To simulate ground motion during time-history analysis,

Software automatically applies [acceleration load] to all elements and joints of a structural model. These loads are determined through d’Alembert’s principal. As the

[_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 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 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 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

, 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 then 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, as shown in Figure 3:

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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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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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