Answer
$$P = 6.752 \space kip$$
Work Step by Step
1) Setup:
Cross sectional Areas:
$A_{AB} = \frac{1}{4} \pi d_{AB}^2 = \frac{1}{4} \pi * 1.25^2 = 1.227 \space in^2$
$A_{BC} = \frac{1}{4} \pi d_{BC}^2 = \frac{1}{4} \pi *0.75^2 = 0.4418 \space in^2$
2)Internal Forces
$\bullet AB:$
Taking a cut between points A and B and applying the equilibrium equations:
$\sum {F_y} = 0; \space \space N_{AB} -12 - P = 0; \space\space N_{AB} = 12+P$
$\bullet BC:$
Taking a cut between points B and C and applying the equilibrium equations:
$\sum {F_y} = 0; \space \space N_{BC} - P = 0; \space\space N_{BC} =P$
3)Average Axial Stress
Average axial stress is defined as follows:
$\sigma_{AVG} = \frac{N_{INT}}{A}$
The stated requirement in the problem is that $\sigma_{AB} = \sigma_{BC}$
Substituting the average stress equation into the above condition, we get:
$$\frac{N_{AB}}{A_{AB}} = \frac{N_{BC}}{A_{BC}}$$
$$\frac{12+P}{A_{AB}} = \frac{P}{A_{BC}}$$
Solving for P:
$$P = \frac{12*A_{BC}}{A_{AB}-A_{BC}} = \boxed{6.752 \space kip} \space\space \leftarrow ANS$$
