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)


Form of section








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


[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



Form of section








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


[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



Form of section








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


� cos a

" #

xc ¼ R sin a

Ix ¼R4

4a� sin a cos aþ 2 sin

3 a cos a�16 sin


" #

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


[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



Form of section








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


[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)

Form of section








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

�

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