first moment of the area - Blog at UNY dot AC dot ID

Geometrical properties of
cross-sections
Strength of Materials
Introduction
The strength of a component of a
structure is dependent on the
geometrical properties of its cross
section in addition to its material and
other properties.
 For example, a beam with a large cross
section will, in general, be able to resist a
bending moment more readily than a
beam with a smaller cross-section.

Example Shapes
Centroid


The position of the
centroid of a crosssection is the centre of
the moment of area of
the cross section.
If the cross-section is
constructed from a
homogeneous material,
its centroid will lie at
the same position as
its centre of gravity.
First moment of Area
Consider an area A
located in the x-y plane.
Denoting by x and y the
coordinates of an
element of area d.A, we
define the first moment of
the area A with respect
to the x axis as the
integral
 Similarly, the first moment
of the area A with respect
to the y axis is defined as
the integral

Sx 
 y  dA
A
Sy 
 x  dA
A
First moment of Area

It can be conclude
that if x and y passes
through the centroid
of the area of A, then
the first moment of
the area of Sx and Sy
will be zero.
 x  dA 
Ax  Sy  Ax
A
 y  dA 
A
 x
Sy
A
 y
Sx
A
A y  Sx  A y
Second moment of Area

The second
moments of area of
x - x and y - y axes,
respectively, are given
by
Second moment of Area

From the theorem of
Phytagoras :

known as the
perpendicular axes
theorem which states that
the sum of the second
moments of area of two
mutually perpendicular
axes of a lamina is equal to
the polar second moment
where these two axes
cross.
Parallel axes theorem
known as the parallel axes
theorem, which states that
the second moment of
area
about the X-X axis is equal
to the second moment of
area about the x-x axis +
h2 x A, where x-x
and X-X are parallel.
Example 01

Determine the second moment of area of the
rectangular section about its centroid (x-x) axis and
its base (X-X ) axis. Hence or otherwise, verify the
parallel axes theorem.