How to evaluate the mass moment of inertia in ETABS

Mass moment of inertia. Photo taken from http://www.physics.louisville.edu

Good times, good times.

The project I am currently involved in will undergo a wind tunnel test in order to derive a more realistic wind loading for the building.

But before they can generate results, they asked for the following data to be furnished to them :

1. Modal displacements per floor. We need to provide the displacements for the first 3 modes of vibration.
2. Seismic mass, center of mass, and center of rigidity per floor.
3. Mass moments of inertia per floor.

The last one was relatively new to me because honestly, I haven’t given much thought about it. I cannot find any sense of urgency to learn the said concept, and so I’m glad to be assisted by a design manager when the time came. Luckily, he is a part-time nerd like me who has a knack for deep technical discussions such as this topic on the mass moment of inertia and its role in dynamic analysis and in the overall structural response.

By definition from http://www.engineersedge.com, the “mass moment of inertia, usually denoted I, measures the extent to which an object resists rotational acceleration about an axis, and is the rotational analogue to mass.” This should not be confused with the stiffness used in defining the global and local stiffness matrices of members used for structural analysis that is:

Hooke’s law. Taken from http://www.hendrix2.uoregon.edu

and that it is not the same with the K matrix of the equation of motion:

Engineering Dynamics Equations. Taken from http://www.formulasheet.com

The stiffness matrices used for modal analysis are the translational and rotational stiffnesses. Note again that the rotational stiffness stated is a function of mass and it’s distance to the center of rigidity while the rotational stiffness due to its rigidity is based on its geometric properties.

ETABS generated mass moment of inertia…

ETABS produces a mass moment of inertia about its center of mass (by the way, you need to set all diaphragms to “Rigid” for it to generate this calculation). This is what I’ll call the local rotational stiffness. But since we’re dealing with a system, we need to get the global rotational inertia which is the building twist about a point called the center of rigidity. And so just like moment of inertia where we need to transfer it to a certain location other than its centroidal axes, the mass needs to be transferred to the center of rigidity as well.

Center of Rigidity, CT and center of mass, CM. Photo from http://www.debug.pi.gr

So how did we figure out that one? By manual calcs. How? By summing the mass of individual elements (floor elements, walls, beams and columns including the applied line loads and floor loads that constitute the seismic mass or the “mass source”) multiplied by the square of the distance of its centroid to the center of rigidity. It’s an onerous process but there’s really no other way to verify the result from ETABS. Luckily, based on our calculations, when we added the centroidal mass moment of inertia to the product of mass and the square of its distance to the center of rigidity, the results can be safely considered acceptable (discrepancy was kept to a minimum).

Table above are values from ETABS while the mass moments below are based on manual calculations. The results yielded a relative error of +/- 2.19%

Were you able to follow?

Another eureka moment indeed, thanks to the guidance of a very patient design manager. I cannot hide my satisfaction and elation in being a part of decoding that one, it’s like finding gold in my backyard. But as usual, not a lot would share my excitement. Mostly, I’m given the blank are-you-going-nuts look. And most of the time, I just try to understand but sometimes it is kinda frustrating when no one seems to share that joy with you.

Hell I’m still happy despite all that. That’s why I’ve written it here…