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CSI Software uses d’Alembert’s principal to automatically determine [acceleration loads|Acceleration load] at each joint and element of a structure. These loads are used to simulate ground acceleration during time-history analysis. During time-history analysis, the acceleration record is automatically applied to all supports. The [CSI Analysis Reference Manual|doc:Analysis Reference Manual] (chapter: Load Cases, section: Acceleration Loads) explains this topic in greater detail.

*To manually input* ground motion at supports, it is necessary to convert the acceleration record into 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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Integration of acceleration and velocity, at each time step, 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 (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.


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h1. Previous page content:

The acceleration loads automatically calculated by the program are intended to apply ground acceleration. For example, translational acceleration loads are calculated as the negative translational joint mass at a joint times the input acceleration. CSI Analysis Reference Manual, chapter Load Patterns, section Acceleration Loads, has additional information on this.

To apply the acceleration load at the water level, you would need to convert the acceleration time history record to a displacement time history record. Once you have the displacement record, define a Load pattern in which you apply a unit value of joint displacements in the direction of acceleration at the affected joint. For this to affect the structure, you also need to restrain these joints in the direction of loading only. Then create a time history function that will describe a variation of these displacements in time. Finally, define a time history Load case that will be specified using both the displacement and the time function.

h1. FAQ

h1. Converting acceleration time history records to displacement time history records

Would it be appropriate for this user to use equations J.2 (provided in Appendix J of the book Static & Dynamic Analysis of Structures by Prof. Wilson) to convert acceleration time history records to displacement time history records?

Yes, that equation can be used to generate displacement history from the ground acceleration history. However, it should be borne in mind that the double integration of ground acceleration records should produce zero displacements at the end of the record. If this does not happen then it may be necessary to apply a base line correction to the displacement record.

Is the base line correction the same procedure as the "Algorithm to Set Displacements at End of Records to Zero" summarized in Table J.1 of Prof. Wilson's book?

Yes, that is one of the various methods available for base line correction.


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