Thermal postbuckling analysis of laminated cross-ply truncated circular conical shells

Abstract

In this work, thermoelastic postbuckling behavior of cross-ply laminated composite conical shells under presumed uniform temperature distribution is studied. The finite deflection analysis is carried out to determine the relationship between the maximum deflection and the temperature rise, and to evaluate the minimum temperature parameter that causes the bifurcation of shell deformation from axisymmetric deformation mode to asymmetric one. The formulation is based on first-order shear deformation theory that accounts for the transverse shear. The governing equations, derived using minimum total potential energy principle, are solved using semi-analytical finite element approach. The critical temperature parameter values corresponding to the onset of bifurcation are compared with those evaluated from linear eigenvalue analysis. The detailed study is carried out to highlight the influences of length-to-radius and radius-to-thickness ratios, semi-cone angle, number of layers and the boundary conditions on the nonlinear prebuckling/postbuckling thermoelastic response of the laminated circular conical shells. The participation of axisymmetric and asymmetric modes in the total response of the shells is also highlighted.

Introduction

The conical shells are often used as transition elements between cylinders of different diameter and/or end closures and sometimes as stand-alone components in various engineering applications such as tanks and pressure vessels, missiles and spacecraft, submarines, nuclear reactors, jet nozzles, and such other civil, chemical, mechanical, marine and aerospace engineering structures. Therefore, these shells may be regarded as elementary shell geometry together with cylinders and spheres. The potential of using the directional dependence of composite properties in designing tailored structures to improve structural performance together with their high specific strength/stiffness, damping properties and low coefficient of expansion especially in the fiber direction has received increasing attention in the recent years. The modern aeronautical and other engineering structures will often operate at elevated temperatures. In this category belong the state of the art future transportation systems such as supersonic and hypersonic aircraft, rockets, satellites, as well as electronic equipment, nuclear components etc. The advances in composite technology have lead to the application of elevated-temperature composite cylindrical/conical shells tailored for the required performance in the design of more and more sophisticated futuristic structures. However, the analysis of such structures is a complex task, compared with conventional single layer metallic structures, because of the exhibition of coupling among membrane, torsion and bending strains; weak transverse shear rigidities; and discontinuity of the mechanical characteristics along the thickness of the laminates.

The temperature rise may introduce compressive membrane pre-stress state due to boundary restraints in the shells leading to thermoelastic buckling failures. An estimate of the critical buckling temperature can be made through a linear eigenvalue analysis, whereas the sensitivity to the imperfections and postbuckling behavior is evaluated by means of nonlinear analysis. For the later analyses, generally in the literature, an imperfection affine to the critical buckling mode evaluated from eigenvalue analysis is assumed to be present in the shell structures.

The thermoelastic buckling analysis of composite laminated shells has received limited attention in the literature compared to that of isotropic shells and laminated panels [1], [2], [3]. The available studies on laminated shells are mostly dealing with the stability characteristics of circular cylindrical shells based on linear buckling [4], [5], [6], [7], [8], [9], [10], nonlinear/postbuckling [11], [12], [13], [14], [15], [16], [17] analyses, and incorporating the dynamic effects [18], [19], [20], [21], [22]. Thermal buckling analysis of orthotropic cylindrical shells considering uniform temperature rise is studied without incorporating the prebuckling rotation and employing Rayleigh–Ritz method [4], and with prebuckling rotations included using parametric differentiation in conjunction with homogeneous integration approach [5]. The stability characteristics of laminated composite cylindrical shells under thermal and mechanical loading situations are examined using Semiloof shell finite element for asymmetric circumferential modes [6] and for axisymmetric ones [7]. The improved Donnell’s equations incorporating circumferential shear and independent normal rotations are employed to study the thermal buckling of cylindrical shells of composite materials under uniform temperature rise, radial and axial temperature differences [8]. The influence of piezoelectric actuators on the thermal buckling characteristics of laminated cylindrical shells [9] and thermal buckling of composite cylindrical shells conveying hot fluid [10] are examined using semi-analytical finite element approach based on first-order shear deformation theory.

The study on the buckling and postbuckling response of composite shells subjected to high temperature is carried out by Birman and Bert [11] employing nonlinear thermoelastic version of Love’s first approximation theory. The qualitative snap-through behavior of flat plates and cylindrical shells under constant compression with increasing temperature and with increasing compression keeping temperature constant is brought out in Ref. [11]. The postbuckling analyses for perfect and imperfect, unstiffened and stiffened multilayered cylindrical shells under thermo-mechanical loading situations have been performed using von Karman–Donnell theory and employing perturbation approach [12], [13], [14], [15], [16]. The nonlinear thermal buckling behavior of laminated cylindrical shells assuming the presence of near the surface delamination is studied using energy principles [17] and it is brought out that the linear solution of the critical buckling temperature gives a higher value than that of the nonlinear consideration. The dynamic effects on the stability characteristics of laminated cylindrical shells arising due to the sudden application of the heat load [18], [19], [20] and the presence of periodic time dependent temperature filed [21], [22] are also investigated in the literature.

The literature on the investigation of thermoelastic instability problems of conical shells is scarce [23], [24], [25], [26], [27], [28], [29]. This may be attributed to the inherent complexity of the basic equations in curvilinear circular conical coordinates that are a system of nonlinear partial differential equations with variable coefficients. Bendavid and Singer [23] studied buckling of truncated isotropic conical shells heated along an axial strip using Rayleigh–Ritz method, whereas Lu and Chang [24] and Chang and Lu [25] examined the thermoelastic buckling based on linear [24] and nonlinear [25] analyses considering axisymmetrically or nonaxisymmetrically heated isotropic cones using Galerkin method. The effects of the axisymmetric initial deflection [26] and internal pressure [27] are examined by Tani [26], [27] on the thermal buckling of shallow, truncated isotropic conical shells under uniform heating. The Donnel-type shell equations with nonlinear prebuckling deflections are solved using a semi-analytical finite difference method. Thermoelastic buckling [28] and thermally induced dynamic instability [29] of laminated composite conical shells are investigated recently employing perturbation approach to solve the linear three-dimensional equations of motion in terms of incremental stresses perturbed from the state of neutral equilibrium. The initial prebuckling thermal stresses are evaluated directly by multiplying thermal strains (product of thermal expansion coefficients and temperature rise) with constitutive matrix in Refs. [28], [29]. However, it has been brought out in the literature referred above that the prebuckling state of stress has to be determined using the deformation filed obtained from the static analysis of the shells subjected to assumed thermal load distribution.

In order to fully exploit the strength and load carrying capacity of laminated composite conical shells at elevated temperatures, accurate prediction of their thermal buckling/postbuckling characteristics is essential. However, to the authors’ knowledge, the study on the thermoelastic buckling/postbuckling behavior of laminated conical shells based nonlinear analysis appears to be scarce in the literature. Furthermore, the participation of axisymmetric/asymmetric modes and presence of secondary bifurcation is not brought out in the limited number of analytical investigations available on postbuckling analysis of laminated shells. Since the knowledge of cross-ply laminate behavior sheds light on the characteristics of more complicated laminates such as angle-ply and laminates with mixed laminate orientations the study here is dealt with cross-ply case.

In the present work, the thermoelastic buckling/postbuckling characteristics of cross-ply truncated circular conical/cylindrical shells subjected to uniform temperature rise are studied through nonlinear static analysis employing semi-analytical finite element approach. The critical temperature parameter values corresponding to the onset of bifurcation are also compared with those evaluated from linear eigenvalue analysis. The presence of asymmetric perturbation in the form of small magnitude lateral pressure spatially proportional to the linear buckling mode shape is assumed to initiate the bifurcation of the shell deformation from axisymmetric mode to asymmetric one. The detailed parametric study is carried out to highlight the influences of length-to-radius and radius-to-thickness ratios, semi-cone angle, number of layers and the boundary conditions on the nonlinear prebuckling/postbuckling thermoelastic response of the laminated circular conical shells. The participation of axisymmetric and asymmetric modes in the total response of the shells is brought out through the mode shape analysis.