After cutting through all the spam, you’ll find some good posts on the OpenSees message board. In one such post, Ahmet Alper Parker asked about the large mass method (LMM) and if it can be implemented in OpenSees. I was not familiar with the LMM, so Ahmet pointed me to this paper.

The basic idea of the LMM is you input the ground acceleration as a force acting on a very large mass which acts as the base of the structural model. For a uniform excitation, the LMM is an alternative to applying effective earthquake forces to the dynamic DOFs.

Based on the paper and other sources online, the large mass, *M*, should be about 10^{5} times larger than the total mass of the structural model. The large mass also needs to be restrained, e.g., with a soft spring, so that the model doesn’t float off into space.

My initial thought was the LMM sounds a lot like a soil column, which is commonly used for propagating ground motion through an SSI model. I asked a couple geotechs–they hadn’t heard of the LMM either and they also brought up soil columns.

Regardless, here’s an OpenSees model of the two DOF example from the paper. The script loops through fixed-based and LMM analyses for the Tabas ground acceleration record.

```
import openseespy.opensees as ops
m = 10
k = 100
a = 1e5
M = a*(m+m)
Tfinal = 50.0
dt = 0.01
Nsteps = int(Tfinal/dt)
for model in ['EF','LM']:
ops.wipe()
ops.model('basic','-ndm',1,'-ndf',1)
ops.node(0,0); ops.fix(0,1)
ops.node(1,0)
if model == 'EF':
ops.fix(1,1)
if model == 'LM':
ops.mass(1,M)
ops.node(2,0); ops.mass(2,m)
ops.node(3,0); ops.mass(3,m)
ops.uniaxialMaterial('Elastic',0,k/100) # Soft spring
ops.uniaxialMaterial('Elastic',1,k)
ops.element('zeroLength',0,0,1,'-mat',0,'-dir',1)
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')
if model == 'EF':
ops.pattern('UniformExcitation',1,1,'-accel',1)
if model == 'LM':
ops.pattern('Plain',1,1)
ops.load(1,M)
ops.analysis('Transient')
ops.analyze(Nsteps,dt)
```

The eigenvalues and eigenvectors of the two models are the same; however, the large mass produces an additional mode with essentially zero frequency. The displacement response histories for the three DOFs in each model are shown below.

One cool thing about the LMM is you get the ground displacement “for free” because the big mass DOF is part of the numerical solution. For this simple example, after you subtract the motion of the large mass, the LMM gives the same response as the effective force approach.

Although this analysis did not use damping, note that mass proportional Rayleigh damping may have unintended consequences in LMM models.

Based on online discussion (here and here), it seems the LMM may have been a workaround for older FEA software that didn’t have functionality for support excitation. However, Google Scholar searching reveals some recent papers that explore how the LMM can be used for structural models with non-uniform, multi-support excitation. In his post, Ahmet pointed out that the LMM is able to dissipate energy in models of base isolated structures. Also, the LMM seems to be a physically sound way to model shake table experiments, but I feel like I’m repeating the past.

This post was meant to show a simple example of the LMM, not to dive deep into the method’s pros and cons. Further investigation is required.

So, OpenSees readers, what do you think? Have you used the LMM? Leave your thoughts in the comments section below.

Thanks for this post and the link to the paper. I came to know about this method in the discussion forum of code-aster (https://www.code-aster.org/forum2/viewtopic.php?id=19223). It is nice to see the OS results that you have produced. In harmonic excitation, it should be possible to use displacement as input too, by using the relation between displacement and acceleration. I wonder, for linear analysis, can we use Fourier Spectrum of a ground motion as an input in this method to see which are the important modes of the structure for a given ground motion?

Best regards.

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I haven’t used the large mass method, but I have used it’s sibling, the large spring method (probably not the real name) before. The large spring method is for achieving pseudo-displacement control. You add a linear elastic spring with high stiffness to your system in the direction you want to control. Then you apply load to get the displacement you want (based on f = Kd) noting that the contribution to K from the system will be small. It has been a while, so I don’t remember how well it worked.

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I can say it is better in base isolation. In relative displacement method, you apply earthquake acceleration (in some sort) to structure, not base. In total (or large mass) method, you apply your earthquake to base and you have the correct base results. We are working on a paper on this (Thank you very much, this is something important)

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We (Edward L. Wilson, James Kelley and I) are working also on nonlinear structures (P-Delta). Even pdelta has significantly different results compared classic relative displacement approach

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If there is large mass (acceleration) and large spring (displacement) methods, there should be large damping (velocity) method

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I used this method unknowingly 2 years ago. I used large mass at the base, results were same. But i didnt know this was different method. :):)

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Suppose if I want to model base isolation, then can I replace the soft spring by the stiffness of base isolator?

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No

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You would have to put your base isolation spring above the large mass.

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In the 1990 paper by Léger, Idé, and Paultre (https://www.sciencedirect.com/science/article/pii/004579499090224P), they conclude that the LMM is a nice way to avoid the need for double integration of the ground motion acceleration to obtain the ground motion displacement (therefore also avoiding a potential baseline correction). Nowadays, I believe both ground motion acceleration and displacement are easily found in GM databases (such as the PEER NGA-W2). So I think this feature may not be so appealing. Nevertheless, compared to a relative motion method based on modal superposition (which cannot handle the nonlinear response of the structure), the LMM can be employed with a step-by-step time integration method to obtain the nonlinear response. Again, a total motion method (which I believe is the method implement in OpenSees) works as well for nonlinear problems. Personally, I have already worked with linear examples and the results were really close to both total motion method or relative motion method. I still need to test it on nonlinear problems for my own curiosity though. Either way, I think it’s a pretty elegant solution to multiple support excitation problems.

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