Non-uniform In-plane Loading Research Articles

Stability analysis of graphene based laminated composite sheets under non-uniform inplane loading by nonlocal elasticity

Size dependent buckling of composite laminates made of isotropic graphene layers interlaid with bonding agents is considered. Nonlocal theory of elasticity is used in the buckling analysis to reflect the size scale effects on the critical buckling loads which is discussed in detail. The method is capable of predicting the relative buckling modes for non-uniform inplane loading applied through the thickness of the laminate. All modes of buckling in which the layers may displace together or opposite one another are investigated to study their scale dependent effects. Displacement or load controls are implemented through independent parameters as constraints to form special combination of buckling modes. Each graphene sheet is considered as a Kirchhoff plate model. The interlaid bonding agent is laterally treated as Winkler elastic foundation between graphene layers while neglecting their other load carrying capacities. Various numerical results are obtained reflecting the nonlocality effects. It is observed that in cases of higher load ratios and simpler buckling modes, the effect of nonlocality tends to drastically increase. The results of simpler examples studied are verified by another reference.

Buckling and Postbuckling Behavior of Cross-Ply Composite Plate Subjected to Nonuniform In-Plane Loads

This paper presents a study of buckling and postbuckling behaviour of simply supported composite plates subjected to nonuniform in-plane loading. The mathematical model is based on higher order shear deformation theory incorporating von Kármán nonlinear strain displacement relations. Because the applied in-plane edge load is nonuniform, in the first step the plane elasticity problem is solved to evaluate the stress distribution within the prebuckling range. Using these stress distributions, the governing equations for postbuckling analysis of composite plates are obtained through the theorem of minimum potential energy. Adopting Galerkin’s approximation, the governing nonlinear partial differential equations are reduced into a set of nonlinear algebraic equations in the case of postbuckling analysis, and homogeneous linear algebraic equations in the case of buckling analysis. The critical buckling load is obtained from the solution of associated linear eigenvalue problem. Postbuckling equilibrium paths are obtained by solving nonlinear algebraic equations employing the Newton-Raphson iterative scheme. Explicit expressions for the plate in-plane stress distributions within the prebuckling range are reported for isotropic and composite plates subjected to parabolic in-plane edge loading. Buckling loads are determined for three plate aspect ratios (a/b=0.5, 1, 1.5) and three different types of in-plane load distributions. The effect of shear deformation on the buckling loads of composite plate is reported. The present buckling results are compared with previously published results wherever possible.

Postbuckling analysis of cross-ply laminated cylindrical shell panels under parabolic mechanical edge loading

Postbuckling equilibrium paths of simply supported cross-ply laminated cylindrical shell panels subjected to non-uniform (parabolic) inplane loads are traced in this paper. Love's shell theory with higher order shear deformation theory and von Kármán nonlinear strain–displacement relations are used in the mathematical formulation of the problem. In the first step, the plate membrane problem is solved to evaluate the stress distribution within the prebuckling range as the applied inplane edge load is non-uniform. The governing shell panel postbuckling equations are derived from the principle of minimum total potential energy using the above stress distributions. Adopting multi-term Galerkin's approximation, the governing equations are reduced into a set of non-linear algebraic equations. Newton–Raphson method in conjunction with Riks approach is employed to plot the postbuckling paths through limit points. Numerical results are presented for symmetric (0/90/0) crossply laminated cylindrical shell panels under parabolic inplane load, lateral distributed load and initial imperfections. Limit loads and snap-through behavior of shell panels are studied.

Buckling of rectangular plates with various boundary conditions loaded by non-uniform inplane loads

In the present paper, buckling loads of rectangular composite plates having nine sets of different boundary conditions and subjected to non-uniform inplane loading are presented considering higher order shear deformation theory (HSDT). As the applied inplane load is non-uniform, the buckling load is evaluated in two steps. In the first step the plane elasticity problem is solved to evaluate the stress distribution within the prebuckling range. Using the above stress distribution the plate buckling equations are derived from the principle of minimum total potential energy. Adopting Galerkin's approximation, the governing partial differential equations are converted into a set of homogeneous linear algebraic equations. The critical buckling load is obtained from the solution of the associated linear eigenvalue problem. The present buckling loads are compared with the published results wherever available. The buckling loads obtained from the present method for plate with various boundary conditions and subjected to non-uniform inplane loading are found to be in excellent agreement with those obtained from commercial software ANSYS. Buckling mode shapes of plate for different boundary conditions with non-uniform inplane loadings are also presented.

Combination Resonance Instability of Curved Panels with Cutout Subjected to Nonuniform Loading with Damping

The dynamic instability of doubly curved panels with a centrally located circular cutout, subjected to nonuniform compressive in-plane harmonic edge loading is investigated. The present work deals with the problem of the occurrence of combination resonances in contrast to single (mode) resonances in parametrically excited doubly curved panels with a central circular cutout. The method of multiple scales is used to obtain analytical expressions for the single (mode) and combination resonance instability regions. It is shown that other cases of the combination resonance can be of major importance and yield a significantly enlarged instability region in comparison to the principal instability region. The effects of nonuniform edge loading, centrally located circular cutout, damping, the static, and dynamic load factors on dynamic instability behavior of simply supported doubly curved panels are studied. The results show that under localized edge loading, combination resonance zones are as important as single (mode) resonance zones. The effects of damping show that there is a finite critical value of the dynamic load factor for each instability region below which doubly curved panels cannot become dynamically unstable. A central circular cutout has the destabilizing effect on the dynamic stability behavior of doubly curved panels subjected to nonuniform edge loading. This example of simultaneous excitation of two modes, each oscillating steadily at its own natural frequency, may be of considerable interest in vibration testing of actual structures.

Dynamic instability analysis of stiffened shell panels subjected to partial edge loading along the edges

The dynamic instability characteristics of stiffened shell panels subjected to partial in-plane harmonic edge loading are investigated in this paper. The eight-noded isoparametric degenerated shell element and a compatible three-noded curved beam element are used to model the shell panels and the stiffeners, respectively. As the usual formulation of degenerated beam element is found to overestimate the torsional rigidity, an attempt has been made to reformulate it in an efficient manner. Moreover, the new formulation for the beam element requires five degrees of freedom per node as that of shell element. The method of Hill's infinite determinant is applied to analyze the dynamic instability regions. Numerical results are presented through convergence and comparison with the published results from the literature. The effects of parameters like loading type and shell geometry are considered in the dynamic instability analysis of stiffened panels subjected to non-uniform in-plane harmonic loads along the boundaries. The tension buckling aspect of the stiffened panels are also considered and the dynamic stability behavior due to tensile in-plane edge loading is studied for the concentrated load.

Buckling of orthotropic plates under various inplane loads

When a laminated composite plate is subjected to a compressive edge force lying in the plane of the plate, failure can occur by buckling at a stress below the strength of the material. Critical buckling loads of orthotropic plates are usually calculated using the analytical solutions which are based on the assumption of uniform end loads, despite the fact that real structures are subjected to various non-uniform inplane loads. This paper examines the buckling behavior of the orthotropic plates under non-uniform inplane loads with different boundary conditions (all edges simply supported, two loaded edges clamped and others simply supported, andall edges clamped) using the finite element method (FEM). Numerical results show that the existing formulas for buckling load basedon the uniform load assumption leads to overestimation of buckling loads.

PARAMETRIC COMBINATION RESONANCE CHARACTERISTICS OF DOUBLY CURVED PANELS SUBJECTED TO NON-UNIFORM HARMONIC EDGE LOADING

This paper is concerned with the problem of occurrence of combination resonances in parametrically excited doubly curved panels. The dynamic instability of doubly curved panels, subjected to non-uniform in-plane harmonic loading is investigated. Sander's first-order shear deformation theory is used to model the doubly curved panels, considering the effects of transverse shear deformation and rotary inertia. The theory can be reduced to Love's and Donnell's theories by means of tracers. Analytical expressions for the instability regions are obtained at Ω = ωm+ ωn(Ω is the excitation frequency and ωmand ωnare the natural frequencies of the system), by using the method of multiple scales. It is shown that besides the principal instability region at Ω =2ω1, where ω1is the fundamental frequency, other cases of Ω = ωm+ ωnwhich are related to other modes, can be of major importance and yield a significantly enlarged instability region. The results show that under localized edge loading, combination resonance zones are as important as simple resonance zones. The effects of edge loading, curvature, shallowness ratio, edge length to thickness ratio, aspect ratio, boundary conditions and the static load factor on dynamic instability regions are considered.

Simple and combination resonances of rectangular plates subjected to non-uniform edge loading with damping

The effects of damping on the behaviour of simple and combination resonances of simply-supported rectangular plates subjected to non-uniform in-plane periodic loading are studied. Finite element formulation is applied to obtain the equilibrium equation for the plate. The relations for the boundaries of parametric instability regions are obtained by using the method of multiple scales. The results show that under localized edge loading, combination resonance zones are as important as simple resonance zones. For nearly uniform loading, the combination resonance zones are very small in width and may disappear in the presence of slight damping. The effects of damping show that there is a finite critical value of the dynamic load factor for each zone below which the plate cannot become dynamically unstable. It is also shown that the effects of damping on the combination resonances may be destabilizing under certain conditions.

An Energy Solution to the Buckling of Rectangular Plates Under Non-Uniform In-Plane Loading

Circumstances can arise in which a rectangular panel, forming part of a structure, is subject to non-uniform in-plane compression, and in which it is important that instability of the panel should not occur. Problems of this nature occur, for example, in the structure of modern highspeed aircraft subject to the transient thermal stresses that can arise from rapid changes of speed or altitude.

Finite Difference Accuracy in Structural Analysis

An accuracy study is made of central finite difference methods for solving boundary value problems in structural analysis which are governed by equations with variable coefficients leading to odd order derivatives. Two methods are studied through application to beam-columns with nonuniform inplane loads and nonuniform stiffness. Definitive expressions for the error in each method are obtained by using Taylor series to derive the differential equations which exactly represent the finite difference approximations. The resulting differential equations are accurately solved by a perturbation technique which yields the error directly. A half station method, which corresponds to making finite difference approximations before expanding derivatives of function production in the beam-column differential equations, is found clearly superior to a whole station method which corresponds to expanding such products first.