During time-history analysis, an acceleration record is automatically applied to all restraint supports.

Software then uses d’Alembert’s principal to translate the time history into [acceleration loads] which are applied to structural joints. This process is explained further in the

[_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 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.

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 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 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.

Alternatively, users may implement the mathematical formulation which is summarized in Appendix J of Dr. Edward L. Wilson’s text

, and outlined below:

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

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

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Evaluation of these expressions at t = ∆t yields a set of recursive equations (Figure 3).

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These expressions may then be used to translate an 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 exists, it is necessary to apply a base line correction (Figure 4).

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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] article.

References

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