Question

The compatibility equation in terms of stress components in polar coordinates are given by ____________

(a) \((\frac{∂^2}{∂r^2} +\frac{1}{r} \frac{∂}{∂r}+\frac{1}{r^2}   \frac{∂^2}{∂θ^2} )(σ_r+σ_θ )=0\)

(b) \((\frac{∂^2}{∂r^2} +\frac{1}{r} \frac{∂}{∂r}+\frac{1}{r^2}   \frac{∂^2}{∂θ^2} )(σ_θ )=0\)

(c) \((\frac{∂^2}{∂r^2} +\frac{1}{r} \frac{∂}{∂r}+\frac{1}{r^2}   \frac{∂^2}{∂θ^2} )(σ_r )=0\)

(d) \((\frac{∂^2}{∂r^2} +\frac{1}{r} \frac{∂}{∂r}+\frac{1}{r^2}   \frac{∂^2}{∂θ^2} )(σ_r+σ_θ )=1\)

I had been asked this question in a national level competition.

The query is from Stress Distribution in division Stress Distribution of Geotechnical Engineering I

Answer 1

The correct option is (a) \((\frac{∂^2}{∂r^2} +\frac{1}{r} \frac{∂}{∂r}+\frac{1}{r^2}   \frac{∂^2}{∂θ^2} )(σ_r+σ_θ )=0\)

For explanation: The compatibility equation is the additional equation to solve the stress problem. The compatibility equation in terms of stress components in polar coordinates are given by,

\((\frac{∂^2}{∂r^2} +\frac{1}{r} \frac{∂}{∂r}+\frac{1}{r^2}   \frac{∂^2}{∂θ^2} )(σ_r+σ_θ )=0.\)