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.

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.

On the Quest for the Best Timoshenko Shear Coefficient

2003 ◽  
Vol 70 (1) ◽  
pp. 154-157 ◽  
Author(s):  
M. B. Rubin
Keyword(s):  
Correction Factor ◽  
Timoshenko Beam ◽  
Theoretical Basis ◽  
Natural Frequencies ◽  
Beam Theory ◽  
Vibrational Frequencies ◽  
Timoshenko Beam Theory ◽  
Cosserat Theory ◽  
Shear Correction Factor ◽  
Shear Coefficient

Classical Timoshenko beam theory includes a shear correction factor κ which is often used to match natural vibrational frequencies of the beam. In this note, a number of static and dynamic examples are considered which provide a theoretical basis for specifying κ=1. Within the context of Cosserat theory, natural frequencies of the beam can be matched by appropriate specification of the director inertia coefficients with κ=1.

scholarly journals On the Accuracy of the Timoshenko Beam Theory Above the Critical Frequency: Best Shear Coefficient

2016 ◽  
Vol 32 (5) ◽  
pp. 515-518 ◽  
Author(s):  
J. A. Franco-Villafañe ◽  
R. A. Méndez-Sánchez
Keyword(s):  
Experimental Data ◽  
Timoshenko Beam ◽  
Critical Frequency ◽  
Beam Theory ◽  
Timoshenko Beam Theory ◽  
Least Square ◽  
Shear Coefficient ◽  
Least Square Fitting

AbstractWe obtain values for the shear coefficient both below and above the critical frequency by comparing the results of the Timoshenko beam theory with experimental data published recently. The best results are obtained, by a least-square fitting, when different values of the shear coefficient are used below and above the critical frequency.

A New Methodology for Modeling and Free Vibrations Analysis of Rotating Shaft Based on the Timoshenko Beam Theory

2016 ◽  
Vol 138 (2) ◽  
Author(s):  
S. H. Mirtalaie ◽  
M. A. Hajabasi
Keyword(s):  
Free Vibration ◽  
Cross Section ◽  
Timoshenko Beam ◽  
Variable Coefficient ◽  
Slenderness Ratio ◽  
Beam Theory ◽  
Free Vibrations ◽  
Timoshenko Beam Theory ◽  
Euler Angles ◽  
Shear Deformations

The linear lateral free vibration analysis of the rotor is performed based on a new insight on the Timoshenko beam theory. Rotary inertia, gyroscopic effects, and shear deformations are included, but the torsion is neglected and a new dynamic model is presented. It is shown that if the total rotation angle of the beam cross section is considered as one of the degrees-of-freedom of the Timoshenko rotor, as is common in the literature, some terms are missing in the modeling of the global dynamics of the system. The total deflection of the beam cross section is divided into two steps, first the Euler angles relations are employed to establish the curved geometry of the beam due to the elastic deformation of the beam centerline and then the shear deformations was superposed on it. As a result of this methodology and the mutual interaction of shear and Euler angles some variable coefficient terms appeared in the kinetic energy of the system which makes the problem be classified as the parametrically excited systems. A linear coupled variable coefficient system of differential equations is derived while the variable coefficient terms have been missing in all previous studies in the literature. The free vibration behavior of parametrically excited system is investigated by perturbation method and compared with the common Rayleigh, Timoshenko, and higher-order shear deformable spinning beam models in the rotordynamics. The effects of rotating speed and slenderness ratio are studied on the forward and backward natural frequencies and the critical speeds of the system are examined. The study demonstrates that the shear and Euler angles interaction affects the high-frequency free vibrations behavior of the spinning beam especially for higher slenderness ratio and rotating speeds of the rotor.

scholarly journals Timoshenko Beam Theory Exact Solution For Bending, Second-Order Analysis, and Stability

2020 ◽  
Author(s):  
Valentin Fogang
Keyword(s):  
Exact Solution ◽  
Timoshenko Beam ◽  
Beam Theory ◽  
Second Order ◽  
Timoshenko Beam Theory ◽  
Bernoulli Beam ◽  
Closed Form Solutions ◽  
Order Analysis ◽  
Stiffness Matrices ◽  
Order Element

This paper presents an exact solution to the Timoshenko beam theory (TBT) for bending, second-order analysis, and stability. The TBT covers cases associated with small deflections based on shear deformation considerations, whereas the Euler–Bernoulli beam theory neglects shear deformations. A material law (a moment-shear force-curvature equation) combining bending and shear is presented, together with closed-form solutions based on this material law. A bending analysis of a Timoshenko beam was conducted, and buckling loads were determined on the basis of the bending shear factor. First-order element stiffness matrices were calculated. Finally second-order element stiffness matrices were deduced on the basis of the same principle.

scholarly journals A new method to determine the shear coefficient of Timoshenko beam theory

2011 ◽  
Vol 330 (14) ◽  
pp. 3488-3497 ◽  
Author(s):  
K.T. Chan ◽  
K.F. Lai ◽  
N.G. Stephen ◽  
K. Young
Keyword(s):  
Timoshenko Beam ◽  
Beam Theory ◽  
Timoshenko Beam Theory ◽  
New Method ◽  
Shear Coefficient
Sign in / Sign up

Export Citation Format

Share Document