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Find the maximum and minimum stress intensity of the base of uniform circular chimney having external and internal diameter 5m and 3m respectively.

Height of the chimney is 25m and subjected to wind pressure $1.5 KN/m^2$ , the density of measure is $21 KN/m^3$.

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$\textit {Wind pressure} = 1.5\hspace{0.05cm}KN/m^2\\ \textit{density} \hspace{0.05cm}(\rho) = 21\hspace{0.05cm}KN/m^3\\ y_1 = \frac{25}{2} = 12.5\\ y_2 = \frac{5}{2} = 2.5$

(i) $\textit{Projected Area} = 25\hspace{0.05cm}\times\hspace{0.05cm}5\\ \hspace{02cm} = 125\hspace{0.05cm}m^2$

(ii)$\textit{Wind force}\hspace{0.05cm}(P) = \textit{Wind pressure X Projected Area}\\ \hspace{02cm} = 1.5\hspace{0.05cm}\times\hspace{0.05cm}125\\ \hspace{02cm} = 187.5\hspace{0.05cm}KN = 1875\hspace{0.05cm}\times\hspace{0.05cm}10^2\hspace{0.05cm}N$\

(iii) $\textit{Self Weight} = \rho\hspace{0.05cm}\times\hspace{0.05cm}Volume\\ \hspace{02cm} = \rho\hspace{0.05cm}\times\hspace{0.05cm}A\hspace{0.05cm}\times\hspace{0.05cm}L\\ \hspace{02cm} = 21\hspace{0.05cm}\times\hspace{0.05cm}\frac{\pi}{4}(D^2 - d^2)\hspace{0.05cm}\times\hspace{0.05cm}25\\ \hspace{02cm} = 6594\hspace{0.05cm}N$

Direct Stress $\sigma_d = \frac{W}{A} = \frac{6594}{12.56}\\ \hspace{0.25cm} = 525\hspace{0.05cm}KN/m^2$

$M = P . y_1\\ \hspace{0.25cm} = 187.5\hspace{0.05cm}\times\hspace{0.05cm}12.5\\ \hspace{0.25cm} = 2343.75\hspace{0.05cm}KN/m^2$

$I_{yy} = I_{yy(outer)} - I_{yy(inner)}\\ \hspace{0.25cm} = \frac{\pi}{64} (D^4 -d^4)\\ \hspace{0.25cm} = 26.70\hspace{0.05cm}m64$

$\sigma_b = \frac{M.y}{I}\\ \hspace{0.25cm} = \frac{2343.75\hspace{0.05cm}\times\hspace{0.05cm}2.5}{26.70}\\ \hspace{0.25cm} = 219.45\hspace{0.05cm}KN/m^2$

$\sigma_{max} = \sigma_d + \sigma_b\\ \hspace{0.25cm} = 525 + 219.45\\ \hspace{0.25cm} = 744.45\hspace{0.05cm}KN/m^2$

$\sigma_{min} = \sigma_d - \sigma_b\\ \hspace{0.25cm} = 525 - 219.45\\ \hspace{0.25cm} = 305.55\hspace{0.05cm}KN/m^2$

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