11

Class #29Class #29

Moments of InertiaMoments of Inertia

Second Moments of AreasSecond Moments of Areas

Radius of GyrationRadius of Gyration

StaticsStatics

Spring 2006Spring 2006

Dr. PickettDr. Pickett

22

22ndnd MOMENTS OF A PLANE AREAMOMENTS OF A PLANE AREAB & J 7B & J 7thth, Sections: 9.1, 9.2, 9.3 , Sections: 9.1, 9.2, 9.3 ρρ

ρρ

Y

lengthOriginal

lengthinChangeStrainAxial X

.

...

33

INTERNAL RESISTING MOMENT to INTERNAL RESISTING MOMENT to an applied Force:an applied Force:

Y

lengthOriginal

lengthinChangeStrainAxial X

.

...

44

55

66

RADIUS OF GYRATIONRADIUS OF GYRATIONB & J 7B & J 7thth, Section: 9.5, Section: 9.5

2

22

2

L

rE

rL

EBuckle

A

P

77

A

Ir yY

A

Ir XX

ININ

IN

A

Ir

2

4

Xr

Yr = distance away from Y-axis, that an equivalent area should be placed, to give the same second moment of area ( Iy ) about Y-axis, as the real area

= distance away from X-axis, that an equivalent area should be placed, to give the same second moment of area ( Ix ) about X-axis, as the real area.

88

PARALLEL AXIS THEOREM FOR 2PARALLEL AXIS THEOREM FOR 2ndnd MOMENTS OF AREAMOMENTS OF AREAB & J 7B & J 7thth, Sections: 9.6, 9.7 ., Sections: 9.6, 9.7 .

The centroidal axis of the plate is BB’. Centroid = CThe centroidal axis of the plate is BB’. Centroid = C

The elemental area The elemental area ΔΔa is located Y’ a is located Y’ from centroidal axis BB’, from centroidal axis BB’, and located Y from axis AA’. and located Y from axis AA’.

axisYCentroidalaboutAreaofmomentaXI

and

BBaxisXCentroidalaboutAreaofmomentaYIDefine

nd

A

YC

nd

A

XC

......2

',......2)'(......

2

2

axiscentroidalthetorespectwithcentroidtheofmomentstthemeansthisce

aYbut

adaYdII

adaYdaYadYYI

axisAAaboutareaofmomentondfordYYngSubstituti

AA

XCAAX

A A AA

AAX

...................1........sin

0'......

'2

2

'.......sec......'....

2'.

222

'.

Δa

A’A

B B’Cd

Y’Y

99

The Parallel Axis Theorem is valid The Parallel Axis Theorem is valid only with respect to the only with respect to the

Centroidal AxisCentroidal Axis

A

XCAAX

AA

XCAAX

A A AA

AAX

adII

thus

axiscentroidalthetorespectwithcentroidtheofmomentstthemeansthisce

aYbut

adaYdII

adaYdaYadYYI

dYYngSubstituti

2'.

2'.

222

'.

0

...................1........sin

0'......

'2

2

'....

1010

22ndnd MOMENTS OF COMPOSITE AREASMOMENTS OF COMPOSITE AREASB & J 7B & J 7thth, Section: 9.7, Section: 9.7

233

3332

22

3222

11

311

321

121212YA

hbYA

hbYA

hb

IIII XXXX

1111

RADIUS OF GYRATIONRADIUS OF GYRATIONB & J 7B & J 7thth,9.5,9.5

2

22

2

L

rE

rL

EBuckle

A

P

A

Ir

1212

Prob. #9.32, B&J 5Prob. #9.32, B&J 5thth ed. ed.

"75.3

"5.72

1

Y

AREA

2

2

2

6875.1"25.2"75.0

625.5"5.7"75.0

6875.1"25.2"75.0

IN

IN

IN

x

x

x

2

0.9 ININ

IN

IN

oix

125.12

"25.2

625.22

"75.0"25.2

125.12

"25.2

3

2

1

3

3

3

3

562.18

898.1

766.14

898.1

IN

IN

IN

IN

OYM

IN

IN

INO

Y

A

MX 062.2

9

562.182

3

4

3

4

2

444

1

2

1

3.19

37.26012

5.775.0

3.1922.19079.0

2

"5.7

2

"75.06875.1

12

"75.0"25.2

22

3

23

INX

INX

INININX

INX

I

dAI

I

I

4

97.64 INXTOTALI

1313

Prob. #9.32, B&J 5Prob. #9.32, B&J 5thth ed., continued ed., continued

Column will buckle about the weakest axisColumn will buckle about the weakest axis( axis with smallest r )( axis with smallest r )

4

2

44

2

2

4

3

4

1

44

2

1

06.2

7956.12637.0

94.02

75.0625.5

12

"75.0"5.7

187.2

187.2

475.1712.0

2

25.206.26875.1

12

"25.2"75.0

23

23

INY

ININ

INY

INY

INY

ININ

INY

I

I

I

I

I

4

434.6 INYTOTALI

22

2

2

846.0

715.00.9

434.6

69.2

22.70.9

97.64

2

2

4

2

2

4

L

rE

rL

E

r

A

Ir

r

A

Ir

Buckle

INy

IN

IN

INY

y

INx

IN

IN

INX

x

1414

RADIUS OF GYRATIONRADIUS OF GYRATIONB & J 7B & J 7thth,9.5,9.5

2

22

2

L

rE

rL

EBuckle

A

P

A

Ir

1515

Prob. #9.33, B&J 5Prob. #9.33, B&J 5thth ed. ed.

XIFind : and YI about

the

the centroid

of whole tionsec

centroidFind : with respect to side AB

AfrommmX

mmY

A

MY

MAyAY

ABFrom

mm

mmAB

ABii

X

X

90

44

000,12

000,5282

3

1616

Prob. #9.33, B&J 5Prob. #9.33, B&J 5thth ed. ed. continued continued

46

23

23

222

211

1088.13

000,888,13

)080,622000,560,2200,147,4000,960(

36608012

80602418040

12

40180

4

4

21

mmxI

xxI

dAIdAII

X

mm

mm

mmmmmmmmmm

mmmmmmmmmm

X

ccXccXX yy

1717

Prob. #9.33, B&J 5Prob. #9.33, B&J 5thth ed., continued ed., continued

mmr

xx

x

A

Ir

r

xx

x

A

Ir

y

mm

mm

mm

Yy

mmx

mm

mm

mm

Xx

7.41

1074.11012

1088.20

34

101573.11012

10888.13

2

2

4

2

2

4

3

3

6

3

3

6

46

23

23

222

211

1088.20

000,880,20

000,440,1000,944,1

0608012

6080018040

12

18040

4

44

21

mmxI

xxI

dAIdAII

Y

mm

mmmm

mmmmmmmmmm

mmmmmmmmmm

Y

ccYccYY xx

•But axis thru C1 and C2 which are parallel to Y-axis thru C have an X displacement of zero between C1 and C2 and c

1818

RADIUS OF GYRATIONRADIUS OF GYRATIONB & J 7B & J 7thth,9.5,9.5

2

22

2

L

rE

rL

EBuckle

A

P

A

Ir