Topic: Strength of Materials
A short column of external diameter D and internal diameter d carries an external load W. The greatest ecentricity which the load can have without producing tension on the cross section of the column is-
- A. $(D+d)/8$
- B. $(D^{2}+d^{2})/8$
- C. $(D^{2}+d^{2})/8D$
- D. $(D^{2}+d^{2})/8d$
Answer: C. $(D^{2}+d^{2})/8D$
Explanation: External diameter = D, Internal diameter = d, External load = W. For no tension, Stress due to bending moment ($\sigma_{b}$) = Stress due to axial load ($\sigma_{a}$). $\sigma_{b}=\sigma_{a} \Rightarrow \frac{M}{Z}=\frac{W}{A} \Rightarrow \frac{W \times e}{\frac{\pi}{32} \times \frac{(D^{4}-d^{4})}{D}} = \frac{4W}{\pi(D^{2}-d^{2})} \Rightarrow \frac{32 \times W \times e \times D}{\pi(D^{4}-d^{4})}=\frac{4W}{\pi(D^{2}-d^{2})} \Rightarrow e=\frac{D^{2}+d^{2}}{8D}$.