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# Beam Loads

The reference mechanical load is the point load; all the other mechanical load types can be established by integrating the results for a point load over the loaded part of the beam.

The basic approach to calculating the load on the beam for a force at position $a$ is to consider the beam split at $a$ into two separate beams. Flexibility matrices can be established for axial, torsional and flexural loading

$f_{ax} = \frac{l}{EA},\qquad f_{tor} = \frac{l}{GJ},\qquad f_{flex} = \frac{1}{EI}\begin{bmatrix} \frac{l}{4} & \frac{l^{2}}{4} \\ \frac{l^{2}}{6} & \frac{l^{3}}{12} + \frac{EIl}{GA_{s}} \\ \end{bmatrix}$

There must be continuity of displacement and rotation between the two beams and the forces and moment must balance the applied load. This allows a set of equations to be set up for the sub-beams $a$ and $b$ which can be solved for the shear force and bending moment at the loaded point.

$\mathbf{f}_{a}\mathbf{w}_{a} = \mathbf{f}_{b}\mathbf{w}_{b}$

Where the vector $\mathbf{w}$ is respectively for unit force and unit moment

$\mathbf{w}_{f} = \begin{Bmatrix} 1 \\ 0 \\ \end{Bmatrix}\qquad\mathbf{w}_{m} = \begin{Bmatrix} 0 \\ 1 \\ \end{Bmatrix}$

Once the force and moment at the loaded point have been established the end forces and moments (and hence the equivalent nodal forces) result from equilibrium of the two sub-beams.

The general distributed loading in the patch load, varying in linearly in intensity from position $a$ to position $b$. The nodal forces and moments are then given by integrating the results for a point load

$\mathbf{f} = \int_{a}^{b}\left( w_{1} + w_{2}x \right)\mathbf{f}_{p}(x)dx$

where $\mathbf{f}_{p}(x)$ is the force due to a point load at $x$ and

$w_{1} = \frac{w_{a}b - w_{b}a}{b - a}\qquad w_{2} = \frac{w_{b} - w_{a}}{b - a}$

The tri-linear load option is simply a repeated set of patch loads.

Note: In GSA, loads can be applied to a list of members or elements. Any loads applied onto members will be automatically expanded into the appropriate elements loads in the solver in order to analyse the model.