Question

The in situ stress at a point in a dry sandstone terrain is as follows: σ₁= 12 MPa and σ₃=4 MPa. The pore water pressure (p_w) increases by the construction of a reservoir. The failure criterion of the sandstone is given by \(σ'_1\)= 3.48 MPa +3σ'₃, where of and \(σ'_1\) are the effective maximum and minimum principal stresses, respectively. Assuming that the failure occurs at peak stress, the minimum value of P_w (in MPa) that will cause the sandstone to fail in situ is______  to two decimal places). 

Hint:

Use effective stress: \(\sigma'=\sigma-p_w\). Substitute directly into the failure relation and solve for \(p_w\). Always check units and round at the end.

Answer 1

Correct Answer: 1.74

Step 1: Effective stresses.
\(\sigma_1' = \sigma_1 - p_w = 12 - p_w,\quad \sigma_3' = \sigma_3 - p_w = 4 - p_w.\) Step 2: Apply failure criterion.
\(12 - p_w = 3.48 + 3(4 - p_w) \Rightarrow 12 - p_w = 3.48 + 12 - 3p_w.\) Step 3: Solve for \(p_w\).
\(-p_w + 3p_w = 15.48 - 12 \Rightarrow 2p_w = 3.48 \Rightarrow p_w = 1.74\ \text{MPa}.\) \[ \boxed{p_w = 1.74\ \text{MPa}} \]