Overview

Big Picture

Loading, deformation, stress, and strain are all related. A diagram illustrating the connections between these concepts is shown below.

Big Picture of Solid Mechanics

General Stress Tensor

For a beam, assume the given coordinate system, where the x direction is axial as shown below.

Beam Coordinate System

The components of normal and shear stress from different kinds of simple loading can be combined into the stress tensor to represent the overall loading scenario. This is a symmetric matrix representing the state of stress with respect to three directions.

Stress tensor.

$$ T= \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\ \sigma_{yx}& \sigma_{yy} & \sigma_{yz} \\ \sigma_{zx}&\sigma_{zy}&\sigma_{zz} \end{bmatrix} \ $$