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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 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).
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{center-text}Figure 1 - Ground acceleration record{center-text}
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At each time step, integration of acceleration and velocity then yields expressions for ground velocity and displacement (Figure 2).
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{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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{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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{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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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:
- Create a simple SAP2000 model.
- Apply the acceleration time history using the given time step (perhaps 0.02).
- Set the output time step to 1/10 of this value (0.002).
- Extract the displacement results from a support restraint.
- Correct for zero initial and final displacement and velocity using a + bt.
- Use this refined displacement function as the ground-motion input for the actual model.
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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 Structures. The conversion is given in Figure 3, and its formulation is described as follows:
Ground acceleration is idealized as linear within each time increment, as shown in Figure 1:
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Figure 1 - Ground acceleration record
Acceleration and velocity are integrated at each time step to generate expressions for velocity and displacement, as shown in Figure 2:
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Figure 2 - Expressions for a, v, and d, derived through integration
These expressions are evaluated at t = ∆t to produce the set of recursive equations shown in Figure 3:
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Figure 3 - Recursive equations characterizing ground motion
An acceleration record is then translated into its corresponding displacement record using these expressions.
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:
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Figure 4 - Algorithm for zero displacement at record ends
- 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
- Ground motion at specific supports – Manual multi-support excitation
References
- 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