**Given:**
- Span of the beam = 20 m
- Uniformly distributed load = 20 kN/m
- Length of the load = 5 m

**To find:**
- Maximum bending moment

**Assumptions:**
- The beam is simply supported, which means it is supported at both ends and has no fixed or pinned supports.
- The beam is linearly elastic, which means it obeys Hooke's law and does not undergo plastic deformation.
- The load is evenly distributed along its length.

## **Solution:**

1. **Determine the reactions at the supports:**
- Since the beam is simply supported, the reactions at the supports will be equal.
- Let the reaction force at each support be R.
- The total load on the beam is the product of the distributed load (20 kN/m) and the length of the load (5 m).
- Therefore, the total load on the beam is 20 kN/m * 5 m = 100 kN.
- The sum of the vertical forces at the supports must equal the total load, so each reaction force is R = 100 kN / 2 = 50 kN.

2. **Determine the shear force along the beam:**
- The shear force at any point along the beam can be found by summing the vertical forces to the left or right of that point.
- Since the load is symmetrically placed, the shear force will be constant along the entire length of the beam.
- The shear force at any point is equal to the reaction force at the support minus the portion of the load to the left of that point.
- Therefore, the shear force at any point is V = 50 kN - (20 kN/m * x), where x is the distance from the support.

3. **Determine the bending moment along the beam:**
- The bending moment at any point along the beam can be found by integrating the shear force.
- The bending moment at any point is equal to the integral of the shear force from the support to that point.
- Therefore, the bending moment at any point is M = ∫(50 kN - 20 kN/m * x) dx, where x is the distance from the support.
- Simplifying the integral, we get M = 50 kN * x - 10 kN/m * (x^2) + C, where C is the constant of integration.

4. **Determine the maximum bending moment:**
- The maximum bending moment occurs at the midpoint of the beam, where x = 10 m.
- Substituting x = 10 m into the equation for bending moment, we get M = 50 kN * 10 m - 10 kN/m * (10 m^2) + C.
- The constant of integration, C, can be determined by using the fact that the bending moment at the supports is zero.
- When x = 0, M = 0. Substituting these values into the equation, we get 0 = 50 kN * 0 - 10 kN/m * (0^2) + C.
- Solving for C, we get C = 0.
- Substituting C = 0 back into the equation for bending moment,

320kn.m