Question

If deflection at the free end of a uniformly loaded cantilever beam of length 1 m is equal to 7.5 mm, then slope at the free end is ............

• A. 0.01 radians
• B. 0.015 radians
• C. 0.02 radians
• D. 0.001 radians

Hint:

The slope at the free end of a cantilever beam can be calculated using the formula for the deflection and relating it to the bending moment. The slope decreases with increasing length and increasing beam stiffness.

Answer 1

The Correct Option is A

For a cantilever beam subjected to a uniformly distributed load, the deflection \( \delta \) at the free end and the slope \( \theta \) at the free end can be calculated using standard beam formulas. The deflection at the free end for a uniform load \( w \) is given by: \[ \delta = \frac{wL^4}{8EI} \] where:
- \( w \) is the uniform load per unit length,
- \( L \) is the length of the beam,
- \( E \) is the modulus of elasticity,
- \( I \) is the moment of inertia of the beam’s cross-section.
The slope at the free end is related to the deflection by the following equation: \[ \theta = \frac{d\delta}{dx} \] For a cantilever beam under a uniformly distributed load, the formula for the slope at the free end is: \[ \theta = \frac{wL^3}{6EI} \] Given that the deflection at the free end is \( \delta = 7.5 \) mm = 0.0075 m, and the length of the beam is \( L = 1 \) m, the slope can be estimated using the relationships between deflection and slope. From the deflection equation, using appropriate values for \( E \), \( I \), and \( w \), the slope is calculated as: \[ \theta = 0.01 \text{ radians} \] Thus, the slope at the free end of the beam is \( 0.01 \) radians.