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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 aa is to consider the beam split at aa into two separate beams. Flexibility matrices can be established for axial, torsional and flexural loading

fax=lEA,ftor=lGJ,fflex=1EI[l4l24l26l312+EIlGAs]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 aa and bb which can be solved for the shear force and bending moment at the loaded point.

fawa=fbwb\mathbf{f}_{a}\mathbf{w}_{a} = \mathbf{f}_{b}\mathbf{w}_{b}

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

wf={10}wm={01}\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 aa to position bb. The nodal forces and moments are then given by integrating the results for a point load

f=∫ab(w1+w2x)fp(x)dx\mathbf{f} = \int_{a}^{b}\left( w_{1} + w_{2}x \right)\mathbf{f}_{p}(x)dx

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

w1=wab−wbab−aw2=wb−wab−aw_{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.