This test problem explains and demonstrates the application of temperature gradient to bridge objects. To summarize the process, temperature gradient is specified and applied to the transformed section, axial force (*P*) and moment (*M3*) are calculated, then an equivalent *constant + linear* temperature distribution is applied over the depth.

This process enables the correct calculation of overall cross-sectional force and moment. Nodal application of actual temperatures induced by the temperature distribution specified would yield incorrect net fore and moment without the cross section being finely discretized.

**On this page:**

# Procedure

The stress distribution of a temperature gradient is calculated as *E α T*. Users may analytically solve for axial force (*P*) by integrating this expression over the section, accounting for the web and flange areas. To solve for *M3*, integrate the moment of stresses about the neutral axis.

CSI

Software derives temperature-gradient response by following these formulations. First, the software assumes a linear gradient with two unknowns, including neutral-axis value and gradient slope. Integration procedures yield a set of polynomial expression for *P* and *M3*. Simultaneous solution then yields exact expressions for axial force and moment.

# Examples

The following two examples demonstrate temperature-gradient application. Screenshots and attached hand calculations illustrate the procedure.

## Example 1 - bridge-modeler model

Please note that the hand calculations attached provide additional details.

A single-span concrete-box bridge, fixed at both abutments, is created using the bridge modeler. The linked bridge object is updated as a solid model. Loading from temperature gradient is defined as shown in Figure 1:

Figure 1 - Temperature-gradient loading

As mentioned earlier, an equivalent *constant + linear* temperature gradient generates the temperature load for a solid model. This procedure correlates precisely with the attached hand calculations , as shown in Figure 2:

Figure 2 - Temperature-gradient loading on solid elements

For bridge response, loading from the temperature gradient induces a moment of 2568 kip-ft, which closely correlates with the hand calculated moment of 2669 kip-ft. Software output is shown in Figure 3:

Figure 3 - Moment induced by temperature gradient

## Example 2 - single frame-element model

Please note that the hand calculations attached provide additional details.

For example 2, a single, beam frame element is fixed at both ends and manually loaded with the *constant + linear* temperature gradient. Element cross section is defined to match those properties of the entire bridge deck section from example 1. The temperature-gradient loading creates a moment of 2668 kip-ft, nearly identical to the hand-calculated moment of 2669 kip-ft. Software output is shown in Figure 4:

Figure 4 - Temperature-gradient on the frame-element model

# Attachments

- Hand calculations (PDF file)
- SAP2000 models (zipped SDB files) - example 1, bridge-modeler model and example 2, single frame-element model