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To simulate ground motion during [time-history analysis|kb:Time-history analysis], {new-tab-link:http://www.csiberkeley.com/}CSI{new-tab-link} Software automatically applies [acceleration loading|Acceleration load] to all [joints|kb:Joint] and elements of a structural model. These loads are determined through d’Alembert’s principal. As the [_CSI Analysis Reference Manual_|doc:Analysis Reference Manual] (Acceleration Loads, page 284) explains in greater detail, acceleration loads are derived from the application of the acceleration record to all supports.

To manually input ground motion at specific supports, it is necessary to convert the acceleration record to its corresponding *displacement time-history record*. This process, summarized in Appendix J of Dr. Edward L. Wilson’s text _Static and Dynamic Analysis of Structures_, is outlined below:

First, ground acceleration is idealized, within each time increment, as linear (Figure 1).

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!Figure 4.png|align=center,border=1!
{center-text}Figure 1 - Ground acceleration record{center-text}

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At each time step, integration of acceleration and velocity then yields expressions for ground velocity and displacement (Figure 2).

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!Figure 5.png|align=center,border=1!
{center-text}Figure 2 - Expressions for a, v, and d, derived through integration{center-text}

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Evaluation of these expressions at _t = ∆t_ yields a set of recursive equations, as shown in Figure 3:

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!Figure 6.png|align=center,border=1!
{center-text}Figure 3 - Recursive equations characterizing ground motion{center-text}

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These expressions may then be used to translate a ground acceleration record into its corresponding displacement record.

This double integration procedure should produce zero displacement at either end of the record. If non-zero displacement does exist, it is then necessary to apply a base line correction. Figure 4 presents a formulation for this process.

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!Figure 7.png|align=center,border=1!
{center-text}Figure 4 - Algorithm for zero displacement at record ends {center-text}

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Once the displacement time-history record has been produced, users may continue to manually input ground motion at supports by following the process outlined in the [Multi-support excitation|Multi-support excitation] article.


h1. References

* Wilson, Dr. Edward L. _Static & Dynamic analysis of Structures_. 4th ed. Berkeley: Computers and Structures, Inc., 2004.
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