## 2D Element Displacements

The element displacements are the displacements of the flexible part of the element. In GSA displacements are calculated at degrees of freedom which for 2D elements are at nodes. However when elements are offset the nodal displacements need a transformation to the flexible part of the element.

## 2D Element Force and Moment Results

When dealing with elements with no thickness (such as fabrics) or composite materials (such as reinforced concrete) it is more useful to work with stress resultants or forces, than with stresses. For concrete the stress values are based on the properties of an equivalent isotropic material. The checks used for stress results noted above can also be applied to force results. For fabric elements the force resultants are calculated directly but for elements with thickness they are calculated from the in-plane and bending stresses and the element thickness. Details of these are given in the GSA Theory.

## 2D Element Stress Results

The method of calculating the stresses and forces in 2D elements depends to a certain extent on the solution method. The stresses are based on the strains, which in turn are based on the displacement gradients in the element. The stresses are calculated at the Gaussian integration points and then extrapolated to the nodes. Depending on the solution GSA either calculates the force and moment results from the stresses or by direct integration through the thickness of the element.

## 3D Element Displacements

The element displacements are the displacements at each node on the element. 3D elements have only translational degrees of freedom so there are no element rotations.

## 3D Element Stress Results

The stresses in 3D elements are based on the strains, which in turn are based on the displacement gradients in the element. The stresses are calculated at the Gaussian integration points and then extrapolated to the nodes.

## Analysis Diagnostics: Error norm

Static analysis

## Analysis Stage Data Display Options

| Data type | Components (all unless noted) | Default axes (n/a unless noted) | Axes may be transformed | Output views | Diagrams | Contours | Labels |

## Beam Stresses

Beam stresses are calculated from the forces, moments and the shape of the section. The assumption is that the stresses are linear through the thickness, so the stresses are incorrect for elements in which yield has taken place.

## Bridge Data Display Options

| Data type | Components (all unless noted) | Default axes (n/a unless noted) | Axes may be transformed | Output views | Diagrams | Contours | Labels |

## Bridge Results Display Options

| Data type | Components (all✓* unless noted) | Default axes (n/a unless noted) | Axes may be transformed | Output views | Diagrams | Contours | Labels |

## Cross Strain Energy Density

The strain energy density for a beam is a measure of how hard the beam is working. The definition of strain energy density is:

## Displacements

Displacements (or mode shapes) are calculated at the nodes. When viewing results graphically it is useful to be able to see deflections at intermediate points on elements. The displacements at these intermediate points are not stored by calculated as required, from the end conditions on the element and any load on the element. The number of intermediate points at which these are calculated can be selected by the user, but GSA will add points that are significant e.g. points at which distortion loads are applied.

## Displacements and Rotations

Displacements reported are element displacements; element end values may differ from nodal displacements when the element has end releases, is offset, or shear deformation is included in the analysis due to its section property having a shear area. Such differences are reported in the Beam End Relative Displacement output, contours, and diagrams if the element has an end release.

## Displaying Data and Results

Data and results are displayed in the following formats:

## Forces

Forces are calculated at the end of beam, bar, tie, strut, cable and spring elements, from the displacements at the nodes.

## Load Data Display Options

| Data type | Components (all unless noted) | Default axes (n/a unless noted) | Axes may be transformed | Output views | Diagrams | Contours | Labels |

## Modal Analysis Results

The motivation for a modal dynamic analysis is to characterise the dynamic response of a structure in discrete modes. Along with the mode shape (the eigenvectors) are the load factors (the natural frequencies). The “Global Results – Buckling Details” reports the following data:

## Model Data Display Options

| Data type | Components (all unless noted) | Default axes (n/a unless noted) | Axes may be transformed | Output views | Diagrams | Contours | Labels |

## Numeric formats

The numeric format governs the way that numbers are output in GSA. This is best illustrated by an example of output of $E$, $ν$ and $ρ$ for a standard material in the three different formats.

## Reactions

Reactions are calculated at constrained nodes. The constraint can be due to a restraint or a spring support.

## Results Display Options

| Data type | Components (all unless noted) | Default axes (n/a unless noted) | Axes may be transformed | Output Views | Diagrams | Contours | Labels |

## Static Analysis Results

The basic results of an analysis are the displacements and rotations at nodes, generally referred to as displacements. All other results are derived from the displacements. The forces or stresses in elements depend on the loading on the elements and the displacements at nodes. The force and stress results available will depend on the element types. The other results at nodes are the reactions. Reactions at nodes are calculated by looking at all the element forces and the load applied to the node. For an unrestrained node these should be in equilibrium but for a restrained node the out-of-balance force must be balanced by the reaction force.

## Strain Energy Density

The strain energy density for a beam is a measure of how hard the beam is working. Strain energy density may be output for Beam, Bar, Tie and Strut element types.

## Stress Averaging in 2D elements

The formulation of 2D elements used in GSA means that there will be stress discontinuities between elements. Large discontinuities indicate that the mesh is not as refined as it should be. However, in the cases where the discontinuities are relatively small it is still convenient for viewing purposes to average the stresses at the nodes to give smooth contours between elements.

## Torce Lines

Torce lines are identical to thrust lines in plane frames, but in generalising thrust lines for space frames it is necessary to combine the effects of torsion with the effects of axial force (hence the name).

## User Modules Display Options

| Data Type | Components (all unless noted) | Default axes (n/a unless noted) | Axes may be transformed | Output Views | Diagrams | Contours | Labels |