Thermal post-buckling behaviour of laminated plates using a shear-flexible element based on coupled-displacement field V. Sita Thankam a , Gajbir Singh b , G. Venkateswara Rao a, * , A.K. Rath a a Structural Engineering Group, Vikram Sarabhai Space Centre, Trivandrum-695022, India b Structural Analysis Division, ISRO Satellite Centre, Bangalore-560017, India Abstract Postbucklingbehaviourofrectangularlaminatedplatessubjectedtothermalloadsisinvestigatedinthispaper.Forthispurpose, a four-node, lock-free, rectangular composite plate finite element having six degrees of freedom per node viz three translations, two bending rotations about x- and y -axes and a twist is developed. The element is based on a bicubic representation of the transverse displacement field. The field descriptions for other variables are derived using equilibrium equations of strips along x-and y -axes of theplate.Asaresultfielddescriptionsinvolvematerialpropertiesapartfromtheusualgeometricvariables.3  3GuassQuadrature formula is employed to compute the elemental matrices. Though an exact integration rule has been employed, the element is free of shear locking even in the extreme thin plate regimes. The effect of boundary conditions, aspect ratio, number of layers and lay-up sequenceonthepost-bucklingbehaviorisstudiedindetail.Thenumericalexamplessolvedhereinrevealthepossibilityofsecondary bifurcations from the primary post-buckling path. Ó 2003 Elsevier Science Ltd. All rights reserved. Keywords: Coupled-field finite element; Thermal buckling; Thermal post-buckling 1. Introduction The structural components of launch vehicles, spacecraft and aircraft mainly constitute beams, plates and shells. These structural components are being fast replaced with fiber-reinforced composite materials due totheirhighspecificstrengthandspecificstiffness.These structural elements are subjected to extreme thermal environments during their service life. The thickness of the majority of such structural components being very low,theyarepronetobuckling.Itiswellknownthatthe plate structures are capable of carrying a much in- creased load beyond buckling without failure. In order to fully exploit their strength, an accurate prediction of their post-buckling load carrying capacity is essential. Theimportanceofthefieldhasattractedtheattentionof numerous researchers and as a result one finds a large number of publications on the subject of buckling and post-buckling. The closed-form/analytical solutions to such problems, if not impossible, are cumbersome; therefore finite element method solution is preferred. However, the finite element solutions are also not free from problems. In order to address the problem in to- tality, the literature review is split into two parts i.e. (i) four-node shear flexible finite elements and (ii) buckling and post-buckling of laminated plates. 1.1. Four-node shear-flexible plate elements Due to the low ratio of transverse shear to the in- plane moduli of fiber-reinforced composite plates, the shear flexibility plays a predominant role for the accu- rate prediction of their response predictions. To ap- propriately account for the transverse shear flexibility, Yang et al. [1] generalized the Reissner–Mindlin thick plate theory to arbitrarily laminated anisotropic plates. The proposed Yang–Norris–Stavsky (YNS) theory is consideredsufficientlyaccurateforpredictingtheoverall behaviour such as transverse deflections, first few fre- quenciesandbucklingloadsoflayeredcompositeplates. * Corresponding author. Tel.: +91-471-564181; fax: +91-471- 415224. E-mail address: gv_rao@vssc.org (G. Venkateswara Rao). 0263-8223/03/$ - see front matter Ó 2003 Elsevier Science Ltd. All rights reserved. PII:S0263-8223(02)00243-X Composite Structures 59 (2003) 351–359 www.elsevier.com/locate/compstruct