Bending analysis of thin annular sector plates using extended Kantorovich method

Highly accurate approximate closed-form solution is presented for bending of thin sector plates with clamped edges subjected to uniform and non-uniform loading using the extended Kantorovich method (EKM). Successive application of EKM converts the governing equation which is a forth-order partial differential equation (PDE) to two separate ordinary differential equations (ODE) in terms of r and θ. The obtained ODE systems are then solved iteratively with very fast convergence. In every iteration, exact closed-form solutions are obtained for both ODE systems. It is shown that the method provides sufficiently accurate results not only for deflections but also for stress components. Comparison of the deflection and stresses at various points of the plate shows very good agreement with results of other analytical and numerical analyses.     

A semi-analytical analysis of the elastic buckling of cracked thin plates under axial compression using actual non-uniform stress distribution

An improved hybrid semi-analytical method for calculating elastic buckling load of a thin plate with a central straight through-thickness crack subject to axial compression is proposed. In the study, the actual non-uniform in-plane stress distribution is firstly conducted by using Muskhelishvili's complex variable formulation in conjunction with boundary collocation method. A deflection shape function, satisfying not only the outer boundary conditions but also the inner boundary conditions of the crack edges, is obtained by using domain decomposition method. Finally the buckling load of a cracked plate using Raleigh–Ritz energy method is calculated based on the actual in-plane stress distribution and the reasonable deflection shape function obtained. The effects of crack length, plate's aspect ratio are studied for thin plates with different boundary conditions. Results obtained from the proposed method are in good agreement with the existing numerical results and experimental ones. It is finally shown that the proposed method, based on a correct non-uniform in-plane stress distribution, is more accurate than the few existing analytical methods based on a uniform in-plane stress distribution.     

Approximate large-deflection analysis of simply supported rectangular plates under transverse loading using plate post-buckling solutions

This paper describes an approximate analytical method to determine the large-deflection behaviour of rectangular simply supported thin plates under transverse loading. The method is based on a simple trial function to describe the shape of the initial and total deflections of the plate, corresponding to the first buckling mode shape of the plate when subjected to uniform in-plane compression. Therefore, the large-deflection behaviour of the plate under transverse loading can be expressed as a function of the ratio of the pre- to post-buckling in-plane stiffness of the plate, which is given in literature for plates with various membrane boundary conditions of the plate edges. Due to the simplicity of the trial function flexural and membrane actions of the plate are decoupled. It was found that the approximate method gives good insight into the large-deflection behaviour. The results of the approximate method have been compared to the results of finite element solutions and solutions available in literature. For uniformly distributed loads the approximate method gives deflections which differ less than about 10% from the finite element simulation values.     

Exact stress functions implementation in stability analysis of plates with different boundary conditions under uniaxial and biaxial compression

Analytical approach used for critical load determination is based on Ritz energy technique in which two factors are crucial for the accuracy of results. First factor is deflection function. Herein, double Fourier series are used to represent buckled shape of the plates under arbitrary external loads. The second and the most important factor is adoption of adequate realistic stress distribution within plate prior to buckling. Based on the Baker&Pavlović&Tahan and Pavlović&Liu investigations, exact stress distributions were introduced herein. In that way, with the adequate deflection functions and precise stress distribution, Ritz energy method can produce highly accurate results.Through several examples for the plates with different boundary conditions, for which available literature offers very few analytical solutions, accuracy of the presented analytical approach is proved. Results obtained by analytical approach are reaffirmed by numerical finite-element runs.     

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.     

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.     

Differential quadrature buckling analyses of rectangular plates subjected to non-uniform distributed in-plane loadings

The new version of differential quadrature method (DQM), proposed by the senior author recently, is used to obtain buckling loads of thin rectangular plates under non-uniform distributed in-plane loadings for the first time. Two steps are involved: (1) solve a problem in plane-stress elasticity to obtain the in-plane stress distributions and (2) solve the buckling problem under the loads obtained in step (1). The methodology and procedures are worked out in detail and buckling problems with loadings of non-uniform distributions are studied. Numerical studies are performed and the DQ results are compared well with analytical solutions and finite element results. This fact indicates that the DQ method can be employed for obtaining buckling loads of plates with other combinations of boundary conditions subjected to non-uniform distributed loadings.     

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.     

Numerical study on the permissible gap of intermittent fillet welds of longitudinally stiffened plates under in plane axial compression

Intermittent fillet welding of stiffeners to plates and its influence on the collapse behaviour of stiffened plates is investigated applying the finite element method. Special attention is paid to the modelling of the fillet welds at the plate-to-stiffener junction. Some available experimental results are simulated in order to obtain reliable numerical results. A series of numerical analyses of stiffened steel plates subjected to an in plane axial compressive load has been performed. Stiffened plates are selected from the deck structure of real sea-going ships and inland waterway vessels. Complete equilibrium paths are traced up to collapse for the non-linear elastoplastic response of stiffened plates. Finally a proposal is presented for the permissible gap of welds in intermittent fillet welding of stiffened plates.     

Semi-analytical models for the post-buckling analysis and ultimate strength prediction of isotropic and orthotropic plates under uniaxial compression with the unloaded edges free from stresses

This paper introduces two semi-analytical models developed for the nonlinear analysis of stability of isotropic and orthotropic plates under uniaxial compression. The possibility of considering fully free in-plane displacements at longitudinal edges (or unloaded edges) is the innovation of these models over existing models, where these displacements are always assumed constrained to remain straight. Contributions for the large deflection theory of plates related to the derivation of analytical solutions for the Airy stress function which satisfy Marguerre׳s equations for isotropic and orthotropic plates are presented. Namely, the extension of the Coan and Urbana solution for isotropic plates in order to consider all the terms of the unknown amplitudes of the out-of-plane displacements and the derivation of a solution for orthotropic plates. Comparisons between the semi-analytical model and nonlinear finite element model results are presented in order to discuss the effect of in-plane displacement boundary conditions on behaviour and strength of plates similar to bottom flanges used in steel box girder bridges. This study shows that the semi-analytical models have a clear potential to provide accurate solutions, requiring only a short computer time. It is also shown that the in-plane displacement boundary conditions for the unloaded edges significantly influence the behaviour and strength of plates and this problem cannot be neglected in the definition of the design rules.