Stability criteria for rectangular plates subjected to intermediate and end inplane loads

This paper is concerned with the elastic buckling of rectangular plates subjected to intermediate and end uniaxial inplane loads, whose direction is parallel to two simply supported edges. The aforementioned buckling problem is solved by decomposing the plate into two subplates at the location where the intermediate uniaxial load acts. Each subplate buckling problem is solved exactly using the Levy approach and the two solutions brought together by matching the continuity equations at the separated interface. It is worth noting that there are five possible solutions for each subplate and consequently there are 25 combinations of solutions to be considered. For different boundary conditions, the buckling solutions comprise of different combinations. For each boundary condition, the correct solution combination depends on the ratio of the intermediate load to the end load. The exact stability criteria, presented both in tabulated and in graphical forms, should be useful for engineers designing walls or plates that have to support intermediate floors/loads.     

Buckling of circular plates with an internal ring support and elastically restrained edges

Presented herein are the exact buckling solutions of circular plates with a concentric internal ring support and elastically restrained edges. This study shows the existence of buckling mode switching with respect to the radius of the internal ring support. Generally the plate buckles in an axisymmetric mode but when the ring support radius becomes small, the plate may buckle in an asymmetric mode. The cross-over ring support radius varies from 0.081 to 0.152 times the plate radius, depending on the rotational stiffness of the elastic restraint at the edges.     

Buckling analysis of skew laminate plates subjected to uniaxial inplane loads

Elastic stability of skew composite laminate plates subjected to uniaxial inplane compressive forces has been studied. The critical buckling loads of the skew laminate plates are carried out by the bifurication buckling analysis implemented in finite element program ABAQUS. The effects of skew angles, laminate layups, plate aspect ratios, plate thicknesses, central circular cutouts, and edge conditions on the buckling resistance of skew composite laminate plates are presented.     

Buckling analysis of a cable-stayed circular frame

Large parabolic dish concentrators have been widely employed in solar thermal applications. The supporting structure of a solar dish concentrator consists of a circular frame, a central post, and front and rear cables connecting the frame to the post. The tensions in the cables cause compressive stresses in the circular frame and the central post, and this support structure must be designed for stability. In this paper, the nonlinear buckling behavior of the supporting structure of a cable-stayed circular frame is studied in detail. A three-dimensional finite element model of the supporting structure is developed to predict the critical cable tensions that would cause buckling of the circular frame and to determine the associated buckling mode shapes of the supporting structural system. The results show that the buckling load of a cable-stayed circular frame depends not only upon the cross-section of the frame, but also upon the number of cables and the inclinations of the cables. In all cases, in-plane buckling modes are predominant. The concentrated torques resulting from unbalanced cable tensions tend to induce the out-of-plane buckling modes.     

Buckling of imperfect functionally graded plates under in-plane compressive loading

Buckling behavior of rectangular functionally graded plates with geometrical imperfections is studied in this paper. The equilibrium, stability, and compatibility equations of an imperfect functionally graded plate are derived using the classical plate theory. It is assumed that the nonhomogeneous mechanical properties of the plate, graded through thickness, are described by a power function of the thickness variable. The plate is assumed to be under in-plane compressive loading. Simultaneous solving of the stability and compatibility equations in conjunction with the equilibrium equations leads to the buckling relation of the plate. The critical buckling load of a sample plate is obtained and compared for different geometrical ratios. The results are reduced and compared with the results of perfect functionally graded and imperfect isotropic plates.     

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 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 analysis of stepped plates using modified buckling mode shapes

A new approximate procedure for buckling analysis of simply supported rectangular stepped or perforated plates subjected to uniform edge stresses is formulated. The procedure uses energy method based on modified buckling mode shapes. The change of thickness within a plate is characterized by introducing a stepping index. It is shown that the buckling (vibrational) mode shapes of stepped plates can be predicted by linear combination of various mode shapes of the equivalent flat plates. These buckling mode shapes, in turn, are incorporated to evaluate buckling loads of stepped plates. Some case studies are carried out to demonstrate the accuracy and the versatility of the proposed method by comparing them to the results presented by other researchers.     

Parametric instability of edge cracked plates

The presented paper deals with the parametric instability behavior of a simply supported rectangular plate with a crack emanating from one edge, subjected to in-plane compressive periodic edge loading. The problem is reduced to computing the free vibration frequencies and the corresponding mode shapes and substituting them into an integral equation based formula, which leads to a compact matrix form. Once the components of this matrix are found, the rest of the computation, i.e., establishing regions of instability, buckling loads and modified frequencies, is straightforward and fast. Several plates, each with a different dimension and crack length size are analyzed using this approach. The comparison of results with those of finite element models is found to be in close agreement.     

Ultimate strength of dented steel plates under edge shear loads

The aims of this paper are to investigate the ultimate shear strength reduction characteristics of steel plates due to local impacts, and also to develop the ultimate shear strength design formulae of dented steel plates. The ANSYS nonlinear finite element code is used to investigate the effects of shape, size (depth, diameter), and location of the denting on the ultimate strength behavior of simply supported steel plates under edge shear loads. A closed-form expression for predicting the ultimate shear strength of dented steel plates is derived by the regression analysis based on the computed results. The results and insights developed from the present study will be very useful for damage tolerant design of steel plated structures with local denting.     

Buckling analysis of thin circular FG plates made of saturated porous-soft ferromagnetic materials in transverse magnetic field

This study presents the buckling analysis of soft ferromagnetic FG circular plates made of poro material. Equilibrium and stability equations of a poro circular plate in transverse magnetic field are derived. This study analyzes the poroelastic instability of clamped edge ferromagnetic plates subjected to magnetic loadings. The geometrical nonlinearities are considered in the Love–Kirchhoff hypothesis sense. In this paper the effect of pore pressure on critical magnetic field of plate and the effect of important parameters of poroelastic material on buckling capacity are investigated. Also the compressibility of fluid and porosity on the buckling strength are being investigated.     

Buckling of intermediate ring supported cylindrical shells under axial compression

This paper is concerned with the elastic buckling of axially compressed, circular cylindrical shells with intermediate ring supports. The simple Timoshenko thin shell theory and the more sophisticated Flügge thin shell theory have been adopted in the modeling of the cylindrical shells. We used these two representative theories to examine the sensitivity of the buckling solutions to the different degree of approximations made in shell theories. By dividing the shell into segments at the locations of the ring supports, the state-space technique is employed to derive the solutions for each shell segment and the domain decomposition method utilized to impose the equilibrium and compatibility conditions at the interfaces of the shell segments. First-known exact buckling factors are obtained for cylindrical shells of one and multiple intermediate ring supports and various combinations of boundary conditions. Comparison studies are carried out against benchmark solutions and independent numerical results from ANSYS and p-Ritz analyses. The influence of the locations of the ring supports on the buckling behaviour of the shells is examined.     

Buckling of cracked thin-plates under tension or compression

Plates are easily susceptible to buckling under compression, in particular when plate's thickness becomes sufficiently small with respect to others plate's sizes; such a mode of failure is often prevalent with respect to strength failure. The buckling phenomena under tension loading can also occur, especially in plates containing defects such as cracks or holes; when the buckling load is reached, complex wrinkling deflection patterns in compressed regions develops around such imperfections.In the present paper, the buckling analysis of variously cracked rectangular elastic thin-plates under tension and compression is considered. A short explanation of the buckling phenomena in plates is recalled and several numerical analyses, carried out by using the Finite Element Method (FEM), are performed in order to determine the critical load multiplier, both in compression and in tension, by varying some plates' parameters. In particular, the critical load multiplier is determined for different relative crack length, crack orientation and Poisson's coefficient of the plate's material which is made to range between 0.1 and 0.49.Moreover a simple approximate theoretical model to explain and predict the buckling phenomena in cracked plates under tension is proposed and some comparisons are made with FE numerical results in order to assess its reliability in predicting buckling load multipliers.Finally, the obtained results are graphically summarised (in dimensionless form) in several graphs and some interesting conclusions are drawn.     

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.     

Elastic stability of bi-axially loaded rectangular plates with a single circular hole

The finite-element method has been employed to determine the elastic buckling stresses of a bi-axially loaded perforated rectangular plate with dimensions a and b in the x- and y-directions, respectively. The considered perforation is a single circular hole whose center is located along the longitudinal axis of the plate. The considered plate has simply supported edges in the out-of-plane direction and is subjected to bi-axial uniformly distributed end loads (compressive load σ_x in the x-direction, and compressive or tensile load σ_y in the y-direction). Parameters considered in the study are the plate's aspect ratio a/b, the stress ratio ξ between the applied stresses in the y- and x-directions (ξ=σ_y/σ_x), the circular hole size d and location e_x.The study shows that, in most of the considered cases, the bigger the hole size d, the lesser the plate stability and the lesser the buckling stresses. It also shows that the plate aspect ratios, a/b, between 0.6 and 1.2 should be avoided for plates with large holes and negative ξ, due to the large reduction in the buckling stresses. The hole location should also be selected to be away from the loaded edge of the plate as much as possible (better to have e_x/b>1.0) to increase the buckling stresses and improve stability. The study demonstrates also that the increase in tension in the y-direction in bi-axially loaded plate with large hole (d/b>0.4) reduces its stability. This is in contrary to the expected increase in the stability due to the increase in tension which can be seen clearly in the cases of solid plates and plates with small holes.     

Elastoplastic buckling of circular annular plates under uniform in-plane loading

This paper deals with the elastoplastic buckling of a circular annular plate, with various axially symmetric boundary conditions and uniform axially symmetric in-plane radial loads on the inner and outer edge. The analysis is based on the standard linear buckling equations and the material behaviour is modelled by the small strain J₂ flow and deformation theories of plasticity where an elastic linear hardening rheological model of the material is considered. The solutions are obtained using the equilibrium approach where the governing differential equation is solved by the finite difference method which leads to the determination of eigenvalues of a homogeneous system of linear equations. Elastoplastic buckling loads for axially symmetric and asymmetric buckling shape modes with m waves in the circumferential direction are calculated and compared for both theories of plasticity. For one case, an experiment was performed and the results were compared with theoretical predictions.     

Ultimate strength of perforated steel plates under combined biaxial compression and edge shear loads

The present paper is a sequel to the author's papers [Paik JK, Ultimate strength of perforated steel plates under edge shear loading. Thin-Walled Structures 2007; 45: 301–6, Paik JK Ultimate strength of perforated steel plates under axial compressive loading along short edges. Ships Offshore Struct, 2007; 2(3): (in press)]. In contrast to the previous papers with the focus on edge shear or uniaxial compressive loads, the aim of the present study is to investigate the ultimate strength characteristics of perforated steel plates under combined biaxial compression and edge shear loads, which is a typical action pattern of steel plates arising from cargo weight and water pressure together with hull girder motions in ships and ship-shaped offshore structures. The plates are considered to be simply supported along all (four) edges, keeping them straight. The cutout is circular and located at the center of the plate. A series of ANSYS nonlinear finite element analyses (FEA) are undertaken with varying the plate dimension (thickness). Based on the FEA results obtained, closed-form empirical formulae of the ultimate strength interaction relationships of perforated plates between combined loads, which can be useful for first-cut estimations of the ultimate strength in reliability analyses or code calibrations, are derived.     

Vibrations of rectangular plates with elastic intermediate line-supports and edge constraints

A set of static beam functions, which are the solutions of an elastically point-supported beam under a Fourier series of static sinusoidal loads distributed along the length of the beam, are developed as the admissible functions to analyze the vibrations of orthotropic rectangular plates with elastic intermediate line-supports using the Rayleigh–Ritz method. Both the elastic rotational and the elastic translational constraints along the edges of the plate are also considered simultaneously. Unlike conventional admissible functions, this set of static beam functions not only can automatically adjust to the stiffnesses of the intermediate line-supports but also can properly describe the discontinuity of shear forces at the line-supports so that higher accuracy and faster convergence can be expected for the dynamic analysis of such plates. The suggested approach is effective even for various limiting cases by letting the corresponding stiffnesses approach their natural limits of zero or infinity. The present method is theoretically sound and mathematically simple, with each of the static beam functions being only a third-order polynomial plus a sine function. A common and efficient computational program can be compiled because of the fact that a change of the line-support parameters (locations, number and stiffnesses) and the boundary conditions of the plate only results in a corresponding change of the coefficients of the polynomial in the static beam functions. Several numerical examples are presented and the results obtained, where possible, are compared with the known solutions in literature. The present method has proved to be extremely effective for solving the aforementioned problems.     

A generalized analytical approach to the buckling of simply-supported rectangular plates under arbitrary loads

An analytical approach for the elastic stability of simply-supported rectangular plates under arbitrary external loads is presented which, for the first time, may be described as ‘exact’. This is achieved through the use of exact solutions for the in-plane stresses and the adoption of double Fourier series for the buckled profiles which, together, ensure that accurate results are obtained in the Ritz energy technique. Several cases of plate buckling under direct, shear and bending loads (or their combinations) are studied to show the generality of the proposed approach, with the ensuing results compared with existing data (if available) and with numerical FE results.