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The purpose of this tutorial is to explain how SAP2000 applies temperature gradient to bridge objects. SAP2000 is applying the specified [temperature gradient|kb:temperature gradient] to the transformed section, calculating the generated axial force P and moment M3, then applying an equivalent constant+linear temperature distribution over the depth. The emphasis here is on getting the correct overall section force and moment on the cross section. If we were to apply the actual temperatures at the nodes as gotten from the specified temperature distribution, the net force and moment would be wrong unless the section was finely discretized.

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h1. Procedure

For the original gradient, you can calculate the stress distribution simply as E α T. Integrate this analytically over the section, accounting for the webs and flanges to get the P. Integrate the moment of the stresses about the neutral axis to get M3.

Now assume a linear temperature gradient with two unknowns: the value at the neutral axis, and the gradient. Integrate this over the sections, equate P and M3 to the previous results, and solve for the two unknowns. This is be equal to the values used by SAP2000.

h1. Examples

The general approach described above is illustrated using the two below examples. Refer to the attached hand calculations and the screenshots included for each example for additional details.

h2. Example 1 - Bridge Modeler Model

The [bridge modeler|kb:bridge modeler] was used to create a single span concrete box bridge fixed at both abutments. The linked bridge object was updated as solid model. Temperature gradient loading is defined as shown in the figure below:

The program calculates solid temperature loads based on equivalent constant plus linear temperature gradient:

The temperature gradient loading causes a moment of 2568 kip-ft, which compares closely to hand calculated moment of 2669 kip-ft: