LINEAR SOLUTION IN BEAM THEORY FORM FOR ORTHOTROPIC ELASTIC BEAM-COLUMN OF NARROW RECTANGULAR CROSS SECTION SUBJECTED TO UNIFORMLY DISTRIBUTED LOADS

2008 ◽  
Vol 73 (625) ◽  
pp. 405-408 ◽  
Author(s):  
Koichiro HEKI
Keyword(s):  
Cross Section ◽  
Rectangular Cross Section ◽  
Elastic Beam ◽  
Beam Theory ◽  
Linear Solution ◽  
Uniformly Distributed Loads

Contribution to beam theory based on 3-D energy principle

2017 ◽  
Vol 23 (5) ◽  
pp. 775-786 ◽  
Author(s):  
Erick Pruchnicki
Keyword(s):  
Potential Energy ◽  
Cross Section ◽  
Rectangular Cross Section ◽  
Three Dimensional ◽  
Elastic Beam ◽  
Beam Theory ◽  
Displacement Fields ◽  
Energy Principle ◽  
Lateral Boundary Conditions ◽  
The First Variation

This paper presents a general elastic beam theory, which is consistent with the principle of stationary three-dimensional potential energy. For the sake of simplicity we consider the case of a rectangular cross section. The series expansion of the displacement field up to fourth-order in h (dimension of the cross section) is defined by 45 unknowns. The first variation of the potential energy must be zero but we only impose that each term guarantees an [Formula: see text]error. By adding supplementary lateral boundary conditions and on two extremities end cross section of the beam, we finally arrive at a well posed system of unidimensional differential equations. A linear algebraic dependence with respect to 16 displacement fields allows us to reduce the unknown to 19 displacement fields. To our knowledge this work is the first contribution to this end when the beam problem is completely three-dimensional.

Bending Vibrations of Variable Section Beams

1956 ◽  
Vol 23 (1) ◽  
pp. 103-108
Author(s):  
E. T. Cranch ◽  
Alfred A. Adler
Keyword(s):  
Cross Section ◽  
Rectangular Cross Section ◽  
Circular Cross Section ◽  
Beam Theory ◽  
Cantilever Beams ◽  
Constant Depth ◽  
Bending Vibrations ◽  
Variable Section

Abstract Using simple beam theory, solutions are given for the vibration of beams having rectangular cross section with (a) linear depth and any power width variation, (b) quadratic depth and any power width variation, (c) cubic depth and any power width variation, and (d) constant depth and exponential width variation. Beams of elliptical and circular cross section are also investigated. Several cases of cantilever beams are given in detail. The vibration of compound beams is investigated. Several cases of free double wedges with various width variations are discussed.

Distribution of Stress in Built-In Beams of Narrow Rectangular Cross Section

1942 ◽  
Vol 9 (3) ◽  
pp. A108-A116
Author(s):  
F. B. Hildebrand ◽  
Eric Reissner
Keyword(s):  
Cross Section ◽  
Elementary Theory ◽  
Rectangular Cross Section ◽  
Beam Theory ◽  
Degree Polynomial ◽  
Height Ratio ◽  
Load Curve ◽  
Normal Stress Distribution ◽  
Stress Curve ◽  
Distribution Of Stress

Abstract This paper deals with the problem of the distribution of stress in cantilever beams of narrow rectangular flanged cross section with one end of the beam rigidly built-in. Since an exact solution of this plane-stress problem appears difficult to obtain, an approximate solution is derived by applying the principle of least work. Instead of the linear normal stress distribution of the elementary beam theory, a third-degree polynomial is assumed, and the spanwise variation of this stress curve is determined by means of the calculus of variations. Numerical results are obtained with regard to the stresses at the built-in end of the beam, in their dependence upon (a) the span-height ratio of the beam, (b) the flange area-web area ratio of the beam, (c) Poisson’s ratio of the material, and (d) the distribution of load along the span. It is found that the deviations from the results of the elementary theory may be appreciable when the distance of the center of gravity of the load curve from the built-in end of the beam is less than twice the height of the cross section of the beam.

A Refinement of the Theory of Buckling of Rings Under Uniform Pressure

1955 ◽  
Vol 22 (1) ◽  
pp. 95-102
Author(s):  
A. P. Boresi
Keyword(s):  
Cross Section ◽  
Classical Result ◽  
Rectangular Cross Section ◽  
Finite Thickness ◽  
Beam Theory ◽  
Elastic Stability ◽  
External Pressure ◽  
Uniform Pressure ◽  
Variational Theory ◽  
Uniform External Pressure

Abstract A general variational theory of elastic stability that was originated by E. Trefftz (1) is applied to the problem of buckling of rings of rectangular cross section subjected to uniform external pressure. The theory is believed to be more rigorous than previous treatments of the problem, since it avoids conventional assumptions of curved-beam theory, such as the assumptions that plane sections remain plane and that radial stresses vanish. The classical result of Levy (2) is confirmed for a ring of infinitesimal thickness. New results are obtained which show the effect of the finite thickness of a ring on the coefficients in the buckling formula.

scholarly journals Hydroelastic waves propagating along a frozen channel with non-uniform thickness of ice

2021 ◽  
Vol 2057 (1) ◽  
pp. 012021
Author(s):  
K N Zavyalova ◽  
K A Shishmarev ◽  
E A Batyaev ◽  
T I Khabakhpasheva
Keyword(s):  
Cross Section ◽  
Elastic Plate ◽  
Rectangular Cross Section ◽  
Normal Modes ◽  
Elastic Beam ◽  
Wave Profile ◽  
Constant Thickness ◽  
Ice Thickness ◽  
Thin Elastic Plate ◽  
Uniform Thickness

Abstract Hydroelastic waves propagating along a channel covered with ice of non-uniform thickness are considered. The channel has a rectangular cross section. The fluid in the channel is inviscid and incompressible. The ice is modeled as a thin elastic plate. The ice thickness changes linearly. The problem is reduced to the problem of the wave profile across the channel, which is solved using the normal modes of an elastic beam with non-uniform thickness. It is shown that with the decrease in the change in the ice thickness, the modes approach the normal modes of an elastic beam with a constant thickness. The behavior of the dispersion relations of the hydroelastic waves depending on the parameter describing the change in the ice thickness is studied.

A simple single variable shear deformation theory for a rectangular beam

2016 ◽  
Vol 231 (24) ◽  
pp. 4576-4591 ◽  
Author(s):  
Rameshchandra P Shimpi ◽  
Rajesh A Shetty ◽  
Anirban Guha
Keyword(s):  
Shear Deformation ◽  
Cross Section ◽  
Governing Equation ◽  
Deformation Theory ◽  
Rectangular Cross Section ◽  
Shear Deformation Theory ◽  
Beam Theory ◽  
Single Variable ◽  
Bernoulli Beam ◽  
Euler Bernoulli Beam

This paper proposes a simple single variable shear deformation theory for an isotropic beam of rectangular cross-section. The theory involves only one fourth-order governing differential equation. For beam bending problems, the governing equation and the expressions for the bending moment and shear force of the theory are strikingly similar to those of Euler–Bernoulli beam theory. For vibration and buckling problems, the Euler–Bernoulli beam theory governing equation comes out as a special case when terms pertaining to the effects of shear deformation are ignored from the governing equation of present theory. The chosen displacement functions of the theory give rise to a realistic parabolic distribution of transverse shear stress across the beam cross-section. The theory does not require a shear correction factor. Efficacy of the proposed theory is demonstrated through illustrative examples for bending, free vibrations and buckling of isotropic beams of rectangular cross-section. The numerical results obtained are compared with those of exact theory (two-dimensional theory of elasticity) and other first-order and higher-order shear deformation beam theory results. The results obtained are found to be accurate.

scholarly journals Deformation of a cantilever curved beam with variable cross section

2021 ◽  
Vol 16 (1) ◽  
pp. 23-36
Author(s):  
István Escedi ◽  
Attila Baksa
Keyword(s):  
Cross Section ◽  
Elastic Beam ◽  
Beam Theory ◽  
Variable Cross Section ◽  
The Other ◽  
Curved Beam ◽  
Variable Cross ◽  
Cross Sectional ◽  
Euler Bernoulli Beam

This paper deals with the determination of the displacements and stresses in a curved cantilever beam. The considered curved beam has circular centerline and the thickness of its cross section depends on the circumferential coordinate. The kinematics of Euler-Bernoulli beam theory are used. The curved elastic beam is fixed at one end and on the other end is subjected to concentrated moment and force; three different loading cases are considered. The paper gives analytical solutions for radial and circumferential displacements and cross-sectional rotation and circumferential stresses. The presented examples can be used as benchmark for the other types of solutions as given in this paper.

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