Stresses in a Perforated Strip

1957 ◽  
Vol 24 (3) ◽  
pp. 365-375
Author(s):  
Chih-Bing Ling
Keyword(s):  
Stress Function ◽  
Harmonic Functions ◽  
Analytic Solution ◽  
Complete Solution ◽  
Classical Problem ◽  
Stress System ◽  
Infinite Strip ◽  
Numerical Examples ◽  
Biharmonic Functions ◽  
Transverse Bending

Abstract This paper presents an analytic solution of the classical problem dealing with the stresses in an infinite strip having an unsymmetrically located perforating hole. The solution is applicable to any stress system acting in the strip, which is symmetrical with respect to the line of symmetry of the strip. The required stress function is constructed by using four series of biharmonic functions and a bihamonic integral. The four series of biharmonic functions are formed from a class of periodic harmonic functions specially constructed for the purpose. The solution can be regarded as a complete solution of the problem in the sense that, unlike the previous solutions by Howland, Stevenson, and Knight for a symmetrically perforated strip, it is valid in the entire strip. Numerical examples are given for the fundamental cases of longitudinal tension and transverse bending.

On Invariant Perforation in an Infinite Strip

1959 ◽  
Vol 26 (3) ◽  
pp. 422-431
Author(s):  
Chih-Bing Ling
Keyword(s):  
Finite Group ◽  
Stress Function ◽  
Harmonic Functions ◽  
Infinite Group ◽  
The Other ◽  
Infinite Strip ◽  
Method Of Images ◽  
Numerical Examples ◽  
Sigma Function ◽  
Working Out

Abstract The invariant perforation in an infinite strip can be classified into two groups. One is the finite group and the other is the infinite group. There are five cases in the finite group and nine cases in the infinite group. All the cases can be solved by the method of images. This method has, in fact, been used by the author to solve the stresses in an infinite strip containing either an unsymmetrically located single hole or a series of uniformly distributed equal holes. The solution is illustrated by working out in detail one of the cases in the infinite group, in which the strip contains two series of equal holes symmetrically staggered along the strip. The stress function is constructed by using a class of periodic harmonic functions derived from Weierstrass’ sigma function. Numerical examples also are given to show the effect of such a perforation on the stresses in the strip.

On the Stresses in a Rotating Disk of Variable Thickness

1952 ◽  
Vol 19 (3) ◽  
pp. 263-266
Author(s):  
Ti-Chiang Lee
Keyword(s):  
Variable Thickness ◽  
Stress Function ◽  
Hypergeometric Functions ◽  
Analytic Solution ◽  
Radial Distance ◽  
Rotating Disk ◽  
Numerical Examples ◽  
Confluent Hypergeometric Functions ◽  
Method Of Solution ◽  
Two Parameters

Abstract This paper presents an analytic solution of the stresses in a rotating disk of variable thickness. By introducing two parameters, the profile of the disk is assumed to vary exponentially with any power of the radial distance from the center of the disk. In some respects this solution may be considered as a generalization of Malkin’s solution, but it differs essentially from the latter in the method of solution. Here, the stresses are solved through a stress function instead of being solved directly. The required stress function is expressed in terms of confluent hypergeometric functions. Numerical examples are also shown for illustration.

Stresses in a Stretched Slab Having a Spherical Cavity

1959 ◽  
Vol 26 (2) ◽  
pp. 235-240
Author(s):  
Chih-Bing Ling
Keyword(s):  
Boundary Conditions ◽  
Stress Function ◽  
Spherical Cavity ◽  
Analytic Solution ◽  
Linear Equations ◽  
Hankel Transform ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Biharmonic Functions ◽  
Infinite Slab

Abstract This paper presents an analytic solution for an infinite slab having a symmetrically located spherical cavity when it is stretched by an all-round tension. The required stress function is constructed by combining linearly two sets of periodic biharmonic functions and a biharmonic integral. The sets of biharmonic functions are derived from two fundamental functions specially built up for the purpose. The arbitrary functions involved in the biharmonic integral are first adjusted to satisfy the boundary conditions on the surfaces of the slab by applying the Hankel transform of zero order. Then the stress function is expanded in spherical co-ordinates and the boundary conditions on the surface of the cavity are satisfied by adjusting the coefficients of superposition attached to the sets of biharmonic functions. The resulting system of linear equations is solved by the method of successive approximations. The solution is finally illustrated by numerical examples for two radii of the cavity.

Stresses in a Perforated Wedge

1973 ◽  
Vol 40 (3) ◽  
pp. 759-766 ◽  
Author(s):  
Chih-Bing Ling ◽  
Chang-Ming Hsu
Keyword(s):  
Boundary Conditions ◽  
Circular Hole ◽  
Stress Function ◽  
The Other ◽  
Numerical Examples ◽  
Biharmonic Functions ◽  
Method Of Solution ◽  
The Given ◽  
Infinite Wedge

This paper presents a method of solution for an infinite wedge containing a symmetrically located circular hole. The solution is formulated separately according to the given in-plane edge tractions being even or odd with respect to the axis of the wedge. In either case, the stress function is constructed as the sum of four parts of biharmonic functions, two in the form of integrals and the other two in the form of series, in addition to a basic stress function for an otherwise unperforated wedge. The four parts as a whole give no traction along the edges and no stress at infinity of the wedge. Together with the basic stress function, the boundary conditions of no traction at the rim of hole are adjusted. Complex expressions are used in adjusting the boundary conditions. Finally, numerical examples are given for illustration.

Generalized plane stress in a semi-infinite strip under arbitrary end-load

1964 ◽  
Vol 281 (1385) ◽  
pp. 184-206 ◽  
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Boundary Conditions ◽  
Fourth Order ◽  
Plane Stress ◽  
Stress Function ◽  
Infinite Strip ◽  
Order Ordinary Differential Equation ◽  
Orthogonal Functions

The problem involves the determination of a biharmonic generalized plane-stress function satisfying certain boundary conditions. We expand the stress function in a series of non-orthogonal eigenfunctions. Each of these is expanded in a series of orthogonal functions which satisfy a certain fourth-order ordinary differential equation and the boundary conditions implied by the fact that the sides are stress-free. By this method the coefficients involved in the biharmonic stress function corresponding to any arbitrary combination of stress on the end can be obtained directly from two numerical matrices published here The method is illustrated by four examples which cast light on the application of St Venant’s principle to the strip. In a further paper by one of the authors, the method will be applied to the problem of the finite rectangle.

Reconnexion of lines of force in rotating spheres and cylinders

1966 ◽  
Vol 291 (1424) ◽  
pp. 60-72 ◽  
Keyword(s):  
Magnetic Field ◽  
Uniform Magnetic Field ◽  
Magnetic Force ◽  
Analytic Solution ◽  
Transient Phase ◽  
General Behaviour ◽  
Closed Loops ◽  
Numerical Examples ◽  
Conducting Sphere ◽  
Approximate Expressions

A solid conducting sphere or cylinder begins to rotate suddenly from rest in an initially uniform magnetic field. An analytic solution for this problem is obtained and it is shown that lines of magnetic force reconnect to form closed loops during the transient phase. The general behaviour of the system is investigated for all conductivities. Numerical examples are given and approximate expressions derived in the limiting cases of large conductivity and time.

The Hybrid Boundary Element Method: An Alliance Between Mechanical Consistency and Simplicity

1989 ◽  
Vol 42 (11S) ◽  
pp. S54-S63 ◽  
Author(s):  
N. A. Dumont
Keyword(s):  
Finite Element Method ◽  
Boundary Element ◽  
Complete Solution ◽  
Computational Effort ◽  
Element Formulation ◽  
Numerical Examples ◽  
Hybrid Finite Element Method ◽  
Hybrid Finite Element ◽  
Force Displacement ◽  
Element Method

The most important features of a new numerical method are outlined. The mechanical, or variational consistency of the hybrid finite element method is extended to the conventional boundary element formulation, giving rise to naturally established symmetric force-displacement relations. The computational effort for the complete solution of a given problem, according to this method, is in some cases only a small fraction of the effort needed with traditional methods. This paper also outlines briefly the types of analyses which may be advantageously performed with this new method, many of which are already being implemented by the author and co-workers. Some numerical examples are provided.

Solution of Plane Problems of Elasticity Utilizing Partitioning Concepts

1973 ◽  
Vol 40 (3) ◽  
pp. 767-772 ◽  
Author(s):  
O. L. Bowie ◽  
C. E. Freese ◽  
D. M. Neal
Keyword(s):  
Analytic Function ◽  
Power Series ◽  
Collocation Method ◽  
Stress Function ◽  
Function Theory ◽  
Series Expansions ◽  
Two Dimensional ◽  
Numerical Examples ◽  
Power Series Expansions

A partitioning plan combined with the modified mapping-collocation method is presented for the solution of awkward configurations in two-dimensional problems of elasticity. It is shown that continuation arguments taken from analytic function theory can be applied in the discrete to “stitch” several power series expansions of the stress function in appropriate subregions of the geometry. The effectiveness of such a plan is illustrated by several numerical examples.

Notes on problems in hexagonal aeolotropic materials

1951 ◽  
Vol 47 (2) ◽  
pp. 401-409 ◽  
Author(s):  
R. T. Shield
Keyword(s):  
General Solution ◽  
Stress Function ◽  
Harmonic Functions ◽  
Present Note ◽  
Three Dimensional ◽  
Stress Distributions ◽  
Axially Symmetric ◽  
Isotropic Materials ◽  
Stress Functions ◽  
Single Stress

Three-dimensional stress distributions in hexagonal aeolotropic materials have recently been considered by Elliott(1, 2), who obtained a general solution of the elastic equations of equilibrium in terms of two ‘harmonic’ functions, or, in the case of axially symmetric stress distributions, in terms of a single stress function. These stress functions are analogous to the stress functions employed to define stress systems in isotropic materials, and in the present note further problems in hexagonal aeolotropic media are solved, the method in each case being similar to that used for the corresponding problem in isotropic materials. Because of this similarity detailed explanations are unnecessary and only the essential steps in the working are given below.

A two-step iterative method based on diagonal and off-diagonal splitting for solving linear systems

Filomat ◽  
2017 ◽  
Vol 31 (5) ◽  
pp. 1441-1452
Author(s):  
Mehdi Dehghana ◽  
Marzieh Dehghani-Madisehb ◽  
Masoud Hajarianc
Keyword(s):  
Numerical Analysis ◽  
Linear Systems ◽  
Iterative Methods ◽  
Classical Problem ◽  
Coefficient Matrix ◽  
New Method ◽  
Step Method ◽  
Numerical Examples ◽  
Special Cases ◽  
Two Parameter

Solving linear systems is a classical problem of engineering and numerical analysis which has various applications in many sciences and engineering. In this paper, we study efficient iterative methods, based on the diagonal and off-diagonal splitting of the coefficient matrix A for solving linear system Ax = b, where A ? Cnxn is nonsingular and x,b ? Cnxm. The new method is a two-parameter two-step method that has some iterative methods as its special cases. Numerical examples are presented to illustrate the effectiveness of the new method.

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