Line theory: Calculation stage 3 shear forces

Having calculated the bend moments at each end of the segment, the shear force in the segment can be then calculated.

Each line segment is a straight stiff rod in which the bend moment vector varies from $\vec{m}_1$ at one end (the end nearest end A of the line) to $\vec{m}_2$ at the other end. Because the segment is stiff in bending, the bend moment varies linearly along it; the shear force in the segment is then the constant vector representing the rate of change of bend moment along the length. The shear force $f_\mathrm{s}$ is therefore given by \begin{equation} \vec{f}_\mathrm{s} = \vec{s}_\mathrm{z} \times \frac1l(\vec{m}_2 - \vec{m}_1) \end{equation} where $l$ is the instantaneous length of the segment. Note that $\vec{s}_\mathrm{z}$, $\vec{m}_1$ and $\vec{m}_2$ are vectors, as defined in calculation stage 2 bend moments, so this is a vector formula that defines both the magnitude and direction of the shear force.

This shear force vector is applied (with opposite signs) to the nodes at each end of the segment.