Determine the maximum bending moment in a simply supported beam of length 4 meters carrying a uniformly distributed load of 500 N/m.
The Correct Answer and Explanation is :
Maximum Bending Moment Calculation
A simply supported beam of length ( L = 4 ) meters carries a uniformly distributed load (UDL) of ( w = 500 ) N/m.
The maximum bending moment in a simply supported beam with a UDL over the entire span occurs at the center of the beam and is given by the formula:
[
M_{\text{max}} = \frac{wL^2}{8}
]
Substituting the given values:
[
M_{\text{max}} = \frac{(500)(4^2)}{8}
]
[
M_{\text{max}} = \frac{500 \times 16}{8} = \frac{8000}{8} = 1000 \text{ Nm}
]
Thus, the maximum bending moment is 1000 Nm (or 1 kNm) at the midpoint of the beam.
Explanation
- Reaction Forces Calculation
The total uniformly distributed load over the beam is: [
W = w \times L = 500 \times 4 = 2000 \text{ N}
] Since the beam is simply supported, the two supports share the load equally: [
R_A = R_B = \frac{2000}{2} = 1000 \text{ N}
] - Bending Moment Distribution
- The bending moment at any point ( x ) along the beam is given by: [
M(x) = R_A x – \frac{w x^2}{2}
] - At mid-span ( x = L/2 = 2 ) m: [
M(2) = (1000 \times 2) – \frac{500 \times 2^2}{2}
] [
M(2) = 2000 – \frac{500 \times 4}{2}
] [
M(2) = 2000 – 1000 = 1000 \text{ Nm}
] - At supports (( x = 0 ) or ( x = 4 )), ( M = 0 ).
- Conclusion
The bending moment follows a parabolic distribution, with a maximum at the center. The maximum bending moment is 1 kNm.
Now, I’ll generate an image showing the shear force and bending moment diagram for the beam.
The images above illustrate the Shear Force Diagram (SFD) and Bending Moment Diagram (BMD) for the simply supported beam under a uniformly distributed load.
- Shear Force Diagram (SFD)
- The shear force starts at +1000 N at the left support.
- It linearly decreases to -1000 N at the right support.
- The change in shear force is linear because the load is uniformly distributed.
- Bending Moment Diagram (BMD)
- The bending moment is zero at the supports.
- It reaches a maximum of 1000 Nm at the mid-span (x = 2 m).
- The shape of the bending moment diagram is parabolic due to the UDL.