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A cantilever is 2m long and is subjected to a UDL of 2kN/m. The cross section of cantilever is T section with flange 80mm X 10mm and web of 10mm X 120mm such that its total depth is 130mm.

The flange is at top and the web is vertical. Determine the maximum tensile and the compressive stress developed and locate their position.

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$a_1 = 80\hspace{0.05cm}\times\hspace{0.05cm}10 = 80\hspace{0.05cm}mm^2\\ a_2 = 10\hspace{0.05cm}\times\hspace{0.05cm}120 = 1200\hspace{0.05cm}mm^2\\ y_1 = 125\\ y_2 = 60\\ y = \frac{a_1y_1 + a_2y_2}{a_1 + a_2} = 86\hspace{0.05cm}mm$

$M_{max} = \frac{WL^2}{2}\\ \hspace{0.5cm} = \frac{2\hspace{0.05cm}\times\hspace{0.05cm}2^2}{2}\\ \hspace{0.5cm} = 4\hspace{0.05cm}KNm\\ \hspace{0.5cm} = 4\hspace{0.05cm}\times\hspace{0.05cm}10^6\hspace{0.05cm}Nmm$

$\frac{M}{I} = \frac{\sigma}{y}$

Now, $I_{xx} = I_{NA}\\ I_{NA} = I_{Flange} + I_{Web}\\ I_{Flange} = \frac{bd^3}{12} + Ah^2\\ \hspace{0.5cm} = \frac{80\hspace{0.05cm}\times\hspace{0.05cm}10^3}{12} + (80\hspace{0.05cm}\times\hspace{0.05cm}10)\hspace{0.05cm}\times\hspace{0.05cm}39^2\\ \hspace{0.5cm} = 1.2\hspace{0.05cm}\times\hspace{0.05cm}10^6\hspace{0.05cm}mm^4$

$I_{Web} = \frac{bd^3}{12} + Ah^2\\ \hspace{0.5cm} = \frac{10\hspace{0.05cm}\times\hspace{0.05cm}120^3}{12} + (10\hspace{0.05cm}\times\hspace{0.05cm}120)\hspace{0.05cm}\times\hspace{0.05cm}26^2\\ \hspace{0.5cm} = 2.25\hspace{0.05cm}\times\hspace{0.05cm}10^6\hspace{0.05cm}mm^4$

$I_{NA} = I_{Flange} + I_{Web}\\ \hspace{0.25cm} = 3.47\hspace{0.05cm}\times\hspace{0.05cm}10^6\hspace{0.05cm}mm^4$

$\frac{M}{I} = \frac{\sigma_t}{y_t}\\ \frac{4\hspace{0.05cm}\times\hspace{0.05cm}10^6}{3.47\hspace{0.05cm}\times\hspace{0.05cm}10^6} = \frac{\sigma_t}{44}\\ \sigma_t = 50.66\hspace{0.05cm}N/mm^2$

$\frac{M}{I} = \frac{\sigma_c}{y_c}\\ \frac{4\hspace{0.05cm}\times\hspace{0.05cm}10^6}{3.47\hspace{0.05cm}\times\hspace{0.05cm}10^6} = \frac{\sigma_c}{86}\\ \sigma_c = 99.13\hspace{0.05cm}N/mm^2$

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