# Multiple-Support Excitation

Structural systems typically have different ground accelerations at supports separated by long distances. These systems can be a single structure such as a long span bridge or multiple structures in a region.

While most OpenSees analyses use uniform excitation with effective earthquake forces applied to the dynamic DOFs of the model, the framework also accommodates multi-support excitation. The basic idea is that the locations of support excitation become DOFs where the ground displacement is input–much like putting sp constraints inside a load pattern.

For more details on the equation of motion for multiple-support excitation, see Ch. 9 of Chopra, 5th edition. Although Chopra presents the equation of motion for linear-elastic models, the extension to nonlinear models is straightforward.

But rather than get into details of the equation of motion, I’d like to show how to apply multiple-support excitation in OpenSees. As usual, so we don’t get mired in the details of specific element and material models, I’ll use a simple 1D model.

After defining the model, we define time series for two ground motions, then impose the ground motions at the two supports.

import openseespy.opensees as ops

k = 60
m = 2
g = 386.4

ops.wipe()
ops.model('basic','-ndm',1,'-ndf',1)

ops.node(1,0); ops.fix(1,1)
ops.node(2,0); ops.mass(2,m)
ops.node(3,0); ops.fix(3,1)

ops.uniaxialMaterial('Elastic',1,k)

ops.element('zeroLength',1,1,2,'-mat',1,'-dir',1)
ops.element('zeroLength',2,2,3,'-mat',1,'-dir',1)

ops.timeSeries('Path',1,'-dt',0.02,'-filePath','tabasFN.txt','-factor',g)
ops.timeSeries('Path',2,'-dt',0.02,'-filePath','tabasFP.txt','-factor',g)

ops.pattern('MultipleSupport',1)
ops.groundMotion(1,'Plain','-accel',1)
ops.imposedMotion(1,1,1) # node, dof, gmTag
ops.groundMotion(2,'Plain','-accel',2)
ops.imposedMotion(3,1,2)

The default integration for obtaining ground velocity and displacement from ground acceleration is the trapezoidal rule. You can also use Simpson’s rule.

In addition, the groundMotion and imposedMotion objects must be defined after the load pattern. In Tcl speak, these objects should be inside braces.

timeSeries Path 1 -dt 0.02 -filePath tabasFN.txt -factor \$g
timeSeries Path 2 -dt 0.02 -filePath tabasFP.txt -factor \$g

pattern MultipleSupport 1 {
groundMotion 1 Plain -accel 1
imposedMotion 1 1 1 ;# node, dof, gmTag
groundMotion 2 Plain -accel 2
imposedMotion 3 1 2
}

After performing dynamic analysis, the displacement response histories of the mass and the two supports are shown below.

We see that the supports have different displacement response histories, as expected, while the dynamic DOF vibrates. Multiple-support excitation is fairly straightforward in OpenSees.

The remainder of this post deals with two ways to apply uniform excitation to the simple 1D model. First, we can impose the same ground motion at the supports.

ops.timeSeries('Path',1,'-dt',0.02,'-filePath','tabasFN.txt','-factor',g)

ops.pattern('MultipleSupport',1)
ops.groundMotion(1,'Plain','-accel',1)
ops.imposedMotion(1,1,1) # node, dof, gmTag
ops.imposedMotion(3,1,1)

Alternatively, we can apply a standard uniform excitation.

ops.timeSeries('Path',1,'-dt',0.02,'-filePath','tabasFN.txt','-factor',g)

ops.pattern('UniformExcitation',1,1,'-accel',1) # tag, dir, tsTag

These two approaches give the same response for the dynamic DOF (relative to the ground displacement). But of course we lose the ground displacement when doing the standard uniform excitation.

## 5 thoughts on “Multiple-Support Excitation”

1. Prabakaran Kesavan says:

Dear Dr. Scott,
Thank you for such an informative post. I consider every post from you as a gem. I have a question to you. In OS wiki page, it says at groundMotion page that “For earthquake excitations it is important that the user provide the displacement time history, as the one generated using the trapezoidal method will not provide good results.”, Why is it so? Do we have to be careful about our accelerogram input? Kindly throw some light on this point. Thank you again.

Liked by 1 person

1. Hello Prabakaran,
Thank you for the question. I’m not sure how important that is for earthquake ground motion, but I’ve never looked at the issue.
PD

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1. Prabakaran Kesavan says:

Sure Dr. Scott. Thanks for your reply.

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2. Thank you for a fantastic article. I’d would like to ask you a question. Is Multiple-Support Excitation better for high-rise structures? Because I believe uniform excitation causes a uniform acceleration across all nodes of the structure, which may be true for rigid and short structures but not for high-rise structures (of course, this is my opinion). What are your thoughts on this?

Liked by 1 person

1. Hello MSB,
Thank you! I don’t know if multiple-support excitation would be better for high rise structures.
PD

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