materiales.azc.uam.mxmateriales.azc.uam.mx/gjl/Clases/E10_I/Prop_Areas.pdf · Form of section Area...
Transcript of materiales.azc.uam.mxmateriales.azc.uam.mx/gjl/Clases/E10_I/Prop_Areas.pdf · Form of section Area...
TABLE A.1 Properties of sectionsNOTATION: A ¼ area ðlengthÞ
2; y ¼ distance to extreme fiber (length); I ¼ moment of inertia ðlength
4Þ; r ¼ radius of gyration (length); Z ¼ plastic section modulus ðlength
3Þ; SF ¼ shape factor. See
Sec. 8.15 for applications of Z and SF
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
1. Square A ¼ a2
yc ¼ xc ¼a
2
y0c ¼ 0:707a cosp4� a
Ix ¼ Iy ¼ I 0x ¼ 1
12a4
rx ¼ ry ¼ r0x ¼ 0:2887a
Zx ¼ Zy ¼ 0:25a3
SFx ¼ SFy ¼ 1:5
2. Rectangle A ¼ bd
yc ¼d
2
xc ¼b
2
Ix ¼ 112
bd3
Iy ¼ 112
db3
Ix > Iy if d > b
rx ¼ 0:2887d
ry ¼ 0:2887b
Zx ¼ 0:25bd2
Zy ¼ 0:25db2
SFx ¼ SFy ¼ 1:5
3. Hollow rectangle A ¼ bd � bidi
yc ¼d
2
xc ¼b
2
Ix ¼bd3 � bid
3i
12
Iy ¼db3 � dib
3i
12
rx ¼Ix
A
� �1=2
ry ¼Iy
A
� �1=2
Zx ¼bd2 � bid
2i
4
SFx ¼Zxd
2Ix
Zx ¼db2 � dib
2i
4
SFy ¼Zyb
2Iy
802
FormulasforStressandStrain
[APP.A
4. Tee section A ¼ tb þ twd
yc ¼bt2 þ twdð2t þ dÞ
2ðtb þ twdÞ
xc ¼b
2
Ix ¼b
3ðd þ tÞ3 �
d3
3ðb � twÞ � Aðd þ t � ycÞ
2
Iy ¼tb3
12þ
dt3w
12
rx ¼Ix
A
� �1=2
ry ¼Iy
A
� �1=2
If twd5 bt, then
Zx ¼d2tw
4�
b2t2
4tw
þbtðd þ tÞ
2
Neutral axis x is located a distance ðbt=tw þ dÞ=2
from the bottom.
If twd4 bt, then
Zx ¼t2b
4þ
twdðt þ d � twd=2bÞ
2
Neutral axis x is located a distance ðtwd=b þ tÞ=2
from the top.
SFx ¼Zxðd þ t � ycÞ
I1
Zy ¼b2t þ t2
wd
4
SFy ¼Zyb
2Iy
5. Channel section A ¼ tb þ 2twd
yc ¼bt2 þ 2twdð2t þ dÞ
2ðtb þ 2twdÞ
xc ¼b
2
Ix ¼b
3ðd þ tÞ3 �
d3
3ðb � 2twÞ � Aðd þ t � ycÞ
2
Iy ¼ðd þ tÞb3
12�
dðb � 2twÞ3
12
rx ¼Ix
A
� �1=2
ry ¼Iy
A
� �1=2
If 2twd5 bt, then
Zx ¼d2tw
2�
b2t2
8tw
þbtðd þ tÞ
2
Neutral axis x is located a distance
ðbt=2tw þ dÞ=2 from the bottom.
If 2twd4 bt, then
Zx ¼t2b
4þ twd t þ d �
twd
b
� �
Neutral axis x is located a distance twd=b þ t=2
from the top.
SFx ¼Zxðd þ t � ycÞ
Ix
Zy ¼b2t
4þ twdðb � twÞ
SFy ¼Zyb
2Iy
APP.A]
Propertie
sofaPlaneArea
803
TABLE A.1 Properties of sections (Continued)
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 flanges
A ¼ 2bt þ twd
yc ¼d
2þ t
xc ¼b
2
Ix ¼bðd þ 2tÞ3
12�ðb � twÞd
3
12
Iy ¼b3t
6þ
t3wd
12
rx ¼Ix
A
� �1=2
ry ¼Iy
A
� �1=2
Zx ¼twd2
4þ btðd þ tÞ
SFx ¼Zx yc
Ix
Zy ¼b2t
2þ
t2wd
4
SFy ¼Zyxc
Iy
7. Equal-legged angle A ¼ tð2a � tÞ
yc1 ¼0:7071ða2 þ at � t2Þ
2a � t
yc2 ¼0:7071a2
2a � t
xc ¼ 0:7071a
Ix ¼a4 � b4
12�
0:5ta2b2
a þ b
Iy ¼a4 � b4
12where b ¼ a � t
rx ¼Ix
A
� �1=2
ry ¼Iy
A
� �1=2
Let yp be the vertical distance from the top corner to
the plastic neutral axis. If t=a5 0:40, then
yp ¼ at
a�ðt=aÞ2
2
" #1=2
Zx ¼ Aðyc1 � 0:6667ypÞ
If t=a4 0:4, then
yp ¼ 0:3536ða þ 1:5tÞ
Zx ¼ Ayc1 � 2:8284y2pt þ 1:8856t3
8. Unequal-legged angle A ¼ tðb þ d � tÞ
xc ¼b2 þ dt � t2
2ðb þ d � tÞ
yc ¼d2 þ bt � t2
2ðb þ d � tÞ
Ix ¼ 13½bd3 � ðb � tÞðd � tÞ3 � Aðd � ycÞ
2
Iy ¼ 13½db3 � ðd � tÞðb � tÞ3 � Aðb � xcÞ
2
Ixy ¼ 14½b2d2 � ðb � tÞ2ðd � tÞ2 � Aðb � xcÞðd � ycÞ
rx ¼Ix
A
� �1=2
ry ¼Iy
A
� �1=2
804
FormulasforStressandStrain
[APP.A
9. Equilateral triangle A ¼ 0:4330a2
yc ¼ 0:5774a
xc ¼ 0:5000a
y0c ¼ 0:5774a cos a
Ix ¼ Iy ¼ Ix0 ¼ 0:01804a4
rx ¼ ry ¼ rx0 ¼ 0:2041a
Zx ¼ 0:0732a3; Zy ¼ 0:0722a3
SFx ¼ 2:343; SFy ¼ 2:000
Neutral axis x is 0:2537a from the base.
10. Isosceles triangle A ¼bd
2
yc ¼23d
xc ¼b
2
Ix ¼ 136
bd3
Iy ¼ 148
db3
Ix > Iy if d > 0:866b
rx ¼ 0:2357d
ry ¼ 0:2041b
Zx ¼ 0:097bd2; Zy ¼ 0:0833db2
SFx ¼ 2:343; SFy ¼ 2:000
Neutral axis x is 0:2929d from the base.
11. Triangle A ¼bd
2
yc ¼23d
xc ¼23b � 1
3a
Ix ¼ 136
bd3
Iy ¼ 136
bdðb2 � ab þ a2Þ
Ixy ¼ 172
bd2ðb � 2aÞ
yx ¼1
2tan�1 dðb � 2aÞ
b2 � ab þ a2 � d2
rx ¼ 0:2357d
ry ¼ 0:2357ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 � ab þ a2
p
12. Parallelogram A ¼ bd
yc ¼d
2
xc ¼12ðb þ aÞ
Ix ¼ 112
bd3
Iy ¼ 112
bdðb2 þ a2Þ
Ixy ¼ � 112
abd2
yx ¼1
2tan�1 �2ad
b2 þ a2 � d2
rx ¼ 0:2887d
ry ¼ 0:2887ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 þ a2
p
APP.A]
Propertie
sofaPlaneArea
805
TABLE A.1 Properties of sections (Continued)
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 ¼bd
2
yc ¼d
2
xc ¼b
2
Ix ¼ 148
bd3
Iy ¼ 148
db3
rx ¼ 0:2041d
ry ¼ 0:2041b
Zx ¼ 0:0833bd2; Zy ¼ 0:0833db2
SFx ¼ SFy ¼ 2:000
14. Trapezoid A ¼d
2ðb þ cÞ
yc ¼d
3
2b þ c
b þ c
xc ¼2b2 þ 2bc � ab � 2ac � c2
3ðb þ cÞ
Ix ¼d3
36
b2 þ 4bc þ c2
b þ c
Iy ¼d
36ðb þ cÞ½b4 þ c4 þ 2bcðb2 þ c2Þ
� aðb3 þ 3b2c � 3bc2 � c3Þ
þ a2ðb2 þ 4bc þ c2Þ
Ixy ¼d2
72ðb þ cÞ½cð3b2 � 3bc � c2Þ
þ b3 � að2b2 þ 8bc þ 2c2Þ
15. Solid circle A ¼ pR2
yc ¼ R
Ix ¼ Iy ¼p4
R4
rx ¼ ry ¼R
2
Zx ¼ Zy ¼ 1:333R3
SFx ¼ 1:698
806
FormulasforStressandStrain
[APP.A
16. Hollow circle A ¼ pðR2 � R2i Þ
yc ¼ R
Ix ¼ Iy ¼p4ðR4 � R4
i Þ
rx ¼ ry ¼ 12
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 þ R2
i
qZx ¼ Zy ¼ 1:333ðR3 � R3
i Þ
SFx ¼ 1:698R4 � R3
i R
R4 � R4i
17. Very thin annulus A ¼ 2pRt
yc ¼ R
Ix ¼ Iy ¼ pR3t
rx ¼ ry ¼ 0:707R
Zx ¼ Zy ¼ 4R2t
SFx ¼ SFy ¼4
p
18. Sector of solid circle A ¼ aR2
yc1 ¼ R 1 �2 sin a
3a
� �
yc2 ¼2R sin a
3a
xc ¼ R sin a
Ix ¼R4
4aþ sin a cos a�
16 sin2 a
9a
!
Iy ¼R4
4ða� sin a cos aÞ
ðNote: If a is small; a� sin a cos a ¼ 23a3 � 2
15a5Þ
rx ¼R
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ
sin a cos aa
�16 sin
2 a9a2
s
ry ¼R
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 �
sin a cos aa
r
If a4 54:3�, then
Zx ¼ 0:6667R3 sin a�a3
2 tan a
� �1=2" #
Neutral axis x is located a distance
Rð0:5a= tan aÞ1=2 from the vertex.
If a5 54:3�, then
Zx ¼ 0:6667R3ð2 sin3 a1 � sin aÞ where the
expression 2a1 � sin 2a1 ¼ a is solved for the value
of a1.
Neutral axis x is located a distance R cos a1 from
the vertex.
If a4 73:09�, then SFx ¼Zxyc2
Ix
If 73:09� 4a4 90�, then SFx ¼Zxyc1
Ix
Zy ¼ 0:6667R3ð1 � cos aÞIf a4 90� , then
SFy ¼ 2:6667 sin a1 � cos a
a� sin a cos aIf a5 90� , then
SFy ¼ 2:66671 � cos a
a� sin a cos a
APP.A]
Propertie
sofaPlaneArea
807
TABLE A.1 Properties of sections (Continued)
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 R2ða� sin a cos aÞ
yc1 ¼ R 1 �2 sin
3 a3ða� sin a cos aÞ
" #
yc2 ¼ R2 sin
3 a3ða� sin a cos aÞ
� cos a
" #
xc ¼ R sin a
Ix ¼R4
4a� sin a cos aþ 2 sin
3 a cos a�16 sin
6 a9ða� sin a cos aÞ
" #
Iy ¼R4
12ð3a� 3 sin a cos a� 2 sin
3 a cos aÞ
rx ¼R
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ
2 sin3 a cos a
a� sin a cos a�
16 sin6 a
9ða� sin a cos aÞ2
s
ry ¼R
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 �
2 sin3 a cos a
3ða� sin a cos aÞ
s
20. Segment of solid circle
(Note: Do not use if
a > p=4Þ
A ¼ 23R2a3ð1 � 0:2a2 þ 0:019a4Þ
yc1 ¼ 0:3Ra2ð1 � 0:0976a2 þ 0:0028a4Þ
yc2 ¼ 0:2Ra2ð1 � 0:0619a2 þ 0:0027a4Þ
xc ¼ Rað1 � 0:1667a2 þ 0:0083a4Þ
Ix ¼ 0:01143R4a7ð1 � 0:3491a2 þ 0:0450a4Þ
Iy ¼ 0:1333R4a5ð1 � 0:4762a2 þ 0:1111a4Þ
rx ¼ 0:1309Ra2ð1 � 0:0745a2Þ
ry ¼ 0:4472Rað1 � 0:1381a2 þ 0:0184a4Þ
21. Sector of hollow circle A ¼ atð2R � tÞ
yc1 ¼ R 1 �2 sin a
3a1 �
t
Rþ
1
2 � t=R
� �� �
yc2 ¼ R2 sin a
3að2 � t=RÞþ 1 �
t
R
� �2 sin a� 3a cos a
3a
� �
xc ¼ R sin a
Ix ¼ R3t 1 �3t
2Rþ
t2
R2�
t3
4R3
� ��
� aþ sin a cos a�2 sin
2 aa
!
þt2 sin
2 a3R2að2 � t=RÞ
1 �t
Rþ
t2
6R2
� �#
Iy ¼ R3t 1 �3t
2Rþ
t2
R2�
t3
4R3
� �ða� sin a cos aÞ
rx ¼
ffiffiffiffiffiIx
A
r; ry ¼
ffiffiffiffiffiIy
A
r(Note: If t=R is small, a can
exceed p to form an
overlapped annulus)
808
FormulasforStressandStrain
[APP.A
Note: If a is small:
sin aa
¼ 1 �a2
6þ
a4
120; a� sin a cos a ¼
2
3a3 1 �
a2
5þ
2a4
105
� �;
sin2 aa
¼ a 1 �a2
3þ
2a4
45
� �
cos ¼ 1 �a2
2þ
a4
24; aþ sin a cos a�
2 sin2 a
a¼
2a5
451 �
a2
7þ
a4
105
� �
22. Solid semicircle A ¼p2
R2
yc1 ¼ 0:5756R
yc2 ¼ 0:4244R
xc ¼ R
Ix ¼ 0:1098R4
Iy ¼p8
R4
rx ¼ 0:2643R
ry ¼R
2
Zx ¼ 0:3540R3; Zy ¼ 0:6667R3
SFx ¼ 1:856; SFy ¼ 1:698
Plastic neutral axis x is located a distance 0:4040R
from the base.
23. Hollow semicircle
Note: b ¼R þ Ri
2
t ¼ R � Ri
A ¼p2ðR2 � R2
i Þ
yc2 ¼4
3pR3 � R2
i
R2 � R2i
or
yc2 ¼2b
p1 þ
ðt=bÞ2
12
" #
yc1 ¼ R � yc2
xc ¼ R
Ix ¼p8ðR4 � R4
i Þ �8
9pðR3 � R3
i Þ2
R2 � R2i
or
Ix ¼ 0:2976tb3 þ 0:1805bt3 �0:00884t5
b
Iy ¼p8ðR4 � R4
i Þ
or
Iy ¼ 1:5708b3t þ 0:3927bt3
Let yp be the vertical distance from the bottom to the
plastic neutral axis.
yp ¼ ð0:7071 � 0:2716C � 0:4299C2 þ 0:3983C3ÞR
Zx ¼ ð0:8284 � 0:9140C þ 0:7245C2
� 0:2850C3ÞR2t
where C ¼ t=R
Zy ¼ 0:6667ðR3 � R3i Þ
24. Solid ellipse A ¼ pab
yc ¼ a
xc ¼ b
Ix ¼p4
ba3
Iy ¼p4
ab3
rx ¼a
2
ry ¼b
2
Zx ¼ 1:333a2b; Zy ¼ 1:333b2a
SFx ¼ SFy ¼ 1:698
APP.A]
Propertie
sofaPlaneArea
809
TABLE A.1 Properties of sections (Continued)
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 a
xc ¼ b
Ix ¼p4ðba3 � bia
3i Þ
Iy ¼p4ðab3 � aib
3i Þ
rx ¼1
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiba3 � bia
3i
ab � aibi
s
ry ¼1
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiab3 � aib
3i
ab � aibi
s
Zx ¼ 1:333ða2b � a2i biÞ
Zy ¼ 1:333ðb2a � b2i aiÞ
SFx ¼ 1:698a3b � a2
i bia
a3b � a3i bi
SFy ¼ 1:698b3a � b2
i aib
b3a � 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 K1
a � b
a þ b
� �2" #
where
K1 ¼ 0:2464 þ 0:002222a
bþ
b
a
� �
yc ¼ a þt
2
xc ¼ b þt
2
Ix ¼p4
ta2ða þ 3bÞ 1 þ K2
a � b
a þ b
� �2" #
þp16
t3ð3a þ bÞ 1 þ K3
a � b
a þ b
� �2" #
where
K2 ¼ 0:1349 þ 0:1279a
b� 0:01284
a
b
2
K3 ¼ 0:1349 þ 0:1279b
a� 0:01284
b
a
� �2
For Iy interchange a and b in the expressions
for Ix;K2, and K3
Zx ¼ 1:3333taða þ 2bÞ 1 þ K4
a � b
a þ b
� �2" #
þt3
3
where
K4 ¼ 0:1835 þ 0:895a
b� 0:00978
a
b
2
For Zy interchange a and b in the expression for Zx
and K4.
See the note on maximum
wall thickness in case 27.
810
FormulasforStressandStrain
[APP.A
27. Hollow semiellipse with
constant wall thickness t.
The midthickness
perimeter is an ellipse
(shown dashed).
0:2 < a=b < 5
Note: There is a limit on the
maximum wall thickness
allowed in this case. Cusps
will form in the perimeter at
the ends of the major axis
if this maximum is exceeded.
Ifa
b4 1; then tmax ¼
2a2
b
Ifa
b5 1; then tmax ¼
2b2
a
A ¼p2
tða þ bÞ 1 þ K1
a � b
a þ b
� �2" #
where
K1 ¼ 0:2464 þ 0:002222a
bþ
b
a
� �
yc2 ¼2a
pK2 þ
t2
6paK3
where
K2 ¼ 1 � 0:3314C þ 0:0136C2 þ 0:1097C3
K3 ¼ 1 þ 0:9929C � 0:2287C2 � 0:2193C3
Using C ¼a � b
a þ b
yc1 ¼ a þt
2� yc2
xc ¼ b þt
2
IX ¼p8
ta2ða þ 3bÞ 1 þ K4
a � b
a þ b
� �2" #
þp32
t3ð3a þ bÞ 1 þ K5
a � b
a þ b
� �2" #
where
K4 ¼ 0:1349 þ 0:1279a
b� 0:01284
a
b
2
K5 ¼ 0:1349 þ 0:1279b
a� 0:01284
b
a
� �2
Ix ¼ IX � Ay2c2
For Iy use one-half the value for Iy in case 26.
Let yp be the vertical distance from the bottom to the
plastic neutral axis.
yp ¼ C1 þC2
a=bþ
C3
ða=bÞ2þ
C4
ða=bÞ3
� �a
where if 0:25 < a=b4 1, then
C1 ¼ 0:5067 � 0:5588D þ 1:3820D2
C2 ¼ 0:3731 þ 0:1938D � 1:4078D2
C3 ¼ �0:1400 þ 0:0179D þ 0:4885D2
C4 ¼ 0:0170 � 0:0079D � 0:0565D2
or if 14a=b < 4, then
C1 ¼ 0:4829 þ 0:0725D � 0:1815D2
C2 ¼ 0:1957 � 0:6608D þ 1:4222D2
C3 ¼ 0:0203 þ 1:8999D � 3:4356D2
C4 ¼ 0:0578 � 1:6666D þ 2:6012D2
where D ¼ t=tmax and where 0:2 < D4 1
Zx ¼ C5 þC6
a=bþ
C7
ða=bÞ2þ
C8
ða=bÞ3
� �4a2t
where if 0:25 < a=b4 1, then
C5 ¼ �0:0292 þ 0:3749D1=2 þ 0:0578D
C6 ¼ 0:3674 � 0:8531D1=2 þ 0:3882D
C7 ¼ �0:1218 þ 0:3563D1=2 � 0:1803D
C8 ¼ 0:0154 � 0:0448D1=2 þ 0:0233D
or if 14a=b < 4, then
C5 ¼ 0:2241 � 0:3922D1=2 þ 0:2960D
C6 ¼ �0:6637 þ 2:7357D1=2 � 2:0482D
C7 ¼ 1:5211 � 5:3864D1=2 þ 3:9286D
C8 ¼ �0:8498 þ 2:8763D1=2 � 1:8874D
For Zy use one-half the value for Zy in case 26.
APP.A]
Propertie
sofaPlaneArea
811
TABLE A.1 Properties of sections (Continued)
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
28. Regular polygon with n
sidesA ¼
a2n
4 tan a
r1 ¼a
2 sin a
r2 ¼a
2 tan a
If n is odd
y1 ¼ y2 ¼ r1 cos an þ 1
2
� ��p2
� �
If n=2 is odd
y1 ¼ r1; y2 ¼ r2
If n=2 is even
y1 ¼ r2; y2 ¼ r1
I1 ¼ I2 ¼ 124
Að6r21 � a2Þ
r1 ¼ r2 ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi124ð6r2
1 � a2Þ
q For n ¼ 3, see case 9. For n ¼ 4, see cases 1 and 13.
For n ¼ 5, Z1 ¼ Z2 ¼ 0:8825r31. For an axis perpen-
dicular to axis 1, Z ¼ 0:8838r31. The location of this
axis is 0.7007a from that side which is perpendicular
to axis 1. For n56, use the following expression for a
neutral axis of any inclination:
Z ¼ r31 1:333 � 13:908
1
n
� �2
þ 12:5281
n
� �3" #
29. Hollow regular polygon
with n sidesA ¼ nat 1 �
t tan aa
� �
r1 ¼a
2 sin a
r2 ¼a
2 tan a
If n is odd
y1 ¼ y2 ¼ r1 cos an þ 1
2�p2
� �If n=2 is odd
y1 ¼ r1; y2 ¼ r2
If n=2 is even
y1 ¼ r2; y2 ¼ r1
I1 ¼ I2 ¼na3t
8
1
3þ
1
tan2 a
� �
� 1 � 3t tan a
aþ 4
t tan aa
� �2
�2t tan a
a
� �3" #
r1 ¼ r2 ¼affiffiffi8
p
�
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
3
� �þ
1
tan2 a1 � 2
t tan aa
þ 2t tan a
a
� �2" #vuut
812
FormulasforStressandStrain
[APP.A