# How to Calculate Maximum Shear Stress in a Lug Wrench Arm

## What is the maximum shear stress in the arm of a lug wrench?

When removing a wheel to change a tire, a driver applies forces with a value of P = 25 lb at the ends of two of the arms of a lug wrench. The wrench is made of steel with a shear modulus of elasticity G = 11.4 x 106 psi. Each wrench arm is 9'' long and has a cross section solid circular with diameter d = 0.5 in. Determine the maximum shear stress in the arm that rotates the drive nut screw (arm A).

## Answer:

The maximum shear stress in the arm of the lug wrench, taking into account the applied forces and the physical characteristics of the wrench, is calculated to be 13,746.8 psi.

When dealing with mechanical tools like lug wrenches, understanding the stress they can endure is crucial for their proper use and maintenance. In this case, calculating the maximum shear stress in the arm of a lug wrench is important for ensuring its structural integrity.

To determine the maximum shear stress, we first need to calculate the torque on the wrench arm due to the applied force. The torque is the product of the force and the length of the arm. Using the given values of P = 25 lb and L = 9'', we can calculate the torque as 18.75 lb-ft.

Next, we need to convert this torque into shear stress. The formula for shear stress due to a force applied at the edge of a circular cross section is τ/A, where A is the area of the cross section. By calculating the area of the circular cross section with diameter d = 0.5 in, we find it to be 0.19635 sq in.

Subsequently, we can determine the shear stress by dividing the torque by the area of the cross section. After converting the unit to psi, we get the maximum shear stress in the arm of the lug wrench as 13,746.8 psi.

By calculating and understanding the maximum shear stress, we can ensure that the lug wrench is used within its structural limits to prevent any potential failures or safety hazards during use. It is important to always consider the physical characteristics and applied forces when analyzing the stress on a mechanical tool like a lug wrench.