Mechanics of Solids / Strength of Materials

Question

I have shared a question from mechanics of solids and my solution with it, as you can see I am trying to take the derivative of the stress equations to find the min, max points but its going wrong. I need to know why my solution is wrong.

Answer 1

Let us assume that the greatest stress in any segment is at its top. (You can verify this on your own.)

Let us further assume that if the maximum stresses in the segments are not all equal, then we can adjust the geometry (here, $b$) to reduce the larger value. (You can verify this on your own.)

In other words, we've optimized the geometry when the maximum stresses in all segments are equal. (The limit for a continuously changing cross section, i.e., an infinite number of segments, would be a constant stress throughout the hanging rod. This problem comes up in the context of space elevator tethers.)

The maximum stresses in the upper and lower section are therefore

$$\sigma_\mathrm{max,\,upper}=\frac{\mathrm{load}}{\mathrm{cross}\mbox{-}\mathrm{sectional\,area}}=\rho\left(\frac{A_bb+A_a(100\,\mathrm{m}-b)}{A_a}\right)=\rho\left[\left(\frac{A_b}{A_a}-1\right)b+100\,\mathrm{m}\right]$$

and

$$\sigma_\mathrm{max,\,lower}=\frac{\mathrm{load}}{\mathrm{cross}\mbox{-}\mathrm{sectional\,area}}=\frac{\rho A_bb}{A_b}=\rho b,$$

respectively, where $A_i$ is the cross-sectional area of segment $i$, and we wish to equate these stresses to determine the optimum geometry. The result is

$$b_\mathrm{optimum}=\frac{100\,\mathrm{m}}{2-A_b/A_a},$$

and the minimum maximum stress is $\rho_\mathrm{optimum} b$.

Does this result in the answer you were looking for?