Finite element analysis of laminated shells of revolution

A finite element formulation for the analysis of axisymmetric fibre reinforced laminated shells subjected to axisymmetric load is presented. The formulation includes arbitrary number of bonded layers each of which may have different thicknesses, orientation of elastic axes, and elastic properties. Superparamatric curved elements[17] having four degrees of freedom per node including the normal rotation, are used. Stress-strain relation for an arbitrary layer is obtained from the consideration of three dimensional aspect of the problem. The element stiffness matrix has been obtained by using Gauss quadrature numerical integration, even though the elasticity matrix is different for different layers. The formulation is checked for a cylindrical tube subjected to internal pressure and axial tension, and the results are found to compare very well with the elastic solution [9].     

Stability analysis of ring stiffened shells of revolution

A non-linear, large deformation, finite element analysis is presented for the general instability of ring stiffened shells of revolution subjected to combined end compression and circumferential pressure loading. The circumferential pressure loading is treated as a follower force. A combined load-increment and iteration method is used for solving the resulting equilibrium equations. Effect of varying degrees of initial imperfection in shell geometry on the load-displacement behaviour is reported     

Stability analysis of ring stiffened shells of revolution

A finite element formulation is presented for the general instability of ring stiffened shells of revolution subjected to external pressure. Linear bifurcation buckling theory is used. A rigorous derivation for the potential due to the hydrostatic loading including follower force effect is presented. Comparison with results obtained by earlier research workers in this field is given. Substantial reduction in buckling pressures due to follower force effect is reported.     

Vibration analysis of laminated anisotropic shells of revolution

An efficient computational procedure is presented for the free vibration analysis of laminated anisotropic shells of revolution, and for assessing the sensitivity of their response to anisotropic (nonorthotropic) material coefficients. The analytical formulation is based on a form of the Sanders-Budiansky shell theory including the effects of both the transverse shear deformation and the laminated anisotropic material response. The fundamental unknowns consist of the eight stress resultants, the eight strain components, and the five generalized displacements of the shell. Each of the shell variables is expressed in terms of trigonometric functions in the circumferential coordinate and a three-field mixed finite element model is used for the discretization in the meridional direction.The three key elements of the procedure are: (a) use of three-field mixed finite element models in the meridional direction with discontinuous stress resultants and strain components at the element interfaces, thereby allowing the elimination of the stress resultants and strain components on the element level; (b) operator splitting, or decomposition of the material stiffness matrix of the shell into the sum of an orthotropic and nonorthotropic (anisotropic) parts, thereby uncoupling the governing finite element equations corresponding to the symmetric and antisymmetric vibrations for each Fourier harmonic; and (c) application of a reduction method through the successive use of the finite element method and the classical Bubnov-Galerkin technique.The potential of the proposed procedure is discussed and numerical results are presented to demonstrate its effectiveness.     

Non-linear analysis of shells of revolution under arbitrary loads

A new finite element formulation is presented for the non-linear analysis of elastic doubly curved segmented and branched shells of revolution subject to arbitrary loads. The circumferential variations of all quantities are described by truncated Fourier series with an appropriate number of harmonic terms. A coupled harmonics approach is employed, in which coupling between different harmonics is dealt with directly rather than by the use of pseudo-loads. Key issues in the formulation, such as non-linear coupling and growth of harmonic modes, are carefully and systematically explained. This coupled harmonics approach allows an easy implementation of the arc-length method. As a result, post-buckling load–deflection paths can be traced efficiently and accurately. The formulation also employs a non-linear shell theory more complete than existing classical theories. The results from the present study are independently verified using ABAQUS, while those from other studies are found to be inaccurate in general.     

Sensitivity analysis for the dynamic response of viscoplastic shells of revolution

A computational procedure is presented for evaluating the sensitivity coefficients of the dynamic axisymmetric response of viscoplastic shells of revolution. The analytical formulation is based on Reissner's large deformation shell theory with the effects of transverse shear deformation, rotatory inertia and moments turning around the normal to the middle surface included. The material model is chosen to be isothermal viscoplasticity, and an associated flow rule is used with a von Mises effective stress. A mixed formulation is used with the fundamental unknowns consisting of six stress resultants, three generalized displacements and three velocity components. Spatial discretization is performed using finite elements, with discontinuous stress resultants across element interfaces. The temporal integration is performed by using an explicit central difference scheme (leap-frog method) with an implicit constitutive update. The sensitivity coefficients are evaluated using a direct differentiation approach. Numerical results are presented for a spherical cap subjected to step loading, and a circular plate subjected to impulsive loading. The sensitivity coefficients are generated by evaluating the derivatives of the response quantities with respect to the thickness, mass density, Young's modulus, and two of the material parameters characterizing the viscoplastic response. Time histories of the response and sensitivity coefficients are presented, along with spatial distributions of these quantities at selected times.     

Finite element analysis of bimodulus composite stiffened thin shells of revolution

This paper is a sequel to the work published by the first and third authors[l] on stiffened laminated shells of revolution made of unimodular materials (materials having identical properties in tension and compression). A finite element analysis of laminated bimodulus composite thin shells of revolution, reinforced by laminated bimodulus composite stiffeners is reported herein. A 48 dot doubly curved quadrilateral laminated anisotropic shell of revolution finite element and it's two compatible 16 dof stiffener finite elements namely: (i) a laminated anisotropic parallel circle stiffener element (PCSE) and (ii) a laminated anisotropic meridional stiffener element (MSE) have been used iteratively.The constitutive relationship of each layer is assumed to depend on whether the fiberdirection strain is tensile or compressive. The true state of strain or stress is realized when the locations of the neutral surfaces in the shell and the stiffeners remain unaltered (to a specified accuracy) between two successive iterations. The solutions for static loading of a stiffened plate, a stiffened cylindrical shell. and a stiffened spherical shell, all made of bimodulus composite materials, have been presented.     

Direct dynamic analysis of shells of revolution using high-precision finite elements

For the transient dynamic analysis of structural systems, the direct numerical integration of the equations of motion may be regarded as an alternative to the mode superposition method for linear problems and a necessity for nonlinear problems. When compared to a modal superposition solution, the direct integration approach is attractive in that the eigenvalue problem is avoided. Depending on the amount of information required from the dynamic analysis, e.g. frequencies, frequencies and mode shapes, and/or a complete time history, a direct integration scheme may prove to be more efficient than a modal superposition solution for some linear problems as well.The purpose of this study is to develop and demonstrate a direct integration algorithm which is compatible with an existing high-precision rotational shell finite element. Excellent comparative efficiency for static problems was achieved with this element by the incorporation of the exact geometry, the utilization of high-order interpolation polynomials and, yet, the retention of only a minimum number of nodal variables in the global formulation. Likewise, accurate and efficient results for the free vibration analysis of rotational shells were facilitated by the inclusion of a consistent mass matrix and the utilization of a rationally justified kinematic condensation procedure. The approach to the direct integration stage is strongly tempered by the established characteristics of this element which enable a given shell to be modeled accurately in the spatial domain with a comparatively coarse discretization.The equations of motion for a shell of revolution under conservative loading are derived from Hamilton's variational principle and specialized for the discretization of a rotational shell into curved shell elements. Degrees of freedom in excess of those required to establish minimum (C°) continuity at the nodal circles are eliminated through kinematic condensation. Some guidance as to the proper order of the polynominal approximations for a dynamic analysis is provided by earlier free vibration studies. Whereas the condensation is exact for static problems, it is only approximate for dynamic response and it was found that the accuracy of the eigenvalues obtained for the reduced problem decreases with increasing order of the condensed functions. This tendency is counted by the desirability of using sufficiently high-order interpolations so as to permit accurate stress computations, both at the nodal circles and between nodes since a coarse discretization is necessary to realize maximum efficiency. It was found that cubic polynomials were generally satisfactory from both standpoints, except in localized regions of high stress gradients where quintic polynomials were employed. The finite element discretizations for the direct integration studies were selected on this basis.For the high-precision finite element at hand, the efficiencies achieved in the space domain are demonstrated by the ability to achieve precise solutions with relatively coarse discretization patterns. The resulting comparatively large elements are not subject to accurate representation by diagonal mass matrices so that an implicit, consistent mass approach is followed. Efficiency in the time domain as well rests on the successful modeling of rotational shells subject to dynamic loading using coarse discretitations in space and large increments in time. Computational efficiency and accuracy are demonstrated for various problems documented in the literature, including a shallow spherical cap subject to a step pulse and a hyperboloidal shell under a simulated dynamic wind pressure.     

Sensitivity analysis for the dynamic response of thermoviscoplastic shells of revolution

A computational procedure is presented for evaluating the sensitivity coefficients of the dynamic axisymmetric, fully-coupled, thermoviscoplastic response of shells of revolution. The analytical formulation is based on Reissner's large deformation shell theory with the effects of large-strain, transverse shear deformation, rotatory inertia and moments turning around the normal to the middle surface included. The material model is chosen to be viscoplasticity with strain hardening and thermal hardening, and an associated flow rule is used with a von Mises effective stress. A mixed formulation is used for the shell equations with the fundamental unknowns consisting of six stress resultants, three generalized displacements and three velocity components. The energy-balance equation is solved using a Galerkin procedure, with the temperature as the fundamental unknown.Spatial discretization is performed in one dimension (meridional direction) for the momentum and constitutive equations of the shell, and in two dimensions (meridional and thickness directions) for the energy-balance equation. The temporal integration is performed by using an explicit central difference scheme (leap-frog method) for the momentum equation; a predictor-corrector version of the trapezoidal rule is used for the energy-balance equation; and an explicit scheme consistent with the central difference method is used to integrate the constitutive equations. The sensitivity coefficients are evaluated by using a direct differentiation approach. Numerical results are presented for a spherical cap subjected to step loading. The sensitivity coefficients are generated by evaluating the derivatives of the response quantities with respect to the thickness, mass density, Young's modulus, two of the material parameters characterizing the viscoplastic response and the three parameters characterizing the thermal response. Time histories of the response and sensitivity coefficients are presented, along with spatial distributions of some of these quantities at selected times.     

Transient response of inelastic shells of revolution

The paper presents a transient response finite difference solution in space and time for the shell of revolution equations derived by Sanders. Inelastic constitutive relationships and nonlinear kinematics are included. The spatial differencing technique is based on a checkerboard half-spacing mesh configuration that allows the use of lower order difference expressions than those normally used for accuracy. Several example problems are considered which illustrate convergence for elastic and elastic-plastic problems. Experimental results for cylindrical shells are compared with predictions based on linear and nonlinear kinematics, various mesh spacings, and constitutive differences.     

Natural structural shapes for shells of revolution in the membrane theory of shells

Natural Structural Shapes are derived for axisymmetrically loaded shells of revolution within the membrane theory of shells. The concept of natural structural shapes is based on the simultaneous minimization of the mass and the strain energy of the loaded structure, a multicriteria optimization problem with Edgeworth-Pareto optimality as the basic optimality concept. The problem is formulated as a multicriteria control problem and necessary conditions for arbitrary loading and boundary are derived. Exact and numerical results are obtained for both the case of uniform pressure and that of a ring load with zero surface loads.Presented at the IUTAM Symposium on Structural Optimization in Melbourne, Feb. 1988.     

Shape optimization of dynamically loaded viscoelastic cylindrical shells

Axisymmetric deflections of cylindrical shells of variable thickness are examined. The shell material is linear viscoelastic. The loading is of the impulsive type—it induces inside the shell a radial velocity field. The amount of kinetic energy is prescribed. The thickness function includes some design parameters, which must be calculated so that deflections of the beam are minimal. Only designs with a given volume are considered.For solving this optimization problem the space variable and the time will be separated. For evaluating the minimum of the objective function the Nelder-Mead technique has been used. Computations show that the viscosity effect is essential only for very short shells. Some numerical examples are presented.     

Application of the initial value method to analysis of elastic-plastic plates and shells of revolution

Basic idea of a modification of initial value methods is shown and its application to computation of elastic-plastic shells and plates of revolution is presented.

Analysis of large deflections of spherical caps made of material with isotropic and/or kinematic strain-hardening is carried out. Limit points for external pressure are computed for full-walled and equivalent ideal sandwich cross-sections.

Perturbating the membrane state the bifurcation loads of Shanley's either Karman's type are computed for annular plates under uniform radial compression. Limit points are also computed for different boundary conditions.

Analysis of spherical caps subject to changes of temperature and pressure is presented. For load control programs determined in the plane load parameters (pressure and temperature) neutral domain curves of elastic unloadings are evaluated.     

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.     

Application of DSISR program to recessed shells of revolution

Application of the DSISR program to recessed shells of revolution is illustrated on the examples of a cylinder with two diametrically-opposite rectangular cutouts and a sphere with a single trapezoidal cutout. The program which is suitable for a wide range of static and dynamic problems was developed for general linear analysis of shells of revolution with arbitrary stiffness and mass density distributions. In the analysis, the equations of motion are derived with the aid of Sanders' theory, and the numerical solution procedure is based on Fourier expansion in the circumferential direction, on finite differences in the meridional direction, and on Houbolt's method in the time domain.     

Large deflection analysis of plates and shells by discrete energy method

The discrete energy method—a special form of finite difference energy approach—is presented as a suitable alternative to the finite element method for the large deflection elastic analysis of plates and shallow shells of constant thickness. Strain displacement relations are derived for the calculation of various linear and nonlinear element stiffness matrices for two types of elements into which the structure is discretized for considering separately energy due to extension and bending and energy due to shear and twisting. Large deflection analyses of plates with various edge and loading conditions and of a shallow cylindrical shell are carried out using the proposed method and the results compared with finite element solutions. The computational efforts required are also indicated.     

A doubly curved quadrilateral finite element for the analysis of laminated anisotropic thin shells of revolution

The details of development of the stiffness matrix for a doubly curved quadrilateral element suited for static and dynamic analysis of laminated anisotropic thin shells of revolution are reported. Expressing the assumed displacement state over the middle surface of the shell as products of one-dimensional first order Hermite polynomials, it is possible to ensure that the displacement state for the assembled set of such elements, is geometrically admissible. Monotonic convergence of total potential energy is therefore possible as the modelling is successively refined. Systematic evaluation of performance of the element is conducted, considering various examples for which analytical or other solutions are available.     

Analysis of multicircuit shells of revolution by the field method

The field method, presented previously for the solution of linear boundary-value problems defined on one-dimensional open branch domains, is extended to one-dimensional domains which contain circuits. This method converts the boundary-value problem into two successive numerically stable initial-value problems, which may be solved by standard forward integration techniques. Also presented is a new treatment of singular boundary conditions (kinematic constraints) — this is problem independent with respect to both accuracy and efficiency. The method has been implemented in a computer program which calculates the asymmetric response of ring-stiffened orthotropic multicircuit shells of revolution.     

Large displacement analysis of plates by a stress-based finite element approach

A mixed-hybrid incremental variational formulation, involving orthogonal rigid rotations and a symmetric stretch tensor, is proposed for finite deformation analysis of thin plates and shells. Isoparametric eight-noded elements are based upon the Kirchhoff-Love hypotheses, the assumption of plane stress, and the moderately large rotations of Von Karman plate theory. Semilinear elastic isotropic material properties are assumed, and the right polar decomposition of the deformation gradient is used. The symmetrized Biot-Luré (Jaumann) stress measure gives a unique complementary energy density and a set of variational principles with a priori satisfaction of linear momentum balance, a posteriori angular momentum balance, and interelement traction reciprocity by means of Lagrange multipliers. The incremental modified Newton-Raphson solution procedure is generated by a truncated Taylor series expansion of the functional in a total Lagrangian formulation. The theory is applied to laterally loaded and buckled thin plates, and numerical results are compared with truncated series solutions.