MOMENTS OF AREAS APPENDIX A MOMENTS OF AREAS A.1 First Moment of An Area; Centroid First Moments of...

APPENDIX A

MOMENTS OF AREASMOMENTS OF AREASA.1 First Moment of An Area; Centroid

First Moments of the Area A Aboutthe x- and y-Axis are Defined As

AxQ ydAAyQ xdA

x y

The centroid of the area A is defined as the point C of coordinates

and which satisfy the relations

/yx Q A/xy Q A

APPENDIX A

MOMENTS OF AREASMOMENTS OF AREAS The first moment of an area about its symmetric axis is zero, so, the centroid of the area must be on the symmetric axis. When an area possesses a center of symmetry O, the first moment of the area about any axis through O is zero. In other words, O is the centroid of the area.

APPENDIX A

MOMENTS OF AREASMOMENTS OF AREAS

21 1

2 2xQ Ay bh h bh

21 1

2 2yQ Ax bh b b h

If an area has two symmetric axes, the inter-section of the two axes must be the centroid of the area.

APPENDIX A

MOMENTS OF AREASMOMENTS OF AREASSample Problem A.1For the triangular area of Fig. (a), determine (a) the first moment Qx of the area with respect to the x-axis, (b) the

coordinate y of the centroid of the area.

Fig. (a) Fig. (b)

APPENDIX A

MOMENTS OF AREASMOMENTS OF AREAS

dA=udy, h y

u bh

2

0 6Ax

h h y bhydA

hQ y bdy

2 / 6

/ 2 3xQ bh h

yA bh

APPENDIX A

MOMENTS OF AREASMOMENTS OF AREASComplement ProblemDetermine the first moment of a semi-circular area about the x-axis, (b) the coordinate of the centroid of the semi-circle.

x AQ ydA 2 22dA R y dy

2 2

0

2 2 3/ 23

0

2

( ) 2

3/ 2 3

R

x

R

Q y R y dy

R yR

3

2

2 4

3( / 2) 3xQ R R

yA R

y

APPENDIX A

MOMENTS OF AREASMOMENTS OF AREASA.1 Determination of The First Moment And Centroid of A Composite Area

1 2 3x A A A A

Q ydA ydA ydA ydA 1 1 2 2 3 3xQ A y A y A y

x i ii

Q A y y i ii

Q A xi i

i

ii

A xX

A

i ii

ii

A yY

A

APPENDIX A

MOMENTS OF AREASMOMENTS OF AREASSample Problem A.2Locate the Centroid C of the area A shown in Fig. (a).

Fig. (a) Fig. (b)

APPENDIX A

MOMENTS OF AREASMOMENTS OF AREAS

Solution:

1 1 2 2

1 2

y A y AY

A A

A1=80×20=1600 mm2,

A2=60×40=2400 mm2

70 1600 30 240046mm

1600 2400Y

APPENDIX A

MOMENTS OF AREASMOMENTS OF AREASSample Problem A.3 Referring to the area A of Sample Problem A.2, we consider the horizontal x axis which is through its centroid C. (Such an axis is called a centroidal axis.) Denoting by A the portion of A located above that axis (Fig. a), determine the first moment of A with respect to the x axes.

Fig. (a) Fig. (b) Fig. (c)

APPENDIX A

MOMENTS OF AREASMOMENTS OF AREAS

Solution:

1 1 3 3

3

24 80 20

7 14 40

42320mm

xQ y A y A

4 4

3

23 46 40

42320mm

xQ y A y A

3

0

42320mm

x x x

x x

Q Q Q Y A

Q Q

In fact 0

1 1 2 2

1 2

i iA x A x A xx

A A A

Complement Problem

1 1 2 2

1 2

i iA y A y A yy

A A A

80

10

10

c(19.7;39.7)

x

y

C1

C2

Determine the centroid of the L-shape area.2

1 1 1700 , 45 , 5A mm x mm y mm 2

2 2 21200 , 5 , 60A mm x mm y mm

120 45 700 5 1200

700 1200

19.7( )mm

)(7.391200700

1200607005mm

Alternative method: Negative area method

40 9600 45 ( 7700)19.7( )

9600 7700mm

z

y

21 1 19600 , 40 , 60A mm x mm y mm

22 2 27700 , 45 , 65A mm x mm y mm

2C60 9600 65 ( 7700)

39.7( )9600 7700

mm

1C0C

1 1 2 2

1 2

i ic

A x A x A xx

A A A

1 1 2 2

1 2

i iA y A y A yy

A A A

80

120

10

10

x

y

x

APPENDIX A

MOMENTS OF AREASMOMENTS OF AREASA.3 Second Moment, or Moment of Inertia, of An Area; Radius of Gyration

Moment of Inertia of A With Respect To the And x Axis And y Axis are Defined, Respectively, As

2x A

I y dA2

y AI x dA

Define the Polar Moment of Inertia of the Area A With Respect To Point O As the Integral :

2 2 2O x yA A A

J dA y dA x dA I I

APPENDIX A

MOMENTS OF AREASMOMENTS OF AREASRadii of Gyration of An Area A with respect to the x and y axis:

2x xI r A

2y yI r A

xx

Ir

A y

y

Ir

A

2O OJ r A O

O

Jr

A

2 2 2O x yr r r

Radii of Gyration With Respect To the Origin O

APPENDIX A

MOMENTS OF AREASMOMENTS OF AREASSample Problem A.4For the rectangular area of Fig.(a). determine (a) the moment of inertia Ix of the area with respect to the centroidal x axis. (b) the corresponding radius of gyration rx.

Fig. (a) Fig. (b)

2

/ 2 2

/ 2

3

12

x A

h

h

I y dA

y bdy

bh

12x

hr

APPENDIX A

MOMENTS OF AREASMOMENTS OF AREASSample Problem A.5

For the circular area of Fig. (a), determine (a) the polar moment of inertia JO, (b) rectangular moments of inertia Ix and Iy.

Fig. (a) Fig. (b)

4 42 2

2 32O A

r dJ d

APPENDIX A

MOMENTS OF AREASMOMENTS OF AREASA.4 Parallel-Axis TheoremA.4 Parallel-Axis Theorem

2x A

I y dA 2 2( )x A AI y dA y d dA

2 2( ) 2x A A AI y dA d y dA d dA

2x xI I Ad

APPENDIX A

MOMENTS OF AREASMOMENTS OF AREASA.5 Determination of The Moment of Inertia of a Composite Area.

APPENDIX A

MOMENTS OF AREASMOMENTS OF AREASSample Problem A.6Determine the moment of inertia xI of the area shown with

respect to the centroidal x axis (Fig. a).

Fig. (a) Fig. (b)

APPENDIX A

MOMENTS OF AREASMOMENTS OF AREAS

1 2( ) ( )x x A x AI I I

1

21 1

32 4

( )

80 2080 20 24 974933mm

12

x A xI I A d

2

22 2

32 4

( )

40 6040 60 16 1334400mm

12

x A xI I A d

41 2( ) ( ) 974933 1334400 2309333mmx x A x AI I I

Solution:Solution:

APPENDIX A

MOMENTS OF AREASMOMENTS OF AREASA.6 Product of Inertia for An Area.product of inertia for an element of area located at point (x, y)

is defined as

xy AI xydAxydI xydA

0xyI

If either x or y axis is a symmetric axis, Ixy=0.

APPENDIX A

MOMENTS OF AREASMOMENTS OF AREAS

Parallel-Axis theorem

( )( )xy x yI x d y d dA x y x yxydA d ydA d xdA d d dA

xy xy x yI I Ad d

APPENDIX A

MOMENTS OF AREASMOMENTS OF AREASSample Problem A.7Determine the product of inertia Ixy of the triangle shown in Fig. (a).

APPENDIX A

MOMENTS OF AREASMOMENTS OF AREASSample Problem A.7

APPENDIX A

MOMENTS OF AREASMOMENTS OF AREAS

xdA hdx

b

23

2

1

2 2y

x x hdI xydA x h h dx x dx

b b b

23

20 2

b

xy xyA A

hI xydA dI x dx

b

2 2

8xy

b hI

Solution:Solution:

APPENDIX A

MOMENTS OF AREASMOMENTS OF AREASSample Problem A.8 Compute the product of inertia of the beam’s cross-sectional area, shown in Fig. (a), about the x and y centroidal axes.

APPENDIX A

MOMENTS OF AREASMOMENTS OF AREAS

Solution:Solution:

80 (300)(100)( 250)(200) 15 10xy xy x yI I Ad d

0 0 0xy xy x yI I Ad d

80 (300)(100)(250)( 200) 15 10xy xy x yI I Ad d

8 8 8( 15 10 ) 0 ( 15 10 ) 30 10xyI

Rectangle B

Rectangle D

The product of inertia for the entire cross section is

mm4

Rectangle A

mm4

mm4

mm4

APPENDIX A

MOMENTS OF AREASMOMENTS OF AREASA.7 Moments of Inertia for An Area About Inclined Axes

cos sinu x y

cos sinv y x

2 2

2 2

( cos sin )

( cos sin )

( cos sin )( cos sin )

u

v

uv

dI v dA y x dA

dI u dA x y dA

dI uvdA x y y x dA

APPENDIX A

MOMENTS OF AREASMOMENTS OF AREAS2 2cos sin 2 sin cosu x y xyI I I I

2 2sin cos 2 sin cosv x y xyI I I I 2 2sin cos sin cos 2 cos sinuv x y xyI I I I

cos 2 sin 22 2

x y x yu xy

I I I II I

cos 2 sin 22 2

x y x yv xy

I I I II I

sin 2 cos 22

x yuv xy

I II I

0 u v x yJ I I I I

APPENDIX A

MOMENTS OF AREASMOMENTS OF AREASPrincipal Moments of Inertia

2 sin 2 2 cos 2 02

x yuxy

I IdII

d

2tan 2 xy

x y

I

I I

1

1

sin 2

cos 22

xyp

x yp

I

RI I

R

2

2

sin 2

cos 22

xyp

x yp

I

RI I

R

APPENDIX A

MOMENTS OF AREASMOMENTS OF AREAS

maxmin

2

2

2 2x y

xyx y I II I

I I Ra

2

2

2x y

xy

I IR I

2x yI I

a

APPENDIX A

MOMENTS OF AREASMOMENTS OF AREASSample Problem A.9 Determine the principal moments of inertia for the beam’s cross-sectional area shown in Fig. (a) with respect to an axis passing through the centroid.

APPENDIX A

MOMENTS OF AREASMOMENTS OF AREASA.8 Mohr’s Circle for Moments of Inertia

2 2

2 2

2 2x y x y

u uv xy

I I I II I I

2 2 2u uvI a I R

APPENDIX A

MOMENTS OF AREASMOMENTS OF AREASSample Problem A.10Using Mohr’s circle, determine the principal moments of inertia for the beam’s cross-sectional area, shown in Fig. (a), with respect to an axis passing through the centroid.

APPENDIX A

MOMENTS OF AREASMOMENTS OF AREAS