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The tale of shear coefficients in Timoshenko–Ehrenfest beam theory: 130 years of progress

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Abstract

The first-order shear deformable beam theory should be named after Stephen Timoshenko and Paul Ehrenfest in recognition of the significant contribution of both of them. The Timoshenko–Ehrenfest beam model can predict the flexure mechanics of short stubby beams with adequate accuracy if it has been enriched with a proper shear coefficient. The shear coefficient lies at the heart of the Timoshenko–Ehrenfest beam theory was apparently first introduced by Friedrich Engesser in the nineteenth century. Detecting the appropriate formula of the shear coefficient for solid rectangular cross-sections was surprisingly a challenging issue in the literature. A stationary variational framework of the Timoshenko–Ehrenfest beam, founded on the elasticity theory, is conceived and applied to set forth the variationally consistent shear coefficient of a prismatic beam of a solid rectangular cross-section. Evidence of efficacy of the introduced shear coefficient is illustrated as the intrinsic anomalies of the counterpart shear coefficients are thoroughly discussed. The present study may pave the way ahead in appreciating the significance of implementing the apposite formula of the shear coefficient associated with the Timoshenko–Ehrenfest beam theory.

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Abbreviations

A :

Cross-sectional area

\({\mathfrak{A}}\) :

Rectangular cross-section

\({\mathbf{b}}\) :

Body force

b :

Half width of cross-section

\({\mathcal{B}}\) :

General bounded region

\(\partial {\mathcal{B}}\) :

Closed boundary of the region

\({\mathbf{E}}\) :

Strain field

E :

Young’s modulus

\(\Im\) :

Stationary variational functional

G :

Shear modulus

h :

Half height of cross-section

I :

Second area moment of the cross-section

\(k\) :

Shear coefficient

\(\ell\) :

Length of beam

m :

Distributed bending couple

M :

Flexural moment field

q :

Distributed transversal loading

\({\mathbb{S}}\) :

Compliance tensor of elasticity

\({\mathbf{t}}\) :

Surface traction

\({\mathbf{T}}\) :

Stress field of the inflected elastic beam

\(u\) :

Centroidal deflection

\({\mathbf{u}}\) :

Displacement field of the beam

\(V\) :

Distribution of the shear force

\(x,y,z\) :

Cartesian orthogonal coordinates

\(\chi\) :

Auxiliary parameter

\(\nu\) :

Poisson’s ratio

\(\rho\) :

Destiny of the beam

\(\psi\) :

Total rotation in the x–z plane

\(\zeta\) :

Aspect ratio of the rectangle

\(\Upsilon\) :

Shear flexibility of the cross-section

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Authors and Affiliations

  1. Department of Mechanical Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran

    S. Ali Faghidian

  2. Department of Ocean and Mechanical Engineering, Florida Atlantic University, Boca Raton, FL, USA

    Isaac Elishakoff

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Faghidian, S.A., Elishakoff, I. The tale of shear coefficients in Timoshenko–Ehrenfest beam theory: 130 years of progress. Meccanica 58, 97–108 (2023). https://doi.org/10.1007/s11012-022-01618-1

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