Concentrated-Force Problems in Plane Strain, Plane Stress, and Transverse Bending of Plates

1946 ◽  
Vol 13 (3) ◽  
pp. A183-A197
Author(s):  
P. S. Symonds
Keyword(s):  
Boundary Conditions ◽  
Plane Stress ◽  
Biharmonic Equation ◽  
Infinite Plate ◽  
Outer Edge ◽  
Transverse Force ◽  
Concentrated Force ◽  
Transverse Bending ◽  
Complementary Function ◽  
Concentrated Forces

Abstract A general method is described for the solution of problems of transverse bending of thin plates acted on by concentrated normal forces, and of problems of plane stress or plane strain, in which concentrated forces are applied to the boundaries. The solution is taken in two parts: (a) The special functions which give the stresses or deflections in the neighborhood of the concentrated forces. (b) A complementary function, satisfying the appropriate biharmonic equation, such that the complete solution satisfies the boundary conditions of the problem. For certain types of boundaries, this complementary function can be determined by expanding the concentrated-force functions as infinite trigonometric series. Then by addition of general solutions of the appropriate biharmonic equation, the required boundary conditions may be satisfied. The method is first illustrated by solving the plate-bending problem, for which the solution is known, of a clamped circular disk loaded by a transverse force at any point. It is then applied to the problem of an infinite plate fixed at an inner circular boundary, with outer edge free, and loaded by a transverse force at any point. This solution is obtained in finite form, and typical curves of deflection, bending moments, and shear forces are given in Figs. 3 to 8, inclusive. Using this result, solutions are next obtained for ring-shaped plates of finite outer radius, with the force applied either at the outer edge or at any point between the inner clamped edge and the outer free edge. The former case was previously solved by H. Reissner. Curves comparing the maximum moments and shears in the infinite plate with those of the annular plate with force either at the outer edge, or inside the ring are given in Figs. 9 to 12, inclusive. Finally, a solution is given of the problem in plane stress of a large plate containing an elliptical hole, which is loaded by line forces at the ends of the minor axes of the ellipse. Curves showing results of this solution are given in Figs. 14 and 15.

Stress Distribution in a Uniformly Rotating Equilateral Triangular Shaft

1955 ◽  
Vol 22 (2) ◽  
pp. 255-259
Author(s):  
H. T. Johnson
Keyword(s):  
Approximate Solution ◽  
Stress Distribution ◽  
Boundary Conditions ◽  
Plane Strain ◽  
Cross Section ◽  
Plane Stress ◽  
Biharmonic Equation ◽  
Approximate Solutions ◽  
Distribution Of Stresses ◽  
General Method

Abstract An approximate solution for the distribution of stresses in a rotating prismatic shaft, of triangular cross section, is presented in this paper. A general method is employed which may be applied in obtaining approximate solutions for the stress distribution for rotating prismatic shapes, for the cases of either generalized plane stress or plane strain. Polynomials are used which exactly satisfy the biharmonic equation and the symmetry conditions, and which approximately satisfy the boundary conditions.

A Matrix Solution for the Vibration Modes of Nonuniform Disks

1956 ◽  
Vol 23 (1) ◽  
pp. 109-115
Author(s):  
F. F. Ehrich
Keyword(s):  
Boundary Conditions ◽  
Vibration Frequency ◽  
Outer Edge ◽  
Transverse Force ◽  
Constant Thickness ◽  
Vibration Modes ◽  
Matrix Solution ◽  
Efficient Manner ◽  
Polar Moment ◽  
Matrix Vector

Abstract An arbitrary disk is represented by a simulated disk composed of circumferential strips. Alternate strips are considered to be massless, constant-thickness elements with the average local elastic properties of the actual disk. Intermediate strips are considered to have the properties of local mass and polar moment of inertia, but to have no physical dimensions or elasticity. A matrix vector, formed of the local antinodal value of deflection, slope, moment, and transverse force, may be operated on by matrices representative of the elastic strips and by matrices representative of the vibratory inertia loading, centrifugal inertia loading, internal stress, and external supports at the mass strips. Thus the influence of boundary conditions at the outer edge on conditions at the inner edge may be calculated in a simple efficient manner. Successive guesses of vibration frequency lead to final satisfaction of all boundary conditions. Concise treatment of all types of boundary conditions and numerical values of required matrices are given in tables. The results of a sample calculation are compared with exact analytic results.

A New Approach to Analyzing Reactions and Deflections of Beams: Formulation and Examples

2006 ◽  
Author(s):  
I. C. Jong ◽  
J. J. Rencis ◽  
H. T. Grandin
Keyword(s):  
Boundary Conditions ◽  
Shear Force ◽  
Flexural Rigidity ◽  
Bending Moment ◽  
Bending Moments ◽  
Shear Forces ◽  
Concentrated Force ◽  
New Approach ◽  
Concentrated Forces ◽  
The Right

This paper is aimed at developing a new approach to analyzing statically indeterminate reactions at supports, as well as the slopes and deflections, of beams. The approach uses a set of four general formulas, derived using singularity functions. These formulas are expressed in terms of shear forces, bending moments, distributed loads, slopes, and deflections of a beam having a constant flexural rigidity and carrying typical loads. These loads include (a) a bending moment and a shear force at the left, as well as at the right, end of the beam; (b) a concentrated force, as well as a concentrated moment, somewhere on the beam; and (c) a uniformly, as well as a linearly varying, distributed force over a portion of the beam. The approach allows one to treat reactions at supports (even supports not at the ends of a beam) as concentrated forces or moments, where corresponding boundary conditions at the points of supports are to be imposed. This feature allows one to readily determine reactions at supports as well as slopes and deflections of beams. A beam needs to be divided into segments for study if it contains discontinuities in slope at hinge connections or different flexural rigidities in different segments. Several examples are included to illustrate the new approach.

scholarly journals Point-loaded discs and blocks applicable to tensile testing of brittle materials

1972 ◽  
Vol 7 (3) ◽  
pp. 178-185 ◽  
Author(s):  
F J Appl
Keyword(s):  
Residual Stresses ◽  
Boundary Conditions ◽  
Plane Strain ◽  
Plane Stress ◽  
Brittle Materials ◽  
Compression Members ◽  
Elasticity Problems ◽  
Concentrated Forces ◽  
Stress Boundary Conditions ◽  
Stress Boundary

A method of numerically approximating the solutions of plane-stress or plane-strain elasticity problems with boundary conditions consisting of concentrated forces or distributed loads is presented herein. The effect of each concentrated force (commonly termed a point load) that acts on the boundary is represented by a Flamant solution. Usually, the combined effect of these Flamant solutions indicates the presence of distributed loadings or ‘residual stresses’ on some portions of the boundary that are not consistent with the actual boundary conditions. The negatives of these ‘residual stresses’ are used as stress boundary conditions in a singular integral method of numerical analysis that is applicable to plane elasticity problems involving distributed loadings on the boundaries. Since the method presented herein involves only stress boundary conditions, the solutions are valid for both plane stress and plane strain. The accuracy of this superposition method is demonstrated by consideration of a circular disc or cylinder subjected to diametrically opposed concentrated forces for which accuracy to within 0.2 per cent of the exact solution is obtained. Parametric analyses of rectangular and elliptical compression members subjected to point loads are presented. Results determined herein are found to compare relatively well with those determined in previous numerical and experimental investigations of specific cases. These results make possible the design and analysis of compression members used to evaluate the tensile fracture strength of brittle materials.

Biharmonic navigation using radial basis functions

Robotica ◽  
2021 ◽  
pp. 1-12
Author(s):  
Xu-Qian Fan ◽  
Wenyong Gong
Keyword(s):  
Finite Difference ◽  
Boundary Conditions ◽  
Radial Basis Functions ◽  
Real World ◽  
Biharmonic Equation ◽  
Finite Difference Methods ◽  
Basis Functions ◽  
Large Size ◽  
Radial Basis ◽  
Difference Methods

Abstract Path planning has been widely investigated by many researchers and engineers for its extensive applications in the real world. In this paper, a biharmonic radial basis potential function (BRBPF) representation is proposed to construct navigation fields in 2D maps with obstacles, and it therefore can guide and design a path joining given start and goal positions with obstacle avoidance. We construct BRBPF by solving a biharmonic equation associated with distance-related boundary conditions using radial basis functions (RBFs). In this way, invalid gradients calculated by finite difference methods in large size grids can be preventable. Furthermore, paths constructed by BRBPF are smoother than paths constructed by harmonic potential functions and other methods, and plenty of experimental results demonstrate that the proposed method is valid and effective.

scholarly journals Bending of an incomplete annular plate and related problems

1979 ◽  
Vol 14 (3) ◽  
pp. 103-109 ◽  
Author(s):  
J R Barber
Keyword(s):  
Stress Concentration ◽  
Closed Form ◽  
Circular Plate ◽  
Annular Plate ◽  
Infinite Plate ◽  
Concentration Data ◽  
Closed Form Solutions ◽  
Concentrated Forces

Closed-form solutions and stress-concentration data are obtained for the problem of a sector of an annular plate subjected to moments and transverse forces on its radial edges. Closed-form solutions are also given for a semi-infinite plate or a circular plate subjected to a system of concentrated forces and/or moments at the edge.

Sign in / Sign up

Export Citation Format

Share Document