# Calculate the Maximum Shearing Stress in a Cross Section For a Particular State of Stress

Identify the stress state. Determine the maximum shearing stress that a solid shaft can withstand. For example, a solid shaft can withstand the maximum shearing stress when the load is applied to its centre of mass. A cantilever box is another example. A cross section of a cantilever box is a third example. The maximum shearing stress depends on the design requirements of the cantilever box.

## Calculating maximum shearing stress in a cantilever box

A cantilever box is a flexible beam that is hung from a central axis. Its maximum shearing stress is 50% greater than its average stress. The distance from the center of the beam to the extremities of the limbs determines the shearing stress at each of the extremities. The distance between the two extremities is the centroidal distance.

The shearing force and bending moment are two components of the moment diagram. The shear force is the applied force while the bending moment is the elongation of the top of the beam. The sign of the bending moment and the shear force is the same. The moment diagram has a straight slope and a point where the maximum and minimum moments occur.

In some cases, the area at which the angle of twist occurs is the maximum shearing stress. This angle is called the angle of twist. If this angle exceeds the maximum shearing stress of the beam, the stress in the cantilever box will be zero. For larger angles, the angle of twist equals the maximum shearing stress of the beam.

## Calculating maximum shearing stress in a solid shaft

To calculate the maximum shearing stress in a solid motor shaft, you need to know its shear modulus. This stress is the bending moment that is applied to the shaft when it twists in the direction of the torque. The ratio of the shear modulus to the torque is the shear strength. The maximum shearing stress of a solid motor shaft is equal to its torque times the angle of twist.

You can also use the polar moment of inertia to find the resistance to twisting deformation. These forces are similar to rotational kinetics and bending of beams. General formulas are provided in Textbook Appendix C. Once you have determined the polar moment, you will know the maximum shearing stress in a solid motor shaft.

The shear stresses on the x and y faces are represented by black lines in Mohr’s circle. The radial stresses on these faces are equal and opposite. Therefore, you will get the maximum and minimum shearing stresses on each face. The normal stress is the maximum stress and the minimum is the shearing stress. Once you have calculated both, you can determine the angle of rotation of the shaft and the maximum and minimum shearing stress of that shaft.

## Calculating maximum shearing stress in a cross section

In the following example, we will see how to calculate the maximum shearing stress in a cross-section for a particular state of stresses. We will take a cold-formed section, subjected to a torque of 50 Nm, and calculate the maximum shear stress in the cross-section. The resulting maximum stress is 220.6 N/mm2.

The stress element surfaces are arranged so that Mohr’s circle passes through them on opposite sides. The radii of the circles indicate the maximum and minimum normal stress of a surface. The radii of the circles are the maximum and minimum shear stress. The maximum and minimum normal stresses are equal when the planes are perpendicular to one another.

Now let’s use a simple example of transverse shear to show how shear stress is calculated in a cross section. Consider a rectangular beam with a span of 1.8 m and a concentrated load of P=150 kN at the midpoint. The maximum shear stress of the beam is tmax / d2 – where sx is the beam diameter and tmax is its transverse shear force.

Now we have a thin-walled cantilever beam with a closed rectangular cross section subjected to constant torque of 90500 N m at a point R. Moreover, the torque is uniformly distributed along the length of the beam. The Bredt-Batho theory of torsion allows us to calculate the maximum shear stress in a cross section using the maximum twist and shear modulus. Moreover, we can even find the distribution of twist and warping displacement within the cross-section.