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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
.
...
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
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
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