Graphical-Numerical Solution of Problems of Saint-Venant Torsion and Bending

1953 ◽  
Vol 20 (3) ◽  
pp. 321-326
Author(s):  
B. A. Boley
Keyword(s):  
Cross Section ◽  
Stress Function ◽  
Arbitrary Cross Section ◽  
Successive Approximations ◽  
Shear Stresses ◽  
Shearing Stress ◽  
First Approximation ◽  
Difference Form ◽  
Bending Of Beams

Abstract A simple successive-approximations procedure for the solution of the problems of Saint-Venant torsion and bending of beams of arbitrary cross section is presented. The shear stresses in a cross section of the beam are first calculated from the formulas valid for thin-walled sections, on the basis of an assumed set of lines of shearing stress. From these a first approximation to the stress function of either the torsion or the bending problem is found. The second approximation to the stress function is then obtained from the governing equation of the problem, expressed in finite-difference form; this in turn allows the determination of an improved set of lines of shearing stress, and hence of the shearing stress itself. The procedure can be repeated until the results of two successive steps are sufficiently close. Applications are presented for a beam cross section for which the exact solutions are known, and it is shown that no further difficulties arise in applications to more complicated shapes.

scholarly journals Bars under Torsional Loading: A Generalized Beam Theory Approach

2013 ◽  
Vol 2013 ◽  
pp. 1-39 ◽  
Author(s):  
Evangelos J. Sapountzakis
Keyword(s):  
Cross Section ◽  
Practical Interest ◽  
Beam Theory ◽  
Arbitrary Cross Section ◽  
Shear Stresses ◽  
Theory Approach ◽  
3D Fem ◽  
Mass Model ◽  
Dynamic Analyses ◽  
Load Vector

In this paper both the static and dynamic analyses of the geometrically linear or nonlinear, elastic or elastoplastic nonuniform torsion problems of bars of constant or variable arbitrary cross section are presented together with the corresponding research efforts and the conclusions drawn from examined cases with great practical interest. In the presented analyses, the bar is subjected to arbitrarily distributed or concentrated twisting and warping moments along its length, while its edges are supported by the most general torsional boundary conditions. For the dynamic problems, a distributed mass model system is employed taking into account the warping inertia. The analysis of the aforementioned problems is complete by presenting the evaluation of the torsion and warping constants of the bar, its displacement field, its stress resultants together with the torsional shear stresses and the warping normal and shear stresses at any internal point of the bar. Moreover, the construction of the stiffness matrix and the corresponding nodal load vector of a bar of arbitrary cross section taking into account warping effects are presented for the development of a beam element for static and dynamic analyses. Having in mind the disadvantages of the 3D FEM solutions, the importance of the presented beamlike analyses becomes more evident.

scholarly journals An analytical solution for static problems of curved composite beams

2019 ◽  
Vol 6 (1) ◽  
pp. 105-116 ◽  
Author(s):  
István Ecsedi ◽  
Ákos József Lengyel
Keyword(s):  
Analytical Solution ◽  
Cross Section ◽  
Radial Displacement ◽  
Fundamental Solutions ◽  
Composite Beams ◽  
Shear Stresses ◽  
Equilibrium Equations ◽  
Euler Beam ◽  
Internal Forces

AbstractAn analytical solution is presented for the determination of deformation of curved composite beams. Each cross-section is assumed to be symmetrical and the applied loads are acted in the plane of symmetry of curved beam. In-plane deformations are considered of composite curved beams. Assumed form of the displacement field assures the fulfillment of the classical Bernoulli-Euler beam theory. The curvature of beam is constant and the internal forces in a cross-section is replaced by an equivalent forcecouple system at the origin of the cylindrical coordinate system used. The internal forces are expressed in terms of two kinematical variables, which are the radial displacement and the rotation of the cross-sections. The determination of the analytical solutions of the considered static problems are based on the fundamental solutions. Linear combination of the fundamental solutions which are filling to the given loading and boundary conditions, gives the total solution. Closed form formulae are derived for the radial displacement, cross-sectional rotation, nomral and shear forces and bending moments. The circumferential and radial normal stresses and shear stresses are obtained by the integration of equilibrium equations. Examples illustrate the developed method.

Eigenvalues and eigenfunctions of finite-difference operators

1961 ◽  
Vol 57 (3) ◽  
pp. 532-546 ◽  
Author(s):  
W. G. Bickley ◽  
John McNamee
Keyword(s):  
Continuous System ◽  
Successive Approximations ◽  
Difference Operators ◽  
Difference System ◽  
Eigenvalues And Eigenfunctions ◽  
First Approximation ◽  
The Difference ◽  
The Given ◽  
Moderate Length

Numerical solution of differential and integral equations is concerned mainly with the determination of the wanted function at a finite number of discrete points which are, in general, uniformly spaced. A first approximation to the solution can be obtained if the given differential or integral system is replaced by a difference system. Any differential or integral operator can be expressed as an infinite series of difference operators and the difference system of the first approximation is obtained by neglecting all but the first few terms of the infinite expansions. We shall distinguish two processes for improving the approximation: the first uses a tabular interval of moderate length but the approximation to the given continuous system is improved by including in the difference system as many terms of the infinite expansions as are necessary or practicable; the second process uses an unvarying difference system of low accuracy, but the tabular interval is reduced in successive approximations, the process being continued until two successive approximations agree within the accuracy required. We regard these processes as essentially distinct. If the solutions obtained by the two processes approach limits, these limits need not coincide.

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