Project ForgeStrength of Materials
Bending of Beams
GATE 2013
Hard
A simply supported beam of length L is subjected to a varying distributed load sin(3πx/L) N/m, where the distance x is measured from the left support. The magnitude of the vertical reaction force in N at the left support is
- Azero
- L/(3π)
- CL/π
- D2L/π
Explanation
Taking moments about the right support: R_left·L = ∫₀ᴸ(L−x)w(x)dx = L∫₀ᴸw(x)dx − ∫₀ᴸx·w(x)dx, with w(x)=sin(3πx/L). ∫₀ᴸsin(3πx/L)dx = [−L/(3π)cos(3πx/L)]₀ᴸ = −L/(3π)[cos3π−1] = 2L/(3π). ∫₀ᴸx·sin(3πx/L)dx, evaluated via integration by parts, equals L²/(3π). So R_left·L = L·(2L/(3π)) − L²/(3π) = L²/(3π), giving R_left = L/(3π).
Source: GATE ME 2013 (Booklet Code A) (2013)
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