The shear coefficient for quartz crystal of rectangular cross section in Timoshenko's beam theory

2002 ◽  
Author(s):  
H. Kawashima
Keyword(s):  
Cross Section ◽  
Quartz Crystal ◽  
Rectangular Cross Section ◽  
Beam Theory ◽  
Shear Coefficient

On the Transverse Vibration of Beams of Rectangular Cross-Section

1986 ◽  
Vol 53 (1) ◽  
pp. 39-44 ◽  
Author(s):  
J. R. Hutchinson ◽  
S. D. Zillmer
Keyword(s):  
Exact Solution ◽  
Cross Section ◽  
Plane Stress ◽  
Timoshenko Beam ◽  
Transverse Vibration ◽  
Rectangular Cross Section ◽  
Beam Theory ◽  
Timoshenko Beam Theory ◽  
Shear Coefficient ◽  
Stress Solution

An exact solution for the natural frequencies of transverse vibration of free beams with rectangular cross-section is used as a basis of comparison for the Timoshenko beam theory and a plane stress approximation which is developed herein. The comparisons clearly show the range of applicability of the approximate solutions as well as their accuracy. The choice of a best shear coefficient for use in the Timoshenko beam theory is considered by evaluation of the shear coefficient that would make the Timoshenko beam theory match the exact solution and the plane stress solution. The plane stress solution is shown to provide excellent accuracy within its range of applicability.

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.

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.

Timoshenko Beam Theory Is Not Always More Accurate Than Elementary Beam Theory

1977 ◽  
Vol 44 (2) ◽  
pp. 337-338 ◽  
Author(s):  
J. W. Nicholson ◽  
J. G. Simmonds
Keyword(s):  
Cross Section ◽  
Plane Stress ◽  
Timoshenko Beam ◽  
Virtual Work ◽  
Rectangular Cross Section ◽  
Weighted Average ◽  
Beam Theory ◽  
Vertical Displacement ◽  
Timoshenko Beam Theory ◽  
Displacement Fields

A counterexample involving a homogeneous, elastically isotropic beam of narrow rectangular cross section supports the assertion in the title. Specifically, a class of two-dimensional displacement fields is considered that represent exact plane stress solutions for a built-in cantilevered beam subject to “reasonable” loads. The one-dimensional vertical displacement V predicted by Timoshenko beam theory for these loads can be regarded as an approximation to either the exact vertical displacement v at the center line, or a weighted average of v over the cross section, or a quantity defined to make the virtual work of beam theory equal to that of plane stress theory. Regardless of the interpretation of V and despite the presence of an adjustable shear factor, Timoshenko beam theory for this class of problems is never more accurate than elementary beam theory.

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