Question

In analysis of mechanical stresses, which of the following is the correct form of the 3D equation of stress equilibrium?

(a) \(\frac{\partial \sigma x}{\partial x}+\frac{\partial \tau xy}{\partial y}+\frac{\partial \tau xz}{\partial z}\)=0

(b) \( \frac{\partial \tau xy}{\partial x}+\frac{\partial \sigma y}{\partial y}+\frac{\partial \tau yz}{\partial z}\)=0

(c) \( \frac{\partial \tau xz}{\partial x}+\frac{\partial \tau yz}{\partial y}+\frac{\partial \sigma z}{\partial x}\)=0

(d) No such equation exists

I had been asked this question in an international level competition.

This question is from Single Variable Problems in chapter Single Variable Problems of Finite Element Method

Answer 1

Right choice is (a) \(\frac{\partial \sigma x}{\partial x}+\frac{\partial \tau xy}{\partial y}+\frac{\partial \tau xz}{\partial z}\)=0

For explanation: For Cartesian problems in three dimensions, the stresses σx, τxy, and τxz act on the face normal to X-axis and the body is said to be in equilibrium if\(\frac{\partial \sigma x}{\partial x}+\frac{\partial \tau xy}{\partial y}+\frac{\partial \tau xz}{\partial z}\)=0 on that face.