M. Wahhaj Uddin

A finite-difference scheme for mixed boundary value problems of arbitrary-shaped elastic bodies

Abstract This paper presents a program based on a finite-difference technique, which solves plane stress and plane strain problems of arbitrary shaped elastic bodies with mixed boundary conditions. A new formulation of governing equations in terms of the displacement potential function ψ, as introduced by Uddin (Finite difference solution of two-dimensional elastic problems with mixed boundary conditions, MSc Thesis, Carleton University, Canada, 1966), has been used. This formulation has the capability to handle problems of mixed boundary condition, which is beyond the ability of the conventional formulations in terms of Airys stress function φ. Results found with this program for classical problems are in very good agreement with known solutions. This program can handle practical boundary conditions very efficiently.

Generalized mathematical model for the solution of mixed-boundary-value elastic problems

This paper describes a new mathematical formulation, specifically suitable for finite-difference analysis of stresses and displacements of three-dimensional mixed-boundary-value elastic problems. Earlier, mathematical models of elasticity were very deficient in handling three-dimensional practical stress problems. In the present model, a new scheme of reduction of unknowns is used to formulate the three-dimensional problem in terms of a single potential function, defined in terms of the three displacement components. Compared to the conventional models, the present model provides numerical solution of higher accuracy in a shorter period of computational time. The application of the potential function formulation is demonstrated here through a number of classical problems of solid mechanics, and the results are compared with the available solutions in the literature. The comparison of the results establishes the rationality of the present approach.

An efficient algorithm for finite-difference modeling of mixed-boundary-value elastic problems

The paper describes an efficient numerical scheme for the solution of displacements and stresses in mixed-boundary-value elastic problems of solid mechanics. A new variable reduction scheme is used to develop the computational model. Solution of both two- and three-dimensional problems of linear elasticity is considered in the present paper. In the present approach, the problems are formulated in terms of a potential function, defined in terms of the displacement components. Compared to the conventional computational approaches, the present scheme is capable of providing numerical solution of higher accuracy with less computational effort. Application of the present finite-difference modeling scheme is demonstrated through the solutions of a number of practical stress problems of interest, and the results are compared with those obtained by the standard method of solution. The comparison of the results establishes the rationality as well as suitability of the present variable reduction scheme.

The stability of conical end caps with spherical tips as end closures for pressure vessels

Abstract The instability under pressure of conical end caps with spherical tips when used as end closures to pressure vessels is studied in this paper. The spherical tip of the conical end cap was assumed to be attached in such a way that continuity of the slope at the cone/sphere junction was maintained. The geometrical parameters of the spherical-tip conical end closure are the ratio r R ; the apex angle Ψ of the conical frustum; and the thickness ratio R h , where r and R are respectively the radius of the cone at the sphere/cone and vessel/cone junctions and h is the thickness of the shell. Governing nonlinear differential equations of axisymmetric deformation which ensure the unique states of lowest potential energy under given pressure have been solved by using the method of multisegment integration, developed by Kalnins and Lestingi.1 The results show that the critical pressure for the end closure decreases with increasing apex angle at constant values of R h and r R . At constant values of Ψ and R h , the critical pressure remains constant over a considerable range of r R and then decreases to a minimum value at r R = 1·0 which corresponds to a purely spherical end cap without the conical extension.

THREE-DIMENSIONAL SOLUTION OF A DEEP BEAM USING AN EFFICIENT FINITE-DIFFERENCE SCHEME

The application of a new numerical method of solution is described for 3-D analysis of an elastic deep beam. More specifically, an efficient finite-difference scheme has been developed based on a potential function formulation to solve the 3-D deep beam. In the present approach, a new scheme of reduction of unknowns is used to formulate the 3-D beam problem in terms of a single potential func- tion, defined in terms of the three displacement components. Compared to the conventional computational approaches, the present method provides numerical solution of higher accuracy with reduced computational effort. The suitability and reliability of the method has been verified through the comparison of results with those obtained by the usual method of solution.

A NEW MATHEMATICAL MODEL FOR FINITE-DIFFERENCE SOLUTION OF THREE-DIMENSIONAL MIXED-BOUNDARY-VALUE ELASTIC PROBLEMS

This paper describes a new mathematical formulation, specifically suitable for finite-difference analysis of stresses and displacements of three-dimensional mixed-boundary-value elastic problems. Earlier, mathematical models of elasticity were very deficient in handling three-dimensional practical stress problems. In the present model, a new scheme of reduction of unknowns is used to formulate the three-dimensional problem in terms of a single potential function, defined in terms of the three displacement components. Compared to the conventional models, the present model provides numerical solution of higher accuracy in a shorter period of computational time. The application of the potential function formulation is demonstrated here through a number of classical problems of solid mechanics, and the results are compared with the available solutions in the literature. The comparison of the results establishes the rationality of the present approach.