CSI Software uses d’Alembert’s principal to automatically determine [acceleration loads] 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] (chapter: Load Cases, section: Acceleration Loads) explains this topic in greater detail.
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).
Figure 1 - Ground acceleration record
At each time step, integration of acceleration and velocity then yields expressions for ground velocity and displacement (Figure 2).
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Figure 2 - Expressions for a, v, and d, derived through integration
Evaluation of these expressions at t = ∆t yields a set of recursive equations, as shown in Figure 3:
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Figure 3 - Recursive equations characterizing ground motion
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 4 - Algorithm for zero displacement at record ends
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] article.