12 S.A.M. Ghannadour, H.R. Ovesy and M. Nassirnia rogressive end shortening. They also consider the ost-buckling behavior of late structures subjected

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1 JAST, Vol. 8, No. 1, 11{19 c Iranian Aerosace Society, Winter-Sring 2011 Nonlinear Post-Buckling Analysis of Isotroic Plates Using Finite Stri Methods S.A.M. Ghannadour 1, H.R. Ovesy 2 and M. Nassirnia 3 This aer resents the theoretical develoments of two nite stri methods, i.e. semi-analytical and full-analytical, for the ost-buckling analysis of isotroic lates. In the semi-analytical nite stri aroach, all the dislacements are ostulated by the aroriate shae functions while in the develoment rocess of the full-analytical aroach, the Von-Karman's equilibrium equation is solved exactly to obtain the buckling loads and the outof-lane buckling deection modes. The investigation of the lates' buckling behaviour is then extended totheost-buckling study with the assumtion that the deected form after the buckling is a combination of the rst, second and higher (if required) modes of buckling. Thus, the full-analytical ostbuckling study is an eective multi term analysis. In this method, the Von- Karman's comatibility equation is used together with a consideration of the total strain energy of the strut. Through the solution of the comatibility equation, the in-lane dislacement functions, which are themselves related to the Airy stress function, are develoed in terms of the unknown coecients in the assumed out-of-lane deection function. The in-lane and out-of-lane deection functions are substituted in the total strain energy exressions and the theorem of minimum total otential energy is alied to solve for the unknown coecients. INTRODUCTION Prismatic lates and late structures are increasingly used as structural comonents in various branches of engineering, articularly in aerosace and marine engineering. These structures are often emloyed in situations where they are subjected to in-lane comressive loading. In aerosace engineering, in articular, the quest for ecient, light-weight structures often leads to allowing for the ossibility of local buckling and ost-local-buckling at design load levels. Thus, it is 1. Corresonding Author, Assistant Professor, Det. of Aerosace Eng., Faculty of New Technologies and Engineering, Shahid Beheshti University, G.C., Tehran, Iran, A_ghannadour@sbu.ac.ir. 2. Professor, Det. of Aerosace Eng., Center of Excellence in Comutational Aerosace Eng., Amirkabir Univ. of Tech., Tehran, Iran. 3. M.Sc. Student, Det. of Aerosace Eng., Center of Excellence in Comutational Aerosace Eng., Amirkabir Univ.ofTech., Tehran, Iran. imortant to accurately redict the buckling and ostbuckling behavior of such structures. There are a lot of aers concerning linear buckling and vibration analysis of comosite laminated lates and late structures formed of comosite materials having very general material roerties [1-5]. They have extensively used the Finite Stri Method (FSM) based on the use of Classical Plate Theory (CPT), rst-order Shear Deformation Plate Theory (SDPT) and Higher-order Shear Deformation Plate Theory (HSDPT). The ost-local-buckling behavior of elastic lates or late structures is a geometric non-linear roblem. The non-linearity occurs as a result of relatively large out-of-lane deections, which necessitates the inclusion of non-linear terms in the strain-dislacement equations. The early works concerned with the use of the FSM in redicting the geometrically non-linear resonse of single rectangular late and rismatic late structures are those of Graves Smith and Sridharan [6], [7] and Hancock [8]. These authors consider the ost-buckling behavior of lates with simly suorted ends when subjected to

2 12 S.A.M. Ghannadour, H.R. Ovesy and M. Nassirnia rogressive end shortening. They also consider the ost-buckling behavior of late structures subjected to uniform or linearly varying end shortening with each comonent late of the structure having simly suorted ends. The elastic ost-buckling resonse of channel section struts and rectangular box columns have beeninvestigated by Graves Smith and Sridharan. Hancock has used the nite stri method to investigate the ost-buckling behavior of square box and I- section columns. In the nite stri methods develoed by the aforementioned authors, in-lane dislacement elds are ostulated in addition to the out-of-lane dislacement eld. The lengthwise variations in the dislacement elds are trigonometric functions. The crosswise variations in both in-lane and out-of-lane dislacement elds are simle olynomial functions. It is noted that the above-mentioned nite stri methods can be categorized as Semi-analytical FSM (S-a FSM). Ref. [9] rovides a fairly state-of-the-art summary on the use of the semi-analytical nite stri method (Sa FSM) as well as the sline nite stri method. It is worth mentioning that there are two aers, written by Rhodes [10] and Chou & Rhodes [11], which are extremely useful in roviding references on the theoretical (mostly based on the semi-energy method) and exerimental research into thin-walled structures. Khong and Rhodes [12] have set u a comutationally ecient aroach to the ost-buckling analysis of rismatic structural members. In this aroach, a linear nite stri method, develoed for the buckling analysis, based on the Princile of Minimum Potential Energy is emloyed to nd the buckling eigenvector. This eigenvector is then used as the ost-buckled deected shae in a single-term ost-buckling analysis based on the Princile of Minimum Potential Energy. The analysis is simlied by the assumtion that stresses in the direction erendicular to loading and shear stresses have negligible eects. This aroach can be considered as a lower bound method of ostbuckling analysis (i:e: the ost-buckling stiness of the structure is underestimated by this aroach). The method is alied to lain and stiened channel sections as well as Z-sections. Ovesy et:al: [13-15] have develoed a Semi-energy ost-local-buckling FSM in which the out-of-lane dislacement of the nite stri is the only dislacement which is ostulated by a deected form. The develoed Semi-energy FSM (S-e FSM) has been alied to analyze the ost-local-buckling behavior of thin at lates [13], oen channel section [14] and box section struts [15]. More recently, Ghannadour and Ovesy [16-20] have develoed a full-analytical ostlocal-buckling FSM in which thesolution of the Von- Karman's equilibrium equation was substituted into the Von-Karman's comatibility equation. It is noted that in their analysis, only one of the calculated out-oflane buckling deection modes, corresonding to the lowest buckling load, i:e: the rst mode, is used for the initial ost-buckling study. For this reason, their method was designated by the name single-term Fullanalytical Finite Stri Method (F-a FSM). It is worth mentioning that the single-term assumtion within F-a FSM analysis corresonds to the fact that the shae of the late in the ost-buckling region is unchangeable in both longitudinal and transverse directions. Thus, the relationshi between load and end-shortening becomes a linear function. In the current aer, the theoretical develoments of a high accuracy multi-term F-a FSM for the ost-buckling analyses of some isotroic lates are attemted. The stri is develoed based on the concet that it is an eective late, and thus the Von-Karman's equilibrium equation is solved exactly to obtain the buckling loads and the corresonding forms of out-oflane buckling deection modes. In order to solve the Von-Karman's comatibility equation, the deected form after the buckling is assumed as a combination of the rst, second and higher (if required) modes of buckling. Subsequently, the general form of inlane dislacement elds in ost-buckling region can be obtained. This method is characterized by the use of buckling mode shaes, obtained from the Von- Karman's equilibrium equation, as global shae functions for reresenting dislacements in a geometrically non-linear analysis. The in-lane and out-of-lane deection functions are substituted in the total strain energy exressions and the theorem of minimum total otential energy is alied to solve for the unknown coecients. THEORETICAL DEVELOPMENTS OF THE FULL-ANALYTICAL FSM The detailed fundamental elements of the theory have been given in earlier ublications by Ghannadour and Ovesy [16-20]. It is noted that a erfectly at high accuracy stri made u of a linear isotroic material (with a constant modulus of elasticity E and Poisson ratio v) is assumed throughout the aer. The socalled high accuracy nite stri is assumed to be simly suorted out-of-lane at the loaded ends, and to be thin so that the Classical Plate Theory (CPT) is alied in the remainder of the aer. The nite stri, which is schematically shown in Figure 1, is of the length L, widthband thickness t. The Von-Karman's equilibrium and comatibility equations for large deections of late with the assumtion that the normal ressure is zero are given by the following equations, resectively. Dr 4 w t (F yy w xx 2F xy w xy +F xx w yy )=0 r 4 F = E w 2 w xy xx w yy (1a) (1b)

3 Nonlinear Post-Buckling Analysis of Isotroic Plates Using Finite Stri Methods 13 A searable form is assumed for the dislacement w 1 in order to reduce the above artial dierential equation into an ordinary dierential equation. The out-of-lane buckling deection mode w 1 is assumed to vary sinusoidally with x. This assumtion is consistent with the simly suorted boundary conditions at the loaded ends. Thus: w 1 (x y) =f w1 (y): sin(x) (5) Figure 1. A tyical high accuracy nite stri. In this equation, the function F is known as the Airy stress function. The ositive directions of the edge forces/moments and dislacements are shown in Figure 1. It can be seen in Figure 1 that the nodal line forces and moments can be exressed in terms of internal inlane forces and normal dislacements on the edges as follows: P xi = N xy j y=0 P xj = N xy j y= b P yi = N y j y=0 P yj = N y j y= b P zi = D [w yyy +(2 )w xxy ] y=0 P zj = D [w yyy +(2 )w xxy ] y= b M yi = D [w yy +w xx ] y=0 M yj = D [w yy +w xx ] y= b (2) In the above equation, N y and N xy are the membrane direct and shearing stress resultants er unit length. It is noted that in the remainder of the aer, the subscrits are used for re-buckling, buckling and ost-buckling stages, resectively. Buckling analysis The out-of-lane buckling deection mode w 1 is obtained by trying to solve thevon-karman's equilibrium equation, i:e: Eq. (1-a). This gives: Dr 4 w 1 t (F 0 yy w 1 xx 2F 0 xy w 1 xy +F 0 xx w 1 yy )=0 (3) where r 4 w 1 = w 1 xxxx +2 w 1 xxyy +w 1 yyyy. In the re-buckling stage, the stri is subjected to the comressive axial stress and thus: D(w 1 xxxx +2 w 1 xxyy +w 1 yyyy )+tw 1 xx =0 (4) where = n=l and arameter n in the above dislacement function is merely an integer which reresents the number of buckle half-wavelengths along the stri, and f w1 (y) reresents the shae function in the transverse direction y. Substituting out-of-lane buckling deection mode w 1 from Eq. (5) into Eq. (4) and rearranging it leads to the following fourth-order ordinary dierential equation: f 0000 w1 2 2 f 00 w1 + 4 (1 )f w1 =0 (6) where = t= 2 D and the suerscrit 0 denotes dierentiation with resect to y, i:e: () 0. The solution of Eq. (6) deends on whether, which is clearly ositive, is greater than, less than, or equal to unity. For >1, the solution can be written as: f w1 (y) =C 1 w1 cosh(y)+c2 w1 sinh(y) + C 3 w1 cos(y)+c4 = while for 0 <<1, q 1+ = w1 sin(y) q 1+ (7) f w1 (y) =C 1 w1 cosh(y)+c2 w1 sinh(y) + C 3 w1 cosh(y)+c4 = q 1+ = w1 sinh(y) q 1 and for =1 (8) f w1 (y) =C 1 w1 cosh 2 y + C 2 w1 sinh 2 y where C k w1 + C 3 w1 y + C4 w1 (9) (k = ) denote unknown constants. The dislacement boundary conditions for f w1 (y) at the two edges y =0andy = b can be written as: f w1 (0) = w 1i f 0 w1 (0) = 1i f w1 (b) f 0 w1 (b) = w 1j = 1j (10) where the subscrit 1 is used because they are initial buckling quantities and the subscrits i and j denote amlitudes at edges i and j of the stri, resectively.

4 14 S.A.M. Ghannadour, H.R. Ovesy and M. Nassirnia These buckling dislacement amlitudes can be written as the dislacement vector: d 1 = 8 >< >: w 1i 1i w 1j 1j 9 >= > (11) The four unknown constants C k w1 (k = ) corresonding to a given stri can be fully determined in terms of buckling dislacement amlitudes by substituting the boundary conditions of Eq. (10) into Eq. (7), (8) or (9). Thus, the solution of Eq. (6) can be obtained analytically in terms of the edge dislacements d 1. Substituting Eq. (5) into Eq. (2) yields the force boundary conditions as: ( P 1zi = D f 000 w1 (0) (2 ) 2 1i P 1zj = D f 000 w1 (b) (2 ) 2 1j ( M 1yi = D f 00 w1 (0) 2 w 1i M 1yj = D f 00 w1 (b) (12) 2 w 1j The left-hand sides of Eq. (12) are the amlitudes of the buckling forces and moments at the corresonding edges of the stri. They can be written as the force vector: 1 = 8 >< >: P 1zi M 1yi P 1zj M 1yj 9 >= > (13) The above equation which describes the edge forces in terms of the edge dislacements can be re-arranged as: 1 = k 1 d 1 (14) where k 1 denotes the stri out-of-lane stiness matrix. By alying these exressions to obtain the stiness matrices of individual stris, the exact overall stiness matrix K 1 for the whole structure can be assembled by using the conventional routines of the nite element analysis. The corresonding buckling roblem can nally be exressed as the eigenvalue roblem: K 1() D 1 =0 (15) where the vector D 1 consists of the out-of-lane dislacement amlitudes (w 1 1 ) for each nodal line, and K 1 is the stiness matrix whose coecients include trigonometric and hyerbolic functions involving longitudinal stress as the stri is analyzed exactly by solving its governing dierential equation. It is realized that alication of the exact method to the buckling of structures has resulted in a transcendental eigenvalue roblem in the form of Eq. (15) as distinct from equation (K K G) D = 0 which is encountered when aroximate methods such as the nite stri method are used. In this aer, two secure methods (i:e: bisection method and recursive Newton method) for nding the buckling load and the corresonding buckling mode of the late structures are used [16-20]. Post-buckling analysis After obtaining the exact shaes of buckling modes from the buckling analysis, the analysis of ostbuckling behavior can roceed on the assumtion that the deected form after the buckling is a combination of the rst, second and higher (if required) modes of buckling. Thus, the ost-buckling out-of-lane deection function w 2 can be written as: w 2 (x y) = q=1 q w 1q (x y) (16) where q are the deection coecients and n is the number of the required buckling modes. The in-lane boundary conditions at the loaded ends of the stri are summarized as follows: ( 0 x =0 N xy =0 at x =0&L u 2 = "L x = L (17) By adoting the semi-energy ost-buckling rocedure in the manner described in Ref. [16], the outof-lane dislacement w 2 is then substituted in the Von-Karman's comatibility equation in order to nd the corresonding in-lane dislacement functions as follows: u 2 (x y) = "x+ 2 (x y) ="y + where f u2 (y) = 4 Zy g 2 (y) = q=1 s=1 q=1 s=1 q s f u2 (y) sin (2 x) (18) q s g 2 2 (y)+f (y)cos(2 x) f 2 (0) + 2 (0 0) (19) 00 (y)+42 (y) 1 2 f qf s " 2 f q f s + f 0 q f 0 s dy f 2 (y) = (y) 4(2 + )2 0 (y) +f qf 0 s (20)

5 Nonlinear Post-Buckling Analysis of Isotroic Plates Using Finite Stri Methods 15 and function (y) can be found from the following equation: = f 0 q f 0 s f q f 00 s (21) It is noted that the rst term on the right hand side of Eq. (18) reresents the rescribed uniform endshortening strain. The amlitude of the second term on the right hand side of Eq. (18), whilst evaluated at y = 0 and y = b (i:e: f u2 (0) fu2 (b)), reresents the ost-buckling in-lane dislacement arameters u 2i and u 2j, resectively(seefigure1). Also, the rst term on the right handsideofeq. (19) describes the transverse in-lane exansion of the stri, which occurs due to the Poisson's ratio eect. The second term describes the transverse in-lane movement of the longitudinal bers of the stri. This movement, which is constant along the length of a given ber, varies from a minimum value of zero at edge y = 0 to its maximum value at the edge y = b. The third term describes the in-lane waviness of the longitudinal bers. The amlitude of this term, whilst evaluated at y = 0 and y = b (i:e: f 2 2 (0) f (b)), reresents the ost-buckling in-lane dislacement arameters 2i and 2j, resectively (see Figure 1). Finally, the fourth term and the fth term on the right hand side of Eq. (19) reresent values which remain constant at all oints onagiven stri. It is also noted that the ostbuckling in-lane dislacements u 2 and 2 are functions of out-of-lane buckling deection modes (which are functions of critical longitudinal stresses) and deection coecients q. After obtaining the in-lane dislacements, the in-lane shear force and in-lane transverse force can be calculated from Eq. (2) as the following set of linear equations: 2 = k 2 d 2 + f 2 (22) where d 2 = and 2 = f 2 n u 2i 2i u 2j 2j o T (23) n P 2xi P 2yi P 2xj P 2yj o T (24) consists of terms corresonding to the out-oflane dislacement arameters, and k 2 consists of terms corresonding to the in-lane dislacement arameters (i:e: u 2, 2 ) which is the stiness matrix of the stri. The overall stiness equations corresonding to the whole structure are formed by following the conventional nite element assembly rocedure, and noting that the structure is not subjected to any external force thus, 2 vector vanishes during the assembly rocess. The overall stiness equations are: K 2 D 2 = F 2 (25) where matrices K 2, D 2 and F 2 are assembled from their counterarts (i:e: k 2, d 2 and f 2) for each stri. Once Eq. (25) is solved and the ost-buckling in-lane dislacement arameters are obtained, they are then substituted into Eq. (18) and (19) to determine the analytical form of u 2 and v 2 for each stri, resectively. It is noted that the obtained u 2 and v 2 and the assumed w 2 are all determined in terms of the deection coecients q,which will be calculated below. Deection coecients ( q ) calculation For a rescribed uniform end-shortening strain ", the total strain energy of the structure U = P U s, which is simly equal to its total otential energy, can be exressed as follows: U = m 0 " 2 + where + q=1 s=1 q=1 q=1 s=1 s=1 q s b2 "m 2 m 4q s = X 1 32 te4 L( q q s s m 4q s (26) Z b 0 fq f q f s f s q s q s +162 (1 + ) 0 0 q s q s dy) X 1 m 2 = 4 tel2 m 0 = X 1 2 telb Zb 0 f q f s dy b 2 =X 1 4 LD Z 4 f q f s 2 2 (1 )f 0 q f 0 s 2 2 ( 2)f q f 00 s + f 00 q f 00 s dy (27) P Here, the summation relates to all stris. In evaluating the above constants, all integrations are determined analytically. It is emhasized that these constants need to be evaluated once. It is noted that the deection coecients q are the only unknowns in the energy exression. The strain energy is then minimized by dierentiating U with resect to each q. The obtained set of nonlinear equations is solved

6 16 S.A.M. Ghannadour, H.R. Ovesy and M. Nassirnia by using Newton-Rahson rocedure. Once the global equilibrium equations are solved and the deection coecients are found for a articular rescribed endshortening, it is ossible to calculate all dislacements, strains and stresses. THEORETICAL DEVELOPMENTS OF THE SEMI-ANALYTICAL FSM The theoretical develoment of the Semi-analytical FSM (S-a FSM) for the ost-buckling analysis of the isotroic lates is resented in this section. It is emhasized that in the semi-analytical nite stri aroach, all the dislacements are ostulated by the aroriate shae functions while in the develoment rocess of the full-analytical aroach, the dislacement elds were obtained by solving the Von-Karman's equations. The dislacement elds of the single term S-a FSM are exressed as: u = "x + f u(y)sin 2x L = "y + f 0 (y)+f (y)cos w = f w(y)sin x L 2x L (28) (29) (30) The fu f 0 f and fw are transverse olynomial interolation functions of various tyes and orders, involving undetermined dislacement coecients. It is noted that in obtaining the S-a FSM results, in addition to the alication of the linear Lagrange olynomial interolation functions for the in-lane dislacements u and, as noted in Ref. [21], the quadratic olynomial interolation functions are also used. In reresenting w, the cubic Hermitian olynomial is utilized as exlained in Ref. [21]. With the establishment of the nite stri dislacement elds according to the equations mentioned above, the strain energy of the stri which is equal to its total otential energy can ultimately be obtained. For the whole structure, comrising an assembly of nite stris, the total otential energy is simly the summation of the otential energies of the individual nite stris [13-15]. The ertinent structure equilibrium equations are obtained by alying the rincile of minimum otential energy. That is to say, the artial dierentiation of the structure otential energy with resect to each degree of freedom, in turn, gives a set of non-linear equilibrium equations which needs to be modied by alying the aroriate zerodislacement boundary conditions at the longitudinal exterior edges. After the alication of any aroriate zero-dislacement boundary conditions, the equations must be solved. In the resent study, the Newton- Rahson (N-R) iterative rocedure is selected for solving the equations as in the revious section. Once the global equilibrium equations are solved and the degrees of freedom are found for a articular rescribed end-shortening, it is ossible to calculate all dislacements, strains and stresses. RESULTS AND DISCUSSIONS A late with the width to thickness ratio b=t of 100 and the Poisson's ratio of 0.3 is considered. The loaded edges of the late are simly suorted as described earlier whilst the out-of-lane boundary conditions at the two unloaded edges are clamed-simly suorted (SCSS). The in-lane transverse dislacement is allowed to occur at the unloaded edges. In the case of F- a FSM analysis, the late is initially divided into two, four and ten stris of equal width giving three cases for consideration. The investigation of the results has revealed that the critical buckling load and the ostbuckling results are identical among the three cases as exected. Thus, a late can be accurately modelled by alying only one stri to it. However, in the case of S-a FSM analysis, in order to obtain converged results, at least 32 or 16 stris are needed when the linear or quadratic interolation functions are used, resectively. After the buckling analysis carried out by using F-a FSM, the rst ve out-of-lane buckling deection Figure 2. Out-of-lane buckling deection modes in transverse direction for SCSS late. Figure 3. Convergence study of F-a FSM analysis.

7 Nonlinear Post-Buckling Analysis of Isotroic Plates Using Finite Stri Methods 17 modes of this late can be obtained as follows: f 1w1 (y) =0:64112 cosh(0:029y) 0:64113 sinh(0:029y) 0:64112 cos(0:019y)+sin(0:019y) f 2w1 (y) =0:84374 cosh(0:041y) 0:84374 sinh(0:041y) 0:84374 cos(0:035y)+sin(0:035y) f 3w1 (y) =0:91632 cosh(0:055y) 0:91632 sinh(0:055y) 0:91632 cos(0:05y) + sin(0:05y) f 4w1 (y) =0:94866 cosh(0:07y) 0:94866 sinh(0:07y) 0:94866 cos(0:0666y) + sin(0:0666y) Figure 4. Variations of the non-dimensional load-end shortening. f 5w1 (y) =0:96551 cosh(0:085y) 0:96551 sinh(0:085y) 0:96551 cos(0:082y)+sin(0:082y) (31) The variations of these modes in transverse direction are deicted in Figure 2. As can be seen, the out-oflane buckling deection modes of a SCSS late include trigonometric and hyerbolic functions. The non-dimensional load-end shortening variations (P=P Cr "=" Cr ) are deicted in Figure 3 where the convergence study with resect to the number of buckling modes assumed in the F-a FSM analysis is also investigated. It is noted that for the late under consideration, the number of buckle half-wavelengths along the late (i:e: arameter n in Eq. 5) is assumed to be equal to one. Thus, the buckling modes are dierent in their reresentation in the transverse y direction. For examle, the symbol y:(1,3) is used to refer to the case where the rst and the third modes in the y direction are considered. The nonlinear nature of the behavior, which haens due to the eects of change in the deection coecients q, can be clearly seen for the cases where several modes are considered. It is seen that in the advanced stages of ost-buckling, the combination of the rst three buckling modes, i:e: y:(1,2,3), has led to the results with high accuracy. Nevertheless, for further assurance in the remainder of the aer, the results are resented by assuming the rst ve buckling modes in the case of F-a FSM analysis. In order to investigate the validation of the roosed methods, the load-end shortening variations obtained by the develoed F-a FSM and S-a FSM are comared in Figure 4. Two tyes of S-a FSM results, which are obtained by either linear or quadratic interolation functions, are resented. The gure shows that the results of the F-a FSM agree very well, articularly with those obtained by the quadratic S-a FSM. It is noted that, in comarison to the linear S- a FSM, the quadratic S-a FSM has delivered slightly more accurate results by imlementing less stris. Figure 5. Comarison between the FSM and FEM results. As can be seen, the validation of the newly develoed full analytical FSM is attemted by comaring the results with those obtained by the semi-analytical FSM method. But the degree of accuracy of the semianalytical FSM, which is used for the validation, is aroved in the authors' revious ublication [13] by comaring the FSM results with those obtained by FEM. For comleteness and to attain a full understanding of concets in a single aer, this comarison is reeated in Figure 5. In order to obtain the results, a late with a length to width ratio of 2, which was clamed out-of-lane at one unloaded edge, and was free on the other edge (i:e: C-F), was considered. This late had a Poisson's ratio of 1/3. The unloaded edges were assumed to be unconstrained in-lane. In order to obtain the FEM results, the geometric non-linear FEM analysis was carried out for late under consideration emloying the MSC/NASTRAN nite element ackage. The CQUAD4 element was used to model the late. After erforming the aroriate convergence study with resect to the number of elements, a total of 288 square elements were considered to be sucient for delivering accurate results. As can be seen again, the semi analytical FSM results agree well with the FEM results. Thus, the semi-

8 18 S.A.M. Ghannadour, H.R. Ovesy and M. Nassirnia Figure 6. The out-of-lane deected shae across the late at the crest of the buckle. analytical FSM is considered to be a good latform for the validation of the full-analytical FSM results. Figure 6 reresents the deected shae at the crest of the buckle (i:e: at x = L=2) at twice the buckling load. As far as the F-a FSM results are concerned, the contribution of dierent buckling modes is deicted searately. In other words, the linear summation of the deections corresonding to 1 f 1 to 5 f 5 has led to the total deection of the late, which is designated by the symbol y:(1,2,3,4,5). It is seen that the total deection of the late is mainly governed by thecontribution from the rst buckling mode (i:e: 1 f 1 ). It is also seen that the deected shae obtained by the F-a FSM agrees very well with that of S-a FSM. However, it is worth mentioning that the results of F-a FSM are achieved by emloying only one stri and 5 degrees of freedom (i:e: 1 to 5 ), whereas the quadratic S-a FSM analysis is accomlished by alying 16 stris and 130 degrees of freedom. CONCLUSION Theoretical develoments of two nite stri methods (i:e: semi-analytical and full-analytical) for the ostbuckling analysis of some isotroic at lates have been resented here. In the S-a FSM ost-buckling aroach, all the dislacements are ostulated by the aroriate shae functions whilst in the develoment rocess of the F-a FSM aroach, the Von-Karman's equilibrium equation is solved exactly to obtain the buckling loads and the corresonding forms of outof-lane buckling deection modes. The F-a FSM ost-buckling study is then attemted with the assumtion that the deected form after the buckling is the combination of the rst, second and higher (if required) modes of buckling, and the so-called semienergy method is utilized. It is realised that the F- a FSM analysis is beneting from a considerably less comutational eort due to the imlementation of a single stri, as comared to that of S-a FSM which requires at least 16 stris to deliver converged results for the cases considered in this aer. REFERENCES 1. Lau S.C.W, Hancock G.J., \Buckling of Thin Flat- Walled Structures by a Sline Finite Stri Method", Thin-Walled Struct., 4, PP (1986). 2. Dawe D.J., Craig T.J., \Buckling and vibration of shear deformable rismatic late structures by a comlex nite stri method", Int. J. of Mech. Sci., 30, PP 77-79(1988). 3. Wang S., Dawe D.J., \Buckling of comosite shell structures using the sline nite stri method", Comos.: Part B, 30, PP (1999). 4. Zou G.P., Lam S.S.E., \Buckling analysis of comosite laminates under end shortening by higher-order shear deformable nite stris", Int. J. for Numer. Meth. in Eng., 55, PP (2002). 5. Cheung Y.K., Kong J., \The Alication of a new nite stri to the free vibration of rectangular lates of varying comlexity", J. of Sound and Vib., 181, PP (1995). 6. Graves Smith T.R. and Sridharan S., \A nite stri method for the ost-locally-buckled analysis of late structures", Int. J. of Mech. Sci., 20, PP (1978). 7. Sridharan S. and Graves-Smith T.R., \Post-buckling analysis with nite stris", J. Eng. Mech. Div., ASCE, 107, PP (1981). 8. Hancock G.J., \Nonlinear Analysis of Thin Sections in Comression", J. Struct. Div. ASCE, 107, PP (1981). 9. Dawe D.J., \Use of the nite stri method in redicting the behaviour of comosite laminated structures", Comos. Struct., 57, PP 11-36(2002). 10. Rhodes J., \Research into thin-walled structures at the university of Strathclyde - A brief history", Published in the Conference Proceedings of Bicentenary Conference on Thin-Walled Structures, (1996). 11. Chou S.M. and Rhodes J., \Review and comilation of exerimental results on thin-walled structures", Comuters and Structures, , (1997). 12. Khong P.W., Rhodes J., \Linear and Non Linear Analysis on the Micro Using Finite Stri", (1988). 13. Ovesy H.R., Loughlan J. and Ghannadour S.A.M., \Geometric non-linear analysis of thin at lates under end shortening, using dierent versions of the nite stri method", Int. J. of Mech. Sci., 47, PP (2005). 14. Ovesy H.R., Loughlan J. and Ghannadour S.A.M., \Geometric non-linear analysis of channel sections under end shortening, using dierent versions of the nite stri method", Comuters and Structures, 84, PP (2006). 15. Ovesy H.R., Loughlan J., Ghannadour S.A.M. and Morada G., \Geometric non-linear analysis of box sections under end shortening, using three dierent versions of the nite stri method", Thin-Walled Structures, 44, PP (2006).

9 Nonlinear Post-Buckling Analysis of Isotroic Plates Using Finite Stri Methods Ovesy H.R. and Ghannadour S.A.M., \An exact nite stri for the calculation of relative ost-buckling stiness of isotroic lates", Structural Engineering and Mechanics, 312, PP (2009). 17. Ghannadour S.A.M. and Ovesy H.R., \The alication of an exact nite stri to the buckling of symmetrically laminated comosite rectangular lates and rismatic late structures", Comos. Struct., 891, PP (2009). 18. Ghannadour S.A.M. and Ovesy H.R., \An exact nite stri for the calculation of relative ost-buckling stiness of I-section struts", Int. J. of Mech. Sci., 509, PP (2008). 19. Ghannadour S.A.M. and Ovesy H.R., \Exact ostbuckling stiness calculation of box section struts", Engineering Comutations: Int. J. for Comuter- Aided Eng. and Soft., 267, PP (2009). 20. Ovesy H.R. and Ghannadour S.A.M., \An Exact Finite Stri for the Initial Post-Buckling Analysis of Channel Section Struts", Com. Struc., , PP (2011). 21. Ovesy H.R. and Ghannadour S.A.M., \Geometric nonlinear analysis of imerfect comosite laminated lates, under end shortening and ressure loading, using nite stri method", Comosite Structures, 75, PP (2006).