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{excerpt:hidden=true}Displacement time-history records should be obtained from acceleration readings such that ground motion may be manually applied to specific structural supports. Otherwise, time histories are automatically applied to all supports. This article outlines the mathematical formulation for conversion from acceleration to displacement. Visuals are taken from Dr. Wilson’s text Static and Dynamic Analysis of Structures, available for sale through the link provided in the References section.{excerpt}

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During [time-history|kb:Time-history analysis] analysis, an acceleration record is automatically applied to all [restraint|kb:Constraint] supports. {new-tab-link:http://www.csiberkeley.com/}CSI{new-tab-link} Software then uses d’Alembert’s principal to translate the time history into [acceleration loads|Acceleration load] which are applied to structural [joints|kb:Joint]. This process is explained further in the {new-tab-link:http://www.csiberkeley.com/}CSI{new-tab-link} [_Analysis Reference Manual_|doc:Analysis Reference Manual] (Acceleration Loads, page 304).

To manually input ground motion at specific supports, rather than all supports, it is necessary to first convert the acceleration record into its corresponding *displacement time-history record*. Because displacement is piecewise linear, while velocity is piecewise constant, and acceleration is a series of impulse functions at each time step, users should mind [output accuracy|Time-history output-acceleration accuracy] by smoothing the displacement record. This refinement is accomplished by using a smaller time step, possibly one-tenth that of the acceleration record, when transferring ground motion into its corresponding displacement time-history record.

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There are two basic approaches to conversion from acceleration time history to displacement. Users may follow an experimental approach, given as follows:

* Create a simple [SAP2000|sap2000:home] model

* Apply the acceleration time history using the given time step (perhaps 0.02)

* Set the output time step to one-tenth of this value (0.002)

* Extract the displacement results from a [restrained|kb:Constraint] joint

* Correct for zero initial and final displacement and velocity (a + bt)

* Use this smoothed displacement function as the ground-motion input for the actual model. Please note that analysis will be performed at the shorter time step, though output is reported (more accurately) only for each original time step.

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Alternatively, users may implement the mathematical formulation which is summarized in Appendix J of Dr. Edward L. Wilson’s text {link-window:href=http://orders.csiberkeley.com/SearchResults.asp?Cat=2}{_}Static and Dynamic Analysis of Structures_ {link-window}, and 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 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 (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 groundan 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.
Available for purchase on the {new-tab-link:http://www.csiberkeley.com/}CSI{new-tab-link} Products > {link-window:href=http://orders.csiberkeley.com/SearchResults.asp?Cat=2}Books {link-window} page