Distributed traction and edge loads

This section provides basic verification of using constant resultants in a dead load analysis. The constant resultant method has certain advantages when a traction is used to model a distributed load with a known constant resultant.

If you choose not to have a constant resultant, the traction vector is integrated over the surface in the current configuration, a surface that in general deforms in a geometrically nonlinear analysis. The most common example of a traction that should be integrated over the current configuration is a live pressure load defined as $\mathbf{t}=-p\mathbf{n}$, where $\mathbf{n}$ is the normal in the current configuration. The total resultant due to a pressure load depends on the surface area in the current configuration. A live uniform normal surface traction integrated over the current surface is equivalent to applying a uniform pressure load. By default, the traction vector is integrated over the surface in the current configuration.

If you choose to have a constant resultant, the traction vector is integrated over the surface in the reference configuration, which is constant.

The analysis in this section consists of a unit planar membrane structure that is held fixed at the edges by a kinematic coupling constraint. The normal of the flat structure is in the $\left(1,0,1\right)$ direction. A uniform dead traction load (of magnitude 4) is applied in the negative ${\mathbf{e}}_3$-direction. This could be considered a simple model of a sloped roof with a snow load.

Let $S_o=1$ and S denote the total surface area of the plate in the reference and current configurations, respectively. With no constant resultant, the total integrated load on the plate, $\mathbf{f}$, is

In this case a uniform traction leads to a resultant load that increases as the surface area of the plate increases, which is not consistent with a fixed snow load. With the constant resultant method, the total integrated load on the plate is

In the first step the load is applied without a constant resultant. In the second step the structure is unloaded. In the third step the load is applied with a constant resultant.