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?

  • $\begingroup$ Welcome to Engineering Stack Exchange. This one made me dig. Nice question. $\endgroup$
    – Mark
    Jul 26, 2018 at 16:56

1 Answer 1


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.

In most low-frequency applications (like we see today), the effects from k can be ignored, so Timoshenko ignored these values. Instead, he focused on when the Euler equations began to deviate. For example, the Euler equations could not predict the phenomenon of a longitudinal wave. This occurs when a wave travels along the axis of a beam - like when stretching a spring:

enter image description here

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

  • $\begingroup$ Thanks for your reply. Based on the diagram in the book, c is actually the distance from the centroid to the side (c needs to be a length, otherwise the units don't work out in the equation). I'm not concerned with the dynamic model, just the static model. $\endgroup$
    – Bryhed
    Jul 26, 2018 at 23:40

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