Nonlinear axisymmetric static analysis of orthotropic thin circular plates with elastically restrained edge

This study deals with the nonlinear axisymmetric static analysis of elastic orthotropic thin circular plates with elastically restrained edges for rotation as well as in-plane displacement. Von-Karman equations have been employed and spatial discretisation has been done by using the method of orthogonal point collocation. Numerical results have been presented for orthotropic plates with hinged edge, simply supported edge, movable clamped edge and immovable clamped edge. Results are given for the whole range of elastic edge restraints from the movable simply supported edge to the immovable clamped edge. The effect of a prescribed inplane force or an inplane displacement at the edge, on the static response under a uniformly distributed transverse load has been investigated. Several new results are presented. The present results are in good agreement with the available results.     

Nonlinear flexural oscillations of orthotropic shallow spherical shells

The effect of curvature and polar orthotropy on nonlinear dynamic behaviour of a shallow spherical shell is investigated in the present paper. Numerical solutions based on an assumed two-term modeshape for the axisymmetric, forced (uniform pressure) and free vibrations are obtained for different shell geometries and orthotropic material constants. The results, when specialised for the case of isotropic material, are in good agreement with those available in the literature. Based on a one-term modeshape solution, the values of the geometric parameter at which the transition from hardening to softening type of nonlinearity takes place and where the reversal of the softening trend occurs are obtained for different values of the orthotropic constants.     

Dynamic buckling of orthotropic shallow spherical shells

The dynamic axisymmetric behaviour of clamped orthotropic shallow spherical shell subjected to instantaneously applied uniform step-pressure load of infinite duration, is investigated here. The available modal equations, based on an assumed two-term mode shape for the lateral displacement, for the free flexural vibrations of an orthotropic shallow spherical shell is extended now for the forced oscillations. The resulting modal equations, two in number, are numerically integrated using Runge-Kutta method, and hence the load-deflection curves are plotted. The pressure corresponding to a sudden jump in the maximum deflection (at the apex) is considered as the dynamic buckling pressure, and these values are found for various values of geometric parameters and one value of orthotropic parameter. The numerical results are also determined for the isotropic case and they agree very well with the previous available results. It is observed here that the dynamic buckling load increases with the increase in the orthotropic parameter value. The effect of damping on the dynamic buckling load is also studied and this effect is found to increase the dynamic buckling load. It is further observed that this effect is more pronounced with increase in the rise of the shell.     

Axisymmetric static and dynamic buckling of orthotropic shallow spherical cap with circular hole

Nonlinear axisymmetric static and dynamic buckling of clamped isotropic and cylindrically orthotropic elastic cap with central circular hole have been investigated. The governing equations are expressed in terms of normal displacement ω and stress function ψ. The orthogonal point collocation method has been used in the space domain and Newmark-β scheme in the time domain. The cases of shallow cap with free hole and with a hole plugged by rigid central mass are considered. Analysis has been carried out for uniformly distributed load and a ring load at the hole. Dynamic load is taken as a step function load. Detailed new results for static and dynamic buckling loads have been presented for the isotropic and orthotropic cases.     

Nonlinear axisymmetric static and transient analysis of orthotropic thin tapered circular plates

This study deals with the geometrically nonlinear axisymmetric static and transient analysis of cylindrically orthotropic elastic thin tapered circular plates subjected to uniformly distributed and discrete central loads. Differential equations in terms of transverse displacement w and stress function ψ have been employed. The displacement w and stress function ψ are expanded in finite power series. The orthogonal point collocation method in space domain and Newmark-β scheme in time domain have been used. Step function dynamic loads are considered. Static and dynamic results have been presented for isotropic and orthotropic immovable clamped and simply supported plates with linearly varying thickness for three values of taper ratios and the effect of varying thickness has been investigated. A simple approximate method is used to predict the maximum dynamic response to step load from the results for static loads and is found to yield sufficiently accurate results.     

Improved finite-element analysis of nuclear-containment shell structures

The current utilization of prestressed-concrete shells for nuclear containment purposes requires precise and detailed analytical treatment. This paper will present recent developments and applications of finite-element techniques to design problems with particular emphasis on axisymmetric shells. Using classical laminated-shell theory, the concept of a subelement is introduced as a method for modelling posttensioned, prestressed shells within the framework of lower-dimensional analysis. Examples are used to substantiate procedures. A new numerical approach is described for the generation of axisymmetric shell elements having orthotropic material properties and meridional curvature with all properties variable in the meridional coordinate. Application to design-type problems will be illustrated; development efforts have been closely related to actual design problems.     

Nonlinear flexural vibrations of laminated orthotropic plates

Large amplitude free flexural vibrations of laminated orthotropic plates are studied using C⁰ shear flexible QUAD-8 plate element. The nonlinear governing equations are solved using the direct iteration technique. Numerical results are obtained for isotropic, orthotropic and cross-ply laminated plates with simply-supported boundary conditions on immovable edges. It is observed that hardening behaviour is increased for thick plates and orthotropic plates.     

Axisymmetric static and dynamic buckling of orthotropic truncated shallow conical caps

This investigation deals with the axisymmetric static and dynamic buckling of a cylindricaliy orthotropic truncated shallow conical cap with clamped edge. The cases of conical caps with a free central circular hole and with a hole plugged by a rigid central mass have been considered. The governing equations are formulated in terms of normal displacement w and stress function Ψ. The orthogonal point collection method is used for spatial discretisation and the Newmark-β scheme is used for time-marching. Analysis has been carried out for a uniformly distributed conservative load normal to the undeformed surface and a central axial ring load at the hole. Dynamic load is taken as a step function load. The influence of orthotropic parameter β and annular ratio on the buckling loads has been investigated. New results for static and dynamic buckling loads have been presented for the isotropic and orthotropic truncated conical caps. Dynamic buckling loads obtained from static analysis have been found to agree well with the dynamic buckling loads based on transient response.     

A nonlinear,semi-analytical finite element analysis for nearly axisymmetric solids

The presented research is concerned with the development of the theory and accompanying computer program for a semi-analytical finite element analysis of non axisymmetrically loaded, nearly axisymmetric solids. The theoretical basis of the method together with numerical procedures for handling boundary conditions, rigid body motion and the iterative solution process are described. A finite element program for evaluating the analysis is discussed, evaluated for effectiveness and applied to several examples.The range of application of the analysis for inhomogeneous, orthotropic, nonlinear and nearly axisymmetric bodies is demonstrated by a series of examples. The substantial savings in computer time and memory as compared to a conventional finite element analysis is discussed.     

Free vibration analysis and optimisation of axisymmetric plates and shells—I. Finite element formulation

This paper deals with the free vibration analysis of shells of revolution using the finite element method. A family of variable thickness, curved C(0) Mindlin-Reissner axisymmetric finite elements is presented which include shear deformation and rotatory inertia effects. The accuracy, convergence and efficiency of these newly developed elements are explored through a series of free vibration analyses of axisymmetric shell structures and the results are compared with those obtained by other analytical and numerical methods. The comparisons show that the method yields very good results with a relatively small number of elements and that estimates for the higher modes can be obtained without any difficulties. A companion paper will consider the structural optimisation of axisymmetric shells undergoing free vibrations.     

Optimal orientation of orthotropic materials for plates designed against buckling

The paper deals with the optimization of anisotropic plates loaded by in-plane forces and designed against buckling. The internal structure of the plate is seen as a twodimensional locally orthotropic solid and the orientations of the axes of orthotropy are taken as the design variables. The problem of optimization consists in determining the best orientation of the axes of orthotropy from the condition of the maximum behaviour of the critical buckling loads. General properties of the problem, such as multiplicity of the critical parameters and splitting of multiple eigenvalues, are studied. Optimization algorithms are developed to improve the anisotropic properties of the plate. The results of numerically finding the optimal orientation of orthotropic properties are compared with conventional layouts for square and rectangular plates loaded by normal in-plane forces.     

Nonlinear behavior of shells of revolution under cyclic loading

A large deflection elastic-plastic analysis is presented applicable to orthotropic axisymmetric plates and shells of revolution subjected to monotonie and cyclic loading conditions. The analysis is based on the finite-element method. It employs a new higher order, fully compatible, doubly curved orthotropic shell-of-revolution element using cubic Hermitian expansions for both meridional and normal displacements. Both perfectly plastic and strain hardening behavior are considered. Strain hardening is incorporated through use of the Prager-Ziegler kinematic hardening theory, which predicts an ideal Bauschinger effect. Numerous sample problems involving monotonie and cyclic loading conditions are analyzed. The monotonie results are compared with other theoretical solutions. Experimental verification of the accuracy of the analysis is also provided by comparison with results obtained from a series of tests for centrally monotonically-loaded circular plates that are simply supported at their edges.     

Dynamic finite element analysis of nonaxisymmetric structures

A dynamic finite element method of analysis is developed for structural configurations which are derived from axisymmetric geometries but contain definite nonaxisymmetric features in the circumferential direction. The purpose of the analysis is to develop a method which will take into consideration the fact that the stress and strain conditions in these geometries will be related to the corresponding axisymmetrie solution. This analysis is an extension of previously published work in which a similar approach was developed for static structural problems. The analysis is developed in terms of a cylindrical coordinate system r, θ and z. As the first step of the analysis, the geometry is divided into several segments in the r-θ plane. Each segment is then divided into a set of quadrilateral elements in the r-z plane. The axisymmetric displacements are obtained for each segment by solving a related axisymmetric configuration. A perturbation analysis is then performed to match the solutions at certain points between the segments, and obtain the perturbation displacements for the total structure. The total displacement is then the axisymmetric displacement plus the perturbation displacement. The analysis allows for elastic-plastic materials with orthotropic yield criterion based on Hill's yield function and kinematic strain hardening. The finite element dynamic equations are solved by finite differences by dividing the time domain into incremental steps. The solution has been programmed on a computer and applied to a number of examples.     

A geometrically nonlinear shell finite element for tire vibration analysis

An axisymmetric finite element is developed which includes such features as orthotropic material properties, doubly curved geometry, and both the first and second order nonlinear stiffness terms. This element can be used to predict the equilibrium state of an axisymmetric shell structure with geometrically nonlinear large displacements. Small amplitude vibration analysis can then be performed based on this equilibrium state. The nonlinear path is predicted by using the self-correcting incremental procedure and any point on the path can be checked by using the Newton-Raphson iterative scheme. The present formulation and solution procedure are evaluated by analyzing a series of examples with results compared with alternative known solutions. Examples include: free vibration of an isotropic cylindrical shell, a conical frustum, and an orthotropic cylindrical shell; buckling of a cylindrical shell; large deflection of a clamped disk, a spherical cap, and a steel belted radial tire. The final example is a free vibration analysis of the inflated tire and the natural frequencies obtained compared well with published experimental data.     

Stability analysis of skew orthotropic plates by the finite strip method

A stability analysis based on the Finite Strip Method is presented for skew orthotropic plates subjected to in-plane loadings. The straight sides of the plate are simply supported and the other two skewed sides are supported with any combination of fixed, free and simply supported boundaries. The plate is divided into strips, in contradistinction to elements in the Finite Element Method, and the displacement function is so chosen that it satisfies the boundary conditions and also the inter-strip compatibility conditions of an elemental strip. The energy expressions required to formulate the stiffness and stability coefficient matrices are formulated using smalldeflection theory. The buckling load intensity factor is evaluated for different aspect ratios of isotropic and orthotropic skew plates and the results of certain rectangular isotropic cases are compared with earlier investigations.     

The accuracy of Donnell''s equations for axially-loaded, imperfect orthotropic cylinders

The accuracy of Donnell's equations for the buckling analysis of imperfect (limit point instability), circular, cylindrical, thin orthotropic shells under axial compression is investigated. This is accomplished by comparing critical loads obtained by employing Donnell-type kinematic equations with those based on the more accurate Sanders-type. For this purpose, a solution methodology is developed and described in the body of the paper. This methodology is then employed to generate critical loads for several orthotropic geometries, which cover a wide but practical range of parameters. These include cylinder length to radius ratios, radius to thickness ratios, two positions of the strong direction relative to the cylinder axis (θ=0 and 90°) and two shapes of the initial geometric imperfection, axisymmetric and symmetric. Classical simply supported boundaries are used for all configurations for which results are generated.     

Inplane vibration analysis of polar orthotropic annular plates with linearly varying thickness

The natural frequencies of inplane vibrations of polar orthotropic annular plates with linearly varying thickness have been analysed. A semi-analytical method of analysis has been used where the radial and tangential displacements are expanded in the circumferential direction as Fourier series and the radial behaviour is solved using the finite element method. For the sake of comparison, results have been obtained starting with two different kinds of displacement functions. The frequencies have been studied with respect to various boundary conditions. aspect ratios, thickness ratios, ratios of moduli, and two fibre directions.     

Shear vibration and buckling of double-layer orthotropic nanoplates based on RPT resting on elastic foundations by DQM including surface effects

Microsystem Technologies - In this article, shear vibration and buckling of double-layer orthotropic nanoplates resting on elastic foundations are analyzed subjected to in-plane loadings including...     

An optimality criterion for shape optimization in eigenfrequency problems

For a broad class of static problems an optimality criterion of constant energy density at the designed boundary is known. In the present paper we prove a similar criterion for eigenfrequency problems. This optimality criterion serves as the tool for more basic understanding and for idealized reference cases as well as the basis for recursive procedures. Eigenfrequencies for in-plane vibrations as well as for out-of-plane vibrations of plates are optimized. The focus is on simplicity and multiple eigenfrequencies are not considered.