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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.


When an acceleration record is specified for time-history analysis, the ground motion is automatically applied to all support restraints. CSI Software uses d’Alembert’s principle to then translate the time history into acceleration loads which are applied to structural joints. During formulation, since acceleration couples with mass, it is important that joints have mass. Acceleration loads are explained further in the CSI Analysis Reference Manual (Chapter XVII: Load Patterns, Acceleration Loads).

To manually input ground motion at specific supports, it is necessary to first convert the acceleration record into its corresponding displacement time-history record. During conversion, 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 by smoothing the displacement record. A smaller time step, possibly 1/10 that of the acceleration record, will refine the ground motion during conversion.

Experimental conversion

Two basic methods are available for conversion from Acceleration time history to displacement. First, an experimental approach is described as follows:

  1. Create a simple SAP2000 model.
  2. Apply the acceleration time history using the given time step (perhaps 0.02).
  3. Set the output time step to 1/10 of this value (0.002).
  4. Extract the displacement results from a support restraint.
  5. Correct for zero initial and final displacement and velocity using a + bt.
  6. Use this refined displacement function as the ground-motion input for the actual model.

NOTE: Analysis proceeds according to the shorter time step, though output is reported (more accurately) only for each original time step.

Mathematical conversion

As an alternative, mathematical conversion is summarized in Appendix J of Dr. Edward L. Wilson’s text Static and Dynamic Analysis of StructuresThe conversion is given in Figure 3, and its formulation is described as follows:


  1. Ground acceleration is idealized as linear within each time increment, as shown in Figure 1:

    Figure 1 - Ground acceleration record


  2. Acceleration and velocity are integrated at each time step to generate expressions for velocity and displacement, as shown in Figure 2:

    Figure 2 - Expressions for a, v, and d, derived through integration


  3. These expressions are evaluated at t = ∆t to produce the set of recursive equations shown in Figure 3:

    Figure 3 - Recursive equations characterizing ground motion

    An acceleration record is then translated into its corresponding displacement record using these expressions.

  4. This double-integration procedure should produce zero displacement at either end of the displacement record. However, if nonzero displacement does exist, a base-line correction must be applied according to Figure 4:


    Figure 4 - Algorithm for zero displacement at record ends

  5. Displacement ground motion is then input at specific support locations using the option for Ground Displacement Load. This process is described in the Multi-support excitation article.

See Also


  • Wilson, E. L. (2004). Static and Dynamic Analysis of Structures (4th ed.). Berkeley, CA: Computers and Structures, Inc.
    Available for purchase on the CSI Products > Books page