þ s þ s g V þ s z g z þ s z g z Þ dv V () Sustituting Eqs. () and (7) into Eq. () and integrating through the thickness of the plate, the strain energ of the plate can e rewritten as U ¼ 1 Z ð þ þ g þ M k þ M k þ M k Þ d d þ 1 Z ðq z g z þ Q z g z þ M s k s þ Ms k s þ Ms k s Þ d d (1) where the stress resultants, M and Q are defined ð ; ; Þ¼ Z h= h= ðs ; s ; s Þ dz Z h= ðm ; M ; M Þ¼ ðs ; s ; s Þz dz h= Z h= ðm s ; Ms ; Ms Þ¼ ðs ; s ; s Þf dz ðq z ; Q z Þ¼ h= Z h= h= ðs z ; s z Þ dz () Sustituting Eqs. () and (7) into Eq. () and integrating through the thickness of the plate, the stress resultants are related to the generalized displacements (u, v, w, w s ) the relations M M M 3 qu =q ¼ 4 7 qv =q qu =q þ qv =q 3 D D q w =q ¼ 4 D D 7 q w =q D q w =qq

3 RTICLE I PRESS S.-E. Kim et al. / Thin-Walled Structures 47 () M s 3 D D q w s =q M s ¼ 1 D D q w s =q D q w s =qq ( ) ( ) 44 =q ¼ qw s =q M s Q z Q z () where ij and D ij are called etensional and ending stiffness, respectivel, and are defined in terms of the stiffness Q ij as ð ij ; D ij Þ¼ Z h= h= Q ij ð1; z Þ dz ði; j ¼ 1; ; Þ ij ¼ Q ijh ði; j ¼ 4; Þ (13) The work done of the plate applied forces can e written as V ¼ 1 Z " o q ðw þ w s Þ þ o q ðw þ w s Þ q q # þ o q Z ðw þ w s Þ dd qðw qq þ w s Þdd (14) where q and,, are transverse and in-plane distriuted forces, respectivel. The kinetic energ of the plate can e written as T ¼ 1 Z r u ii dv ¼ 1 Z I ð u V þ v þ w þ w s þ w w sþdd þ 1 Z ( " q w I þ q w # " þ I q w s þ q w #) s dd q q 4 q q (1) where r is mass of densit of the plate and I i (i ¼, ) are the inertias defined ði ; I Þ¼ Z h= h= ð1; z Þrdz (1) Hamilton s principle is used herein to derive the equations of motion appropriate to the displacement field and the constitutive equation. The principle can e stated in analtical form as ¼ Z t dðu þ V TÞdt (17) where d indicates a variation with respect to and. Sustituting Eqs. (1), (14) and (1) into Eq. (17) and integrating the equation parts, collecting the coefficients of du, dv, dw, and dw s, the equations of motion for the orthotropic plate are otained as follows: du : q q þ q q ¼ I u dv : q q þ q q ¼ I v q M dw : q þ q M qq þ q M þ q q þ q w q þ q w qq þ q w q ¼ I w I r w " q M s dw s : q þ q M s qq þ q M s q þ qq z q þ qq # z þ q q þ q w q þ q w qq þ q w q ¼ I w I 4 r w s (1) where r ¼ q q þ q q (1) The oundar conditions of a plate (of length a and width ) are given as follows: Clamped clamped oundaries: On edges ¼ and a w ¼ w s ¼ qw q ¼ qw s q ¼ On edges ¼ and w ¼ w s ¼ qw q ¼ qw s q ¼ Simpl supported oundaries: On edges ¼ and a w ¼ w s ¼ q w D q þ D q w q On edges ¼ and w ¼ w s ¼ q w D q þ D q w q Free free oundaries: On edges ¼ and a (a) () ¼ 1 4 D q w s q þ D q w s ¼ q (1a) ¼ 1 4 D q w s q þ D q w s ¼ q (1) q w D q þ D q w ¼ 1 q 4 D q w s q þ D q w s ¼ q q 3 w D q þðd 3 þ 4D Þ q3 w qq ¼ qw s ¼ q 1 4 D q 3 w s q þðd 3 þ 4D Þ q3 w s qq On edges ¼ and (a) q w D q þ D q w ¼ 1 q 4 D q w s q þ D q w s ¼ q ðd þ 4D Þ q3 w q q þ D q 3 w q 3 qw s ¼ 44 q 1 4 ðd þ 4D Þ q3 w s q q þ D q 3 w s ¼ () q 3 3. Buckling of a simpl supported rectangular plate under compressive loads When a plate is sujected to in-plane compressive forces (Fig. 1), and if the forces are small enough, the equilirium of the plate is stale and the plate remains flat until a certain load is reached. t that load, called the uckling load, the stale state of the plate is distured and plate seeks an alternative equilirium configuration accompanied a change in the load-deflection ehavior. The critical uckling loads of simpl supported, orthotropic, rectangular plate will e determined in this paper using the avier solution. The governing equations of plate in case of static uckling are given D q 4 w q 4 ¼ þ ðd þ D Þ q4 w q q þ D " q w q þ g q w 1 q 4 D q 4 w q 4 q 4 w s q þ ðd 4 þ D Þ q4 w s q q

4 RTICLE I PRESS 4 S.-E. Kim et al. / Thin-Walled Structures 47 () 4 4 # q 4 w s q w s þd q 4 44 q þ q w s q q w ¼ q þ g q w q (3) The avier method is onl applied for simpl supported oundar conditions on all four edges of the rectangular plate, as shown in Fig. 1a. The simpl supported oundar conditions on all four edge of the rectangular plate can e epressed as wð; Þ ¼wð; Þ ¼wð; Þ ¼wða; Þ ¼ M ð; Þ ¼M ða; Þ ¼M ð; Þ ¼M ð; Þ ¼ (4a) (4) The following displacement functions w and w s are chosen to automaticall satisf the oundar conditions in Eqs. (4a) and (4) w ¼ X1 w s ¼ X1 X 1 m¼1 n¼1 X 1 m¼1 n¼1 W mn sin a sin W smn sin a sin () where a ¼ mp/a, ¼ np/ and W mn, W smn are coefficients. Sustituting Eq. () into Eq. (3), the following sstem is otained: ( ) k k Wmn ¼ () k k W smn where k ¼½D a 4 þ ðd þ D Þa þ D 4 Š ða þ g Þ k ¼ ða þ g Þ k ¼ 1 4 ½D a 4 þ ðd þ D Þa þ D 4 Šþ a þ 44 ða þ g Þ (7) For nontrivial solution, the determinant of the coefficient matri in Eq. () must e zero. This gives the following epression for uckling load: D ðd=4 þ ¼ a þ 44 Þ a þ g D þðd=4 þ a þ 44 Þ where () D ¼ D a 4 þ ðd þ D Þa þ D 4 () Clearl, when the effect of transverse shear deformation is neglected, the Eq. () ields the result otained using the classical plate theor. It indicates that transverse shear deformation has the effect of reducing the uckling load. For each choice of m and n, there is a corresponsive unique value of. The critical uckling load is the smallest value of (m, n). 4. umerical results and discussion For verification purposes, a simpl supported rectangular plate sujected to the loading conditions, as shown in Fig., is considered to illustrate the accurac of the present theor in predicting the uckling ehavior of the plate. In order to at = w = M = at = w = M = at = w = M = a at = a w = M = a Fig. 1. Rectangular plate: (a) oundar condition and () in-plane forces. Fig.. The loading conditions of square plate for (a) uniaial compression, () iaial compression and (c) tension in the direction and compression in the direction.

5 RTICLE I PRESS S.-E. Kim et al. / Thin-Walled Structures 47 () investigate the effects of side-to-thickness ratio and modulus ratio, the first eample is applied for isotropic and orthotropic square plates. Man shear correction factors (k ¼ /3, k ¼ / and k ¼ 1) are also used for the FSDT in comparison with the present theor. The following engineering constants are used [] E 1 =E varied; G =E ¼ G 13 =E ¼ :; G 3 =E ¼ :; n ¼ : (3) For convenience, the following nondimensional uckling load is used: ¼ cra (31) E h 3 where a is the length of the square plate and h is the thickness of the plate. The results of critical uckling load of simpl supported square plate are presented in Tales 1 3 and Figs. 3. In the case of isotropic plate (Fig. 3a), the results otained and FSDT are in ecellent agreement event though the plate is ver thick. In case of square plate (a ¼ ¼ h), the maimum difference of and FSDT with the shear correction factor / is.4%, as shown in Tale 3. When the orthotropic plate is used, the difference etween and FSDT will increase with respect to the increase of side-to-thickness ratio (Fig. 3) and modulus ratio (Figs. 4 ). s presented in Tale 1, the differences etween and FSDT (k ¼ /), and and FSDT (k ¼ 1) are 1.14% and.4%, respectivel, for the same case of square plate (a ¼ ¼ h and E 1 /E ¼ 4). It can e seen from Tales 1 3 that the difference of critical uckling load etween and FSDT depends on not onl the side-to-thickness and modulus ratios, ut also the in-plane loading conditions (Fig. ). In case of square plate (a ¼ ¼ 1h), the difference etween and FSDT (k ¼ /) is.% for uniaial compression (Fig. 4 and Tale 1),.3% for iaial compression (Fig. and Tale ), and.% for tension in the - direction and compression in the -direction (Fig. and Tale 3). Tale 1 Comparison of nondimensional critical uckling load of square plates sujected to uniaial compression a/h Theories Isotropic n ¼.3 Orthotropic E 1 /E ¼ 1 E 1 /E ¼ E 1 /E ¼ FSDT (k ¼ /3) FSDT (k ¼ /) FSDT (k ¼ 1) Tale Comparison of nondimensional critical uckling load of square plates sujected to iaial compressive load a/h Theories Isotropic n ¼.3 Tale 3 Comparison of nondimensional critical uckling load of square plates sujected to tension in the direction and compression in the direction a/h Theories Isotropic n ¼.3 Orthotropic E 1 /E ¼ 1 E 1 /E ¼ E 1 /E ¼ a 3.33 a 3.4 a FSDT (k ¼ /3) a.733 a.33 a FSDT (k ¼ /) a 3.14 a 3. a FSDT (k ¼ 1) a a 3.73 a a.4 a 7.3 a FSDT (k ¼ /3) a.77 a FSDT (k ¼ /) a.3 a FSDT (k ¼ 1) a 7. a a.14 a FSDT (k ¼ /3) a. a FSDT (k ¼ /) a.34 a FSDT (k ¼ 1) a.7 a a 1.7 a FSDT (k ¼ /3) a 1.1 a FSDT (k ¼ /) a 1.1 a FSDT (k ¼ 1) a 1. a a 1.17 a FSDT (k ¼ /3) a 1.77 a FSDT (k ¼ /) a 1.77 a FSDT (k ¼ 1) a 1. a a a Mode for plate is (m, n) ¼ (1,). Orthotropic E 1 /E ¼ 1 E 1 /E ¼ E 1 /E ¼ a c 4.1 c FSDT (k ¼ /3) a 3.4 d 3.31 e 3.33 e FSDT (k ¼ /) 4.1 a 3.41 c 3.74 c 4.7 d FSDT (k ¼ 1).37 a c 4.73 c 1.4 a 7.73 a FSDT (k ¼ /3).43 a 7. a 7.7. FSDT (k ¼ /).1 a a FSDT (k ¼ 1).73 a.1 a FSDT (k ¼ /3) FSDT (k ¼ /) FSDT (k ¼ 1) a. a FSDT (k ¼ /3) 7.13 a.131 a.44.1 FSDT (k ¼ /) 7.73 a.7 a.1.33 FSDT (k ¼ 1) 7.31 a.37 a FSDT (k ¼ /3) FSDT (k ¼ /) FSDT (k ¼ 1) a. a FSDT (k ¼ /3) 7.47 a.73 a FSDT (k ¼ /) 7.4 a.7 a FSDT (k ¼ 1) 7.4 a.7 a FSDT (k ¼ /3) FSDT (k ¼ /) FSDT (k ¼ 1) FSDT (k ¼ /3) FSDT (k ¼ /) FSDT (k ¼ 1) a.7 a FSDT (k ¼ /3) 7.1 a.3 a FSDT (k ¼ /) 7. a. a FSDT (k ¼ 1) 7. a.1 a a.17 a a Mode for plate is (m, n) ¼ (1,). 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,). e Mode for plate is (m, n) ¼ (1,).

6 RTICLE I PRESS 4 S.-E. Kim et al. / Thin-Walled Structures 47 () FSDT (k = /) FSDT (k = /) a/h a/h FSDT (k = /) 1 FSDT (k = /) 7 1 a/h 7 1 a/h Fig. 3. The effect of side-to-thickness and modulus ratios on the critical uckling load of square plate sujected to uniaial compression: (a) isotropic, () E 1 /E ¼ 1, (c) E 1 /E ¼ and (d) E 1 /E ¼ FSDP (k = /) 4 3 FSDP (k = /) E 1 /E E 1 /E Fig. 4. The effect of modulus ratio on the critical uckling load of square plate sujected to uniaial compression: (a) a ¼ 1h and () a ¼ h. FSDP (k = /) FSDP (k = /) E 1 /E E 1 /E Fig.. The effect of modulus ratio on the critical uckling load of square plate sujected to iaial compression: (a) a ¼ 1h and () a ¼ h. The difference etween and FSDT is also due to the shear correction factors using in FSDT. In case of tension in the direction and compression in the direction (Fig. c), the uckling mode shape (Fig. 7) switches from asmmetric to smmetric or, conversel, from smmetric to asmmetric, depending on the shear correction factors. The net comparison is carried out for the orthotropic rectangular plates sujected to uniaial compression with the variation of aspect ratio and side-to-thickness ratio. The following material properties are used [13]: E =E 1 ¼ :; G =E 1 ¼ G 3 =E 1 ¼ :, G 13 =E 1 ¼ :1; n ¼ :44; n 1 ¼ :3 (3) The results of uckling stress parameter, k ¼ (P cr /E 1 )(/p ) (/h), are shown in Tale 4 and Fig.. It is oserved that the present theor overpredicts the asmptotic of uckling stress parameter and is more accurate than the Mindlin theor. Even the quite thick plate ( ¼ h), the error is onl.1%.

7 RTICLE I PRESS S.-E. Kim et al. / Thin-Walled Structures 47 () FSDP (k = /) 1 1 FSDP (k = /) E 1 /E E 1 /E Fig.. The effect of modulus ratio on the critical uckling load of square plate sujected to tension in the direction and compression in the direction: (a) a ¼ 1h and () a ¼ h. Fig. 7. Buckling mode shapes of orthotropic square plate: (a) (m, n) ¼ (1,1); () (m, n) ¼ (1,); (c) (m, n) ¼ (1,3); (d) (m, n) ¼ (1,4); (e) (m, n) ¼ (1,); (f) (m, n) ¼ (1,).. Concluding remarks n efficient two variale refined plate theor proposed Shimpi and Patel [] has een applied in this paper for uckling ehavior of isotropic and orthotropic plates. The theor takes account of transverse shear effects and paraolic distriution of the transverse shear strains through the thickness of the plate, hence it is unnecessar to use shear correction factors. The governing equations are strong similarit with the classical plate theor in man aspects. It can e concluded that the two variale

8 RTICLE I PRESS 4 S.-E. Kim et al. / Thin-Walled Structures 47 () 4 4 Tale 4 smptotic k ¼ (P cr /E 1 )(/p )(/h) for uckling of orthotropic plate under normal stress P on edges ¼ and ¼ a /h Eact a Mindlin a Error- (%) Error-Mindlin (%) Error-(%) a Take from Ref. [13]. Buckling stress parameter, k 4 3 refined plate theor can accuratel predict the critical uckling loads of the isotropic plates. cknowledgements ( = h) ( = 1h) ( = h) a/ Fig.. Variation of uckling stress parameter k ¼ (P cr /E 1 )(/p )(/h) with respect to /h and a/ for orthotropic plates under load on ¼ and ¼ a. The support of the research reported here Ministr of Commerce, Industr and Energ through Grant 37-- P a P and Korea Ministr of Construction and Transportation through Grant -C are gratefull acknowledged. References [1] Timoshenko SP, Woinowsk-Krieger S. Theor of plates and shells. ew York: McGraw-Hill; 1. [] Timoshenko SP, Gere JM. Theor of elastic stailit. ew York: McGraw-Hill;. [3] Bank L, Yin J. Buckling of orthotropic plates with free and rotationall restrained unloaded edges. Thin Wall Struct 1;4(1):3. [4] Kang JH, Leissa W. Eact solutions for the uckling of rectangular plates having linearl varing in-plane loading on two opposite simpl supported edges. Int J Solids Struct ;4(14):4 3. [] dogdu M, Ece MC. Buckling and viration of non-ideal simpl supported rectangular isotropic plates. Mech Res Commun ;33(4):3 4. [] Hwang I, Lee JS. Buckling of orthotropic plates under various inplane loads. KSCE J Civ Eng ;1():34. [7] Reissner E. The effect of transverse shear deformation on the ending of elastic plates. J ppl Mech-T SME 14;(): 77. [] Mindlin RD. Influence of rotar inertia and shear on fleural motions of isotropic, elastic plates. J ppl Mech-T SME ;1(1):31. [] Levinson M. n accurate simple theor of the statics and dnamics of elastic plates. Mech Res Commun 1;7():343. [1] Redd J. refined nonlinear theor of plates with transverse shear deformation. Int J Solids Struct 14;():1. [] Shimpi RP, Patel HG. two variale refined plate theor for orthotropic plate analsis. Int J Solids Struct ;43( 3):73. [] Redd J. Mechanics of laminated composite plate: theor and analsis. ew York: CRC Press; 17. [13] Srinivas S, Rao K. Bending, viration and uckling of simpl supported thick orthotropic rectangular plates and laminates. Int J Solids Struct 17;():