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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.enter image description here

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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?

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  • $\begingroup$ Thanks, yes it does help me out but there are various point where I am still confused. Firstly why are we forced to assume/ take constant stress in segments, secondly even if we take the stresses constant in segments then we are only have constant stress only at points A and B, in the rest of rod the stress will have variation as each point is only supporting the weight of the rod below it. So the stresses change with the change in force. $\endgroup$ – DemonLordKing Jan 20 '19 at 21:00
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    $\begingroup$ Try to develop an equation for the stress in the lower segment as a function of location. What's the largest it can be as a function of $b$? Now develop an equation for the stress in the upper segment as a function of location. What's the largest it can be as a function of $b$? Now which value of $b$ minimizes the maximum of these two expressions? Try plotting the maximum stresses as a function of $b$. $\endgroup$ – Chemomechanics Jan 20 '19 at 21:33

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