Recently I came across a FEM code of a linear deformation beam element, and it made me wonder what is the correct relation between shear stress and shear strain in a beam element.
In the beam element of the image below, it seems that shear strain is defined as Shear Strain = $\delta_{v}/L$, with L being the length of the element and (if I am not wrong) the shear deformation, $\delta_{v}$, be calculated from $$\delta_{v}=(\delta_{1}-\delta_{2})+(\phi_1+\phi_2)\frac{L}{2},$$ $\delta_{i}$ and $\phi_i$ being the vertical displacement and the rotations of the ends,
in this case, is the Shear Strain, the engineering shear strain, $\gamma$ or is it the tensor shear strain, $\varepsilon_{xy}$?
A comment on the answer:
As stated by a comment below, the definition of shear may not be same for the beam elements as the assumptions in classic mechanics of materials. It seems to me that the definition of engineering strain and tensorial strain becomes vague when it comes to beam elements.
Usually beams have free faces without stress, so the equal shear stresses assumed for an infinitesimal element with small deformation concept is not applicable here. That is the shear stresses change along the height of the beam and thus we need a coefficient to account for these changes. A common solution is that the cross-section area is modified by a beam shear coefficient and a shear area is defined: $$A_s = coef.A$$ So now the shear stiffness, that depends on the area, is also modified by coef.
On less common cases where a similar element is used to simulate a volume (like a lattice network model), I see that they use 2G instead of G (for example see equation (22) in T. Kawai 1978). This is like the idea that they are assuming that the Shear Strain is tensorial shear strain thus 2G should be used instead of G. But another interpretation can be that the coef. in this case is assumed equal to 2 since the beam does not have free surfaces.
(above image adopted from here)
