Postbuckling Range Research Articles

Thermomechanical postbuckling of composite laminated cylindrical shells with local geometric imperfections

The effect of local geometric imperfections on the buckling and postbuckling of composite laminated cylindrical shells subjected to combined axial compression and uniform temperature loading was investigated. The two cases of compressive postbuckling of initially heated shells and of thermal postbuckling of initially compressed shells are considered. The formulations are based on a boundary layer theory of shell buckling, which includes the effects of the nonlinear prebuckling deformation, the nonlinear large deflection in the postbuckling range and the initial geometric imperfection of the shell. The analysis uses a singular perturbation technique to determine buckling loads and postbuckling equilibrium paths. Numerical examples are presented that relate to the performances of cross-ply laminated cylindrical shells with or without initial local imperfections, from which results for isotropic cylindrical shells follow as a limiting case. Typical results are presented in dimensionless graphical form for different parameters and loading conditions.

Experimental and finite element analyses on the post-buckling behaviour of repaired composite panels

This paper presents experimental and numerical investigations on the performance of repaired thin-skinned, blade-stiffened composite panels in the post-buckling range. The results show that under the present repair scheme the strength of the panel can be recovered satisfactorily. Further, the repair scheme is seen capable of restoring the general load path in the panels as well as the general post-buckling behaviour.

Localised formulations for thick "sandwich" laminated and composite structures

A “sandwich” panel under axial compressive loading is studied over its post-buckling range. Alternative formulations were developed to cover solutions other than the deflection shape given by the classical Rayleigh-Ritz analysis. General elasticity principles are used to derive the total potential energy function in terms of the displacement field. Given a constant modal amplitude displacement field, which is introduced in the total potential energy function, the results are discussed having as background models studied earlier. Techniques for the analysis of localized solutions are presented with reference to the study of the beam on nonlinear elastic foundation. The concepts of localization are introduced by using dynamical analogy and the double scale perturbation method and applied to the sandwich panel, to derive the governing amplitude modulation equations. The resulting formulations are discussed using some examples, and by solving analytical and numerically the amplitude differential equations.

Postbuckling analysis of stiffened laminated cylindrical shells under combined external liquid pressure and axial compression

A postbuckling analysis is presented for a stiffened, laminated, thin cylindrical shell of finite length subjected to combined loading of external liquid pressure and axial compression. The formulations are based on a boundary layer theory of shell buckling, which includes the effects of nonlinear prebuckling deformations, nonlinear large deflections in the postbuckling range and initial geometrical imperfections of the shell. The ‘smeared stiffener’ approach is adopted for the stiffeners. The analysis uses a singular perturbation technique to determine the interactive buckling loads and postbuckling equilibrium paths. Numerical examples cover the performances of perfect and imperfect, stiffened and unstiffened cross-ply laminated cylindrical shells. Typical results are presented in dimensionless graphical form.

Postbuckling of imperfect stiffened cylindrical shells under combined external pressure and heating

A postbuckling analysis is presented for a stiffened cylindrical shell of finite length subjected to combined loading of external pressure and a uniform temperature rise. The formulations are based on a boundary layer theory of shell buckling, which includes the effects of nonlinear prebuckling deformations, nonlinear large deflections in the postbuckling range and initial geometrical imperfections of the shell. The “smeared stiffener” approach is adopted for the stiffeners. The analysis uses a singular perturbation technique to determine the interactive buckling loads and the postbuckling equilibrium paths. Numerical examples cover the performances of perfect and imperfect, stringer and ring stiffened cylindrical shells. Typical results are presented in dimensionless graphical form.

A rigorous method for the analysis of localization of axisymmetric buckling patterns in thick cylindrical shells

This paper theoretically investigates the localization of axisymmetric buckling patterns in cylindrical shells because this phenomenon has a large influence on the strength degradation in the post-buckling range. The localization behavior is caused by a plastic bifurcation on the decreasing equilibrium path subsequent to the maximum load point. Thus, we first show a rigorous numerical method based on the finite-element method technique for analysis of plastic bifurcation under two-axial stresses. Then, using this method, we examine how the structural parameters, constitutive models and boundary conditions affect the deformation capacity of cylindrical shells.

Postbuckling analysis of imperfect stiffened laminated cylindrical shells under combined external pressure and thermal loading

A postbuckling analysis is presented for a stiffened laminated cylindrical shell of finite length subjected to combined loading of external pressure and a uniform temperature rise. The formulation is based on a boundary layer theory of shell buckling which includes the effects of nonlinear prebuckling deformations, nonlinear large deflections in the postbuckling range and initial geometrical imperfections of the shell. The “smeared stiffener” approach is adopted for the stiffeners. The analysis uses a singular perturbation technique to determine the interactive buckling loads and the postbuckling equilibrium paths. Numerical examples are presented that relate to the performance of perfect and imperfect, stiffened and unstiffened cross-ply laminated cylindrical shells. Typical results are presented in dimensionless graphical form for different parameters and loading conditions.

Post-buckling behaviour of graphite/epoxy stiffened panels with initial imperfections subjected to eccentric biaxial compression loading

The post-buckling behaviour of anisotropic stiffened panels with initial imperfections is investigated. Since buckling of the skin between the stiffeners often occurs first, a non-linear analysis is developed for symmetric panels under biaxial compression in order to obtain the out-of-plane panel deflection in the post-buckling range. The non-linear differential equations are expressed in terms of the out-of-plane displacement and the Airy function. They are solved with the Galerkin method for various boundary conditions by imposing an edge displacement control. The theoretical and experimental results obtained by the present analysis show that the transverse load can greatly influence the buckling loads and halfwave number. Since no experimental results have been found in the literature, several tests have been carried out on graphite/epoxy blade stiffened panels 900 mm long and 620 mm wide applying simultaneously biaxial compression loads with several combined ratios. An eccentricity results between longitudinal and transverse load, because the longitudinal compression is applied along the centroidal axes of the stiffened section while the transverse compression is applied to the skin panel. The correlation between the experimental and analytical results has been quite good; the experimental results demonstrate the influence of eccentricity of the transverse load on panel deflection in the pre- and post-buckling range.

Vibration Analysis of Stiffened Laminated Plates Including Thermally Postbuckled Deflection Effect

Vibration characteristics of stiffened laminated plates are investigated considering postbuckled deflection effect due to thermal load. The finite element method is used for the analysis of thermal postbuckling and natural vibration of stiffened plates in the postbuckling range. The finite element model is based on the first order shear deformable plate theory (FSDT) for a skin panel and Timoshenko beam theory for stiffeners. Von Karman strain-displacement relation is used to account for a large deflection. Critical buckling temperature and the corresponding mode shape are determined from Euler buckling problem. In order to solve the thermal-postbuckling problem, the initial nonlinear stiffness is determined from an estimated deflection of scaled buckling mode shape. The converged deflection at any temperature change is obtained using the Newton-Raphson method. The vibration analyses of stiffened laminated plates in the postbuckling range are performed using the tangent stiffness obtained from the converged deflection. The effects of the stiffener size, the number of stiffeners, and lamination scheme on vibration characteristics are studied for stiffened laminated plates subject to steady-state uniform temperature increase.

Analytical prediction of seismic behaviour for concentrically-braced steel systems

This paper presents an analytical model for the inelastic response analysis of braced steel structures. A model is first presented for the behaviour of steel struts subjected to cyclic axial load, which combines the analytical formulation of plastic hinge behaviour with empirical formulas developed on the basis of experimental data. The brace is modelled as a pin-ended member, with a plastic hinge located at the midspan. Braces, with other end conditions, are handled using the effective length concept. Step-wise regression analysis is employed, to approximate the plastic conditions for the steel UC section. Verification of the brace model is performed on the basis of quasi-static analyses of individual struts and a one-bay one-storey X-braced steel frame. The comparison of analytical and experimental data has confirmed that the proposed brace model is able to accurately simulate the cyclic inelastic behaviour of steel braces and braced systems. A series of dynamic analyses has been performed on two-storey V- and X-braced frames to study the influence of brace slenderness ratio on the inelastic response, and to look at the redistribution of forces in the post-buckling range of behaviour of CBFs. Recommendations have been made as to the estimation of maximum storey drifts for concentrically-braced steel frames in major seismic event. © 1997 John Wiley & Sons, Ltd.

Local buckling, strain localization, wrinkling and postbuckling response of line pipe

This paper addresses the prediction of the behaviour of buried pipelines when they are considered to be shell structures and subjected to curvatures that exceed their limit points. It surveys work carried out at the University of Alberta over a period of approximately six years. The work considers the development of local buckling, and the softening, strain localization and wrinkling that characterize the behaviour of pipeline structures, and the relation between these phenomena. It offers a phenomenological explanation of the mechanism by which wrinkling develops in the post-buckling buhaviour of structures. It then demonstrates that the experimental and analytical results of studies of pipe behaviour are consistent with this mechanism.The results of finite element analyses are compared with two series of tests on full-sized industrial pipe. The tests provide a database for the behaviour of the pipe against which the validity of the analyses can be established. Comparisons are made with respect to moment-curvature curves and postbuckling configurations. It is concluded that modern finite element analysis when properly done is capable of closely predicting the behaviour of the pipe when it is subjected to large deformations that carry it deeply into the postbuckling range.

Thermomechanical Postbuckling of Stiffened Laminated Cylindrical Shell

A postbuckling analysis is presented for a stiffened laminated cylindrical shell of finite length subjected to combined axial compression and uniform temperature loading. The two cases of compressive postbuckling of initially heated shells and of thermal postbuckling of initially compressed shells are considered. The formulations are based on a boundary layer theory of shell buckling, which includes the effects of the nonlinear prebuckling deformation, the nonlinear large deflection in the postbuckling range and the initial geometrical imperfection of the shell. The “smeared-stiffener” approach is adopted for the stiffeners. The analysis uses a singular perturbation technique to determine buckling loads and postbuckling equilibrium paths. Numerical examples cover the performances of perfect and imperfect, stiffened and unstiffened cross-ply laminated cylindrical shells with or without initial thermal or compressive stress. Typical results are presented in dimensionless graphical form, and they indicate that the postbuckling response of initially compressed laminated cylindrical shells under thermal load is different from that of mechanically loaded shells with and without initial thermal stress.

Post-buckling analysis of imperfect stiffened laminated cylindrical shells under combined external pressure and axial compression

A post-buckling analysis is presented for a stiffened laminated cylindrical shell of finite length subject to combined loading of external pressure and axial compression. The formulations are based on a boundary layer theory of shell buckling which includes the effects of the nonlinear pre-buckling deformation, the nonlinear large deflection in the post-buckling range and the initial geometrical imperfection of the shell. The “smeared stiffener” approach is adopted for the stiffeners. The analysis uses a singular perturbation technique to determine the interactive buckling loads and the post-buckling equilibrium paths. Numerical examples cover the performances of perfect and imperfect, stiffened and unstiffened cross-ply laminated cylindrical shells. Typical results are presented in dimensionless graphical form.

Thermal postbuckling analysis of imperfect stiffened laminated cylindrical shells

A thermal postbuckling analysis is presented for a stiffened laminated cylindrical shell of finite length subjected to a uniform or non-uniform parabolic temperature distribution varying in the circumferential or axial direction. The formulations are based on a boundary layer theory of shell buckling which includes the effects of non-linear prebuckling deformations, non-linear large deflections in the postbuckling range, and initial geometrical imperfections of the shell. The “smeared stiffener” approach is adopted for the stiffeners. The analysis uses a singular perturbation technique to determine thermal buckling loads and postbuckling equilibrium paths. Numerical examples are presented that relate to the performances of perfect and imperfect, stiffened and unstiffened cross-ply laminated cylindrical shells. Typical results are presented in dimensionless graphical form and exhibit two different types of thermal postbuckling response.

From helix to localized writhing in the torsional post-buckling of elastic rods

In this paper we study the competition between helical and localizing modes in the torsional buckling of stretched and twisted elastic rods. Within the Love-Kirchhoff formulation, we make comparative studies of the helical deformation of Love (1927) and the homoclinic localizing solution of Coyne (1990). Plots of the loads against their corresponding deflections allow the energetically preferred mode to be identified: it is found that preference switches from the helix to the localized mode early in the post-buckling range. These plots also allow us to predict the jumps that are observed under a variety of dead and rigid loading processes: these dynamic jumps take the rod from the spatially localized form to the familiar writhing state. Preliminary experiments confirm this preference for the localizing mode. They also reveal a second type of helical deformation, at a shorter wavelength, that is not predicted by the above long-rod analyses. A programme of further experimental and theoretical studies is suggested, and in a companion paper we lay the mathematical foundations for a numerical investigation of the complex and spatially chaotic deformations of a wider class of elastic rods.

Determination of effective breadth and effective width of stiffened plates by finite strip analyses

A finite strip (FS) method is presented for the numerical investigation of two design parameters — effective breadth and effective width — of stiffened plates. For the effective breadth, stiffened plates under bending are studied. Due to the transverse bending loads there is shear transmission through the plate from the stiffener which leads to a non-uniform longitudinal stress distribution across the plate width. This phenomenon, termed as shear lag, can be represented by the ‘effective breadth concept’, and has been extensively studied by analytical methods. A linear FS method is presented which utilizes the advantages of decoupling of Fourier terms on the one hand and, on the other hand, allows the treatment of both webs and flanges using a plate model. A definitely different situation exists for estimating the effectiveness of the plate breadth (or width) of plates in the postbuckling range. The ‘concept of effect width’ is based on the fact that plates with supported longitudinal edges and/or stiffeners can accept additional load after buckling under longitudinal compression, and enables the designer to evaluate the postbuckling strength of plate structures simply by using the design parameter ‘effective width’. Several formulae (most of them empirically derived) exist for an approximative calculation of the load dependent value of the effective width. A nonlinear FS method is developed and applied to the investigation of the postcritical strength of locally buckled structures. An incremental successive iterative procedure is introduced for an effective numerical analysis.

Shear capacity of plate girders with trapezoidally corrugated webs

In this paper, the shear capacity of plate girders with trapezoidally corrugated webs is numerically studied using a non-linear finite element method. Effects of large deflections are taken into account and a perfectly elastic-plastic material model obeying a von Mises yield criterion is assumed. The following geometric parameters that influence the shear capacity of such girders are investigated: (1) the overall dimension of the web panel; (2) the web thickness; (3) the corrugation depth of the web; (4) the corrugation angle; and (5) the width of the plane sub-panel of the web. More specifically, the influence of these parameters both on the ultimate shear capacity and on the remaining shear capacity in the post-buckling range, as well as on the buckling modes, are reported. Based on the numerical results, empirical formulae that were proposed earlier for the prediction of the shear capacity are examined and suggestions for an optimal design of such girders in shear are given.

Postbuckling analysis of cylindrical shells under combined external liquid pressure and axial compression

A postbuckling analysis is presented for a circular cylindrical shell of finite length which is subjected to combined loading of external liquid pressure and axial compression. The formulations are based on a boundary layer theory which includes effects of the nonlinear prebuckling deformation, the nonlinear large deflection in the postbuckling range and the initial geometrical imperfection of shells. The analysis uses a singular perturbation technique to determine interactive buckling loads and postbuckling equilibrium paths. Numerical examples are presented that relate to the performances of perfect and imperfect cylindrical shells. Typical results are presented in dimensionless graphical form.

Damage tolerance and failure analysis of a composite geodesically stiffened compression panel

A geodesically stiffened continuous-filament composite structural concept has been designed for a transport aircraft fuselage application and fabricated using an automated manufacturing process. Both large panels and element specimens derived from these panels have been experimentally and analytically investigated when subjected to axial compression to understand their buckling, postbuckling, and failure responses. The primary failure mode for this structural concept is skin-stiffene r separation in the skin postbuckling load range. The large panels are subjected to low-speed impact damage and tested to failure in axial compression to evaluate the damage tolerance of this structural concept. These results suggest that damage to the stiffener and a stiffener intersection point from the skin side do not influence the failure load or failure mode of this structural concept. Nonlinear finite element analysis using a detailed element specimen model indicates that failure of this specimen may have initiated at the skin-stiffener flange region close to the stiffener intersection. When the skin is in the postbuckling range at four times its initial buckling load, the interlaminar shear stress concentrations at a region where the stiffener makes a 20-deg turn away from the stiffener intersection seem to initiate panel failure.

Thermal post-buckling of linearly tapered moderately thick isotropic circular plates

A general finite element formulation is employed in this note to evaluate the post-buckling behaviour of linearly tapered moderately thick isotropic circular plates subjected to thermal load. The details of finite element formulation and the numerical results in terms of linear critical load and thermal load ratios in the post-buckling range for different thicknesses and taper parameter values are presented.