Timoshenko–Ehrenfest beam theory
The Timoshenko–Ehrenfest beam theory was developed by Stephen Timoshenko and Paul Ehrenfest[1][2][3] early in the 20th century.
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[6] and Rayleigh.
is the angle of rotation of the normal to the mid-surface of the beam, and
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
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
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
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.
Typical boundary conditions are: The strain energy of a Timoshenko beam is expressed as a sum of strain energy due to bending and shear.
axes and positive moments act in the clockwise direction.
) is such that positive bending moments compress the material at the bottom of the beam (lower
coordinates) and positive shear forces rotate the beam in a counterclockwise direction.
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
, leads to The second equation can then be written as Integration and application of the boundary condition
is the angle of rotation of the normal to the mid-surface of the beam, and
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
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.
Being a fourth order equation, there are four independent solutions, two oscillatory and two evanescent for frequencies below
-direction, then the governing equations of a Timoshenko beam take the form where
Any external axial force is balanced by the stress resultant where
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 a review of studies of the shear coefficient.
More recently, experimental data shows that the shear coefficient is underestimated.
Corrective shear coefficients for homogeneous isotropic beam according to Cowper - selection.
