How to Replace Shearing Forces with a Single Force in Mechanics

How can the shearing forces exerted on the cross section of a steel channel be replaced with a single force?

Given data: A 900N vertical force and two 272N horizontal forces are exerted on a steel channel. Replace this force and couple with a single force applied at point C and determine the distance x from C to line BD.

Answer:

To replace the shearing forces with a single force, we need to calculate the resultant force first and then determine the distance x from point C to line BD by considering the moments of the original forces about point C.

When dealing with replacing shearing forces with a single force in mechanics, it is essential to understand the principles of force equilibrium and moment calculation. In this scenario, we are given a vertical force of 900N and two horizontal forces of 272N each.

The first step is to calculate the resultant force by finding the vector sum of the horizontal and vertical forces. The total horizontal force is 272N + 272N = 544N, and the vertical force is 900N. By using the Pythagorean theorem, we can find the magnitude of the resultant force, which is approximately 1050N.

Next, we need to determine the distance x from point C to line BD. To do this, we analyze the moments of the original forces about point C. Let\'s assume the distance from point C to the line of action of the 900N force is denoted as 'd' and the required distance from point C to line BD is 'x'.

The equation for the moment can be expressed as: 900d = 1050x. By solving this equation, we can determine the distance 'x' from point C to line BD.

This process involves applying the principles of mechanics, including force replacement and moment analysis, to simplify the complex system of shearing forces into a single force. Understanding these fundamental concepts is crucial in solving engineering problems related to force distribution and equilibrium.