A beam is of T section, as shown in fig. The beam is simply supported over a span of 4m and
carries a UDL of (1.7+Your Roll No.) kN/m run over the entire span. Determine the maximum
tensile and maximum compressive stress.
Given,
UDL w = (1.7 + Roll number ) kN/m
L = 4 m
Support = Simply supported
Centroid (NA) ,
$\overline{)x}=0\phantom{\rule{0ex}{0ex}}\overline{)y}=\frac{{y}_{1}{A}_{1}+{y}_{2}{A}_{2}}{{A}_{1}+{A}_{2}}\phantom{\rule{0ex}{0ex}}\overline{)y}=\frac{175\left(7500\right)+75\left(7500\right)}{7500+7500}\phantom{\rule{0ex}{0ex}}\overline{)y}=125mm$
Moment of Inertia,
$I={I}_{1}+{I}_{2}\phantom{\rule{0ex}{0ex}}I=\frac{150{\left(50\right)}^{3}}{12}+\left(150\right)\left(50\right){(175-75)}^{2}]+[\frac{50{\left(150\right)}^{3}}{12}+\left(150\right)\left(50\right){(125-75)}^{2}]\phantom{\rule{0ex}{0ex}}I=1562500+7.5\times {10}^{7}+1.40625\times {10}^{7}+1.875\times {10}^{7}\phantom{\rule{0ex}{0ex}}I=1.09375\times {10}^{8}m{m}^{4}$
Maximum Moment,
Suppose your roll number is 1 .
$w=1.7+1\phantom{\rule{0ex}{0ex}}w=2.7kN/m\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}{M}_{max}=\frac{\left(2700\right){\left(4\right)}^{2}}{8}\phantom{\rule{0ex}{0ex}}{M}_{max}=5400Nm\phantom{\rule{0ex}{0ex}}{M}_{max}=5400\times {10}^{3}Nmm$
Maximum Compressive Stress and Tensile stress is at Top fiber and bottom fiber respectively.
${\sigma}_{C}=\frac{{M}_{max}(200-125)}{I}\phantom{\rule{0ex}{0ex}}{\sigma}_{C}=\frac{5400\times {10}^{3}(200-125)}{1.09375\times {10}^{8}}\phantom{\rule{0ex}{0ex}}{\sigma}_{C}=0.049(200-125)\phantom{\rule{0ex}{0ex}}{\mathit{\sigma}}_{\mathbf{C}}\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{675}\mathbf{}\mathit{M}\mathit{P}\mathit{a}\mathbf{}\mathbf{}\mathbf{\left(}\mathit{M}\mathit{a}\mathit{n}\mathit{g}\mathit{i}\mathit{t}\mathit{u}\mathit{d}\mathit{e}\mathbf{\right)}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}{\sigma}_{T}=\frac{5400000\left(125\right)}{1.09375\times {10}^{8}}\phantom{\rule{0ex}{0ex}}{\mathit{\sigma}}_{\mathbf{T}}\mathbf{=}\mathbf{6}\mathbf{.}\mathbf{171}\mathbf{}\mathit{M}\mathit{P}\mathit{a}$