Moments of Inertia

Moments of Inertia

$Learning Objectives$

Distinguish area moment of inertia from mass moment of inertia and identify their different physical uses.
Calculate standard second moments of area about centroidal axes.
Apply the parallel-axis theorem to shift an area moment of inertia to a parallel axis.
Determine composite-section moments of inertia, including cutouts.
Relate polar moment of area and radius of gyration to Cartesian area moments.
Explain product of inertia and principal centroidal axes conceptually.

$Area Moment of Inertia$

The geometric second moment of an area about an axis. It has units of length to the fourth power and appears in bending, deflection, and buckling relations.

$Mass Moment of Inertia$

The second moment of mass about an axis. It has units of mass times length squared and measures rotational inertia in dynamics.

$Area Inertia and Mass Inertia Are Different Quantities$

Area moment of inertia uses $dA$ and units such as $\text{mm}^4$ ; mass moment of inertia uses $dm$ and units such as $\text{kg}\cdot\text{m}^2$ . They are not interchangeable.

Second Moments of Area

Defines the Cartesian area moments of inertia.

I_x=\int_A y^2\,dA, \qquad I_y=\int_A x^2\,dA

Variables

Symbol	Description	Unit
$I_x$	Area moment of inertia about the x-axis	$m^4$
$I_y$	Area moment of inertia about the y-axis	$m^4$
$dA$	Differential area element	$m^2$

$Physical Meaning for Beam Sections$

Because the coordinate distance is squared, area placed farther from a bending axis contributes disproportionately more to $I$ . This is why deep I-shaped sections can achieve high flexural stiffness efficiently: much of their area is concentrated in flanges far from the neutral axis.

For a rectangle of width $b$ and depth $h$ , the centroidal second moment about the horizontal axis is $I_x=bh^3/12$ . The cubic dependence on depth makes orientation important.

Rectangle about Centroidal Axes

Standard centroidal area moments of inertia for a rectangle.

I_x=\frac{bh^3}{12}, \qquad I_y=\frac{hb^3}{12}

Variables

Symbol	Description	Unit
$b$	Rectangle width parallel to the x-axis	m
$h$	Rectangle height parallel to the y-axis	m

$Interactive Exploration$

Change section dimensions in the visualizer and observe how moving area farther from an axis changes the second moment much more strongly than simply adding the same area near the axis.

Simulation

Controls

120 mm

200 mm

10 mm

physical model

Rectangle Ix

8.000e+7 mm⁴

Rectangle Iy

2.880e+7 mm⁴

Circle area

3.142e+2 mm²

Circle centroidal Ix=Iy

7.854e+3 mm⁴

I=I_c+Ad^2,\qquad J_O=I_x+I_y,\qquad k=\sqrt{\frac{I}{A}}

$Parallel-Axis Theorem$

A relation used to transfer a second moment of area from a centroidal axis to a parallel axis a distance

d

away.

Parallel-Axis Theorem for Area

Transfers a centroidal area moment of inertia to a parallel axis.

I=I_c+Ad^2

Variables

Symbol	Description	Unit
$I$	Area moment of inertia about the shifted axis	$m^4$
$I_c$	Area moment of inertia about the parallel centroidal axis	$m^4$
$A$	Area	$m^2$
$d$	Perpendicular distance between axes	m

$Parallel-Axis Exploration$

Use the parallel-axis scenario to see how the transfer term $Ad^2$ grows as the reference axis moves away from the centroidal axis.

Simulation

Controls

120 mm

200 mm

80 mm

physical model

Centroidal Ix

8.000e+7 mm⁴

Transfer Ad²

1.536e+8 mm⁴

Reference-axis Ix

2.336e+8 mm⁴

Signed offset d

80.0 mm

I=I_c+Ad^2,\qquad J_O=I_x+I_y,\qquad k=\sqrt{\frac{I}{A}}

$Composite Sections$

To find $I$ for a composite area about a common axis:

determine the composite centroid when the target axis is centroidal;
calculate each part's centroidal $I_{c,i}$ ;
shift each part using $A_id_i^2$ ;
sum material parts and subtract cutouts consistently.

For a cutout, both its centroidal inertia and its parallel-axis contribution are subtracted.

Composite Area Moment of Inertia

Sums shifted component inertias about a common axis.

I=\sum \left(I_{c,i}+A_i d_i^2\right)

Variables

Symbol	Description	Unit
$I_{c,i}$	Centroidal area moment of inertia of component i	$m^4$
$A_i$	Signed area of component i	$m^2$
$d_i$	Distance from the component centroidal axis to the target axis	m

$Polar Moment of Area$

The second moment of area about a point perpendicular to the plane. For orthogonal in-plane axes through the same point,

J_O=I_x+I_y

Polar-Area Relation

Relates polar and Cartesian second moments about the same point.

J_O=I_x+I_y

Variables

Symbol	Description	Unit
$J_O$	Polar moment of area about point O	$m^4$
$I_x$	Area moment about x through O	$m^4$
$I_y$	Area moment about y through O	$m^4$

$Radius of Gyration$

The distance

k

from an axis at which the entire area could be conceptually concentrated while preserving the same second moment:

I=Ak^2

Area Radius of Gyration

Expresses an area moment of inertia as an equivalent area distribution radius.

k=\sqrt{\frac{I}{A}}

Variables

Symbol	Description	Unit
$k$	Radius of gyration	m
$I$	Area moment of inertia	$m^4$
$A$	Area	$m^2$

$Product of Inertia$

A geometric second-moment quantity that describes coupling between the two coordinate directions and is used to locate principal axes for unsymmetric areas.

Product of Inertia

Defines the area product of inertia for a selected pair of Cartesian axes.

$I_{xy}=\int_A xy\,dA$

Variables

Symbol	Description	Unit
$I_{xy}$	Product of inertia of the area	$m^4$
$x$	Horizontal coordinate of the differential area	m
$y$	Vertical coordinate of the differential area	m
$dA$	Differential area element	$m^2$

$Principal Centroidal Axes$

Principal axes through a point are orientations for which the product of inertia is zero. For areas with an axis of symmetry, that symmetry axis and the perpendicular centroidal axis are principal axes.

Principal-axis concepts become important in unsymmetric bending, where the convenient geometric axes may not be the uncoupled bending axes.

$Mass Moment of Inertia$

Mass moment of inertia is the dynamic analogue based on mass distribution. Moving mass farther from a rotation axis increases the torque required to produce a given angular acceleration. This is conceptually related to area inertia through a second-moment integral, but the underlying measure and units are different.

Mass Moment of Inertia

Defines rotational inertia from the distribution of mass about an axis.

$I_m=\int r^2\,dm$

Variables

Symbol	Description	Unit
$I_m$	Mass moment of inertia about the selected axis	$kg·m^2$
$r$	Perpendicular distance from the axis to the mass element	m
$dm$	Differential mass element	kg

Key Takeaways

Area moment of inertia is a geometric property with units of length to the fourth power; mass moment of inertia is a dynamic mass property.
Area farther from the reference axis contributes strongly because distance is squared.
The parallel-axis theorem adds $Ad^2$ when transferring from a centroidal axis to a parallel axis.
Composite-section inertia requires both centroid location and consistent shifting of every component.
Polar moment satisfies $J_O=I_x+I_y$ for perpendicular axes through the same point.
Radius of gyration repackages the second moment as an equivalent distance.
Product of inertia and principal axes are essential for understanding unsymmetric sections.