Prop Areas

8/20/2019 Prop Areas

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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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Form of section

Area and distances from

centroid to extremities

Moments and products of inertia



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



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Form of section








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



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Form of section








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



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Form of section








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



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

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