CEE 213—Deformable Solids
Please email
[email protected]
for the answer to the below question. You can also email us
for help on any of your assignments, testbanks etc.
CEE 213—Deformable Solids
CP 2
Properties of polygonal plane areas
© Keith D. Hjelmstad 2014

** Subscribe** to view the full document.

CEE 213—Deformable Solids
CP 2—Properties of Areas
Please email
[email protected]
for the answer to the below question. You can also email us
for help on any of your assignments, testbanks etc.
School for Sustainable Engineering and the Built Environment
Ira A. Fulton Schools of Engineering
Arizona State University
The properties of cross sections.
In the study of beams
The properties of cross sections
(including the axial bar and torsion) it becomes evident
that much of the behavior is dictated by the properties of the
cross sectional geometry
. In fact, the resistance to
deformation is always a function of the distribution of material
in a cross section. For the axial bar the
cross sectional area
is
key; for torsion the
polar moment of inertia
shows up; for
flexure the
moment of inertia
about the axis of bending is an
important property. These properties are essential to
determining the deformation under load.
Most of the classical
“tricks of the trade” that
can be used in service of
the computation of
geometric properties
amount to making use
of the properties of
simple geometric
pieces (often rectangles)
to build up the overall
cross sectional properties.
© Keith D. Hjelmstad 2014
Cross Section
x
Longitudinal Axis
z

CEE 213—Deformable Solids
CP 2—Properties of Areas
What we show in this set of notes is that it is possible to derive the integrals
needed to compute the cross sectional properties for a general triangle and
then use that basic foundation to create a method to compute the cross
sectional properties of
any y
closed polygonal region. This approach is based
on two key observations: (1) all integrals can be divided into pieces and (2) it
is possible to add in an integral that is not in the physical region as long as
you subtract it right
Cross Section
back out.
The key cross sectional properties.
Consider the
The key cross sectional properties
general cross section defined at left. We establish a (
y
,
z
)
coordinate system with origin
O
, and imagine another coordinate system (
x
,
h
) passing through the
centroid
C
. For beam theory we need
to compute area and moment of inertia
(about the centroid). To compute the moment of inertia we need the
location of the centroid.
The
area
is simply defined as the sum of the infinitesimal areas that make
up the cross section:
The
centroid
of an area is defined
as follows:
© Keith D. Hjelmstad 2014
y
z
x
h
c
ρ
dA
2
e
3
e
r
O
C
A
A
dA
1
A
dA
A
c
r

** Subscribe** to view the full document.

CEE 213—Deformable Solids
CP 2—Properties of Areas
In other words, it is the constant vector
c
that is the average over the area of the position
r
of the infinitesimal
areas. Finally, the
moment of inertia
(tensor) is defined as
Where the vector
r
is the
distance from the
centroid
C
to the

- Fall '17