Centroids and The Moment of Inertia
Centroids: The concept of the centroid is nearly the same as the center of
mass of an object in two dimensions, as in a very thin plate. The center of
mass is obtained by breaking the object into very small bits of mass dM,
multiplying these bits of mass by the distance to the x (and y) axis, summing
over the entire object, and finally dividing by the total mass of the object to
obtain the Center of Mass - which may be considered to be the point at which
the entire mass of the object may be considered to "act". See Diagram 1.
The only difference between the center of mass and the centroid is that rather
than summing the product of each bit of mass dM and the distance xi (and yi) to
an axis then dividing by the total mass, we instead divided the object into
small bits of areas dA, and then take the sum of the product of each bit of area
dA and the distance xi (and yi) to an axis then divide by the total area of the
object. This results in an Xct. and Yct location for the Centroid (center of area)
of the object. See Diagram 2
Several points to mention. We will assume all our beams have uniform density
and will not consider the case of non-uniform density beams. We will also point
out that for any beam cross section (or object) which is symmetry, the centroid
will simply be at the geometric center of the cross section. Thus for
rectangular beam and I-beams, the centroid is located at the exact center of
the beam. This is not the case for T-beams.
Centroids of Composite Areas:
Some objects or beams may be formed from several simple areas, such as
rectangles, triangles, etc. (See Diagram 3) In this case the centroid of the
compose area may be found by taking the sum of the produce of each simple
area and the distance it's centroid is from the axis, divided by the sum of the
areas. For the composite area shown in Diagram 3, the location of it's x centroid would be given by:
X ct = (A1 * x1 + A2 * x2 + A3 * x3 + A4 * x4)/(A1 +A2 +A3 + A4)
where x1, x2, x3, and x4 are the distances from the centroid of each simple
area to the y-axis as shown in the Diagram 3. The location of the y - centroid
would be given in like manner, although the y distances are not shown in
Diagram 3:
Y ct = (A1 * y1 + A2 * y2 + A3 * y3 + A4 * y4)/(A1 +A2 +A3 + A4)
Moment of Inertia
A second quantity which is of importance when considering beam stresses is the
Moment of Inertia. Once again, the Moment of Inertia as used in Physics
involves the mass of the object. The Moment of Inertia is obtained by breaking
the object into very small bits of mass dM, multiplying these bits of mass by
the square of the distance to the x (and y) axis and summing over the entire
object. See Diagram 4.
For use with beam stresses, rather than using the Moment of Inertia as
discussed above, we will once again use an Area Moment of Inertia. This Area
Moment of Inertia is obtained by breaking the object into very small bits of
area dA, multiplying these bits of area by the square of the distance to the x
(and y) axis and summing over the entire object. See Diagram 5.
The actual value of the moment of inertia depends on the axis chosen to
calculate the moment of the inertia with respect to. That is, for a rectangular
object, the moment of inertia about an axis passing through the centroid of the
rectangle is: I = 1/12 (base * depth3) with units of inches4., while the moment
of inertia with respect to an axis through the base of the rectangle is: I = 1/3
(base * depth3) in4. See Diagram 6. Note that the moment of inertia of any
object has its smallest value when calculated with respect to an axis passing
through the centroid of the object.
Parallel Axis Theorem:
Moments of inertia about different axis may calculated using the Parallel Axis
Theorem, which may be written: Ixx = Icc + Adc-x2 This says that the moment of
inertia about any axis (Ixx) parallel to an axis through the centroid of the object
is equal to the moment of inertia about the axis passing through the centroid
(Icc) plus the product of the area of the object and the distance between the
two parallel axis (Adc-x2).
We lastly take a moment to define several other concepts related to the
Moment of Inertia.
Radius of Gyration: rxx = (Ixx/A)1/2 The radius of gyration is the distance from
an axis which, if the entire area of the object were located at that distance, it
would result in the same moment of inertia about the axis that the object has.
Polar Moment of Inertia J = r2 dA The polar moment of inertia is the sum of
the produce of each bit of area dA and the radial distance to an origin squared.
In a case as shown in Diagram 7, the polar moment of inertia in related to the x
& y moments of inertia by: J = Ixx + Iyy.
One final comment - all the summations shown above become integrations as
we let the dM's and dA's approach zero. And, while this is important and useful
when calculating Centroids and Moments of Inertia, the summation method is
just as useful for understanding the concepts involved