# Timoshenko Beam Theory for Quasistatic Cantilever Beam: Shear Term

I'm trying to determine the effect of shearing force on the deflection of a quasistatic cantilever beam. In his book, Theory of Elasticity, Timoshenko added the following term to the deflection:

$$\frac{Pc^2}{2IG} * (l-x)$$

but the Wikipedia article on Timoshenko beam theory added the term

$$\frac{P(l-x)}{kAG}$$ https://en.wikipedia.org/wiki/Timoshenko_beam_theory

Both of these reduce to the Euler-Bernoulli form when shear is disregarded. What is the difference between these two forms? Should both be included in the deflection?

• Welcome to Engineering Stack Exchange. This one made me dig. Nice question. – Mark Jul 26 '18 at 16:56

Timoshenko originally solved the equation with a $k$ factor. The $c^2$ factor came as a result of Timoshenko's analysis. The goal was to keep $k$ as a constant, which would make the equations easier. Unfortunately, $k$ is actually a function of frequency, which is unknown. This can be seen from a finite element or other elasticity analysis of the beam cross section with dynamic loading, and analyzing the shear stress in the cross sectional area.
The fundamental longitudinal wave frequency, therefore was considered highly important. Since this is a property that is based upon the material itself, as the $c$ in $c^2$ is the "speed of sound in the material", it made sense to find the value for $k$ such that Timoshenko's equations predicted this frequency. In other words, since $k$ is a function of frequency, when Timoshenko's equations were set to describe the longitudinal wave, the value for $k$ would correctly predict the first fundamental frequency. The net result is the $c^2$ term, which is accurate for this desired value of $k$. Other values of k can occur. For more information, there is a declassified resource from the Army Services I was able to consult - AD013061