Support Moment in a Two-Span Beam
Hardthree-momentcontinuous-beam
A continuous prismatic beam spans A–B–C with AB = BC = 4 m and constant EI. A is a pin; B and C are rotationally free vertical supports, with C anchored so it can resist uplift. No support settles. Only span AB carries a downward uniform load of 6 kN/m; BC is unloaded. Using sagging positive, find the bending moment at B.
- A−6.00 kN·m
- B+6.00 kN·m
- C−12.0 kN·m
- D−8.00 kN·m
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Answer: A. −6.00 kN·m
Three-moment equation with MA = MC = 0: 2MB(L₁+L₂) = −6A₁x̄₁/L₁. The UDL reference moment area is A₁ = wL³/12 = 6(64)/12 = 32 kN·m² with x̄₁ = 2 m, so 16MB = −6(32)(2)/4 = −96 and MB = −6.00 kN·m (hogging). Reactions: 10.5, 15.0, −1.5 kN, which is why C must be anchored. Wrong: +6 has the sign wrong (hogging at B is negative); −12 is wL²/8, the propped-cantilever value that treats B as fixed; −8 is the fixed-end moment wL²/12, ignoring the redistribution to span BC.
kN·m
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