The Timoshenko–Ehrenfest beam theory was developed by Stephen Timoshenko and Paul Ehrenfest early in the 20th century. The model takes into account shear deformation and rotational bending effects, making it suitable for describing the behaviour of thick beams, sandwich composite beams, or beams subject to high-frequency excitation when the wavelength approaches the thickness of the beam. The resulting equation is of fourth order but, unlike Euler–Bernoulli beam theory, there is also a second-order partial derivative present. Physically, taking into account the added mechanisms of deformation effectively lowers the stiffness of the beam, while the result is a larger deflection under a static load and lower predicted eigenfrequencies for a given set of boundary conditions. The latter effect is more noticeable for higher frequencies as the wavelength becomes shorter (in principle comparable to the height of the beam or shorter), and thus the distance between opposing shear forces decreases. Rotary inertia effect was introduced by Bresse and Rayleigh. 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 towards Euler–Bernoulli beam theory.
Quasistatic Timoshenko beam
In static Timoshenko beam theory without axial effects, the displacements of the beam are assumed to be given by
u x ( x , y , z ) = − z φ ( x ) ; u y ( x , y , z ) = 0 ; u z ( x , y ) = w ( x ) {\displaystyle u_{x}(x,y,z)=-z~\varphi (x)~;~~u_{y}(x,y,z)=0~;~~u_{z}(x,y)=w(x)}
where ( x , y , z ) {\displaystyle (x,y,z)} are the coordinates of a point in the beam, u x , u y , u z {\displaystyle u_{x},u_{y},u_{z}} are the components of the displacement vector in the three coordinate directions, φ {\displaystyle \varphi } is the angle of rotation of the normal to the mid-surface of the beam, and w {\displaystyle w} is the displacement of the mid-surface in the z {\displaystyle z} -direction. The governing equations are the following coupled system of ordinary differential equations:
d 2 d x 2 ( E I d φ d x ) = q ( x ) d w d x = φ − 1 κ A G d d x ( E I d φ d x ) . {\displaystyle {\begin{aligned}&{\frac {\mathrm {d} ^{2}}{\mathrm {d} x^{2}}}\left(EI{\frac {\mathrm {d} \varphi }{\mathrm {d} x}}\right)=q(x)\\&{\frac {\mathrm {d} w}{\mathrm {d} x}}=\varphi -{\frac {1}{\kappa AG}}{\frac {\mathrm {d} }{\mathrm {d} x}}\left(EI{\frac {\mathrm {d} \varphi }{\mathrm {d} x}}\right).\end{aligned}}}
The Timoshenko beam theory for the static case is equivalent to the Euler–Bernoulli theory when the last term above is neglected, an approximation that is valid when
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