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{live-template:Test Problem}

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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

Please note that the [hand calculations |Temperature-gradient loading for bridge objects^Hand calculations.pdf] attached provide additional details.

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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!Temperature_gradient_loading_for_bridge_object.png|align=center,border=0!

{center-text}Figure 1 - Temperature-gradient loading{center-text}

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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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!Bridge_temperature_gradient_loading_-_temperature_load_applied_to_solid_elements.png|align=center,border=0,width=800pxpx!

{center-text}Figure 2 - Temperature-gradient loading on solid elements{center-text}

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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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!Temperature_gradient_loading_bridge_object_moments.png|align=center,border=0!

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


h2. Example 2 - Single frame-element model

Please note that the [hand calculations |Temperature-gradient loading for bridge objects^Hand calculations.pdf] 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:

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!Temperature_gradient_loading_frame_model.png|align=center,border=0,width=800pxpx!

{center-text}Figure 4 - Temperature-gradient on the frame-element model{center-text}


h1. Attachments

* [Hand calculations |Temperature-gradient loading for bridge objects^Hand calculations.pdf] (PDF file)
* [SAP2000 models |Temperature-gradient loading for bridge objects^SAP2000 V14.0.0 models.zip] (zipped SDB files) - example 1, bridge-modeler model and example 2, single frame-element model