Question

Consider that a force \( P \) is acting on the surface of a half-space (Boussinesq's problem). The expression for the vertical stress \( \sigma_z \) at any point \( (r, z) \), within the half-space is given as, \[ \sigma_z = \frac{3P}{2\pi} \frac{z^3}{(r^2 + z^2)^{5/2}}, \] where \( r \) is the radial distance, and \( z \) is the depth with downward direction taken as positive. At any given \( r \), there is a variation of \( \sigma_z \) along \( z \), and at a specific \( z \), the value of \( \sigma_z \) will be maximum. What is the locus of the maximum \( \sigma_z \)?

• A. \( z^2 = \frac{3}{2} r^2 \)
• B. \( z^3 = \frac{3}{2} r^2 \)
• C. \( z^2 = \frac{5}{2} r^2 \)
• D. \( z^3 = \frac{5}{2} r^2 \)

Hint:

For problems involving Boussinesq's solution, differentiate the stress equation with respect to depth (\( z \)) to find the point of maximum stress. The locus is typically a relation between \( z \) and \( r \).

Answer 1

The Correct Option is A

We are given the expression for vertical stress as: \[ \sigma_z = \frac{3P}{2\pi} \frac{z^3}{(r^2 + z^2)^{5/2}} \] To find the locus of the maximum stress, we need to differentiate \( \sigma_z \) with respect to \( z \) and set it equal to zero. The condition for a maximum occurs when the derivative of \( \sigma_z \) with respect to \( z \) is zero. First, differentiate \( \sigma_z \) with respect to \( z \): \[ \frac{d\sigma_z}{dz} = \frac{d}{dz} \left( \frac{3P}{2\pi} \frac{z^3}{(r^2 + z^2)^{5/2}} \right) \] Using the quotient rule and simplifying the expression, we set \( \frac{d\sigma_z}{dz} = 0 \) to find the critical points. Solving this will give us the locus of maximum \( \sigma_z \).
After solving the equation, we find that the locus of the maximum stress occurs when: \[ z^2 = \frac{3}{2} r^2 \] Thus, the correct answer is (A).
\[ \boxed{\text{The locus of maximum } \sigma_z \text{ is } z^2 = \frac{3}{2} r^2.} \]