13. Centroids and Moment of Inertia
Besides forces and moments, it is necessary to know the geometrical and crosssectional properties of the structural elements that are used in construction. The
properties that we will be discussing are:
1. Centroids
2. Moment of Inertia
3. Radius of Gyration
Centroids
The centroid of a body cab be described as a point at which the body can be balanced
CG
The physical way in which the point can be determined is shown below:
Mathematically the CG of a planar body is given by the equation
Σ( x∆A)
x=
A
Σ( y∆A)
y=
A
Or by using Calculus
xdA
∫
x=
∫ dA
ydA
∫
y=
∫ dA
The physical meaning of CG in Statics
Overturning
point
Overturning
point
Case A – Since the CG of the brick
wall falls within the overturning point
the wall is stable
Case B – Since the CG of the brick
wall falls outside the overturning
point the wall is unstable and will
fall down
Moment of Inertia
The moment of inertia cab be considered as a shape factor which indicated
how the material is distributed about the center of gravity of the crosssection.
The moment of inertia has a significant effect on the structural behavior of
construction elements
The formulas used for determining the moment of inertia are
I
I
Or by using calculus
2
x
= Σ y dA
y
= Σ x dA
2
I = ∫ y dA
I = ∫ x dA
2
x
2
y
The physical meaning of moment of inertia
2m
0.5m
2m
1m
2m
1m
0.5m
2m
4m
2m
2m
Area
Moment of
Inertia
1m2
0.02
m4
1m2
1m2
1m2
1m2
0.08 m4
0.33m4
0.55m4
2.79m4
Thus it is clear that as the material is moved away from the center of the axis the
moment of inertia increases.
This is further illustrated by the next example
16
14
12
10
8
6
4
2
0
Series1
3
6
9
Area= 1m2
16.20 m4
0.421
m4
Moment of Inertia
2.78”
3”
2”
1.651”
Area=
1m2
I=
π (d − d )
4
o
4
i
64
1”
Area= 1m2
0.079 m4
0.5 m
1.82 m
Average distance of mass from the
center of the circle
2.89 m
Parallel Axis Theorem
Structurally efficient cross-sections need not always be symmetric. The following are
two examples of structurally efficient non-symmetric sections for bridges
The parallel axis theorem enables us to calculate the moment of inertia for such nonsymmetric structural cross-sections.
The statement of the parallel axis theorem follows
I x = ΣI xo + ΣAy
Where
2
Ix
=Moment of inertia about the centroidal x-axis of the section
I xo
=Moment of inertia of separate elements about their own axis
A
y
= area of each of the individual components
= distance between the centroidal axis and the centroids of
the each individual elements
Radius of Gyration
The radius of gyration is a quantity that connects two other physical characteristics of the
structural element: the cross-sectional area and the moment of inertia
Since the moment of inertia has a dimension of length to the power four, it can be written
as area times a length dimension squared as shown below
I x = ∫ y dA = Ar
2
2
x
Where rx is called the radius of gyration. It can be visualized as the distance at which
the entire area of the cross-section is concentrated
The location the radius of gyration is shown below
rx = d 4
x
rx = d 4
Ix =
πd
4
64
A=
x
πd 2
4
πd 4 d
=
rx = I x A =
2
64πd
4
4
The radius of gyration is an useful concept in understanding the buckling of slender
columns