To determine the support reactions, we can start by calculating the total load on the beam.
Dead load = 25.2 kN/m * 10 m = 252 kN
Imposed load = 12.8 kN/m * 10 m = 128 kN
Total load = dead load + imposed load = 252 kN + 128 kN = 380 kN
To balance the total load on a simply supported beam, the reactions at the two supports must be equal. Therefore, each support reaction will be 380 kN / 2 = 190 kN.
Next, let's plot the uniformly distributed load, imposed distributed load, and the dead load on the beam.
3 m 4 m 3 m
|---------|---------|---------|---------|
The dead load of 33.6 kN is located 3 m from each end. This load can be considered as a point load acting at the midpoint of each 3 m span. Therefore, each point load will be 33.6 kN / 2 = 16.8 kN.
Applying the point loads, we can now plot the shear force and bending moment diagrams. We'll start from the left end and move towards the right.
At the left support (A):
- There is a reaction of 190 kN pointing upwards.
- There is no horizontal force acting on the beam (since it is laterally restrained).
- The shear force at A is 190 kN pointing upwards.
- The bending moment at A is 0.
Moving towards the right:
- Between the left support (A) and the first point load of 16.8 kN, there is no external force acting on the beam. Hence, the shear force remains constant at 190 kN.
- The bending moment increases linearly from 0 at A to a value of -16.8 kNm (clockwise) just before the point load.
At the first point load of 16.8 kN:
- The shear force suddenly jumps to 190 kN - 16.8 kN = 173.2 kN (pointing downwards).
- The bending moment at this point is -16.8 kNm (clockwise).
Moving further right:
- Between the first point load and the second point load, there is no external force acting on the beam. So, the shear force remains constant at 173.2 kN.
- The bending moment decreases linearly from -16.8 kNm (clockwise) to -50.4 kNm (clockwise) just before the second point load.
At the second point load of 16.8 kN:
- The shear force suddenly jumps to 173.2 kN - 16.8 kN = 156.4 kN (pointing downwards).
- The bending moment at this point is -50.4 kNm (clockwise).
Moving further right:
- Between the second point load and the right support (B), there is no external force acting on the beam. Hence, the shear force remains constant at 156.4 kN.
- The bending moment decreases linearly from -50.4 kNm (clockwise) to -190 kNm (clockwise) just before the right support.
At the right support (B):
- There is a reaction of 190 kN pointing upwards.
- There is no horizontal force acting on the beam (since it is laterally restrained).
- The shear force at B is 156.4 kN pointing downwards.
- The bending moment at B is -190 kNm (clockwise).
To summarize:
Shear force diagram: (from left to right)
A: 190 kN (upwards)
First point load: 173.2 kN (downwards)
Second point load: 156.4 kN (downwards)
B: 156.4 kN (downwards)
Bending moment diagram: (from left to right)
A to first point load: 0 to -16.8 kNm (clockwise)
First point load: -16.8 kNm (clockwise)
First point load to second point load: -16.8 kNm (clockwise) to -50.4 kNm (clockwise)
Second point load: -50.4 kNm (clockwise)
Second point load to B: -50.4 kNm (clockwise) to -190 kNm (clockwise)
Note: The negative sign indicates a clockwise bending moment.
A laterally restrained beam 10m long that is simply supported at both ends support a dead uniformly distributed load of 25.2KN/m and and imposed distributed load of 12.8KN/m. The beam also carries a dead load of 33.6KN at distance 3m from both ends. Determine support reactions and draw shear force and bending moment diagram
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