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The formula to calculate average shear stress τ or force per unit area is: [1] =, where F is the force applied and A is the cross-sectional area.. The area involved corresponds to the material face parallel to the applied force vector, i.e., with surface normal vector perpendicular to the force.
Consider the case where q is constant and does not depend on x or t, combined with the presence of a small damping all time derivatives will go to zero when t goes to infinity. The shear terms are not present in this situation, resulting in the Euler-Bernoulli beam theory, where shear deformation is neglected.
The stress due to shear force is maximum along the neutral axis of the beam (when the width of the beam, t, is constant along the cross section of the beam; otherwise an integral involving the first moment and the beam's width needs to be evaluated for the particular cross section), and the maximum tensile stress is at either the top or bottom ...
This is only the average stress, actual stress distribution is not uniform. In real world applications, this equation only gives an approximation and the maximum shear stress would be higher. Stress is not often equally distributed across a part so the shear strength would need to be higher to account for the estimate. [2]
Stress resultants are simplified representations of the stress state in structural elements such as beams, plates, or shells. [1] The geometry of typical structural elements allows the internal stress state to be simplified because of the existence of a "thickness'" direction in which the size of the element is much smaller than in other directions.
Timoshenko improved upon that theory in 1922 by adding the effect of shear into the beam equation. Shear deformations of the normal to the mid-surface of the beam are allowed in the Timoshenko–Rayleigh theory. The equation for the bending of a linear elastic, isotropic, homogeneous beam of constant cross-section under these assumptions is [7 ...
As described above, in thin-walled structures, the variation along the thickness of the member can be neglected, so the shear stress across the cross section of a beam that is composed of thin-walled elements can be examined as shear flow, or the shear stress multiplied by the thickness of the element. [2]
In that case, the shear stress on each cross-section is parallel to the cross-section, but oriented tangentially relative to the axis, and increases with distance from the axis. Significant shear stress occurs in the middle plate (the "web") of I-beams under bending loads, due to the web constraining the end plates ("flanges").