Consider a bar of varying cross-sectional area . A(x). Here, x denotes the dista

Question

Consider a bar of varying cross-sectional area . A(x). Here, x denotes the distance of a cross-section from the top end. The bar is of length L. has a mass density of p and is subjected to a force P at the bottom. It is also given that the bar is fixed at the top end. Figure 1: A bar of varying cross-sectional area deforming due to its weight and an applied load P. Determine the cross-sectional area A(x) such that the normal stress sigmax is the same at all the cross-sections. Do not use any numbers for this part. Assume that AL is the area of the cross-section at x = L.

Explanation / Answer

dA = -gAdx

ln A = -gx + c

at x= L, A is AL

c = ln AL +gL

ln A = -gx + ln AL +gL

ln (A/AL) = g(L-x)

A = AL * e^(g(L-x)/)

= P/AL

A = AL * e^(gAL(L-x)/P)

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