What is the Mass Moment of Inertia?
The Mass Moment of Inertia is a measure of a body's resistance to rotational (angular) acceleration about a given axis. It plays the same role in rotational dynamics that mass plays in linear dynamics.
Where r is the distance from the mass element dm to the axis of rotation. In plain terms: mass that sits further from the axis contributes more to I than mass sitting close to it, because it has to travel further (and faster) for the same angular rotation.
Units: kg·m² (SI). Not to be confused with the area moment of inertia below, which looks deceptively similar in name and formula.
Common Confusion with the Area Moment of Inertia
These two quantities share the same "second moment" mathematical form, and this is where most of the confusion comes from.
| Area Moment of Inertia | Mass Moment of Inertia | |
|---|---|---|
| Formula | I = ∫ y² dA | I = ∫ r² dm |
| Property of | Cross-section geometry | Distribution of mass |
| Units | m⁴ | kg·m² |
| Governs | Bending stiffness, section strength | Resistance to angular acceleration |
| Used in | Beam/column design (M = EI·κ) | Dynamics (M = I·α) |
The area moment of inertia is purely a geometric property of a cross-section — it has nothing to do with mass, and everything to do with how a section resists bending curvature.
The mass moment of inertia has nothing to do with cross-section shape — it's about how the mass of a body (a floor plate, a structure) is distributed relative to an axis, and how that resists rotational acceleration. Same-sounding name, completely different physics.
Relevance to Wind and Seismic Inertial Loads
In structural dynamics, the rotational equivalent of Newton's second law is:
Where T is the applied torsion, Ip is the mass moment of inertia, and at is the angular (torsional) acceleration.
This becomes directly relevant whenever a building's mass is displaced eccentrically from its centre of stiffness — which is essentially always, to some degree. Under wind or seismic excitation, this eccentricity generates a torsional inertial load, causing the floor plate to want to rotate about its vertical axis as well as translate.
The mass moment of inertia of each floor diaphragm (about the vertical axis) is what governs how that floor resists this angular acceleration. It feeds directly into:
- Torsional modal analysis — torsional mode shapes and periods depend on the rotational mass (MMI), not just the translational mass.
- Seismic torsional irregularity checks (e.g. accidental eccentricity provisions in codes) — larger floor plate MMI means more torque is needed to produce the same angular acceleration, but also more inertial torque generated for a given angular acceleration demand.
- Wind-induced torsional response — particularly for slender towers, where across-wind and torsional excitation can govern serviceability (occupant comfort) design.
Estimating MMI Using Radius of Gyration — Rectangular Floor Plate
For a quick hand-check, a floor plate can be idealised as a uniform rectangular plate, with the vertical (rotational) axis passing through its centroid.
Mass moment of inertia, I = m · r² = m · (a² + b²) / 12
Where:
m = total seismic mass of the floor plate (kg or tonnes)
a, b = plan dimensions of the floor plate (m)
Example Calculation
Consider a rectangular floor plate 24 m × 16 m, with a total seismic mass of 500 tonnes.
Mass moment of inertia, Ip = mr² = 500 tonnes x (8.33 m)² = 34,667 tonnes-m²
This is a reasonable first-pass estimate for a fairly uniform floor. Where mass is concentrated away from the centroid (heavy plant, cores offset from the geometric centre), a more refined calculation — summing r²·dm for discrete mass elements — will be needed.