# Page History

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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.

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h1. 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.

{new-tab-link:http://www.csiberkeley.com/}CSI{new-tab-link} 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.

h1. Examples

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

h2. Example 1 - Bridge-modeler model

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

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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 |Temperature-gradient loading for bridge objects^Hand calculations.pdf], as shown in Figure 2:

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For bridge response, loading from the temperature gradient induces a moment of 2568 kip-ft, which closely correlates with the [hand calculated |Temperature-gradient loading for bridge objects^Hand calculations.pdf] moment of 2669 kip-ft. Software output is shown in Figure 3:

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{center-text}Figure 3 - Moment induced by temperature gradient{center-text}

h2. Example 2 - Single frame-element model

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:

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