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Simple Structural Members
Published in Ansel C. Ugural, Plates and Shells, 2017
Interestingly, the Timoshenko theory of beams constitutes an improvement over the ordinary theory. In static case, the difference between the two hypotheses is that the former includes the effect of shear stresses on the deformation by assuming a constant shear over the beam height. The latter ignores the influence of transverse shear on beam deformation. Thus, the Timoshenko beam theory is an extension of the Bernoulli–Euler theory to allow for the effect of the transverse shear deformation while relaxing the assumption 2 stated above. The Timoshenko theory is highly suited for describing the behavior of short and sandwich beams. In dynamic case, the theory incorporates shear deformation as well as rotational inertia effects, and it will be more accurate for very slender beams.
Theory of Beams and Columns
Published in K.T. Chau, Applications of Differential Equations in Engineering and Mechanics, 2019
where κ is called shear coefficient and has been considered by many, including Timoshenko, Mindlin, Cowper, Stephen, Hutchinson, Kaneko and many others. This shear coefficient is resulted from the fact that the shear deformation effect is replaced by a single parameter β in Timoshenko beam theory. In fact, shear stress must increase from zero at the top surface of the beam and return to zero at the bottom surface of the beam. Clearly, this area average factor κ must be a function of the cross-section (different cross-section will have different shear stress distribution). Depending on the cross-section, Kaneko (1975) suggested the use of the following formulas for κ κ=5(1+ν)6+5ν,rectangular=6+12ν+6ν27+12ν+4ν2,circular $$ \begin{gathered} \kappa = \frac{{5(1 + \nu )}}{{6 + 5\nu }},\quad \quad \quad {\text{rectangular}} \\ = \frac{{6 + 12\nu + 6\nu ^{2} }}{{7 + 12\nu + 4\nu ^{2} }},\quad {\text{circular}} \\ \end{gathered} $$
Continuous Models for Vibration
Published in Haym Benaroya, Mark Nagurka, Seon Han, Mechanical Vibration, 2017
Haym Benaroya, Mark Nagurka, Seon Han
The Timoshenko beam theory, one of his many contributions, takes into account shear deformation and rotational inertia effects, making it suitable for describing the behavior of short beams, sandwich composite beams, and beams subject to high-frequency excitation when the wavelength approaches the thickness of the beam. The added effects lower the stiffness of the beam. If the shear modulus of the beam material approaches infinity -and thus the beam becomes rigid in shear and if rotational inertia effects are neglected, Timoshenko beam theory converges to ordinary beam theory, that is, Bernoulli-Euler theory.
Crack identification in beam-type structural elements using a piezoelectric sensor
Published in Nondestructive Testing and Evaluation, 2021
Goutam Roy, Brajesh Panigrahi, G. Pohit
In the present study, the mathematical model of the cracked beam under static loading is considered based on Timoshenko beam theory. The Timoshenko beam theory deals with two field variables, as it includes the effect of pure bending and the shear. Therefore, two field variables, namely ,are considered, for the flexure and shear, respectively. The deflection of the beam is represented as w(x).Since the slope of the deflection curve at a point represents the rotational displacements, the relationship between w and can be expressed as follows: ;
Optimal control of vibrations of two nonlinear Gao beams connected with a joint
Published in International Journal of Systems Science, 2021
In mathematics, a large number of such physical structure can be abstracted into the beam equations depicted by differential equations (Do, 2020; Tavasoli, 2018). The common beam equations include the Euler–Bernoulli beam and the Timoshenko beam. The Euler–Bernoulli beam theory is used to model the behaviour of long beams that flexure is superior, while the Timoshenko theory applies for shear-dominated short beams. For some mid-length beams, the deterministic displacement responses for the two theories agrees very well (Beck & da Silva, 2011). As the classical beam in engineering, the Euler–Bernoulli beam, is applied on a large scale in practice. Its theory covers the case for small deflections of a beam that are subjected to lateral loads only. It is argued that the Euler–Bernoulli beam theory is a special case of Timoshenko beam theory accordingly (Wikipedia, 2020).
Evaluation of vertically and laterally loaded bucket foundation in clay using a three-dimensional displacement approach
Published in Marine Georesources & Geotechnology, 2019
Since shear deformations are fundamentally important for short and high beams, Timoshenko beam theory (Timoshenko 1921; Krenk 2001) should be adopted. Both bending and shear deformations are defined by the coupled equations given by Timoshenko beam theory in terms of both displacement and rotation. However, it may be difficult to obtain the analytical solutions of the coupled equations for complicated boundary conditions. Therefore, Bernoulli–Euler beam theory is used in this article, considering the contribution of the bending deformation and neglecting the influence of the shear deformation. For a bucket foundation subjected to the lateral force PH(h) at depth h when the vertical load V is compression, Hetenyi (1946) presented the differential equation of the lateral displacement on the basis of Bernoulli–Euler beam theory in the following form:where EBIB = EPIP + ESIS is the bending stiffness combined with the bucket and internal soil plug, in which IP and IS are the moment of inertia of the bucket and that of the internal soil plug, respectively; u(h) is the horizontal displacement of the bucket foundation in the x-direction at the depth h; and the lateral force is determined as