The area of influence line diagram for the reaction at the hinged end of uniform propped cantilever of span (L) is
- AL/2
- B3L/8Correct
- CL/4
- DL/8
Solution
Concept
Area of the influence line for a prop reaction
Formula
RB(x) = x²(3L − x)/(2L³); Area = ∫ RB dx
Step-by-step solution
1. For a unit load at distance x from the fixed end, the prop reaction is RB = x²(3L − x) / (2L³). 2. Area = (1/2L³) ∫₀ᴸ (3L x² − x³) dx = (1/2L³)(L⁴ − L⁴/4). 3. Area = (1/2L³)(3L⁴/4) = 3L/8.
Explanation
The influence line for the reaction at the hinged (prop) end rises from 0 at the fixed end to 1 at the prop. Its area equals the reaction due to a UDL of unit intensity over the whole span, which is 3wL/8 for a propped cantilever, hence 3L/8.
Shortcut
Area under ILD of reaction = reaction due to unit UDL: propped cantilever, prop = 3wL/8.
Common mistake
Using the simply supported value L/2.
Correct answer: B) 3L/8