# Torsional and Shearing Stress Measurement of Axis

When an object is twisted, shearing stress τoccurs. At the same time, tensile stress and compressive stress, which are equivalent to the shearing stress, occur in 2 directions inclined by 45° from the axial line.
In measurement of axial twist under simple shearing stress condition, a strain gage does not directly measure the shearing stress. Instead, a strain gage detects tensile or compressive strain resulting from tensile or compressive stress simultaneously generated with the shearing stress. These stress conditions on a surface of axis are illustrated below.

Shearing stress γ is defined as illustrated below, and the magnitude is calculated through the following equation:

When the axis is twisted, point A moves to point B, thereby initiating torsional angle θ .

## Stress Measurement with Quarter-bridge System

Bond the strain gage on the twisted axis in the direction inclined by 45° from the axial line. The relation between strain ε and stress σ are expressed with the following equation to calculate tensile or compressive stress σ :

## Stress Measurement with Half-bridge or Full-bridge System

Half-bridge or full-bridge system increases strain output by 2 (half-bridge system) or 4 times (full-bridge system), because each strain gage in the half-bridge or full-bridge system detects equal strain. To calculate true strain, divide measured strain by 2 (half-bridge system) or 4 (full-bridge system).

## Application to Torque Measurement

Strain on the surface of the axis is proportional to the torque applied to the axis. Thus, the torque is obtained by detecting the strain on the surface. Shearing stress distributed on the lateral section is balanced with the applied torque T, establishing the following equation:

where, Zp: Polar modulus of section

Converting shearing stress in the above equation to tensile strain produces an equation as follows:

The polar modulus of the section is specific to each shape of the cross-section as follows:

A strain gage torque transducer is designed using the above relational expression of ε0and T. Obtain ε0 from the allowable stress for the material, and determine the width d of the axis which is matched with the magnitude of the applied torque. Then, amplify the strain output with a strain amplifier and read the output voltage with a measuring instrument.