prop areas
TRANSCRIPT
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TABLE A.1 Properties of sectionsNOTATION: A ¼ area ðlengthÞ2; y ¼ distance to extreme fiber (length); I ¼ moment of inertia ðlength4Þ; r ¼ radius of gyration (length); Z ¼ plastic section modulus ðlength3Þ; SF ¼ shape factorSec. 8.15 for applications of Z and SF
Form of section Area and distances from
centroid to extremitiesMoments and products of inertia
and radii of gyration about central axes
Plastic section moduli,
shape factors, and locationsof plastic neutral axes
1. Square A ¼ a2
yc ¼ x c ¼ a
2
y0c ¼ 0:707a cos p
4 a
I x ¼ I y ¼ I 0x ¼ 112 a4
rx ¼ r y ¼ r 0x ¼ 0:2887aZ x ¼ Z y ¼ 0:25a3
SFx ¼ SF y ¼ 1:5
2. Rectangle A ¼ bd
yc ¼ d
2
x c ¼ b
2
I x ¼ 112 bd3
I y ¼ 112 db3
I x > I y if d > b
rx
¼ 0:2887d
r y ¼ 0:2887b
Z x ¼ 0:25bd2
Z y ¼ 0:25db2
SFx ¼ SF y ¼ 1:5
3. Hollow rectangle A ¼ bd bidi
yc ¼ d
2
x c ¼ b2
I x ¼ bd3 bi d3i
12
I y ¼ db3 di b3i
12
rx ¼ I x
A
1=2
r y ¼I y
A
1=2
Z x ¼ bd2 bi d2i
4
SFx ¼ Z x d
2I x
Z x ¼ db2 dib2i
4
SF y ¼Z y b
2I y
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4. Tee section A ¼ tb þ twd
yc
¼
bt2 þ tw dð2t þ dÞ2ðtb
þt
wdÞ
x c ¼ b
2
I x ¼ b
3ðd þ tÞ3 d
3
3 ðb twÞ Aðd þ t ycÞ2
I y ¼ tb3
12 þdt3w12
rx ¼ I x
A
1=2
r y ¼I y
A
1=2
If t wd5bt, then
Z x
¼
d2tw
4
b2t2
4tw þ
btðd þ tÞ
2Neutral axis x is located a distance ðbt=tw þ dÞ=from the bottom.
If t wd4bt, then
Z x ¼ t2b
4 þ tw dðt þ d twd=2bÞ
2
Neutral axis x is located a distance ðtwd=b þ tÞ=from the top.
SFx ¼ Z x ðd þ t ycÞ
I 1
Z y ¼ b2 t þ t2w d
4
SF y ¼Z yb
2I y
5. Channel section A ¼
tbþ
2twd
yc ¼ bt2 þ 2tw dð2t þ dÞ
2ðtb þ 2twdÞ
x c ¼ b
2
I x ¼ b
3 ðd þ tÞ3
d3
3 ðb 2twÞ Aðd þ t ycÞ2
I y ¼ ðd þ tÞb3
12 dðb 2twÞ
3
12
rx ¼ I x
A
1=2
r y ¼I y
A 1=2
If 2tw
d5bt, then
Z x ¼ d2tw
2 b
2t2
8twþ btðd þ tÞ
2
Neutral axis x is located a distance
ðbt=2tw þ dÞ=2 from the bottom.If 2tw d4bt, then
Z x ¼ t2b
4 þ tw d t þ d
twd
b
Neutral axis x is located a distance t wd=b þ t=2from the top.
SFx ¼ Z x ðd þ t ycÞ
I x
Z y ¼ b2 t
4 þ tw dðb twÞ
SF y ¼
Z yb
2I y
TABLE A.1 Properties of sections (Continued )
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TABLE A.1 Properties of sections (Continued )
Form of section
Area and distances from
centroid to extremities
Moments and products of inertia
and radii of gyration about central axes
Plastic section moduli,
shape factors, and locations
of plastic neutral axes
6. Wide-flange beam with
equal fl
anges
A ¼ 2bt þ twd
yc ¼ d2 þ t
x c ¼ b
2
I x
¼
bðd þ 2tÞ312
ðb twÞd312
I y ¼ b3t
6 þ t
3wd
12
rx ¼ I x
A
1=2
r y ¼I y
A
1=2
Z x ¼ tw d
2
4 þ btðd þ tÞ
SFx ¼ Z x yc
I x
Z y ¼ b2t
2 þ t
2w d
4
SF y ¼Z y x c
I y
7. Equal-legged angle A ¼ tð2a tÞ
yc1 ¼ 0:7071ða2 þ at t2Þ
2a t
yc2 ¼ 0:7071a2
2a tx c ¼ 0:7071a
I x ¼ a4 b4
12 0:5ta
2 b2
a þ b
I y ¼ a4 b4
12 where b ¼ a t
rx ¼ I x
A
1=2
r y ¼I y
A
1=2
Let y p be the vertical distance from the top corn
the plastic neutral axis. If t=a50:40, then
y p ¼
a t
a ðt=a
Þ2
2" #
1=2
Z x ¼ Að yc1 0:6667 y pÞIf t=a40:4, then
y p ¼ 0:3536ða þ 1:5tÞZ x ¼ Ayc1 2:8284 y2 pt þ 1:8856t3
8. Unequal-legged angle A
¼ t
ðb
þd
t
Þx c ¼
b2 þ dt t22ðb þ d tÞ
yc ¼ d2 þ bt t22ðb þ d tÞ
I x ¼
1
3 ½bd3
ðb
tÞð
d
tÞ
3
A
ðd
yc
Þ2
I y ¼ 13 ½db3 ðd tÞðb tÞ3 Aðb x cÞ2
I xy ¼ 14 ½b2d2 ðb tÞ2ðd tÞ2 Aðb x cÞðd ycÞ
rx ¼ I x
A
1=2
r y ¼I y
A 1=2
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9. Equilateral triangle A ¼ 0:4330a2
yc ¼ 0:5774ax c ¼ 0:5000a y0
c ¼ 0:5774a cosa
I x ¼ I y ¼ I x 0 ¼ 0:01804a4
rx ¼ r y ¼ rx 0 ¼ 0:2041aZ x ¼ 0:0732a3; Z y ¼ 0:0722a3
SFx ¼ 2:343; SF y ¼ 2:000Neutral axis x is 0:2537a from the base.
10. Isosceles triangle A ¼ bd2
yc
¼ 23
d
x c ¼ b
2
I x ¼ 136 bd3
I y ¼ 148 db3
I x > I y if d > 0:866b
rx ¼ 0:2357dr y ¼ 0:2041b
Z x ¼ 0:097bd2; Z y ¼ 0:0833db2
SFx ¼ 2:343; SF y ¼ 2:000Neutral axis x is 0:2929d from the base.
11. Triangle A ¼ bd2
yc ¼ 23 dx c ¼ 23 b 13 a
I x ¼ 136 bd3
I y ¼
1
36bd
ðb2
ab
þa2
ÞI xy ¼ 172 bd2ðb 2aÞ
yx ¼ 1
2tan1
dðb 2aÞb2 ab þ a2 d2
rx ¼ 0:2357d
r y ¼ 0:2357 ffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi ffi
b2 ab þ a2p
12. Parallelogram A ¼ bd
yc ¼ d
2
x c ¼ 12 ðb þ aÞ
I x ¼ 112 bd3
I y ¼ 112 bdðb2 þ a2ÞI xy ¼ 112 abd2
yx ¼ 1
2tan1
2adb2 þ a2 d2
rx ¼
0:2887d
r y ¼ 0:2887 ffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffi
b2 þ a2p
TABLE A.1 Properties of sections (Continued )
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TABLE A.1 Properties of sections (Continued )
Form of section
Area and distances from
centroid to extremities
Moments and products of inertia
and radii of gyration about central axes
Plastic section moduli,
shape factors, and locations
of plastic neutral axes
13. Diamond A ¼ bd2
yc ¼ d
2
x c ¼ b
2
I x ¼ 148 bd3
I y ¼ 148 db3
rx ¼
0:2041d
r y ¼ 0:2041b
Z x ¼ 0:0833bd2; Z y ¼ 0:0833db2
SFx ¼ SF y ¼ 2:000
14. Trapezoid A ¼ d2
ðb þ cÞ
yc ¼ d3 2b þ cb þ c
x c ¼ 2b2 þ 2bc ab 2ac c2
3ðb þ cÞ
I x ¼ d3
36
b2 þ 4bc þ c2b þ c
I y ¼ d36ðb þ cÞ ½b4 þ c4 þ 2bcðb2 þ c2Þ
aðb3 þ 3b2c 3bc2 c3Þþ a2ðb2 þ 4bc þ c2Þ
I xy ¼ d2
72ðb þ cÞ ½cð3b2 3bc c2Þ
þb3
a
ð2b2
þ 8bc
þ 2c2
Þ15. Solid circle A ¼ pR2
yc ¼ RI x ¼ I y ¼
p
4R4
rx ¼ r y ¼ R
2
Z x ¼ Z y ¼ 1:333R3
SFx ¼ 1:698
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16. Hollow circle A ¼ pðR2 R2i Þ yc ¼ R
I x ¼ I y ¼ p
4ðR4 R4i Þ
rx ¼ r y ¼ 12 ffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi
R2 þ R2iq
Z x ¼ Z y ¼ 1:333ðR3 R3i Þ
SFx ¼ 1:698R4 R3i RR4 R4i
17. Very thin annulus A ¼ 2pRt yc ¼ R
I x ¼ I y ¼ pR3trx ¼ r y ¼ 0:707R
Z x ¼ Z y ¼ 4R2t
SFx ¼ SF y ¼ 4
p
18. Sector of solid circle A ¼ aR2
yc1
¼ R 1
2sin a
3a yc2 ¼
2R sina
3a
x c ¼ R sina
I x ¼ R4
4 aþ sina cosa 16 sin
2a
9a
!
I y ¼ R4
4 ða sin a cosaÞ
ðNote: If a is small; a sin a cosa ¼ 23 a3 215a5Þ
rx ¼ R
2
ffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffi ffiffi1 þ sin a cosa
a 16sin
2a
9a2
s
r y ¼ R
2
ffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi ffiffiffiffi1 sin a cosa
a
r
If a454:3, then
Z x ¼ 0:6667R3 sina a3
2 tan a 1=2
" #
Neutral axis x is located a distance
Rð0:5a= tanaÞ1=2 from the vertex.If a554:3, then
Z x ¼ 0:6667R3ð2sin3 a1 sin aÞ where theexpression 2a1 sin2a1 ¼ a is solved for the vof a1 .
Neutral axis x is located a distance R cosa1 fro
the vertex.
If a473:09, then SFx ¼ Z
x y
c2I x
If 73:09 4 a490 , then SFx ¼ Z x yc1
I x
Z y ¼ 0:6667R3ð1 cosaÞIf a490 , then
SF y ¼ 2:6667 sina 1 cos aa sin a cosa
If a590 , then
SF y ¼ 2:6667 1 cosaa sin a cosa
TABLE A.1 Properties of sections (Continued )
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TABLE A.1 Properties of sections (Continued )
Form of section
Area and distances from
centroid to extremities
Moments and products of inertia
and radii of gyration about central axes
Plastic section moduli,
shape factors, and locations
of plastic neutral axes
19. Segment of solid circle(Note: If a4p=4, use
expressions from case 20)
A ¼ R2
ða sin a cosaÞ
yc1 ¼ R 1 2sin
3a
3ða sin a cosaÞ
" #
yc2 ¼ R 2sin
3a
3ða sin a cosaÞ cos a" #
x c ¼ R sina
I x ¼ R44 a sin a cosa þ 2sin3a cosa 16sin6 a
9ða sin a cosaÞ" #
I y ¼ R4
12ð3a 3 sin a cosa 2sin3 a cosaÞ
rx ¼ R
2
ffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi ffiffi1 þ 2sin
3a cosa
a sin a cosa 16sin
6a
9ða sin a cos aÞ2
s
r y ¼ R2 ffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi
1 2sin3
a cosa3ða sina cosaÞ
s 20. Segment of solid circle
(Note: Do not use if
a > p=4Þ
A ¼ 23
R2a3ð1 0:2a2 þ 0:019a4Þ yc1 ¼ 0:3Ra2ð1 0:0976a2 þ 0:0028a4Þ yc2 ¼ 0:2Ra2ð1 0:0619a2 þ 0:0027a4Þ
x c ¼ Rað1 0:1667a2 þ 0:0083a4Þ
I x ¼ 0:01143R4a7ð1 0:3491a2 þ 0:0450a4ÞI y ¼ 0:1333R4a5ð1 0:4762a2 þ 0:1111a4Þrx ¼ 0:1309Ra2ð1 0:0745a2Þr y ¼ 0:4472Rað1 0:1381a2 þ 0:0184a4Þ
21. Sector of hollow circle A ¼ atð2R tÞ
yc1 ¼ R 1 2sin a
3a 1 t
Rþ 1
2 t=R
yc2 ¼ R 2sin a3að2 t=RÞ þ 1 t
R
2sin a 3a cosa
3a
x c ¼ R sina
I x ¼ R3t 1 3t
2Rþ t
2
R2 t
3
4R3
a
þsin a cosa
2sin2 a
a !þ t
2 sin2a
3R2að2 t=RÞ 1 t
Rþ t
2
6R2
#
I y ¼ R3t 1 3t
2Rþ t
2
R2 t
3
4R3
ða sin a cos aÞ
rx
¼ ffiffiffiffiffiI x Ar ; r y ¼ ffiffiffiffiffi
I y
Ar (Note: If t=R is small, a can
exceed p to form an
overlapped annulus)
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Note: If a is small:
sin a
a ¼ 1 a
2
6 þ a
4
120; a sin a cosa ¼ 2
3a3 1 a
2
5 þ 2a
4
105
;
sin2a
a ¼ a 1 a
2
3 þ 2a
4
45
cos ¼ 1 a22 þ a4
24; a þ sina cosa 2sin2 a
a ¼ 2a5
45 1 a2
7 þ a4
105
22. Solid semicircle A ¼ p2
R2
yc1 ¼ 0:5756R yc2 ¼ 0:4244R
x c
¼ R
I x ¼ 0:1098R4
I y ¼ p
8R4
rx ¼ 0:2643R
r y ¼ R
2
Z x ¼ 0:3540R3; Z y ¼ 0:6667R3
SFx ¼ 1:856; SF y ¼ 1:698Plastic neutral axis x is located a distance 0:404
from the base.
23. Hollow semicircle
Note: b ¼ R þ Ri2
t ¼ R Ri
A ¼ p2
ðR2 R2i Þ
yc2 ¼ 4
3p
R3 R2iR2 R2i
or
yc2 ¼ 2b
p 1 þ ðt=bÞ
2
12
" #
yc1 ¼ R yc2x c ¼ R
I x ¼ p
8ðR4 R4i Þ
8
9p
ðR3 R3i Þ2R2 R2i
or
I x ¼ 0:2976tb3 þ 0:1805bt3 0:00884t5
b
I y ¼ p
8ðR4 R4i Þ
or
I y ¼ 1:5708b3t þ 0:3927bt3
Let y p be the vertical distance from the bottom to
plastic neutral axis.
y p ¼ ð0:7071 0:2716C 0:4299C 2 þ 0:3983C 3ÞRZ x ¼ ð0:8284 0:9140C þ 0:7245C 2
0:2850C 3ÞR2twhere C ¼ t=RZ y ¼ 0:6667ðR3 R3i Þ
24. Solid ellipse A ¼ pab yc ¼ ax c ¼ b
I x ¼ p
4ba3
I y ¼ p
4ab3
rx ¼ a
2
r y ¼
b
2
Z x ¼ 1:333a2b; Z y ¼ 1:333b2 aSFx ¼ SF y ¼ 1:698
TABLE A.1 Properties of sections (Continued )
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TABLE A.1 Properties of sections (Continued )
Form of section
Area and distances from
centroid to extremities
Moments and products of inertia
and radii of gyration about central axes
Plastic section moduli,
shape factors, and locations
of plastic neutral axes
25. Hollow ellipse A ¼ pðab aibiÞ yc ¼ ax c ¼ b
I x ¼ p
4ðba3 bia3i Þ
I y ¼ p
4ðab3 ai b3i Þ
rx ¼ 1
2
ffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi ffiffiffiffiffiba3 bia3iab ai bi
s
r y ¼ 1
2 ffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi ffiffiffiffiffiab3 aib3iab ai bis
Z x ¼ 1:333ða2b a2i biÞZ y ¼ 1:333ðb2 a b2i aiÞ
SFx ¼ 1:698a3b a2i bi aa3 b a3i bi
SF y ¼ 1:698b3a b2i ai bb3a b3i ai
Note: For this case the inner and outer perimeters are both ellipses and the wall
thickness is not constant. For a cross section with a constant wall thickness see
case 26.
26. Hollow ellipse with
constant wall thickness t.
The midthickness
perimeter is an ellipse
(shown dashed).
0:2 < a=b < 5
A ¼ ptða þ bÞ 1 þ K 1a
b
a þ b 2" #
where
K 1 ¼ 0:2464 þ 0:002222 a
bþ b
a
yc ¼ a þ t
2
x c
¼ b
þ
t
2
I x ¼ p4 ta2ða þ 3bÞ 1 þ K 2 a ba þ b
2" #
þ p16
t3ð3a þ bÞ 1 þ K 3a ba þ b
2" #
where
K 2 ¼ 0:1349 þ 0:1279a
b 0:01284 a
b
2
K 3 ¼ 0:1349 þ 0:1279 ba 0:01284 b
a
2For I y interchange a and b in the expressions
for I x ; K 2, and K 3
Z x ¼ 1:3333taða þ 2bÞ 1 þ K 4 a ba þ b 2" # þ t3
3
where
K 4 ¼ 0:1835 þ 0:895a
b 0:00978 a
b
2For Z y interchange a and b in the expression fo
and K 4 .
See the note on maximumwall thickness in case 27.
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TABLE A.1 Properties of sections Continued )
Form of section Area and distances from
centroid to extremitiesMoments and products of inertia
and radii of gyration about central axes
Plastic section moduli,
shape factors, and locationsof plastic neutral axes
28. Regular polygon with n
sides A ¼ a
2n
4tan a
r1 ¼ a
2sin a
r2 ¼ a
2tan a
If n is odd
y1 ¼ y2 ¼ r1 cos a n þ 1
2
p
2
If n=2 is odd
y1 ¼ r1; y2 ¼ r2If n=2 is even
y1 ¼ r2; y2 ¼ r1
I 1 ¼ I 2 ¼ 124 Að6r21 a2Þ
r1 ¼ r2 ¼ ffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi
124
ð6r21 a2Þq For n ¼ 3, see case 9. For n ¼ 4, see cases 1 andFor n ¼ 5, Z 1 ¼ Z 2 ¼ 0:8825r31 . For an axis perp
dicular to axis 1, Z ¼ 0:8838r31. The location ofaxis is 0.7007a from that side which is perpendi
to axis 1. For n56, use the following expression
neutral axis of any inclination:
Z ¼ r31 1:333 13:908 1
n
2þ 12:528 1
n
3" #
29. Hollow regular polygon
with n sides A ¼ nat 1 t tana
a
r1 ¼ a
2sin a
r2 ¼ a
2tan a
If n is odd
y1 ¼ y2 ¼ r1 cos an þ 1
2 p
2
If n=2 is odd
y1 ¼ r1; y2 ¼ r2If n=2 is even
y1 ¼ r2; y2 ¼ r1
I 1 ¼ I 2 ¼ na3t
8
1
3 þ 1
tan2 a
1 3 t tanaa
þ 4 t tanaa
22 t tana
a
3" #
r1 ¼ r2 ¼ a
ffiffiffi8p
ffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi ffiffi
1
3
þ 1
tan2 a1 2 t tana
a þ 2 t tana
a
2" #vuut