# Maximum Shear Stress: Theory & Formula

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• 0:02 What Is Maximum Shear Stress?
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Lesson Transcript
Instructor: Hassan Alsaud

Earned my B.S. in Civil Engineering back in 2011. Have two years of experience in oil and gas fields and two year as a graduate research assistant. Earned my Master degree in Engineering from Tennessee State University in 2016.

In this section, we are going to learn about the maximum shear stress in a cross section of structural members such as beams and how they are distributed. We will also learn how rotating an element in a beam under normal stress would affect the normal and shear stress and how to evaluate the maximum of these values.

## What Is Maximum Shear Stress?

Shear force diagrams show the total shear force at each cross section of a structural member throughout the length of the beam or structural member. However, that force is not evenly distributed throughout the individual cross section of the beam or structural member. The maximum shear stress is the maximum concentrated shear force in a small area.

It is very critical for a structural engineer to locate and evaluate the maximum shear stress in a member in order to design the member in such a way to resist it. This does not apply to shear only; a structural engineer is to evaluate all possible load combinations and evaluate the maximum bending moment, maximum shear stress, maximum tensile stress, etc., and then he or she designs the structure accordingly.

As shown in the figure, a rectangular cross section beam under a known shear force will have a shear stress distributed parabolically, with zero values at extreme ends and the maximum value of the shear stress is located at the neutral axis of the beam. The neutral axis of a cross section is the axis at which the value of the normal stress and strain are equal to zero. Note that most often the neutral axis is located at the centroid of the area, but this is not always the case; as with excessive loads, the neutral axis shifts upwards.

And the value of the shear stress at any of the section is presented by this formula, where:

V = shear force in the cross section (as obtained from the shear force diagram)

Q = the first moment of area of the area above the plane upon which the desired shear stress is to be calculated

Q (for a rectangular beam) = A x Y

A = the area of the section above the desired plane

y = the distance from the centroid of the area to the neutral axis

I = the moment of inertia from the whole section about the centroid (second moment area)

I (for a rectangular section) = h x b = bh3 / 12

b = the width of the cross section

From this formula, the maximum shear stress of a rectangular beam can be written as shown:

## Section Rotated under Axial Load

When the structural member is only subjected to an axial load, taking a small element located in the middle of the member, the free body diagram would look as shown:

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