Page Comparison

{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}

\\

To simulate ground motion during [time-history|kb:Time-history analysis] analysis, {new-tab-link:http://www.csiberkeley.com/}CSI{new-tab-link} Software automatically applies [acceleration loadloads|Acceleration load] to all elements and [joints|kb:Joint] of a structural model. These loads are determined through d’Alembert’s principal. As the {new-tab-link:http://www.csiberkeley.com/}CSI{new-tab-link} [_Analysis Reference Manual_|doc:Analysis Reference Manual] (Acceleration Loads, page 304) explains in greater detail, acceleration load is derived through application of an acceleration record to all supports.

To manually input ground motion at specific supports, it is necessary to first convert the acceleration record into its corresponding *displacement time-history record*. Because displacement is piecewise linear, 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] through input-function refinement. This is done by using a smaller time step, possibly one-tenth that of the acceleration record, when transferring ground motion into a displacement record.

\\

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 regular time step (perhaps 0.02)

* Set the output time step to the refined value of one-tenth the input (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 real 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.

\\

Alternatively, users may follow the mathematical formulation, 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).

\\

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

\\

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

\\

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

\\

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

\\

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

\\

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.

\\

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

\\

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