A 20 mm diameter circular rod carries a pull parallel to the centroidal axis at a distance from it. Find the eccentricity if the maximum stress is 20 percent greater than the mean stress at a section normal to the axis.

Concept:

Consider a circular rod subjected to a pull parallel to the centroidal axis at a distance from it as follows:

Mean stress, \({\sigma _{mean}} = \frac{P}{A} = \frac{{4P}}{{\pi {d^2}}}\)

Now, Maximum stress due to eccentric load,

 \({\sigma _{max}} = \frac{P}{A} + \frac{M}{Z} = \frac{P}{A} + \frac{{P.e}}{Z}\)

Where P = load, A = Area, d = diameter circular rod, e = eccentricity, and Z = sectional modulus = \(\pi d^3 \over 32\)

Calculation:

Given data

d = 20 mm

So, Maximum stress

\({\sigma _{max}} = \frac{{4p}}{{\pi {d^2}}} + \frac{{32P.e}}{{\pi {d^3}}}\)

σmax = 1.2 σmean

\(\begin{array}{l} \frac{{4P}}{{\pi {d^2}}} + \frac{{32P.e}}{{\pi {d^3}}} = 1.2 \times \frac{{4P}}{{\pi {d^2}}}\\ 1 + \frac{{8e}}{d} = 1.2 \Rightarrow \frac{{8e}}{d} = 0.2\\ e = \frac{{0.2d}}{8} = \frac{{0.2 \times 20}}{8} = 0.5\;mm \end{array}\)