Spline FSM postbuckling analysis of shear-deformable rectangular laminates

A spline finite strip method is developed for the prediction of the geometrically non-linear response of rectangular, composite laminated plates to progressive in-plane loading. The development takes place within the context of the use of the first-order shear deformation plate theory and the non-linearity is introduced in the strain-displacement equations in the manner of the von Karman assumption. A number of applications of the new capability is described, involving laminates subjected to progressive uniform end shortening and to progressive in-plane shearing. In all the applications a close comparison of the finite strip results with independent finite element results is demonstrated.     

Buckling and postbuckling behavior of unstiffened slender curved plates under uniform shear

Buckling and postbuckling behavior of curved plates under in-plane shear are investigated. After revisiting classic elastic buckling results, the elastoplastic postbuckling behavior and the effects of curvature parameter and aspect ratio are simulated via geometrical and material nonlinear analyses. Imperfection sensitivity is studied for various imperfection shapes and magnitudes. An increase in curvature parameter raises the elastic buckling load, produces unstable buckling and reduces postbuckling reserves. The buckling load and shear capacity are higher in shorter plates. Small initial imperfections are found to have severe effects on the initial buckling load of plates with large curvature parameter, but little effect on ultimate postbuckling capacity.     

Buckling and postbuckling of unsymmetrically laminated angle-ply plates in uniaxial and biaxial compression

A deflection-type perturbation technique is proposed to solve the nonlinear Karman equations of rectangular unsymmetrically laminated angle-ply composite plates with simply supported edges in uniaxial and biaxial compression. Higher-order asymptotic expansions for postbuckling behavior of the plates are constructed by this approach. Effects of initial imperfection, lamination angle and number of layers on buckling and postbuckling of the plates are examined. In some specific cases numerical results obtained from the present solution are in fairly good agreement with other theoretical predictions.     

Buckling of compressed rectangular plates at non-uniformly elevated temperatures

A perturbation analysis of the influence of non-uniformly elevated temperatures upon the critical buckling loads of uniaxially compressed rectangular plates is presented. Within the framework of the uncoupled theory of thermal stresses a temperature dependence of the modulus of elasticity has been taken into consideration. In the special case of a simply supported square plate, the range of practical applicability of the resulting approximate formula has been checked by means of a variational finite-difference method.     

Thermal postbuckling of laminated composite skew plates with temperature-dependent properties

Thermal postbuckling and consequently, the role of temperature dependence of material properties on the thermal postbuckling behavior of laminated composite skew plates have not been addressed in literature. Hence, this problem is investigated here. The plate governing equations are based on the first-order shear deformation theory (FSDT) and the geometrical nonlinearity is modeled using Green’s strain tensor in conjunction with the von Karman assumptions. Since the problem is geometrically and physically nonlinear, the differential quadrature method (DQM) as an accurate, simple and computationally efficient numerical tool is adopted to discretize the governing equations and the related boundary conditions. Then, a direct iterative method is employed to obtain the critical temperature (bifurcation point) and consequently the nonlinear equilibrium path (the postbuckling behavior) of symmetrically laminated skew plates. After validating the formulation and the method of solution, the effects of temperature dependence of the material properties on the postbuckling characteristic of laminated skew plates with different skew angle, boundary conditions, length-to-thickness ratio, number of layers and ply layout are investigated.     

Buckling and vibration of stepped rectangular plates by element-based differential transform method

This paper is concerned with the determination of exact buckling loads and vibration frequencies of stepped rectangular thin-plates using differential transform method (DTM) which is the modified form or the disguised form of the Taylor's series method. The plates have multi-spans or multi-line hinges having two opposite edges simply supported and the other two edges being clamped, free or simply supported. To develop an analytical approach for buckling or free vibration of multi-span plates, we use the Levy solution that satisfies equilibrium and compatibility conditions at the interfaces. New and exact buckling in-plane loads and the natural frequencies are determined for many cases.     

Buckling and postbuckling investigations of imperfect curved stringer-stiffened composite shells. Part B: Computational investigations

Computational investigations of the buckling and postbuckling behaviour of a stringer-stiffened composite wing torsion box employing the finite element method are presented. Perfect and imperfect configurations — considering geometrical imperfections as well as initial stresses - are discussed. Two different loadcases are investigated: pure axial loading and axial loading with a superimposed constant torsion moment. The buckling behaviour is determined by tracing the load-deflection curves using Riks' path following method. Additional investigations, such as accompanying eigenvalue analyses and first-ply failure calculations, are performed for the first loadcase. Special attention is put on the modelling of the stiffened regions, where eccentrically placed layers have to be taken into account due to the bonding of the stiffeners to the inner surface of the box. The results show interesting phenomena, such as additional buckling points in the postbuckling region, which, however, can hardly be detected by simply considering the load-axial displacement path.     

Buckling of simply supported rectangular plates under combined bending and compression using eigensensitivity analysis

In this paper, an approximate quadratic closed-form expression is presented for the critical buckling analysis of a plate subjected to combined bending and compression. The formula is developed by expanding the eigenvalue, the critical buckling load, for a plate under combined bending and compression in a Mauclaurin's series about a plate subjected only to compression. The general expression can be used for all combinations of simply supported and clamped rectangular plates boundary conditions. An explicit formula in terms of the plate aspect ratio R and plate load parameter α is evaluated for simply supported plates. Compared with the Rayleigh–Ritz method, this approximate expression provides an excellent comparison when the load parameter α1.52 for plate aspect ratio between 0.2R2.8.     

Buckling and postbuckling of imperfect cylindrical shells under external pressure

A comprehensive nonlinear finite element elastic stability analysis was performed with the aim of explaining the effect of a deep single longitudinal initial dent on the load-carrying behaviour of an externally-pressurized cylindrical shell. The numerical results are compared with available test results and agree well, at least qualitatively. Imperfection sensitivity studies were carried out comparing the effect of the single dent with the effect of evenly distributed periodic initial imperfections. Imperfection sensitivity tends to vanish for increasing amplitudes of the idealized case of a single dent.     

Buckling and postbuckling behaviors of imperfect cylindrical shells subjected to torsion

Buckling and postbuckling behaviors of imperfect cylindrical shell subjected to torsion are investigated. The governing equations are based on the Karman–Donnell-type nonlinear differential equations. A boundary layer theory of shell buckling is applied to obtain the analytic solutions that meet the boundary conditions strictly. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. Numerical results reveal that the current theory gives quite good estimates of the postbuckling paths of cylindrical shells. The effects of the geometric parameters on the buckling and postbuckling behaviors of the cylindrical shells are analyzed. It is confirmed that the postbuckling equilibrium paths of cylindrical shells subjected to torsion are unstable and the relatively shorter shells have higher postbuckling equilibrium paths. Finally, the effects of the initial imperfections on the buckling and postbuckling behaviors of the cylindrical shells are clarified. The illustrated results of the imperfect shells with different initial transverse deflections show that extremely small imperfections do indeed reduce the buckling loads and make the postbuckling equilibrium paths be lower. The buckling and postbuckling of cylindrical shells under torsion exhibit obvious imperfect sensitivity. Furthermore, the effects become greater following with the larger imperfections.     

Buckling and postbuckling investigations of imperfect curved stringer-stiffened composite shells. Part A: Experimental investigation and effective width evaluation

A box-like stringer-stiffened thin-walled CFRP structure was subjected to load cases well beyond the limits of local buckling. The development of the deflection pattern was recorded via optical means and analysed numerically. In addition, the structure was modelled and analysed using the MARC FE program in the nonlinear deflection range.

The geometrical imperfections of the test structure were recorded by mechanical scanning and optical methods and introduced into the mathematical model. For the perfect (‘ideal’) and the geometrically imperfect (‘real’) structural model, the results of the FE analyses were compared and used to judge the effect of geometric imperfections on the postbuckling behaviour of the structure.

The effective axial stiffness for the various postbuckling states was evaluated and related to analytical estimates of effective width values for orthotropic sheet-like panels.     

Buckling of laminated skew plates

A general high precision triangular plate bending finite element has been extended to the buckling analysis of laminated skew plates. This procedure involves development of the transformation matrix between global and local degrees of freedom for nodes lying on the skew edges and suitable transformation of the element matrices. The accuracy of the present formulation has been verified against literature values. New results are obtained for antisymmetric angle-ply and cross-ply laminated skew plates. In this analysis, the critical buckling loads for different skew angles with various lamination parameters, such as number of layers, fibre orientation angle, different boundary (simply supported, clamped) and loading (uniaxial, biaxial) conditions, have been presented.     

Buckling of plates with strengthenings

This paper studies the buckling behaviour of simply supported square plates, which have weakening or strengthening bands. The weakening/strengthening bands are equally spaced and their thickness is either decreased or increased. The analysis assumes that the stress state in the plate before and during the buckling process remains in the elastic range. Two cases of plate loading are studied, one with compressive forces and one with tangential forces. The buckling coefficients are calculated for different numbers and thicknesses of strengthening/weakening bands. The thickness of strengthening/weakening bands and the thickness of the remaining plate is varied so that the volume/weight of the plate remains constant. In one case it is found that the buckling load at constant weight of the plate can be increased by 118% if an optimal ratio of the thickness of strengthening bands and the thickness of the remaining plate is used.     

Nonlinear stability of unsymmetrically laminated angle-ply shear-deformable plates in biaxial compression

The generalized Karman–Reissner equations governing large deflection of unsymmetrically laminated angle-ply shear-deformable rectangular plates are presented in this paper, based upon Karman and Reissner plate theories. Much effort has been concentrated on derivation of the Karman–Reissner equations for the laminates taking account of transverse shear effects so that only two unknown functions for nonlinear analysis need to be treated to gain a precise insight into the complexities; otherwise high-order refined plate theories must be applied. An asymptotic series solution is constructed according to the Karman-type refined theory for postbuckling behavior of the plates with the boundary condition of four edges simply supported. Typical numerical examples are presented for comparison with other nonlinear analytical and experimental results. The effects of shear deformation, lamination angle and geometrical imperfection on buckling and postbuckling behaviors of the laminates subjected to a combination of biaxial compressive loadings are examined for industrial application.     

Buckling of unstiffened compression curved plates

Motivated by a specific example of a box arch of a bowstring bridge under erection in Belgium, a parametric study of the non-linear behaviour of curved plates under circumferential compression has been undertaken.The influence of the geometrical parameters, radius of curvature R and slenderness b/t, has been investigated by means of a finite element program and results are presented in the form of behavioural curves.Finally a general design formula for curved plates derived from parametric calculations is proposed for specific ranges of R and b/t that cover the main field of civil engineering structures.     

Buckling and post-buckling of rectangular plates under linearly varying compression and shear. Part 2 — Experimental investigations

This experimental investigation was carried out in order to verify the theoretical results obtained in Part 1. The experimental examination of the problem considered theoretically was performed using two separate experiments in which two different approaches have been employed to fulfil loading and boundary conditions at the plate edges. The aspect ratio of the plate specimens examined in both cases was λ = 1.4. The results are presented in graphical form and the conclusions are drawn from the comparison of the experimental and theoretical findings.     

Buckling and postbuckling behaviour of cylindrical shells under combined external pressure and axial compression

Buckling and postbuckling behaviour of perfect and imperfect cylindrical shells of finite length subject to combined loading of external pressure and axial compression are considered. Based on the boundary layer theory which includes the edge effect in the buckling of shells, a theoretical analysis for the buckling and postbuckling of circular cylindrical shells under combined loading is presented using a singular perturbation technique. Some interaction curves for perfect and imperfect cylindrical shells are given. The analytical results obtained are compared with some experimental data in detail, and it is shown that both agree well. The effects of initial imperfection on the interactive buckling load and postbuckling behaviour of cylindrical shells have also been discussed.     

Static and transient response of rectangular plates

Static and transient analysis of composite plates is presented using a recently proposed shear deformation theory. The dynamic response is obtained by employing the numerical time integration scheme due to Newmark. The results obtained by using the classical plate theory (CPT) and the Mindlintype shear deformation theory (SDT) are compared with those obtained by using the proposed theory. The comparison studies reveal that the linear stress distribution, as assumed in CPT and SDT, differs considerably from the predicted nonlinear distribution of the proposed theory.     

Buckling and post-buckling behaviour of rectangular plates under linearly varying compression and shear: part 1 — Theoretical analysis

The buckling and post-buckling behaviour of thin, rectangular, simply supported flat plates subjected to end compression which varies linearly in the direction of loading and simultaneously loaded in shear is the subject of this paper. Part 1 presents a theoretical analysis of the problem. In this analysis the Rayleigh-Ritz energy method is utilised, and numerical minimisation of the total potential energy is applied to obtain the magnitude of the parameters of the assumed deflection functions. These are then used to define and describe other post-buckling characteristics of the plate. A simplified imperfection analysis is used to examine certain cases. The wide range of results presented covers many aspects of plate post-buckling behaviour.