Calculating Internal Moment on a Beam Supporting Distributed Force

How to Determine the Magnitude of Internal Moment at a Specific Point on a Beam?

Given a beam supporting a distributed force f = 35 kN/m over a length of L = 11.1 m, how can we calculate the internal moment at a specific point, for example, x = 5.04 m?

Calculation of Internal Moment at x = 5.04 m

To determine the magnitude of the internal moment at x = 5.04 m on the beam supporting a distributed force f = 35 kN/m, we can use the equation for the moment at a point on a beam.

The moment at a point on a beam is equal to the sum of the moments of all the forces to one side of that point. Since we are interested in the internal moment at x = 5.04 m, we need to consider the forces acting on the beam to the left of this point.

To calculate the internal moment, we can use the equation:

M = ∫ (force × distance) dx

First, let's determine the equation for the distributed force along the beam. Since the distributed force is f = 35 kN/m, the equation for the force at any point x along the beam is:

force = f × distance from the left end

force = 35 kN/m × x

Now, we can integrate this equation to find the internal moment at x = 5.04 m. The limits of integration will be from 0 to 5.04 m since we are only considering the forces to the left of the point.

M = ∫ (force × distance) dx

M = ∫ (35 kN/m × x) dx

M = 35 kN/m × ∫ x dx

M = 35 kN/m × [(1/2) x^2] from 0 to 5.04

Calculating this expression, we find that the magnitude of the internal moment at x = 5.04 m is approximately 88.236 kNm.