A theoretical and experimental study of hemispherical shells subjected to axial loads between flat plates

The quasi-static loading of an open hemispherical shell along its axis of symmetry through a rigid flat plate is considered with particular reference to large deformations and buckling.Deformation patterns which are symmetric and non-symmetric about the axis are studied analytically and experimentally for models of different materials and radius to thickness ratios varying between 36 and 460.     

Finite-element simulation of the lateral compression of aluminium tube between rigid plates

The lateral compression of aluminium and clad tubes owing to a large deformation is examined by an incremental elasto-plastic finite-element method based on an updated Lagrangian formulation in which a sliding-sticking friction mode is specially considered. It is mainly expected to predict the buckling process and load–deflection curves for energy dissipation capacity during the design stage, before trials. The high non-linearity of the process due to geometric changes, the inelastic constitutive behavior, and the deformation-dependent boundary conditions are taken into account in an incremental manner. A static explicit approach to the solution is applied, tangent stiffness matrix equation is solved without iteration and the r_min technique is employed to limit the step size to linear relation. The simulated load–deflection curve agrees with a published experimental result. The predicted geometries of the compressed tube clearly demonstrate the processes of the formation of buckling until unloading. The effects of various parameters of the process, such as elastic modulus, strain hardening exponent, tube thickness, friction coefficient and configurations of the clad tube, on the occurrence of buckling of tube are discussed and interpreted in simulation. The present work may be expected to improve the understanding of the buckling mechanism of lateral compression.     

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.     

Buckling of moderately thick rectangular composite plates subjected to partial edge compression

The objective of the present paper is to investigate the influence of partial edge compression on the critical loads of moderately thick laminated plates. Towards this, an eight node isoparametric plate element is developed. The element has five degrees of freedom per node. The computer code developed accepts two sets of boundary conditions, one for pre-buckling stress analysis and the second for stability analysis. This flexibility is proposed to exploit the mid-line symmetry conditions. Two types of partial edge compression, viz., (I) uniform partial edge compression near the corners and (II) uniform partial edge compression at the middle of edges are considered. The effect of percentage of loaded edge length on the critical load of thin and thick composite plates with simply supported and clamped edge conditions is studied in detail.     

Nonlinear bending response of functionally graded plates subjected to transverse loads and in thermal environments

Nonlinear bending analysis is presented for a simply supported, functionally graded rectangular plate subjected to a transverse uniform or sinusoidal load and in thermal environments. Material properties are assumed to be temperature-dependent, and graded in the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents. The governing equations of a functionally graded plate are based on Reddy's higher-order shear deformation plate theory that includes thermal effects. Two cases of the in-plane boundary conditions are considered. A mixed Galerkin-perturbation technique is employed to determine the load-deflection and load-bending moment curves. The numerical illustrations concern nonlinear bending response of functional graded rectangular plates with two constituent materials. The influences played by temperature rise, the character of in-plane boundary conditions, transverse shear deformation, plate aspect ratio and volume fraction distributions are studied.     

An analysis of the free vibration of thick-walled isotropic toroidal shells

The equations of motion and strain–displacement equations for toroidal shells are derived using toroidal coordinates and dyadic methods of elasticity. Suitable assumptions are proposed that reduce the governing equations to axisymmetric format while maintaining the three-dimensional character of the solution. The free vibration analysis of toroidal shells is carried out using numerical solutions of the governing equations. A nine-node Lagrangian finite element is formulated in the toroidal coordinate system and solutions are obtained for the case where an axis of symmetry can be assumed at the center of the torus. Results for frequency of vibration are tabulated for solid and thick-walled shells. Mode shapes are shown for a representative thick-walled torus.     

Forced axisymmetric response of linearly tapered circular plates

Forced axisymmetric response of a circular plate of linearly varying thickness, based on the classical theory, is analyzed by the eigen-function method. An exact solution for the free vibration mode shapes is obtained by the Frobenius method. Clamped and simply-supported plates subjected to symmetric uniformly distributed and concentrated impulsive ring and point loads are solved as example problems. Numerical results computed for transverse deflection and radial stress are plotted in the figures.     

Limit loading of cylindrical shells subjected to local circumferential bending moments

Experiments have been carried out on twelve mild steel cylindrical shells each of which had a rigid radial square bar attached at its mid length. Strain and deformation patterns in the shell were measured when a hoop bending moment was applied to it through the attachment. Though some elastic stress distributions were obtained, the observations were mainly concerned with plastic deformations, the determination of plastic limit moments and the onset of instability. Measured deformation patterns were used to suggest a collapse mechanism from which a theoretical upper bound for the plastic limit moment was determined. Theoretical and experimental limit moments were compared for all specimens which had ratios varying between 13·75 and 67·5, and attachment half width/cylinder radius ratios varying between and .     

Plane-strain compression of material obeying the double-shearing model between rotating plates

Nonsteady planar flow of a solid obeying the double-shearing model between two rough, inclined plates rotating about the same motionless line is considered. A semi-analytical solution to the problem is obtained. Attention is devoted to the singular behaviour of a velocity field in the vicinity of the friction surface. It is demonstrated that the velocity component tangent to the friction surface is describable by nondifferentiable functions and that the equivalent strain rate approaches infinity on the friction surface. Such a behaviour of the velocity field is compared with other solutions based on the double-shearing model and with a general result on the singular behaviour of velocity fields in classical plasticity of pressure-independent materials. The concept of strain rate intensity factor introduced in classical plasticity is extended to the double-shearing model of pressure-dependent plasticity. Some velocity distributions and strain intensity factor variations are calculated.     

Free-edge effect in cross-ply laminated hollow cylinders subjected to axisymmetric transverse loads

On the basis of the theory of three-dimensional elasticity, free-edge stresses and displacements of axisymmetrically loaded laminated hollow cylinders are found by using the state space equation method. The study assumes three-dimensional variations of both stresses and displacements and takes into account material constants of all individual material layers. By this approach, composite cylinders may be composed of an arbitrary number of orthotropic layers, each of which may have different material properties and thicknesses. The solution provides a continuous inter-laminar stress field across interfaces and a satisfactory approximation to the singularity occurring in the boundary layer region of a free-edge. Numerical solutions are compared with the results obtained from an alternative numerical method. New results are obtained and parametric studies are carried out.     

Lower-bound shakedown analysis of axisymmetric structures subjected to variable mechanical and thermal loads

This paper is concerned with axisymmetric structures subjected to various combinations of steady and (irregularly) pulsating mechanical and thermal loads. The numerical method is based on the static shakedown theorem, which deals with elastic-perfectly-plastic bodies. Linear temperature dependence of the yield condition in a 4D stress space is considered. The pseudo-residual stress field is simulated by the Temperature Parameter Method and the yield surface is linearized, so that the shakedown analysis is transformed into a linear programming problem and hence the computational difficulties otherwise encountered are overcome. The shakedown analysis of cylinders and spherical shells with and without defects is presented in this paper. Results presented show that the method is beneficial for the shakedown analysis of complicated structures subjected to various combinations of loads.     

Limit loads for cylindrical shells subjected to local longitudinal bending moments

Tests have been carried out on mild steel cylindrical shells, each with a radial square bar attachment welded at its mid length. Loads were applied to each attachment so that the shell was subjected to a longitudinal bending moment. On four specimens, elastic strains were measured so that stress distributions in the shell could be compared with theoretical distributions. Fourteen specimens were loaded so that gross deformation took place. Deformations and strains were measured and on two specimens, crack patterns on an oxide layer were observed. With increasing load, the plastic deformation of specimens developed gradually and not in the sudden manner associated with theoretical rigid-plastic collapse. Experimental plastic limit moments are compared with theoretical values calculated by an upper bound analysis. The range of radius/thickness values investigated was 18·3–53·5, and the half width/radius ratios were between 0·120 and 0·356.     

The elastic-plastic wrinkling of rectangular plates subjected to locally uniform tension

After overcoming the difficulty with discontinuity in boundary conditions, the pre-buckling elastic stress field for a rectangular plate subjected to locally uniform tension is given. An iteration procedure for solving an elastic-plastic plane stress problem is developed, so as to obtain the pre-buckling elastic-plastic stress field for the same plates. The effects of the aspect ratio of plates, tensile non-uniformity and material properties upon the stress field are also discussed. By assuming the shape of wrinkling region and the wrinkling mode in it, the dependences of critical wrinkling load and the size of wrinkling region upon the material and geometrical parameters are revealed. It is found that wrinkling is dominant by a combined parameter in elastic case or in plastic case.     

Buckling and vibration of stiffened plates subjected to partial edge loading

The buckling and vibration characteristics of stiffened plates subjected to in-plane partial and concentrated edge loadings are studied using finite element method. The initial stresses are obtained considering the pre-buckling conditions. Buckling loads and vibration frequencies are determined for different plate aspect ratios, edge conditions and different partial non-uniform edge loading cases. The non-uniform loading may also be caused due to the supports on the edges. The analysis presented determines the stresses all over the region for different kinds of loading and edge conditions. In the structural modelling, the plate and the stiffeners are treated as separate elements where the compatibility between these two types of elements is maintained. The vibration characteristics are discussed and the results are compared with those available in the literature. Buckling results show that the stiffened plate is less susceptible to buckling for position of loading near the supported edges and near the position of stiffeners as well.     

Thermal buckling of functionally graded rectangular plates subjected to partial heating

We present the thermal buckling analysis of functionally graded rectangular plates subjected to partial heating in a plane and uniform temperature rise through its thickness. The plate is simply supported for out-of-plane deformation and perfectly clamped for in-plane deformation. It is assumed that the functionally graded material properties such as the coefficient of linear thermal expansion and Young's modulus are changed individually in the thickness direction of the plate with the power law, while Poisson's ratio is assumed to be constant. Analytical developments consist of two stages. First, the nonuniform in-plane resultant forces are determined by solving a plane thermoelastic problem. Then the critical buckling temperatures of the plates with the predetermined resultant forces are calculated as the generalized eigenvalue problem which is constructed by using the Galerkin method. Finally, the effects of material inhomogeneity, aspect ratio, and heated region on the critical buckling temperatures are examined.     

Limit analysis of circular plates subjected to arbitrary rotational symmetric loadings

The lower bound technique of limit analysis has been implemented in order to calculate the critical load factor for loadings on circular plates. The obtained results are applicable to any rotational symmetric loading on clamped or simply supported plates. Both the square and Tresca yield criteria have been considered in the analysis. It has been shown that the calculated lower bound for the critical load factor is the exact answer for the considered problem. In order to perform the calculations, the loading function is expanded in the form of a power series and the critical load factor calculated by use of a numerical technique. Finally, variation of this factor with the loading function has been illustrated.     

Geometrically nonlinear axisymmetric vibrations of polar orthotropic circular plates

The geometrically nonlinear free vibrations of polar orthotropic circular plates are analyzed by the axisymmetric finite element. The formulation is based on the three-dimensional elasticity with all the nonlinear terms in the strains included and is considered to be more general and precise theoretically compared to the plate-theory-based nonlinear approaches. Numerical results with various geometries, orthotropies, amplitudes and boundary conditions by the present formulation are also shown and compared to those by other approaches.     

Stability of laminated composite plates subjected to various types of in-plane loadings

The aim of the present investigation is to examine the stability characteristics of laminated plates subjected to various types of in-plane loadings. Towards this, a rectangular four node finite element, having fourteen degrees of freedom per node, based on a simple higher-order shear deformation theory is developed. The theory employed herein involves four dependent variables. The element is found to be free of shear lock problems. A series of numerical problems are solved to study the effect of various parameters such as lay-up, side to thickness ratio, aspect ratio, type of loadings (uniaxial/biaxial/positive and negative shear/tension—compression/compression—tension) and boundary conditions. Some interesting observations regarding the considerable difference in the buckling resistance of angle-ply plates when subjected to positive and negative shear loading are made.     

Modes of deformation in aluminium rings subjected to static axial compression

Aluminium rings of varying height/outside diameter/inside diameter ratios machined from extruded solid bars, were subjected to static axial compression. Etched diametral planes of deformed cylinders revealed the existence of shear bands, the configuration of which were found to change with initial specimen geometry and deformation.