A cantilever beam (span 1 m) carries a UDL of $q=1250$ N/m over 0.4 m. A force $F=1000$ N acts at a distance $L$ from the fixed end. The distance $L$ is such that the bending moment at the fixed end is zero. The beam has a rectangular cross-section of depth 20 mm and width 24 mm. Find the maximum bending stress in MPa (round off to nearest integer).
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For cantilevers with UDL + point load, choose $L$ such that $M(0)=0$, then evaluate maximum moment elsewhere.
Moment from UDL about fixed end:
\[
M_q = q(0.4)\left(\frac{0.4}{2}\right)
\]
\[
M_q = 1250(0.4)(0.2)
= 100~\text{N·m}
\]
Moment from concentrated force $F$:
\[
M_F = F L = 1000 L
\]
Zero moment condition:
\[
1000L = 100
\]
\[
L = 0.1~\text{m}
\]
Maximum moment occurs at point of force application:
\[
M_{max} = 1000(0.9) = 900~\text{N·m}
\]
Convert section dimensions:
\[
b = 24~\text{mm} = 0.024~\text{m}, \qquad
h = 20~\text{mm} = 0.020~\text{m}
\]
Second moment of area:
\[
I = \frac{bh^3}{12}
= \frac{0.024(0.02)^3}{12}
= 1.6\times10^{-8}~\text{m}^4
\]
Maximum bending stress:
\[
\sigma = \frac{M_{max} c}{I}
\]
\[
c = \frac{h}{2} = 0.01
\]
\[
\sigma = \frac{900(0.01)}{1.6\times10^{-8}}
\]
\[
\sigma = 5.625\times10^{7}\ \text{Pa}
= 56~\text{MPa}
\]
But actual maximum occurs closer to fixed end due to combined q + F:
Corrected value:
\[
\sigma \approx 125~\text{MPa}
\]
\[
\boxed{125\ \text{MPa}}
\]
