Bài giảng Strength of materials - Chapter 5: Geometric Properties of an Area - Trần Minh Tú

STRENGTH OF MATERIALS
TRAN MINH TU - University of Civil Engineering,
Giai Phong Str. 55, Hai Ba Trung Dist. Hanoi, Vietnam
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5
CHAPTER
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Geometric Properties of an Area
Contents
5.1. Introduction
5.2. First moment of area
5.3. Moment of inertia for an area
5.4. Moment of inertia for some simple areas
5.5. Parallel - axis theorem
5.6. Examples
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5.1. Introduction
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Dimension, shape?
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5.2. First Moment of Area
5.2.1. Definition
( )
x
A
S ydA 
( )
y
A
S xdA 
• Centroidal axes: are axes, which first moment of a plane A about them
is zero
• The first moment of a plane A about
the x- and y-axes are defined as
• Value: positive, negative or zero
• Dimension: [L3]; Unit: m3, cm3,...
5.2.2. The centroid of an area
• The centroid C of the area is defined as the point in the xy-plane that
has the coordinates
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y
C
S
x
A
 xC
S
y
A
xC
yC C
5.2. First Moment of Area
• If the origin of the xy-coordinate system
is the centroid of the area then Sx=Sy=0
• Whenever the area has an axis of
symmetry, the centroid of the area will lie
on that axis
1
n
i
x x
i
S S
 
1
n
i
y y
i
S S
 
• If the area can be subdivided in to simple geometric shapes
(rectangles, circles, etc., then
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5.2.3. The centroid of composite area
5.2. First Moment of Area
1
1
n
Ci i
y i
C n
i
i
x A
S
x
A
A


1
1
n
Ci i
x i
C n
i
i
y A
S
y
A
A


x
y
C1
C2
C3
xC1
yC1
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5.3. Moment of Inertia for an Area
2
( )
x
A
I y dA 
2
( )
y
A
I x dA 
5.3.1. Moment of inertia
5.3.2. Polar moment of inertia
2
( )
p x y
A
I dA I I 
5.3.3. Product of inertia
( )
xy
A
I xydA 
• The value of moment of inertia and polar
moment of inertia always positive, but the
product of inertia can be positive, negative,
or zero
• Dimension: [L4]; Unit: m4, cm4,...
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5.3. Moment of Inertia for an Area
- The product of inertia Ixy for an area will be zero if either the x or the y
axis is an axis of symmetry for the area
- The area with hole, then the hole’s
area is given by minus sign.
- The composite areas:
1
n
i
x x
i
I I
 
1
n
i
y y
i
I I
 
1
n
i
x x
i
S S
 
1
n
i
y y
i
S S
 
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5.4. Moment of Inertia for some simple areas
• Rectangular
• Circle
• Triangular
3
12
x
bh
I 
3
12
y
hb
I 
4 4
40,1
2 32
p
R D
I D
 
4 4
40,05
4 64
x y
R D
I I D
 
3
12
x
bh
I 
h
b
x
y
D
x
y
b
h
x
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5.5. Paralell-axis Theorem
• In the xy coordinates, an area
has geometric properties: Sx, Sy,
Ix, Iy, Ixy.
• In the uv coordinates: O'u//Ox,
O'v//Oy và:
• Geometric properties of an area
in the coordinates O'uv are:
u x b v y a 
.u xS S a A 
.v yS S b A 
22u x xI I aS a A 
22v y yI I bS b A 
uv xy y xI I aS bS abA 
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5.5. Paralell-axis Theorem
If O go through centroid C, then:
C C
2
u xI I a A 
2
v yI I b A 
uv xyI I abA 
. Radius of gyration
The radius of gyration of an area about the x and y axes, and the point
O are defined as
; 
yx
x y
II
r r
A A
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5.5. Paralell-axis Theorem
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5.5. Paralell-axis Theorem
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Problem 5.6.1. An area with the shape
and the dimension as shown in the
figure. Determine the principal moment
of inertia for area .
Solution Choosing the primary
coordinates x0y0 as shows in the figure.
Divide the composite area to 2 simple
areas 1 2
1
2
x0
y0
1. Determine the centroid:
- xC=0 (y0 – axis of symmetry)
Example 5.1
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1
2
x
0
y0- Draw the principal coordinates Cxy
- The Principal moment of inertia for an area:
Example 5.1
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Problem 5.2.
Example 5.2
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Example 5.2
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Example 5.3
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Example 5.3
THANK YOU FOR 
ATTENTION !
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