Computing the local coordinate system

Stress linearization requires the results to be transformed from the global coordinate system to a local coordinate system defined by the stress line.

The following topics are discussed:

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Calculating linearized stresses

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Axisymmetric case

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

[cos⁡θsin⁡θ0-sin⁡θcos⁡θ0001].

Three-dimensional case

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 (X1, Y1, Z1) and (X2, Y2, Z2) is

x=[Δ⁢XΔ⁢YΔ⁢Z]=[X2-X1Y2-Y1Z2-Z1].

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

z​=x×y=det[ijkΔ⁢XΔ⁢YΔ⁢Z010]=[-Δ⁢Z0Δ⁢X].

Therefore, the local y-axis will be

y​=z×x=det[ijk-Δ⁢Z0Δ⁢XΔ⁢XΔ⁢YΔ⁢Z]=[-Δ⁢X⁢Δ⁢YΔ⁢X2+Δ⁢Z-Δ⁢Y⁢Δ⁢Z].

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