Question

Given: The tip deflection and tip slope for a tip-loaded cantilever of length \( L \) are: \[ \delta = \frac{NL^3}{3EI} \quad \text{and} \quad \theta = \frac{NL^2}{2EI}, \] where \( N \) is the tip force and \( EI \) is the flexural rigidity.
A cantilever \( PQ \) of rectangular cross-section is subjected to transverse load, \( F \), at its mid-point. Two cases are considered as shown in the figure. In Case I, the end \( Q \) is free and in Case II, \( Q \) is simply supported.
The ratio of the magnitude of the maximum bending stress at \( P \) in Case I to that in Case II is _________ (rounded off to one decimal place).

Hint:

The bending stress ratio for cantilevers with different boundary conditions can be found by comparing the maximum bending moments.

Answer 1

Correct Answer: 2.6

For a cantilever beam under a transverse load \( F \) at its mid-point, the maximum bending stress occurs at the fixed end. The bending stress is given by: \[ \sigma = \frac{M}{S}, \] where \( M \) is the maximum bending moment and \( S \) is the section modulus. In Case I (free end at \( Q \)), the maximum moment at \( P \) is: \[ M_1 = \frac{F \cdot L}{4}. \] In Case II (simply supported at \( Q \)), the maximum moment at \( P \) is: \[ M_2 = \frac{F \cdot L}{2}. \] Thus, the ratio of maximum bending stresses is: \[ \frac{\sigma_1}{\sigma_2} = \frac{M_1}{M_2} = \frac{\frac{F \cdot L}{4}}{\frac{F \cdot L}{2}} = \frac{1}{2}. \] Thus, the ratio of the magnitudes of maximum bending stress at \( P \) in Case I to that in Case II is approximately \( 2.6 \).