Shear Stress of a Beam Section

When a beam is subjected to nonuniform bending, both bending moments, M, and shear forces, V, act on the cross section. The shear force at any location along the beam can then be used to calculate the shear stress over the beam's cross section at that location.

Before you’ll go through this article to learn how to calculate the shear stress in beam sections, we want to remember that Civils.ai has a free tool where you can easily calculate it. Visit the page later for discovering more!

Let start from a beam of rectangular cross section, assuming that the shear stresses τ act parallel to the shear force V and the distribution of shear stresses is uniform across the width of the beam.

There will be horizontal shear stresses between horizontal layers, as well as, transverse shear stresses on the vertical cross section. These complementary shear stresses are equal in magnitude at any point of the beam.

The average shear stress over the cross section is given by:

The shear stress varies over the height of the cross section, as shown in the figure below:

The equation for shear stress at any point located a distance y1 from the centroid of the cross section is given by:

Where:

V = shear force acting at the location of the cross section

Ic = centroidal moment of inertia of the cross section

b = width of the cross section

Q = first moment of the area bounded by the point of interest and the extreme fiber of the cross section: 

Shear stress in Circular Cross Section

For the Circular Cross Section the shear stress is constant just at the centroid of a circular cross section.

Therefore, we can’t determine the shear stress along the height but only the maximum shear stress in the section (occurring at the centroid). The maximum shear stress is then calculated by:

Q = first moment of the area maximum value 

Shear stress in I-beams

The assumption that the shear stress along the width of the beam is constant is valid over the web of an I-Beam but not for the flanges. However, the web of an I-Beam takes approximately 90% - 98% of the shear force, so the web carries all of the shear force.

The shear stress along the web of the I-Beam is given by:

tw = web thickness

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