# Problem 732 | Cantilever beam supported by a cable at midspan

**Problem 732**

The midpoint of the steel in Fig. P-732 is connected to the vertical aluminum rod. Determine the maximum value of P if the stress in the rod is not to exceed 120 MPa.

**Solution 732**

$P_{al} = (\sigma A)_{al}$

$P_{al} = 120(40)$

$P_{al} = 4800 \, \text{ N}$

$\delta_{al} = \left( \dfrac{\sigma L}{E} \right)_{al}$

$\delta_{al} = \dfrac{120(5)(1000)}{70000}$

$\delta_{al} = \frac{60}{7} \, \text{ mm}$

For the steel beam

$t_{B/C} = \dfrac{1}{EI}(Area_{BC}) \cdot \bar{X}_B$

$t_{B/C} = \dfrac{1}{200\,000(50 \times 10^6)} \left[ \frac{1}{2}(2)(9600)(\frac{4}{3}) - \frac{1}{2}(2)(2P)(\frac{2}{3}) - \frac{1}{2}(2)(4P)(\frac{4}{3}) \right](1000^3)$

$t_{B/C} = \dfrac{1}{10\,000}(25600 - \frac{2}{3}P)$

$t_{B/C} = \frac{64}{25} - \frac{1}{1500}P$

From the figure

$\delta_{al} = -t_{B/C}$

$\frac{60}{7} = -(\frac{64}{25} - \frac{1}{1500}P)$

$\frac{1}{1500}P = \frac{1948}{175}$

$P = 16\,697.14 \, \text{ N}$

$P = 16.7 \, \text{ kN}$ *answer*