In a simply 8 m span supported reinforced concrete beam with sections dimensions 200 mm x 400 mm, the maximum shear force and maximum moment produced in the beam due to its self-weight will be __________ respectively
- A8 kN and 8 kNm
- B16 kN and 8 kNm
- C8 kN and 16 kNmCorrect
- D16 kN and 16 kNm
Solution
Concept
Calculation of maximum shear force and bending moment for a simply supported beam subjected to its own self-weight (UDL).
Formula
Self-weight w = b × d × unit weight of concrete Maximum Shear Force (Vmax) = wL / 2 Maximum Bending Moment (Mmax) = wL² / 8
Step-by-step solution
Step 1: Calculate the self-weight (UDL, w) of the beam per meter length. Assuming unit weight of reinforced concrete = 25 kN/m³. w = 0.200 m × 0.400 m × 25 kN/m3 = 2.0 kN/m. Step 2: Calculate the maximum shear force (Vmax) for a simply supported beam. Vmax = wL / 2 = (2.0 kN/m × 8 m) / 2 = 8 kN. Step 3: Calculate the maximum bending moment (Mmax) at mid-span. Mmax = wL² / 8 = (2.0 kN/m × (8 m)²) / 8 = 16 kNm.
Explanation
The self-weight of the beam acts as a uniformly distributed load (UDL) over the simply supported span. The maximum shear force for a simply supported beam under UDL occurs at the supports and is equal to wL/2, while the maximum bending moment occurs at the mid-span and is equal to wL²/8.
Why the other options are wrong
A: Incorrect shear and moment values. B: Shear force and bending moment values are swapped. D: Incorrect shear force calculation.
Shortcut
Notice that for L = 8 m, wL² / 8 = w(8)² / 8 = 8w, and wL / 2 = 8w / 2 = 4w. Since w = 2 kN/m, Shear Force = 8 kN and Bending Moment = 16 kNm.
Common mistake
Confusing the formulas for maximum shear force (wL/2) and maximum bending moment (wL²/8), or forgetting to calculate the UDL from cross-sectional dimensions and unit weight of concrete.
Correct answer: C) 8 kN and 16 kNm