Geometry Definitions and Example Problem:
This sheet is designed to provide a better understanding of the geometric
cross sections terms used in Mechanics of Materials.
Table 1 – Geometry terms that do not care about the point of interest.
Variable
Name
𝐴
Mathematical
Definition
Area
2nd Moment of
Area about z‐
axis
2nd Moment of
Area about y‐
axis
𝐼
𝐼
Polar Moment
of Area
𝐽
𝑦
𝐼
Units
𝑑𝐴
𝑚
𝑦 𝑑𝐴
𝑚
𝑧 𝑑𝐴
𝑚
𝑧 𝑑𝐴
Figure 1 – Example cross section.
This cross section is a little like a
brick, with a large rectangle and
two ‘holes’.
𝑚
𝐼
To compute the Area of our example shape we take the whole
rectangle and then remove the circles.
𝐴
8𝑖𝑛 4𝑖𝑛
𝜋 2𝑖𝑛
𝜋 2𝑖𝑛
6.867 𝑖𝑛
To compute 𝐼 for our example we compute 𝐼 # for each shape about its own centroid and then use the
parallel axis theorem to find the second moment 𝐼 of that shape about the centroid of the whole. Once
that is done, we can add and subtract each shape as we did for the area calculation. See below.
𝐼
1
4𝑖𝑛 8𝑖𝑛
12
4𝑖𝑛 8𝑖𝑛 0𝑖𝑛
𝜋
1𝑖𝑛
4
𝜋 1𝑖𝑛
2𝑖𝑛
𝜋
1𝑖𝑛
4
𝜋 1𝑖𝑛
2𝑖𝑛
Table 2 – Geometry terms that care about the point of interest.
Variable
𝑦
𝑧
𝑏
𝑏
𝐴∗
Name
Mathematical
Definition
Location of the
point as measured
from the z‐axis.
𝑚
Note: Select the
smaller value at
transition points
Height of the
beam at the point
(excluding voids).
𝑚
𝑚
The area above
the point.
𝑧 𝑦 𝑑𝑦
𝑦∗
The centroid of 𝐴∗
𝑄
1st moment of
area for the area
ABOVE the point.
The area left of
the point.
𝐴∗
For the Example, drawn
for point g.
𝑚
Location of the
point as measured
from the y‐axis.
Width of the
beam at the point
(excluding voids).
Units
𝑚
𝑦 𝑑𝐴∗
𝐴∗ 𝑦 ∗
𝑦 𝑧 𝑑𝑧
𝑧̅ ∗
The centroid of 𝐴∗
𝑄
1st moment of
area for the area
LEFT of the point.
𝑚
𝑚
𝑚
𝑚
𝑧 𝑑𝐴∗
𝐴∗ 𝑧̅ ∗
𝑚
To compute 𝑄 for point g, we compute 𝑄 for each of the pieces. As long as we always measure from
the centroid of the whole, we can add and subtract the areas as needed. See below.
𝑄 can be computed in a similar manner. Just turn your paper ‘sideways’.
As a side note. Because the centroid of the whole area is defined as the point where the first moment is
|𝐴 𝑦 | where 𝐴 is the area below g, and
zero, one could also compute 𝑄 using the area below. 𝑄
𝑦 is the centroid of 𝐴 .
Version 001