Computing the local coordinate system

The computation of the local coordinate system is a trivial procedure when performed for axisymmetric stress linearization. The transformation matrix in this case will be

When performing three-dimensional stress linearization, the local x-axis will be defined by the stress line. The local y- and z-axes are computed by a series of cross products. This procedure is shown below.

The vector between points ($X_1$, $Y_1$, $Z_1$) and ($X_2$, $Y_2$, $Z_2$) is

Assuming the local y-axis lies in the plane of the local x-axis and the global Y-axis, the local z-axis is defined by

Therefore, the local y-axis will be

After normalization, the above three vectors can be combined to create the transformation matrix.