The Timoshenko–Ehrenfest beam theory was developed by Stephen Timoshenko and Paul Ehrenfest[1][2][3] early in the 20th century.[4][5] 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 4th 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[6] and Rayleigh.[7]
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.
In static Timoshenko beam theory without axial effects, the displacements of the beam are assumed to be given by
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 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:
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
where
Combining the two equations gives, for a homogeneous beam of constant cross-section,
The bending moment M x x {\displaystyle M_{xx}} and the shear force Q x {\displaystyle Q_{x}} in the beam are related to the displacement w {\displaystyle w} and the rotation φ φ --> {\displaystyle \varphi } . These relations, for a linear elastic Timoshenko beam, are:
Then, from the strain-displacement relations for small strains, the non-zero strains based on the Timoshenko assumptions are
Since the actual shear strain in the beam is not constant over the cross section we introduce a correction factor κ κ --> {\displaystyle \kappa } such that
The variation in the internal energy of the beam is
Define
Then
Integration by parts, and noting that because of the boundary conditions the variations are zero at the ends of the beam, leads to
The variation in the external work done on the beam by a transverse load q ( x , t ) {\displaystyle q(x,t)} per unit length is
Then, for a quasistatic beam, the principle of virtual work gives
The governing equations for the beam are, from the fundamental theorem of variational calculus,
For a linear elastic beam
Therefore the governing equations for the beam may be expressed as
Combining the two equations together gives
The two equations that describe the deformation of a Timoshenko beam have to be augmented with boundary conditions if they are to be solved. Four boundary conditions are needed for the problem to be well-posed. Typical boundary conditions are:
The strain energy of a Timoshenko beam is expressed as a sum of strain energy due to bending and shear. Both these components are quadratic in their variables. The strain energy function of a Timoshenko beam can be written as,
For a cantilever beam, one boundary is clamped while the other is free. Let us use a right handed coordinate system where the x {\displaystyle x} direction is positive towards right and the z {\displaystyle z} direction is positive upward. Following normal convention, we assume that positive forces act in the positive directions of the x {\displaystyle x} and z {\displaystyle z} axes and positive moments act in the clockwise direction. We also assume that the sign convention of the stress resultants ( M x x {\displaystyle M_{xx}} and Q x {\displaystyle Q_{x}} ) is such that positive bending moments compress the material at the bottom of the beam (lower z {\displaystyle z} coordinates) and positive shear forces rotate the beam in a counterclockwise direction.
Let us assume that the clamped end is at x = L {\displaystyle x=L} and the free end is at x = 0 {\displaystyle x=0} . If a point load P {\displaystyle P} is applied to the free end in the positive z {\displaystyle z} direction, a free body diagram of the beam gives us
and
Therefore, from the expressions for the bending moment and shear force, we have
Integration of the first equation, and application of the boundary condition φ φ --> = 0 {\displaystyle \varphi =0} at x = L {\displaystyle x=L} , leads to
The second equation can then be written as
Integration and application of the boundary condition w = 0 {\displaystyle w=0} at x = L {\displaystyle x=L} gives
The axial stress is given by
In Timoshenko beam theory without axial effects, the displacements of the beam are assumed to be given by
Starting from the above assumption, the Timoshenko beam theory, allowing for vibrations, may be described with the coupled linear partial differential equations:[8]
where the dependent variables are w ( x , t ) {\displaystyle w(x,t)} , the translational displacement of the beam, and φ φ --> ( x , t ) {\displaystyle \varphi (x,t)} , the angular displacement. Note that unlike the Euler–Bernoulli theory, the angular deflection is another variable and not approximated by the slope of the deflection. Also,
These parameters are not necessarily constants.
For a linear elastic, isotropic, homogeneous beam of constant cross-section these two equations can be combined to give[9][10]
From equation (1), assuming appropriate smoothness, we have
Differentiating equation (2) gives
Substituting equation (3), (4), (5) into equation (6) and rearrange, we get
However, it can easily be shown that this equation is incorrect. Consider the case where q is constant and does not depend on x or t, combined with the presence of a small damping all time derivatives will go to zero when t goes to infinity. The shear terms are not present in this situation, resulting in the Euler-Bernoulli beam theory, where shear deformation is neglected.
The Timoshenko equation predicts a critical frequency ω ω --> C = 2 π π --> f c = κ κ --> G A ρ ρ --> I . {\displaystyle \omega _{C}=2\pi f_{c}={\sqrt {\frac {\kappa GA}{\rho I}}}.} For normal modes the Timoshenko equation can be solved. Being a fourth order equation, there are four independent solutions, two oscillatory and two evanescent for frequencies below f c {\displaystyle f_{c}} . For frequencies larger than f c {\displaystyle f_{c}} all solutions are oscillatory and, as consequence, a second spectrum appears.[11]
If the displacements of the beam are given by
where u 0 {\displaystyle u_{0}} is an additional displacement in the x {\displaystyle x} -direction, then the governing equations of a Timoshenko beam take the form
where J = ρ ρ --> I {\displaystyle J=\rho I} and N ( x , t ) {\displaystyle N(x,t)} is an externally applied axial force. Any external axial force is balanced by the stress resultant
where σ σ --> x x {\displaystyle \sigma _{xx}} is the axial stress and the thickness of the beam has been assumed to be 2 h {\displaystyle 2h} .
The combined beam equation with axial force effects included is
If, in addition to axial forces, we assume a damping force that is proportional to the velocity with the form
the coupled governing equations for a Timoshenko beam take the form
and the combined equation becomes
A caveat to this Ansatz damping force (resembling viscosity) is that, whereas viscosity leads to a frequency-dependent and amplitude-independent damping rate of beam oscillations, the empirically measured damping rates are frequency-insensitive, but depend on the amplitude of beam deflection.
Determining the shear coefficient is not straightforward (nor are the determined values widely accepted, i.e. there's more than one answer); generally it must satisfy:
The shear coefficient depends on Poisson's ratio. The attempts to provide precise expressions were made by many scientists, including Stephen Timoshenko,[12] Raymond D. Mindlin,[13] G. R. Cowper,[14] N. G. Stephen,[15] J. R. Hutchinson[16] etc. (see also the derivation of the Timoshenko beam theory as a refined beam theory based on the variational-asymptotic method in the book by Khanh C. Le[17] leading to different shear coefficients in the static and dynamic cases). In engineering practice, the expressions by Stephen Timoshenko[18] are sufficient in most cases. In 1975 Kaneko[19] published an excellent review of studies of the shear coefficient. More recently new experimental data show that the shear coefficient is underestimated.[20][21]
Corrective shear coefficients for homogeneous isotropic beam according to Cowper - selection.
where ν ν --> {\displaystyle \nu } is Poisson's ratio.
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