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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, $\varepsilon_{xy}$?

enter image description here


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.

enter image description here

(above image adopted from here)

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    $\begingroup$ Your definition is wrong. Shear strain is not defined as the "average" deformation over the whole length of the beam.. See any mechanics textbook for the correct definition of the different measures for shear strain. $\endgroup$
    – alephzero
    Jul 24 at 13:37
  • $\begingroup$ @alephzero, Hello, this is not my definition, this is how I understood from the mentioned FEM code assuming a single Gauss point. I updated the question to prevent confusion. You can see the displacement-strain relation matrix, [B], in the code from which I extracted that definition here: [github.com/oofem/oofem/blob/…. I think this simplification makes sense as more than one element with single Gauss points can be used to model whole length of the beam. $\endgroup$
    – Saeid
    Jul 24 at 14:02
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Shear strain is defined as the angular deformation caused due to parallel or shearing force. In structural engineering, there are two cases of shear strains that are of particular concern. The occurrence of each depends on the loading that causes shear deformation as indicated below.

1) Shear Strain due to Pure Shear

enter image description here

2) Shear Strain due to Shear Deformation

enter image description here

Note, shear deformation becomes a concern when L/d (Span/beam depth) $\leq$ 10. Otherwise, it is usually ignored.

Edit:

For typical beams that follow the beam theory (Euler-Bernoulli or Timoshenko), as stated above, the shear deformation is very small thus usually ignored. The measured strain is, therefore, only consisted of the horizontal stretching and shortening of the extreme fibers in the direction of the longitudinal axial axis, which I think is your case.

enter image description here

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    $\begingroup$ To the downvoters, as a courtesy and for the better of engineering, you shall at least leave a comment or provide your answer as a contrast to mine that you seem to have a problem with. Thanks. $\endgroup$
    – r13
    Jul 25 at 1:46
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    $\begingroup$ This article may help. Scroll down and read "Tensor Shear Terms" near the middle of the paper. continuummechanics.org/…. $\endgroup$
    – r13
    Jul 25 at 3:14
  • $\begingroup$ Thanks for the answer, my concern for asking this question is the method to calculate stress from shear strain. It seems that there is a contradictory on how the calculated shear strain is converted into stress. I have seen some papers that use the calculated Shear Strain in my question as engineering strain (gamma in your answer) $𝜏=𝐺.𝛾$, some others (for similar elements for lattice network) use it as tensorial shear strain which for isotropic materials is $\epsilon_{𝑥𝑦}=𝛾/2$. That is they use 2G instead of G. From your answer it seems you believe it should be engineering shear strain. $\endgroup$
    – Saeid
    Jul 25 at 3:16
  • $\begingroup$ @r13 It's a pipedream expecting downvoters to leave intelligent comments sadly. Perhaps the voting system should be changed to upvotes or don't vote, then it would avoid this... But that has been suggested... $\endgroup$
    – Solar Mike
    Jul 26 at 8:49
  • $\begingroup$ @SolarMike I think downvote forces the answerer to look deeper into the problem, which is not a bad thing, but some were abusing the power without the good intention/wish. I think the downvoter should be given the binary options - to write a comment or costing 10 points for downvote without reason. Thanks for sharing your view though. $\endgroup$
    – r13
    Jul 26 at 14:58

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