Approximate Stress Functions for Triangular Wedges

1955 ◽  
Vol 22 (1) ◽  
pp. 123-128
Author(s):  
I. K. Silverman
Keyword(s):  
Differential Equation ◽  
Variational Method ◽  
Variational Calculus ◽  
The Third ◽  
Total Differential ◽  
Stress Functions ◽  
Approximate Function ◽  
Zero Stresses

Abstract The variational method is used to determine expressions for approximate stress functions for determining the effect of a third boundary to an infinite triangular wedge. The approximate function is a solution of a Euler-Lagrange total differential equation of the variational calculus with the constants of integration determined from the condition of least work. The approximate function yields zero stresses on the sloping faces of the wedge but furnishes correction stresses on the third boundary.

Boundary-Value Problems for Third-Order Lipschitz Ordinary Differential Equations

2014 ◽  
Vol 58 (1) ◽  
pp. 183-197 ◽  
Author(s):  
John R. Graef ◽  
Johnny Henderson ◽  
Rodrica Luca ◽  
Yu Tian
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Boundary Value Problems ◽  
Order Differential Equation ◽  
Boundary Value ◽  
Point Boundary ◽  
Third Order ◽  
Optimal Length ◽  
The Third

AbstractFor the third-order differential equationy′″ = ƒ(t, y, y′, y″), where, questions involving ‘uniqueness implies uniqueness’, ‘uniqueness implies existence’ and ‘optimal length subintervals of (a, b) on which solutions are unique’ are studied for a class of two-point boundary-value problems.

The Sector Problem

1957 ◽  
Vol 24 (4) ◽  
pp. 574-581
Author(s):  
G. Horvay ◽  
K. L. Hanson
Keyword(s):  
Variational Method ◽  
Orthogonal Polynomials ◽  
Biharmonic Equation ◽  
Approximate Solutions ◽  
Circular Sector ◽  
Orthonormal Polynomials ◽  
Circular Boundary ◽  
Stress Functions ◽  
Complete Set ◽  
Derivatives Of

Abstract On the basis of the variational method, approximate solutions f k ( r ) h k ( θ ) , f k ( r ) g k ( θ ) , F k ( θ ) H k ( r ) , F k ( θ ) G k ( r ) of the biharmonic equation are established for the circular sector with the following properties: The stress functions fkhk create shear tractions on the radial boundaries; the stress functions fkgk create normal tractions on the radial boundaries; the stress functions FkHk create both shear and normal tractions on the circular boundary, and the stress functions FkGk create normal tractions on the circular boundary. The enumerated tractions are the only tractions which these function sets create on the various boundaries of the sector. The factors fk(r) constitute a complete set of orthonormal polynomials in r into which (more exactly, into the derivatives of which) self-equilibrating normal or shear tractions applied to the radial boundaries of the sector may be expanded; the factors Fk(θ) constitute a complete set of orthonormal polynomials in θ into which shear tractions applied to the circular boundary of the sector may be expanded; and the functions Fk″ + Fk constitute a complete set of non-orthogonal polynomials into which normal tractions applied to the circular boundary of the sector may be expanded. Function tables, to facilitate the use of the stress functions, are also presented.

scholarly journals An Elementary Proof of the Theorems of Cauchy and Mayer

1935 ◽  
Vol 4 (3) ◽  
pp. 112-117
Author(s):  
A. J. Macintyre ◽  
R. Wilson
Keyword(s):  
Differential Equation ◽  
Existence Theorem ◽  
Elementary Proof ◽  
Theoretical Discussion ◽  
Practical Advantage ◽  
Total Differential ◽  
Method Of Solution ◽  
The Subject ◽  
Image Position ◽  
Natural Way

Attention has recently been drawn to the obscurity of the usual presentations of Mayer's method of solution of the total differential equationThis method has the practical advantage that only a single integration is required, but its theoretical discussion is usually based on the validity of some other method of solution. Mayer's method gives a result even when the equation (1) is not integrable, but this cannot of course be a solution. An examination of the conditions under which the result is actually an integral of equation (1) leads to a proof of the existence theorem for (1) which is related to Mayer's method of solution in a natural way, and which moreover appears to be novel and of value in the presentation of the subject.

On Stability of Solutions of Certain Differential Equations of the Third Order

1967 ◽  
Vol 10 (5) ◽  
pp. 681-688 ◽  
Author(s):  
B.S. Lalli
Keyword(s):  
Differential Equation ◽  
Asymptotic Stability ◽  
Lyapunov Function ◽  
Differential Equations ◽  
Global Asymptotic Stability ◽  
Sufficient Conditions ◽  
Stability Of Solutions ◽  
Third Order ◽  
The Third ◽  
Image Position

The purpose of this paper is to obtain a set of sufficient conditions for “global asymptotic stability” of the trivial solution x = 0 of the differential equation1.1using a Lyapunov function which is substantially different from similar functions used in [2], [3] and [4], for similar differential equations. The functions f1, f2 and f3 are real - valued and are smooth enough to ensure the existence of the solutions of (1.1) on [0, ∞). The dot indicates differentiation with respect to t. We are taking a and b to be some positive parameters.

Calculation of Elastic Induced Stress of Surrounding Rocks

2012 ◽  
Vol 170-173 ◽  
pp. 37-40
Author(s):  
Bo Qian
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Boundary Conditions ◽  
Analytical Solutions ◽  
Superposition Principle ◽  
Compatibility Conditions ◽  
Partial Differential ◽  
Induced Stress ◽  
Stress Functions ◽  
Round Section

In accordance with equilibrium differential equations and compatibility conditions of deformation, the partial differential equation of induced stress is achieved for elastic surrounding rocks of tunnels and chambers of round section. By method of the superposition principle, elastic analytical solutions of induced stress of surrounding rocks is derived from the partial differential equation, which is based on stress functions and boundary conditions.

scholarly journals Stress systems in aeolotropic plates. II

1939 ◽  
Vol 173 (953) ◽  
pp. 173-192 ◽  
Keyword(s):  
Differential Equation ◽  
Plane Stress ◽  
Isotropic Material ◽  
Infinite Plate ◽  
Mean Values ◽  
Plane Plate ◽  
Method Of Solution ◽  
Stress Functions ◽  
The Mean ◽  
Fundamental Equations

1. Fundamental equations which can be applied to problems of generalized plane stress in any plane plate of aeolotropic material have been obtained by Huber (1938). When the material has two directions of symmetry at right angles in the plane of the plate the differential equation for the stress function simplifies, and in a previous paper by G. I. Taylor and the present writer (1939) a number of stress functions were obtained which satisfied this equation and also gave single-valued expressions for the mean values of the stresses and displacements. Some of these functions were used to solve the problem of an isolated force in an infinite plate. In the present paper formulae are obtained for generalized plane stress systems in an infinite aeolotropic strip and also in a semi-infinite plate which is bounded by one straight edge. In particular, a solution is given for the general problem of any force acting at any point either within or on the boundary of a strip or semi-infinite plate. The stresses due to any distribution of force over the strip or semi-infinite plate may be deduced by integration. The method of solution is similar to that used by Howland (1929) for stress systems in a strip of isotropic material and Howland’s results may be obtained from our general formulae by a limiting process. When a force acts on the boundary of a semi-infinite plate the stresses may be deduced from our general results. It is, however, easier to evaluate these stresses independently by considering the problem of an isolated force at the vertex of a wedge. This problem was actually solved by Michell (1900) for any aeolotropic plate whose moduluses are not functions of the distance from the vertex of the wedge, but we give a solution here using the methods and notation of this and our previous paper. For the case of a wedge with one straight boundary the results agree with those deduced from the general formulae for a force in a semi-infinite plate.

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