What is the slope at the free end of a cantilever beam of length “l” subjected to a moment M at the free end? EI is flexural rigidity of the beam.
- AMl / 2EI
- BMl / EICorrect
- CMl² / EI
- DMl² / 2EI
Solution
Concept
Slope of Cantilever Beam with End Moment
Formula
θ = Ml / EI
Step-by-step solution
1. Identify the internal bending moment along the beam length as M. 2. Use the differential equation for beam curvature: EI (d²y/dx²) = M. 3. Integrate once with respect to x to get the slope equation: EI (dy/dx) = Mx + C. 4. Apply the boundary condition at the fixed end (x = 0, slope = 0), which gives C = 0. 5. Substitute x = l to find the slope at the free end: θ = Ml / EI.
Explanation
When a cantilever beam is subjected to a concentrated bending moment M at its free end, the internal bending moment is uniform and equal to M throughout the entire length of the beam. Integrating the differential equation for beam deflection (EI × d²y/dx² = M) over the length l gives the slope at the free end as θ = Ml / EI.
Shortcut
For a cantilever beam with an end moment M, the moment diagram is uniform (M everywhere) along length l. - Slope is the integral of moment over EI: θ = Ml / EI (power of l is 1) - Deflection is the integral of slope: δ = Ml² / 2EI (power of l is 2)Remember: Moment is uniform, so slope simply grows linearly with length: θ = Ml / EI.
Common mistake
Students often confuse the formula for slope (θ = Ml / EI) with the formula for maximum deflection (δ = Ml² / 2EI, Option D) or confuse a applied end moment with a point load at the free end.
Correct answer: B) Ml / EI