Question 11.7: For the column of Example Problem 11–6, compute the maximum ...
For the column of Example Problem 11–6, compute the maximum stress and deflection if a load of 4780 N is applied with an eccentricity of 19 mm and the column is initially straight.
Objective Compute the stress and the deflection for the eccentrically loaded column.
Given Data from Example Problem 11–6, but eccentricity = e = 19 mm
Solid circular cross section: D = 18 mm; L = 800 mm
Both ends are pinned; KL = 0.8 m; r = 4.5 mm; c = D/2 = 9 mm
Material: SAE 1040 hot-rolled steel; E = 200 × 10^{9} Pa, s_{y} = 414 MPa
Analysis Use Equation (11–22) to compute maximum stress. Then use Equation (11–24) to compute maximum deflection.
y_{\max }=e\left[\sec \left(\frac{K L}{R} \sqrt{\frac{P}{A E}}\right)-1\right] (11–24)
\sigma_{L / 2}=\frac{P}{A}\left[1+\frac{e c}{r^2} \sec \left(\frac{K L}{2 r} \sqrt{\frac{P}{A E}}\right)\right] (11–22)
Results All terms have been evaluated before. Then the maximum stress is found from Equation (11–22):
\sigma_{L/2} = \frac{4780}{255} \left[1+\frac{(19)(9)}{(4.5)^{2}} sec\left\lgroup\frac{800}{2(4.5)} \sqrt{\frac{4780}{(255)(207 \times10^{9})}}\right\rgroup \right]
\sigma_{L/2} = 168.7 MPa
The maximum deflection is found from Equation (11–24):
y_{max} = 19 \left[sec\left\lgroup\frac{800}{2(4.5)} \sqrt{\frac{4780}{(255)(207 \times10^{9})}}\right\rgroup -1 \right] = 7.4 mm
Comments The maximum stress is 168.7 MPa at the midlength of the column. The deflection there is 7.4 mm.