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.

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 A method for the study of fully developed parallel flow in straight ducts of arbitrary cross section

1991 ◽  
Vol 33 (2) ◽  
pp. 211-239
Author(s):  
J. Mazumdar ◽  
R. N. Dubey
Keyword(s):  
Differential Equation ◽  
Cross Section ◽  
Practical Interest ◽  
Momentum Equation ◽  
Arbitrary Cross Section ◽  
Parallel Flow ◽  
Engineering Research ◽  
Contour Lines ◽  
Method Of Solution ◽  
Newtonian Viscous Fluid

AbstractA method is presented for the study of fully developed parallel flow of Newtonian viscous fluid in uniform straight ducts of very general cross-section. The method is based upon the concept of contour lines of constant velocity in a typical cross section of the duct, and uses the function which describes the contour lines as an independent variable to derive the integral momentum equation. The resulting ordinary integro-differential equation is, in principle, much easier to solve than the original momentum equation in partial differential equation form. Several illustrative examples of practical interest are included to explain the method of solution. Some of these solutions are compared with available solutions in the literature. All details are explained by graphs and tables. The method has several interesting features. The study has relevance to biomedical engineering research for blood and urinary tract flow.

Beam Theory and Its Suitability for the Analysis of Pipes

1991 ◽  
Vol 113 (2) ◽  
pp. 291-296
Author(s):  
H. Fan ◽  
G. E. O. Widera ◽  
P. Afshari
Keyword(s):  
Asymptotic Expansion ◽  
Boundary Conditions ◽  
Cross Section ◽  
Three Dimensional ◽  
Beam Theory ◽  
Arbitrary Cross Section ◽  
Elasticity Equations ◽  
Expansion Technique ◽  
Three Dimensional Elasticity ◽  
Benchmark Solutions

The use of the asymptotic expansion technique when applied to the three-dimensional elasticity equations is outlined and used to demonstrate the development of an asymptotic beam theory and associated boundary conditions. The formulation thus obtained holds for arbitrary cross section shapes and is applied here to pipes. It can be used to provide benchmark solutions to test the suitability of engineering beam and shell theories.

Stresses in a Helical Spring of Arbitrary Cross Section With Consideration of End Effects

1987 ◽  
Vol 109 (3) ◽  
pp. 289-301 ◽  
Author(s):  
K. Nagaya
Keyword(s):  
Cross Section ◽  
Three Dimensional ◽  
Beam Theory ◽  
Theory Of Elasticity ◽  
Arbitrary Cross Section ◽  
Coil Spring ◽  
End Effects ◽  
Free Boundary Conditions ◽  
Resultant Forces ◽  
Stress Free Boundary

This paper presents expressions for stresses in the spring due to compression or tension in the direction of the axis of the cylindrical coil. In deriving stresses in the spring, it is necessary to first obtain resultant forces and resultant moments which occur in the coil spring. Then the analysis first derives exact results based on three-dimensional curved beam theory in which all displacements and all forces in three orthogonal directions of the coil spring are included with consideration of the constraints of the ends of the spring. Since the boundary shape of the spring is irregular, it is also difficult to satisfy stress-free boundary conditions along the surface of the coil, so that this paper applies the Fourier expansion collocation method and obtains stresses in the coil spring based on the theory of elasticity.

scholarly journals SEARCHING SHEAR FORCES FLOWS FOR AN ARBITRARY CROSS-SECTION OF A THIN-WALLED BAR: DEVELOPMENT OF NUMERICAL ALGORITHM BASED ON THE GRAPH THEORY

2019 ◽  
Vol 15 (1) ◽  
pp. 153-170
Author(s):  
Vitalina Yurchenko
Keyword(s):  
Graph Theory ◽  
Cross Section ◽  
Numerical Algorithm ◽  
Cross Sections ◽  
Computer Aided Design ◽  
Arbitrary Cross Section ◽  
Software Implementation ◽  
Shear Stresses ◽  
Thin Walled ◽  
Aided Design

Searching problem of shear stresses on outside longitudinal edges of an arbitrary cross-section (including open-closed multi-contour cross-sections) of a thin-walled bar subjected to the general load case has been considered in the paper. Detail numerical algorithm intended to solve the formulated problem using mathematical apparatus of the graph theory has been proposed by the paper. The algorithm is oriented on software implementation in systems of computer-aided design of thin-walled bar structures.

Computation of waveguide modes for waveguides of arbitrary cross-section

1990 ◽  
Vol 137 (2) ◽  
pp. 145 ◽  
Author(s):  
C.Y. Kim ◽  
S.D. Yu ◽  
R.F. Harrington ◽  
J.W. Ra ◽  
S.Y. Lee
Keyword(s):  
Cross Section ◽  
Arbitrary Cross Section ◽  
Waveguide Modes

First and Second Order Elastoplastic Analyses of Thin-Walled Metal Members using Generalized Beam Theory

2018 ◽  
Author(s):  
Miguel Abambres
Keyword(s):  
Finite Element ◽  
Stainless Steel ◽  
Cross Section ◽  
Beam Theory ◽  
Second Order ◽  
Flow Rule ◽  
Non Linear ◽  
Thin Walled ◽  
Shell Finite Element ◽  
Generalized Beam Theory

Original Generalized Beam Theory (GBT) formulations for elastoplastic first and second order (postbuckling) analyses of thin-walled members are proposed, based on the J2 theory with associated flow rule, and valid for (i) arbitrary residual stress and geometric imperfection distributions, (ii) non-linear isotropic materials (e.g., carbon/stainless steel), and (iii) arbitrary deformation patterns (e.g., global, local, distortional, shear). The cross-section analysis is based on the formulation by Silva (2013), but adopts five types of nodal degrees of freedom (d.o.f.) – one of them (warping rotation) is an innovation of present work and allows the use of cubic polynomials (instead of linear functions) to approximate the warping profiles in each sub-plate. The formulations are validated by presenting various illustrative examples involving beams and columns characterized by several cross-section types (open, closed, (un) branched), materials (bi-linear or non-linear – e.g., stainless steel) and boundary conditions. The GBT results (equilibrium paths, stress/displacement distributions and collapse mechanisms) are validated by comparison with those obtained from shell finite element analyses. It is observed that the results are globally very similar with only 9% and 21% (1st and 2nd order) of the d.o.f. numbers required by the shell finite element models. Moreover, the GBT unique modal nature is highlighted by means of modal participation diagrams and amplitude functions, as well as analyses based on different deformation mode sets, providing an in-depth insight on the member behavioural mechanics in both elastic and inelastic regimes.

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