Transcript ME13A: CHAPTER FIVE - Faculty of Engineering

ME13A: CHAPTER FIVE
CENTROIDS AND
CENTRES OF GRAVITY
5.2. Centroids of Areas
For the homogenous plate: weight of element of plate
w = density x g x volume
Where: 
and
=  g t A
is density (kg/m3); g = m/s2 ; t = thickness of plate (m);
A is the area of the element, m2.
Also: The weight of the whole plate, W =

g t A
where A is the area of the plate.

Substituting for
W and W in moment equations (1) and

dividing by
g t,
We write moment equations for area:
M y : x A  x1A1  x2 A2 .... xn An
M x : y A  y1A1  y2 A2 .... yn An
U sin g Limits:
xA 
z
x dA,
yA 
z
y dA ...........(3)
The co-ordinates x and y define the co-ordinates of a homogenous plate. The point (x,y)
defines the centroid of the area A of the plate. See Figure 5.3.
y
x
C
y
x
5.1
First Moments of Areas
The integral  xdA in equation (3) is defined as the first
moment of the area A wrt the y axis and is denoted as Qy: i.e. Qy = xdA
Likewise: along x-axis: Qx =  ydA .
i.e. Qy =  xdA . And : Qx =  ydA . ………………. (4)
Comparing Equations (3) and (4):
Qy =
i.e.
x
x
A and Qx =y
= Qy/A
x
z
x dA
A
A
and
y
y
z
y dA
A
= Qx/A
.
Examples

Solve Examples Using the method of
integration to locate Centroids.
Centroids of Composite Objects
Centroids of Composite Objects
Contd.
Examples

Solve Examples to Illustrate the centroid of
Composite Objects.
Examples

Solve Examples to illustrate the centroids
of beams with distributed Loads.
Assignment

1. The column is used to support the floor which
exerts a force of 3000 N on top of the column. The
effect of soil pressure along its side is distributed as
shown. Replace this loading by an equivalent
resultant force and specify where it acts along the
column, measured from the base A.
Assignment

2. The wall footing is used to support the
column load of 12,000 N. Determine the
intensities w1 and w2 of the distributed
loading acting on the base of the footing for
equilibrium.