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## Q. 1.29

A weight of 200 N is dropped at mid span of a simply supported beam from a height of 0.5 m. Determine the  instantaneous maximum deflection and maximum stress if the span of the beam is 3 m and the cross-section of beam is 10 cm wide and 20 cm deep. Take E = 200 GPa.

## Verified Solution

The deflection at midspan due to statically applied load is as follows.

$\delta_{s t} =\frac{W L^2}{48 E I}=\frac{200 \times 3000^2}{48 \times 200000 \times 100 \times 200^3 / 12} =0.0844 mm$

Impact factor

$=1+\sqrt{1+\frac{2 h}{\delta_{s t}}}=1+\sqrt{1+\frac{2 \times 500}{0.0844}} =109.85$

$\delta_{\max } =109.85 \times 0.0844=9.27 mm$
Maximum bending moment at mid span due to static load 200 N at midspain,

$M=\frac{W L}{4}=\frac{200 \times 3000}{4}=150,000 N. mm,$

Maximum bending stress at mid span due to static load,
$\delta_{s t}=\frac{M y}{I}=\frac{150000 \times 100}{100 \times 200^3 / 12}=0.225 MPa$

Maximum stress due to dynamic load,

$\delta_{\max }=0.225 \times 109.85=24.7 MPa$