Chapter 5 - Geometric properties of an area. The following will be discussed in this chapter: First moment of area, moment of inertia for an area, moment of inertia for some simple areas, parallel - axis theorem.
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Nội dung Text: Lecture Strength of Materials I: Chapter 5 - PhD. Tran Minh Tu
- STRENGTH OF MATERIALS
1/10/2013 TRAN MINH TU - University of Civil Engineering, 1
Giai Phong Str. 55, Hai Ba Trung Dist. Hanoi, Vietnam
- CHAPTER
5
Geometric Properties of an Area
1/10/2013
- 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
Dimension, shape?
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- 5.2. First Moment of Area
5.2.1. Definition
• The first moment of a plane A about
the x- and y-axes are defined as
Sx
( A)
ydA Sy xdA
( A)
• Value: positive, negative or zero
• Dimension: [L3]; Unit: m3, cm3,...
• Centroidal axes: are axes, which first moment of a plane A about them
is zero
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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- 5.2. First Moment of Area
Sy Sx
xC yC
A A yC C
• If the origin of the xy-coordinate system
xC
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
• If the area can be subdivided in to simple geometric shapes
(rectangles, circles, etc., then
n n
Sx S i
x S y S yi
i 1 i 1
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- 5.2. First Moment of Area
y
5.2.3. The centroid of composite area
n
yC1 C1
Sy x Ci Ai
C2
xC i 1
n
A
A
i
i 1 C3
xC1 x
n
Sx y Ci Ai
yC i 1
n
A
A
i
i 1
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- 5.3. Moment of Inertia for an Area
5.3.1. Moment of inertia
Ix
( A)
y 2dA Iy
( A)
x 2dA
5.3.2. Polar moment of inertia
Ip
( A)
2 dA I x I y
5.3.3. Product of inertia • The value of moment of inertia and polar
moment of inertia always positive, but the
I xy xydA
( A)
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:
n
Sx S
n
S y S yi
i
x
i 1 i 1
n n
Ix I i
x I y I yi
i 1 i 1
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- 5.4. Moment of Inertia for some simple areas
• Rectangular y
bh3 hb3
Ix Iy
12 12 y
• Circle
h
x
R 4
D 4
Ip 0,1D 4 x
2 32
b
R4 D4
Ix I y 0,05D 4
4 64 D
• Triangular
bh3
h
Ix
12 x
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b 10
- 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:
u xb v ya
• Geometric properties of an area
in the coordinates O'uv are:
Su S x a. A Iu I x 2aS x a 2 A
Sv S y b. A I v I y 2bS y b2 A
Iuv I xy aS y bS x abA
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- 5.5. Paralell-axis Theorem
If O go through centroid C, then:
Iu I x a 2 A
I v I y b2 A
Iuv I xy abA
C C
. Radius of gyration
The radius of gyration of an area about the x and y axes, and the point
O are defined as
Ix Iy
rx ; ry
A A
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- 5.5. Paralell-axis Theorem
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- 5.5. Paralell-axis Theorem
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- Example 5.1
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
y0
areas 1 2
1. Determine the centroid:
- xC=0 (y0 – axis of symmetry) 1
2
x0
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- Example 5.1
- Draw the principal coordinates Cxy y0
- The Principal moment of inertia for an area:
1
2
x
0
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- Example 5.2
Problem 5.2.
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- Example 5.2
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- Example 5.3
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- Example 5.3
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