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.     

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.     

Numerical analysis of static performance comparison of friction stir welded versus riveted 2024-T3 aluminum alloy stiffened panels

Most researches on the static performance of stiffened panel joined by friction stir welding（FSW） mainly focus on the compression stability rather than shear stability. To evaluate the potential of FSW as a replacement for traditional rivet fastening for stiffened panel assembly in aviation application, finite element method（FEM） is applied to compare compression and shear stability performances of FSW stiffened panels with stability performances of riveted stiffened panels. FEMs of 2024-T3 aluminum alloy FSW and riveted stiffened panels are developed and nonlinear static analysis method is applied to obtain buckling pattern, buckling load and load carrying capability of each panel model. The accuracy of each FEM of FSW stiffened panel is evaluated by stability experiment of FSW stiffened panel specimens with identical geometry and boundary condition and the accuracy of each FEM of riveted stiffened panel is evaluated by semi-empirical calculation formulas. It is found that FEMs without considering weld-induced initial imperfections notably overestimate the static strengths of FSW stiffened panels. FEM results show that, buckling patterns of both FSW and riveted compression stiffened panels represent local buckling of plate between stiffeners. The initial buckling waves of FSW stiffened panel emerge uniformly in each plate between stiffeners while those of riveted panel mainly emerge in the mid-plate. Buckling patterns of both FSW and riveted shear stiffened panels represent local buckling of plate close to the loading corner. FEM results indicate that, shear buckling of FSW stiffened panel is less sensitive to the initial imperfections than compression buckling. Load carrying capability of FSW stiffened panel is less sensitive to the initial imperfections than initial buckling. It can be concluded that buckling loads of FSW panels are a bit lower than those of riveted panels whereas carrying capabilities of FSW panels are almost equivalent to those of riveted panels with identical geometries. Finite elem     

Influence of stiffener location on the stability of stiffened plates under compression and in-plane bending

The elastic behaviour of stiffened plates under non-uniform edge compression is investigated. As a first stage, an energy formulation, in which the structure is modelled as assembled plate and beam elements, is presented. A Sequential Quadratic Programming (SQP) algorithm is then used to evaluate the buckling load and the associated buckling mode for given plate/stiffener geometric properties. Results are presented showing the influence of the stiffener location on the stability of the structure under combined compression and bending. Then, a strategy is presented for optimum location of the stiffener. The proposed methodology can be used to develop an improved design procedure for efficient design of stiffened plates under this type of loading.     

铝合金搅拌摩擦焊接加筋板剪切稳定性能研究

对2024-T3铝合金搅拌摩擦焊接加筋壁板的剪切稳定性能进行试验研究,得到该结构的剪切失稳形式及屈曲载荷、承载能力及破坏形式。应用有限元法对该焊接加筋壁板结构进行简化建模,对该结构的稳定性和承载能力进行计算,将计算结果与试验结果进行比较。试验及有限元计算结果表明,结构的剪切失稳形式表现为筋条间平板的局部屈曲;屈曲后结构进入张力场受力状态,破坏形式主要表现为平板的塑性变形和边缘撕裂、筋条的弯扭变形以及焊接区的局部脱焊;出现脱焊现象的试验件其承载能力较未出现脱焊现象的试验件下降7.7%;线性和非线性屈曲计算所得屈曲载荷分别比试验平均值高出18.4%和26.2%,而非线性计算所得承载能力比试验平均值高出5.7%,焊接引起的初始缺陷对结构承载能力的影响小于对屈曲载荷的影响;有限元分析得到的结构屈曲形式和失效形式与试验现象吻合,验证了有限元模型的合理性,但其仍需要进一步改进以考虑初始缺陷来减小计算误差。     

Buckling of laminated sandwich plates subjected to partial edge compression

The buckling characteristics of sandwich plates having laminated stiff layers are studied for different types of partial edge loadings using a refined plate theory. With this plate theory, the through thickness variation of transverse shear stresses is represented by piecewise parabolic functions where the continuity of these stresses is satisfied at the layer interfaces by taking jumps in the transverse shear strains at the interfaces. The transverse shear stresses free condition at the plate top and bottom surfaces is also satisfied. It is quite interesting to note that this plate model having all these refined features requires unknown parameters only at the reference plane. To have a generality in the present analysis, finite element technique is adopted and it is carried out with newly developed triangular element, as existing finite elements cannot accommodate this plate model. So far, no solution exists in the literature for the problem of sandwich plate subjected to partial edge loading. The present analysis is first validated for the case of an isotropic plate subjected to partial edge compression and then it is extended to analyze sandwich plates. Few results are presented.     

Sound radiation from strip plates with longitudinal stiffeners using waveguide finite and boundary element methods

In this study, the vibration characteristics and sound radiation of strip plates with finite width and infinite length are investigated numerically in order to analyze the vibration and sound radiation of structures consisting of many stiffened and double-layered plates. The waveguide finite element approach, which is effective for waveguide structures, is applied as a numerical scheme. The sound power and radiation efficiencies for an unstiffened plate are calculated numerically via coupling boundary elements to the WFEs. Longitudinal stiffeners and additional upper plates are included in the plate model to investigate the effect of stiffeners and an upper plate on sound power and radiation efficiency. In this study, it is found that the stiffeners contribute differently to plate vibration and sound radiation, and that the radiation efficiencies of the stiffened and double plates are larger than those of the unstiffened plate due to the presence of the stiffeners.     

General sinusoidal stiffness matrices for buckling and vibration analyses of thin flat-walled structures

This paper is concerned with the derivation of stiffness matrices for the buckling or vibration analysis of any structure consisting of a series of long, thin, flat plates rigidly connected together at their longitudinal edges. Each plate is assumed to be subjected to a basic state of plane stress which is longitudinally invariant, and it is further assumed that the mode of buckling or vibration varies sinusoidally in the longitudinal direction. During buckling or vibration, the edges of any individual plate are subjected to additional systems of forces and moments which are sinusoidally distributed along the edges, and these give rise to sinusoidally varying edge displacements and rotations. Spatial phase differences between the forces and displacements are accounted for by defining them in terms of complex quantities. The sinusoidal edge forces and displacements are split into two uncoupled systems, corresponding to out-of-plane and in-plane displacements, and two stiffness matrices are defined. The out-of-plane stiffness matrix is shown to be in general complex, and Hermitian in form, but the inplane stiffness matrix is real and symmetrical. Explicit expressions are derived for the elements of the matrices, in which all the essential destabilizing effects of the basic stresses, as well as dynamic effects, are included. Finally, it is shown that buckling and vibration phenomena for any structure of this type are closely interrelated.     

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.     

FE analysis of unstiffened and stiffened corrugated panels subjected to blast loading

This paper presents the results of a dynamic analysis on unstiffened and stiffened corrugated panels subjected to hydrocarbon explosion. A parametric study is also conducted on simplified models of the stiffened corrugated panels considering the effect of stiffeners on the compressive flange under different loading levels. The 1/2 symmetry of corrugated panels is modeled. This numerical study is performed using NX Nastran version 7.5. The unstiffened panel produces localized buckling at the center of corrugation and large permanent deformation by increasing the peak pressure. The stiffened panels suppress the structural response, and the vee stiffeners are structurally more effective than the round ones.     

A point receptance array approach for stiffened panel structures

A point receptance array is proposed to study behaviors of panel structures with stiffeners. Although the overall objective is intended to develop a general algorithm for large structural members with stiffeners of arbitrary arrangements, the present study focus on an analytical evaluation of the accuracy of the approximation. To simplify the formulations, a thin plate with two diagonally connected beam stiffeners is selected. By solving governing differential equations for the plate and beam, point receptances that represent the interaction and compatibility conditions at the plate-beam interfaces are analytically obtained. Buckling modes of the stiffened plate are obtained by solving an eigenvalue problem. Accuracy of the approach is verified by a comparison of the analytical solution with data obtained from a direct finite element model. A parametric study was also performed. Good agreement indicates that the point receptance is a viable approach for general applications.     

Computational procedures for a matrix analysis of the stability and vibration of thin flat-walled structures in compression

This paper is concerned with computational problems arising in the application of a previously published matrix analysis of the stability and vibration of structures consisting entirely of a series of thin flat rectangular plates connected together along longitudinal edges. The theory requires that the conditions at the ends of the structure permit a mode which varies sinusoidally in the longitudinal direction, and it is then possible to consider each individual flat plate as one element having four degrees of freedom at each of its two longitudinal edges. The corresponding stiffness matrices are essentially exact within the spirit of thin plate theory. The main computational problem is that of testing whether or not, for a specified wavelength of buckle, any chosen value of the compressive stress is less than or greater than the lowest buckling stress of the structure. This problem is discussed in detail, and a systematic test procedure is derived. Examples of panels with integral unflanged stiffeners, bonded Z-section stiffeners, bonded top-hat stiffeners and corrugated-core sandwich panels are discussed.     

A composite view on Windenburg's problem: Buckling and minimum stiffness requirements of compressively loaded orthotropic plates with edge reinforcements

This paper discusses the buckling behaviour of orthotropic composite plates under uniform uniaxial compression with one free reinforced unloaded edge. A typical application example for use of such a mechanical model is the web of stiffeners and frames attached to the fuselage skin of an aircraft. The considered plates are rectangular and simply supported at the loaded transverse edges. One of the longitudinal unloaded edges is also simply supported, while the second unloaded edge is not supported at all but is reinforced by a flange of arbitrary cross-section. At first, an exact solution for the elastic buckling problem is derived from the governing differential equation by imposing the underlying boundary conditions. Thereafter, two approximate closed-form solutions for the buckling load are derived, which can be conveniently used for practical application purposes. Generic buckling curves using characteristic non-dimensional quantities are also presented. Finally, the question of the required bending stiffness EI_min of the flange is treated, to ensure that the flange withstands buckling and provides simply supported boundary conditions to the free reinforced plate edge.     

The stacking sequence optimization of stiffened laminated curved panels with different loading and stiffener spacing

An efficient procedure to obtain the optimal stacking sequence and the minimum weight of stiffened laminated composite curved panels under several loading conditions and stiffener layouts has been developed based on the finite element method and the genetic algorithm that is powerful for the problem with integer variables. Often, designing composite laminates ends up with a stacking sequence optimization that may be formulated as an integer programming problem. This procedure is applied for a problem to find the stacking sequence having a maximum critical buckling load factor and the minimum weight. The object function in this case is the weight of a stiffened laminated composite shell. Three different types of stiffener layouts with different loading conditions are investigated to see how these parameters influence on the stacking sequence optimization of the panel and the stiffeners. It is noticed from the results that the optimal stacking sequence and lay-up angles vary depending on the types of loading and stiffener spacing.     

Buckling of symmetrically laminated rectangular plates under parabolic edge compressions

Buckling analysis of symmetrically laminated rectangular plates with parabolic distributed in-plane compressive loadings along two opposite edges is performed using the Rayleigh-Ritz method. Classical laminated plate theory is adopted. Stress functions satisfying all stress boundary conditions are constructed based on the Chebyshev polynomials. Displacement functions for buckling analysis are constructed by Chebyshev polynomials multiplying with functions that satisfy either simply supported or clamped boundary condition along four edges. Methodology and procedures are worked out in detail. Buckling loads for symmetrically laminated plates with four combinations of boundary conditions are obtained. The proposed method is verified by comparing results to data obtained by the differential quadrature method (DQM) and the finite element method (FEM). Numerical example also shows that the double sine series displacement for simply supported symmetrically laminated plates having bending-twisting coupling may overestimate the stiffness, thus providing higher buckling loads.     

复合材料加筋板在剪切载荷下的屈曲特性研究

通过对复合材料薄壁加筋板结构进行剪切载荷下的屈曲试验研究,得到结构的屈曲模态、屈曲失稳载荷以及破坏形式,并通过有限元方法对结构的屈曲进行数值分析,分析得到的复合材料薄壁加筋板结构的屈曲模态和试验结果一致,屈曲载荷与试验结果吻合较好.试验还发现复合材料薄壁加筋板结构有较高的后屈曲承载能力,后屈曲过程中由于桁条脱胶会造成屈曲模态的变化.还分析了筋条的连续性对屈曲载荷的影响.     

Strongly nonlinear free vibration of four edges simply supported stiffened plates with geometric imperfections

This article investigated the strongly nonlinear free vibration of four edges simply supported stiffened plates with geometric imperfections. The von Karman nonlinear strain-displacement relationships are applied. The nonlinear vibration of stiffened plate is reduced to a one-degree-of-freedom nonlinear system by assuming mode shapes. The Multiple scales Lindstedt-Poincare method (MSLP) and Modified Lindstedt-Poincare method (MLP) are used to solve the governing equations of vibration. Numerical examples for stiffened plates with different initial geometric imperfections are presented in order to discuss the influences to the strongly nonlinear free vibration of the stiffened plate. The results showed that: the frequency ratio reduced as the initial geometric imperfections of plate increased, which showed that the increase of the initial geometric imperfections of plate can lead to the decrease of nonlinear effect; by comparing the results calculated by MSLP method, using MS method to study strongly nonlinear vibration can lead to serious mistakes.     

Free vibration and buckling response of hat-stiffened composite panels under general loading

The paper presents finite element free vibration and buckling analysis of laminated hat-stiffened shallow and deep shells using arbitrarily oriented stiffener formulation. Modified approach for modelling the curved stiffener is implemented using necessary transformations. A simplified stiffener formulation is presented to accommodate various shapes of stiffener shapes in developing the rigidity matrix for the finite element formulation. Investigation has been carried out on free vibration and buckling analyses of laminated composite stiffened shell structures with laminated open section (rectangular or ‘T’ shaped) and closed section (‘hat’ shaped) stiffeners. Parametric study on the hat-stiffened panels for the free vibration and buckling analyses confirms that the closed section stiffener being torsionally rigid is found to show better performance over open section stiffeners.     

Analysis of stiffened corrugated plates based on the FSDT via the mesh-free method

The elastic bending of unstiffened and stiffened corrugated plates is studied in this paper, and a mesh-free Galerkin method is presented for the analyses. A corrugated plate is treated as an orthotropic plate that has different flexure properties in two perpendicular directions. The equivalent flexure properties are estimated by applying constant curvature conditions to the corrugated sheet. The stiffened corrugated plate is considered as a composite structure of an orthotropic plate with beams. By superimposing the strain energy of the orthotropic plate and the beams, and imposing the displacement compatibility conditions between the plate and the beams, the stiffness matrix of the structure is obtained. Because no mesh is needed in the proposed method, there is no limitation to the position of the stiffeners (beams). Changes in the positions of the stiffeners do not require the re-meshing of the plate. Several numerical examples are employed to show the accuracy and convergence of the proposed method. The computation results demonstrate good agreement with the solutions given by ANSYS, and different profiles of corrugated plates are considered.     

Exact solutions for vibration and buckling of an SS-C-SS-C rectangular plate loaded by linearly varying in-plane stresses

Exact solutions are presented for the free vibration and buckling of rectangular plates having two opposite edges (x=0 and a) simply supported and the other two (y=0 and b) clamped, with the simply supported edges subjected to a linearly varying normal stress σ_x=−N₀[1−α(y/b)]/h, where h is the plate thickness. By assuming the transverse displacement (w) to vary as sin(mπx/a), the governing partial differential equation of motion is reduced to an ordinary differential equation in y with variable coefficients, for which an exact solution is obtained as a power series (the method of Frobenius). Applying the clamped boundary conditions at y=0 and b yields the frequency determinant. Buckling loads arise as the frequencies approach zero. A careful study of the convergence of the power series is made. Buckling loads are determined for loading parameters α=0,0.5,1,1.5,2, for which α=2 is a pure in-plane bending moment. Comparisons are made with published buckling loads for α=0,1,2 obtained by the method of integration of the differential equation (α=0) or the method of energy (α=1,2). Novel results are presented for the free vibration frequencies of rectangular plates with aspect ratios a/b=0.5,1,2 subjected to three types of loadings (α=0,1,2), with load intensities N₀/N_cr=0,0.5,0.8,0.95,1, where N_cr is the critical buckling load of the plate. Contour plots of buckling and free vibration mode shapes are also shown.