Beam shear correction coefficient ($k_s$)

The beam shear correction coefficient $k_s$ adjusts a reduced beam model's shear response for the nonuniform shear stress over a material section. With shear modulus $G$, area $A$, transverse force resultant $V$ and generalized beam shear angle $\gamma$,where $S$ is effective beam shear rigidity. Bower's primary account calls this the Timoshenko shear coefficient and explains that its value depends on geometry, Poisson's ratio and how the approximate theory is matched to the underlying solid.^[ref]

An original energy-matching derivation makes the correction precise for one approximation. Assume uniform $G$ and a chosen shear-stress field $\tau$ with resultant $V={\int }_A\tau dA$. The shear elastic energy per unit beam length and its reduced representation areThusThe Cauchy-Schwarz inequality gives $V^2\le A{\int }_A{\tau }^{2}dA$, hence $k_s\le 1$ for this definition. Equality requires uniform stress. This bound belongs to the stated energy-matching definition rather than every convention called a shear correction coefficient.

For the parabolic rectangular stress approximation used in effective beam shear rigidity, the squared-stress integral is $6V^2/(5A)$, giving $k_s=5/6$. Choose $G=80\mathrm{GPa}$, $A=100\mathrm{m}{\mathrm{m}}^2$ and $V=1000\mathrm{N}$. Then $S=6.667\times 10^6\mathrm{N}$ and $\gamma =1.5\times 10^{-4}$ radians. Omitting the correction gives $1.25\times 10^{-4}$, underestimating this generalized shear angle by one sixth.

The generalized angle also differs from the peak local shear strain. Under the same parabolic approximation, peak stress is $3V/(2A)$, giving peak strain $1.875\times 10^{-4}$. The values describe different reductions of the same chosen stress field. Identifying $k_s$ from a bending test therefore requires the section model and displacement readout, not merely a material modulus. For a three-point bending test, the shear deflection contribution varies as $PL/(4k_{s}GA)$, whereas bending varies as $P{L}^3/(48B)$; a span comparison can help separate them.