TRAN MINH TU - University of Civil Engineering, Giai Phong Str... First moment of area.[r]
(1)STRENGTH OF MATERIALS
TRAN MINH TU -University of Civil Engineering, Giai Phong Str 55, Hai Ba Trung Dist Hanoi, Vietnam
(2)5
CHAPTER
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(3)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
(4)5.1 Introduction
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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
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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 n i y y i S S
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5.2.3 The centroid of composite area
5.2 First Moment of Area
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5.3 Moment of Inertia for an Area
2 ( )
x
A
I y dA
( ) 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
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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:
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5.4 Moment of Inertia for some simple areas
• Rectangular • Circle • Triangular 12 x bh I 12 y hb I 4 0,1 2 32 p R D
I D
4 4 0,05 4 64 x y R D
I I D