Gerald Wempner

Discrete approximations related to nonlinear theories of solids

Abstract Interpolation and extrapolation are employed to approximate the fields in a nonlinear theory of solid bodies. Nodal points are employed in the space of position and load, and the continuous fields are essentially replaced by nodal values. Interpolation between nodes (extrapolation in load) defines a continuous approximation. The differential equations of the continuum are replaced by algebraic equations of the discrete system. Nonlinear equations are replaced by a succession of linear equations, as a nonlinear path is approximated by linear segments. Variational theorems are used as the bases of the algebraic formulations which govern the discrete approximation. The algebraic equations are related to their differential counterparts. A generalized arc-length is introduced in the configuration-load space in order to facilitate the incremental computations near limit points. The arc-length is used as the loading parameter in some illustrative problems. An appendix describes the viewpoint of finite elements and the continuity conditions which insure the equivalence of the methods.

General theory of sandwich plates with dissimilar facings

Abstract Equations are derived for the large deflections of sandwich plates with weak cores and presented in an invariant form. The general equations include the bending resistance of the facings and transverse extension of the core. Equations for buckling and for small deflections are obtained as special cases. An example illustrates the use of these equations to predict buckling loads.

A simple finite element for elastic-plastic deformations of shells

Abstract The quadrilateral element of this paper is especially suited to the behavior of elastoplastic shells. It combines the simplicity needed in nonlinear analyses with a competitive accuracy. The elemental approximation shares some common features with an earlier one applied to hookean shells [1], but incorporates an important modification which leads to further simplifications and provides a mechanism for the progressive yielding within an element. The element is derived via the Hu-Washizu functional, which admits independent approximations of all basic variables. The elimination of strains and stresses results in a displacement formulation with all the desired properties, a positive-definite matrix and convergence. Some properties of the elemental matrices are outlined and special considerations are given to plasticity. Numerical tests upon plates and shells demonstrate the efficiency of the element. Finally, some distinctions are drawn between our model and certain ‘mixed’ elements.

The linear isoparametric triangular element: theory and application

Abstract The aim of the present paper is to set forth, in a simple and consistent manner, the mechanical and computational features of the linear isoparametric triangular element. Emphasis is placed upon those mechanical attributes which are important in deriving finite element formulations. The linear isoparametric triangle is treated in a very direct manner. The kinematic features of the linear fields are identified. Homogeneous and higher order deformation modes are uncoupled by appropriate approximations of the stresses and strains. Consistent relations are obtained via the Hu-Washizu theorem. The resulting model is fully consistent with the Reissner continuum theory and is well suited for thick plates and shells. By means of discrete constraints the shear-deformable element is reduced to one which aorresponds to the Kirchhoff-Love continuum theory.

A derived theory of elastic-plastic shells

Abstract The key to a theory for elastic-plastic shells is the formulation of constitutive equations. Here, incremental equations are derived from the Hooke, Prandtl-Reuss equations of elastic, plastic deformations. The theory does not embody an initial yield condition, but admits immediate, though gradual, evolution of inelastic strain. Consequently, the abrupt transitions and interfaces between elastic and plastic regions are nonexistant. Legendre polynomials are employed to approximate the distribution of stresses; the polynomials of first and second degree are identified with the active forces and couples. Higher polynomials represent residual stresses. The balance of work and rate of dissipation serve to establish the constitutive equations and conditions of loading.

Theory for moderately large deflections of sandwich shells with dissimilar facings

Abstract This paper presents the differential equations and boundary conditions for sandwich shells with moderately large rotations. The theory includes the bending resistance of the facings, transverse extension and shear deformation of the core. Approximations and simplifications are described.

Mechanical Behavior of Filament-Wound Composite Tubes.

Abstract : The main component of a lightweight missile launcher is a cylindrical tube which is formed by a helical winding of glass filaments embedded in epoxy resin. In service, the tube is subjected to extreme internal pressure. Consequently, knowledge concerning the mechanisms of failure under pressure is vital to the analysis and design of these tubes. This report identifies the mechanisms of failure and offers appropriate methods to compute the ultimate pressure. (Author)

Elasto-plasticity of the club sandwich

Abstract The kinematics and dynamics of thin shells are well established. The constitutive equations of Hookean shells are linear in 6 strains and 6 stresses, but the equations of elastoplastic shells are incremental and require additional internal variables, notably stresses. In typical computations, the shell is divided into N layers: With the usual hypotheses, 6 strains and 3 N stresses are required. The storage of 3N stresses poses practical limitations. The ideal sandwich of 2 layers is useful for limit analyses, but inadequate for general purposes. The club sandwich of 4 layers offers a practical alternative: Relative strengths, stiffnesses and positions of the layers are selected to fit the conditions of yielding under actions of membrane and bending stresses. Here, the mechanics of the club sandwich are presented and the behavior is compared with that of the multi-layered shell.

A simple model of elastic-plastic plates

Abstract The elastic-plastic plate is modelled by a sandwich of four layers and by two-dimensional quadrilateral elements. Piecewise constant approximations of the stresses within subregions provide a mechanism for the stepwise progression of yielding.

Finite Rotations in the Approximation of Shells

In the spirit of this colloquium, our attention is focused upon finite rotations in structures and, specifically, their role in the approximation of thin shells. Here, the theory of shells is recast; the motion is decomposed into strains and rotations with no restrictions on their magnitudes. With a view toward the further approximation via finite elements, the general theory is couched in alternative forms: A potential admits variations of displacements whereas a complementary functional admits variations of displacements, stresses and strains. To admit very simple approximations, transverse-shear deformations are included. Interelement continuity is then preserved even when kinks occur in the surface.