Question

For both plane stress as well as plain strain case the equilibrium equation in x-direction is _______

(a) \(\frac{∂σ_x}{∂x}+\frac{∂τ_{yx}}{∂y}=0\)

(b) \(\frac{∂σ_x}{∂x}+\frac{∂τ_{yx}}{∂y}+\frac{∂τ_{zx}}{∂z}+X=1\)

(c) \(\frac{∂σ_x}{∂x}+\frac{∂τ_{yx}}{∂y}+X=0\)

(d) \(\frac{∂σ_x}{∂x}+\frac{∂τ_{zx}}{∂z}=0\)

This question was posed to me by my school principal while I was bunking the class.

I want to ask this question from Elasticity Elements topic in division Elements of Elasticity of Geotechnical Engineering I

Answer 1

The correct option is (d) \(\frac{∂σ_x}{∂x}+\frac{∂τ_{zx}}{∂z}=0\)

To elaborate: The equilibrium equation in x-direction is  \(\frac{∂σ_x}{∂x}+\frac{∂τ_{yx}}{∂y}+\frac{∂τ_{zx}}{∂z}+X=0\)

In the plain strain case, one dimension (y) is very large in comparison to the other two directions. So, the strain components in this direction are zero. Also in plain stress condition, the stresses in y-direction are considered as zero.

∴ The equation reduces to  \(\frac{∂σ_x}{∂x}+\frac{∂τ_{zx}}{∂z}=0\)