2542x_appa

799

Appendix

AProperties of a Plane Area

Because of their importance in connection with the analysis of bending

and torsion, certain relations for the second-area moments, commonly

referred to as moments of inertia, are indicated in the following

paragraphs. The equations given are in reference to Fig. A.1, and

the notation is as follows:

A area of the section

X ;Y rectangular axes in the plane of the section at arbitrary pointO

x; y rectangular axes in the plane of the section parallel to X ;Y ;respectively with origin at the centroid, C, of the section

Figure A.1 Plane area.

z polar axis through C

x0; y0 rectangular axes in the plane of the section, with origin at C,inclined at a counterclockwise angle y from x; y

1, 2 principal axes at C inclined at a counterclockwise angle ypfrom x; y

r the distance from C to the dA element, r x2 y2

pBy definition,

Moments of inertia: Ix A

y2 dA; Iy A

x2 dA

Polar moment of inertia:

Iz J A

r2 dA Ix Iy Ix0 Iy0 I1 I2Product of inertia: Ixy

A

xy dA

Radii of gyration: kx Ix=A

p; ky

Iy=A

pParallel axis theorem:

IX Ix Ay2c ; IY Iy Ax2c ; IXY Ixy AxcycTransformation equations:

Ix0 Ix cos2 y Iy sin2 y Ixy sin 2yIy0 Ix sin2 y Iy cos2 y Ixy sin 2yIx0y0 12 Ix Iy sin 2y Ixy cos 2y

Principal moments of inertia and directions:

I1;2 12 Ix Iy Iy Ix2 4I2xy

q ; I12 0;

yp 12 tan12Ixy

Iy Ix

!

Upon the determination of the two principal moments of inertia, I1and I2, two angles, 90

apart, can be solved for from the equation for yp.It may be obvious which angle corresponds to which principal moment

of inertia. If not, one of the angles must be substituted into the

equations Ix0 and Iy0 which will again yield the principal moments of

inertia but also their orientation.

Note, if either one of the xy axes is an axis of symmetry, Ixy 0; withIx and Iy being the principal moments of inertia of the section.

If I1 I2 for a set of principal axes through a point, it follows thatthe moments of inertia for all x0y0 axes through that point, in the sameplane, are equal and Ix0y0 0 regardless of y: Thus the moment ofinertia of a square, an equilateral triangle, or any section having

800 Formulas for Stress and Strain [APP. A

two or more axes of identical symmetry is the same for any central

axis.

The moment of inertia and radius of gyration of a section with

respect to a centroidal axis are less than for any other axis parallel

thereto.

The moment of inertia of a composite section (one regarded as made

up of rectangles, triangles, circular segments, etc.) about an axis is

equal to the sum of the moments of inertia of each component part

about that axis. Voids are taken into account by subtracting the

moment of inertia of the void area.

Expressions for the area, distances of centroids from edges,

moments of inertia, and radii of gyration are given in Table A.1 for a

number of representative sections. The moments of products of inertia

for composite areas can be found by addition; the centroids of compo-

site areas can be found by using the relation that the statical moment

about any line of the entire area is equal to the sum of the statical

moments of its component parts.

Although properties of structural sectionswide-flange beams,

channels, angles, etc.are given in structural handbooks, formulas

are included in Table A.1 for similar sections. These are applicable to

sections having a wider range of web and flange thicknesses than

normally found in the rolled or extruded sections included in the

handbooks.

Plastic or ultimate strength design is discussed in Secs. 8.15 and

8.16, and the use of this technique requires the value of the fully

plastic bending momentthe product of the yield strength of a ductile

material and the plastic section modulus Z. The last column in Table

A.1 gives for many of the sections the value or an expression for Z and

the location of the neutral axis under fully plastic pure bending. This

neutral axis does not, in general, pass through the centroid,

but instead divides the section into equal areas in tension and

compression.

APP. A ] Properties of a Plane Area 801

TABLE A.1 Properties of sectionsNOTATION: A area length2; y distance to extreme fiber (length); I moment of inertia length4; r radius of gyration (length); Z plastic section modulus length3; SF shape factor. SeeSec. 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 a2

yc xc a

2

y0c 0:707a cosp4 a

Ix Iy I 0x 112 a4

rx ry r0x 0:2887aZx Zy 0:25a3

SFx SFy 1:5

2. Rectangle A bd

yc d

2

xc b

2

Ix 112 bd3

Iy 112 db3

Ix > Iy if d > b

rx 0:2887dry 0:2887b

Zx 0:25bd2

Zy 0:25db2

SFx SFy 1:5

3. Hollow rectangle A bd bidi

yc d

2

xc b

2

Ix bd3 bid3i

12

Iy db3 dib3i

12

rx IxA

1=2

ry Iy

A

1=2

Zx bd2 bid2i

4

SFx Zxd

2Ix

Zx db2 dib2i

4

SFy Zyb

2Iy

802

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4. Tee section A tb twd

yc bt2 twd2t d

2tb twd

xc b

2

Ix b

3d t3 d

3

3b tw Ad t yc2

Iy tb3

12 dt

3w

12

rx IxA

1=2

ry Iy

A

1=2

If twd5 bt, then

Zx d2tw

4 b

2t2

4tw btd t

2

Neutral axis x is located a distance bt=tw d=2from the bottom.

If twd4 bt, then

Zx t2b

4 twdt d twd=2b

2

Neutral axis x is located a distance twd=b t=2from the top.

SFx Zxd t yc

I1

Zy b2t t2wd

4

SFy Zyb

2Iy

5. Channel section A tb 2twd

yc bt2 2twd2t d

2tb 2twd

xc b

2

Ix b

3d t3 d

3

3b 2tw Ad t yc2

Iy d tb3

12 db 2tw

3

12

rx IxA

1=2

ry Iy

A

1=2

If 2twd5 bt, then

Zx d2tw

2 b

2t2

8tw btd t

2

Neutral axis x is located a distance

bt=2tw d=2 from the bottom.If 2twd4 bt, then

Zx t2b

4 twd t d

twd

b

Neutral axis x is located a distance twd=b t=2from the top.

SFx Zxd t yc

Ix

Zy b2t

4 twdb tw

SFy Zyb

2Iy

APP.A]

Propertie

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803

TABLE A.1 Properties of sections (Continued)


Form of section








6. Wide-flange beam with

equal flanges

A 2bt twd

yc d

2 t

xc b

2

Ix bd 2t3

12 b twd

3

12

Iy b3t

6 t

3wd

12

rx IxA

1=2

ry Iy

A

1=2

Zx twd

2

4 btd t

SFx Zx yc

Ix

Zy b2t

2 t

2wd

4

SFy Zyxc

Iy

7. Equal-legged angle A t2a t

yc1 0:7071a2 at t2

2a t

yc2 0:7071a2

2a txc 0:7071a

Ix a4 b4

12 0:5ta

2b2

a b

Iy a4 b4

12where b a t

rx IxA

1=2

ry Iy

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 at

a t=a

2

2

" #1=2

Zx Ayc1 0:6667ypIf t=a4 0:4, then

yp 0:3536a 1:5tZx Ayc1 2:8284y2pt 1:8856t3

8. Unequal-legged angle A tb d t

xc b2 dt t22b d t

yc d2 bt t22b d t

Ix 13 bd3 b td t3 Ad yc2

Iy 13 db3 d tb t3 Ab xc2

Ixy 14 b2d2 b t2d t2 Ab xcd yc

rx IxA

1=2

ry Iy

A

1=2

804

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9. Equilateral triangle A 0:4330a2

yc 0:5774axc 0:5000ay0c 0:5774a cos a

Ix Iy Ix0 0:01804a4

rx ry rx0 0:2041aZx 0:0732a3; Zy 0:0722a3

SFx 2:343; SFy 2:000Neutral axis x is 0:2537a from the base.

10. Isosceles triangle A bd2

yc 23 d

xc b

2

Ix 136 bd3

Iy 148 db3

Ix > Iy if d > 0:866b

rx 0:2357dry 0:2041b

Zx 0:097bd2; Zy 0:0833db2

SFx 2:343; SFy 2:000Neutral axis x is 0:2929d from the base.

11. Triangle A bd2

yc 23 dxc 23 b 13 a

Ix 136 bd3

Iy 136 bdb2 ab a2Ixy 172 bd2b 2a

yx 1

2tan1

db 2ab2 ab a2 d2

rx 0:2357d

ry 0:2357b2 ab a2

p

12. Parallelogram A bd

yc d

2

xc 12 b a

Ix 112 bd3

Iy 112 bdb2 a2Ixy 112 abd2

yx 1

2tan1

2adb2 a2 d2

rx 0:2887d

ry 0:2887b2 a2

p

APP.A]

Propertie

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805



Form of section








13. Diamond A bd2

yc d

2

xc b

2

Ix 148 bd3

Iy 148 db3

rx 0:2041dry 0:2041b

Zx 0:0833bd2; Zy 0:0833db2

SFx SFy 2:000

14. Trapezoid A d2b c

yc d

3

2b cb c

xc 2b2 2bc ab 2ac c2

3b c

Ix d3

36

b2 4bc c2b c

Iy d

36b c b4 c4 2bcb2 c2

ab3 3b2c 3bc2 c3 a2b2 4bc c2

Ixy d2

72b c c3b2 3bc c2

b3 a2b2 8bc 2c2

15. Solid circle A pR2

yc RIx Iy

p4

R4

rx ry R

2

Zx Zy 1:333R3

SFx 1:698

806

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16. Hollow circle A pR2 R2i yc R

Ix Iy p4R4 R4i

rx ry 12R2 R2i

qZx Zy 1:333R3 R3i

SFx 1:698R4 R3i RR4 R4i

17. Very thin annulus A 2pRtyc R

Ix Iy pR3trx ry 0:707R

Zx Zy 4R2t

SFx SFy 4

p

18. Sector of solid circle A aR2

yc1 R 1 2 sin a

3a

yc2 2R sin a

3a

xc R sin a

Ix R4

4a sin a cos a 16 sin

2 a9a

!

Iy R4

4a sin a cos a

Note: If a is small; a sin a cos a 23a3 2

15a5

rx R

2

1 sin a cos a

a 16 sin

2 a9a2

s

ry R

2

1 sin a cos a

a

r

If a4 54:3, then

Zx 0:6667R3 sin aa3

2 tan a

1=2" #

Neutral axis x is located a distance

R0:5a= tan a1=2 from the vertex.If a5 54:3, then

Zx 0:6667R32 sin3 a1 sin a where theexpression 2a1 sin 2a1 a is solved for the valueof a1.Neutral axis x is located a distance R cos a1 fromthe vertex.

If a4 73:09, then SFx Zxyc2

Ix

If 73:094a4 90, then SFx Zxyc1

Ix

Zy 0:6667R31 cos aIf a4 90 , then

SFy 2:6667 sin a1 cos a

a sin a cos aIf a5 90 , then

SFy 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, useexpressions from case 20)

A R2a sin a cos a

yc1 R 1 2 sin

3 a3a sin a cos a

" #

yc2 R2 sin

3 a3a sin a cos a cos a" #

xc R sin a

Ix R4

4a sin a cos a 2 sin3 a cos a 16 sin

6 a9a sin a cos a

" #

Iy R4

123a 3 sin a cos a 2 sin3 a cos a

rx R

2

1 2 sin

3 a cos aa sin a cos a

16 sin6 a

9a sin a cos a2

s

ry R

2

1 2 sin

3 a cos a3a sin a cos a

s

20. Segment of solid circle

(Note: Do not use if

a > p=4

A 23R2a31 0:2a2 0:019a4

yc1 0:3Ra21 0:0976a2 0:0028a4yc2 0:2Ra21 0:0619a2 0:0027a4xc Ra1 0:1667a2 0:0083a4

Ix 0:01143R4a71 0:3491a2 0:0450a4Iy 0:1333R4a51 0:4762a2 0:1111a4rx 0:1309Ra21 0:0745a2ry 0:4472Ra1 0:1381a2 0:0184a4

21. Sector of hollow circle A at2R t

yc1 R 1 2 sin a

3a1 t

R 1

2 t=R

yc2 R2 sin a

3a2 t=R 1 t

R

2 sin a 3a cos a

3a

xc R sin a

Ix R3t 1 3t

2R t

2

R2 t

3

4R3

a sin a cos a 2 sin2 a

a

!

t2 sin

2 a3R2a2 t=R 1

t

R t

2

6R2

#

Iy R3t 1 3t

2R t

2

R2 t

3

4R3

a sin a cos a

rx IxA

r; ry

Iy

A

r(Note: If t=R is small, a canexceed p to form anoverlapped annulus)

808

Form

ulasforStre

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in[A

PP.A

Note: If a is small:sin aa

1 a2

6 a

4

120; a sin a cos a 2

3a3 1 a

2

5 2a

4

105

;

sin2 aa

a 1 a2

3 2a

4

45

cos 1 a2

2 a

4

24; a sin a cos a 2 sin

2 aa

2a5

451 a

2

7 a

4

105

22. Solid semicircle A p2

R2

yc1 0:5756Ryc2 0:4244Rxc R

Ix 0:1098R4

Iy p8

R4

rx 0:2643R

ry R

2

Zx 0:3540R3; Zy 0:6667R3

SFx 1:856; SFy 1:698Plastic neutral axis x is located a distance 0:4040R

from the base.

23. Hollow semicircle

Note: b R Ri2

t R Ri

A p2R2 R2i

yc2 4

3pR3 R2iR2 R2i

or

yc2 2b

p1 t=b

2

12

" #

yc1 R yc2xc R

Ix p8R4 R4i

8

9pR3 R3i 2

R2 R2ior

Ix 0:2976tb3 0:1805bt3 0:00884t5

b

Iy p8R4 R4i

or

Iy 1:5708b3t 0:3927bt3

Let yp be the vertical distance from the bottom to the

plastic neutral axis.

yp 0:7071 0:2716C 0:4299C2 0:3983C3RZx 0:8284 0:9140C 0:7245C2

0:2850C3R2twhere C t=RZy 0:6667R3 R3i

24. Solid ellipse A pabyc axc b

Ix p4

ba3

Iy p4

ab3

rx a

2

ry b

2

Zx 1:333a2b; Zy 1:333b2aSFx SFy 1:698

APP.A]

Propertie

sofaPlaneArea

809



Form of section








25. Hollow ellipse A pab aibiyc axc b

Ix p4ba3 bia3i

Iy p4ab3 aib3i

rx 1

2

ba3 bia3iab aibi

s

ry 1

2

ab3 aib3iab aibi

s

Zx 1:333a2b a2i biZy 1:333b2a b2i ai

SFx 1:698a3b a2i biaa3b a3i bi

SFy 1:698b3a b2i aibb3a 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 pta b 1 K1a ba b 2" #

where

K1 0:2464 0:002222a

b b

a

yc a t

2

xc b t

2

Ix p4

ta2a 3b 1 K2a ba b 2" #

p16

t33a b 1 K3a ba b 2" #

where

K2 0:1349 0:1279a

b 0:01284 a

b

2

K3 0:1349 0:1279b

a 0:01284 b

a

2For Iy interchange a and b in the expressions

for Ix;K2, and K3

Zx 1:3333taa 2b 1 K4a ba b 2" #

t3

3

where

K4 0:1835 0:895a

b 0:00978 a

b

2For Zy interchange a and b in the expression for Zxand K4.

See the note on maximum

wall thickness in case 27.

810

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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

2a2

b

Ifa

b5 1; then tmax

2b2

a

A p2

ta b 1 K1a ba b 2" #

where

K1 0:2464 0:002222a

b b

a

yc2 2a

pK2

t2

6paK3

where

K2 1 0:3314C 0:0136C2 0:1097C3

K3 1 0:9929C 0:2287C2 0:2193C3

Using C a ba b

yc1 a t

2 yc2

xc b t

2

IX p8

ta2a 3b 1 K4a ba b 2" #

p32

t33a b 1 K5a ba b 2" #

where

K4 0:1349 0:1279a

b 0:01284 a

b

2

K5 0:1349 0:1279b

a 0:01284 b

a

2

Ix 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 C1 C2a=b

C3a=b2 C4

a=b3

a

where if 0:25 < a=b4 1, then

C1 0:5067 0:5588D 1:3820D2

C2 0:3731 0:1938D 1:4078D2

C3 0:1400 0:0179D 0:4885D2

C4 0:0170 0:0079D 0:0565D2or if 14a=b < 4, then

C1 0:4829 0:0725D 0:1815D2

C2 0:1957 0:6608D 1:4222D2

C3 0:0203 1:8999D 3:4356D2

C4 0:0578 1:6666D 2:6012D2where D t=tmax and where 0:2 < D4 1

Zx C5 C6a=b

C7a=b2 C8

a=b3

4a2t

where if 0:25 < a=b4 1, then

C5 0:0292 0:3749D1=2 0:0578DC6 0:3674 0:8531D1=2 0:3882DC7 0:1218 0:3563D1=2 0:1803DC8 0:0154 0:0448D1=2 0:0233D

or if 14a=b < 4, thenC5 0:2241 0:3922D1=2 0:2960DC6 0:6637 2:7357D1=2 2:0482DC7 1:5211 5:3864D1=2 3:9286DC8 0:8498 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 a

2n

4 tan a

r1 a

2 sin a

r2 a

2 tan a

If n is odd

y1 y2 r1 cos an 1

2

p

2

If n=2 is odd

y1 r1; y2 r2If n=2 is even

y1 r2; y2 r1

I1 I2 124 A6r21 a2

r1 r2 1246r21 a2

q For n 3, see case 9. For n 4, see cases 1 and 13.For n 5, Z1 Z2 0:8825r31. For an axis perpen-dicular to axis 1, Z 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 aneutral axis of any inclination:

Z r31 1:333 13:9081

n

2 12:528 1

n

3" #

29. Hollow regular polygon

with n sidesA nat 1 t tan a

a

r1 a

2 sin a

r2 a

2 tan a

If n is odd

y1 y2 r1 cos an 1

2 p

2

If n=2 is odd

y1 r1; y2 r2If n=2 is even

y1 r2; y2 r1

I1 I2 na3t

8

1

3 1

tan2 a

1 3 t tan aa

4 t tan aa

22 t tan a

a

3" #

r1 r2 a8

p

1

3

1

tan2 a1 2 t tan a

a 2 t tan a

a

2" #vuut

812

Form

ulasforStre

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in[A

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Table of ContentsAppendix A: Properties of a Plane AreaAppendix B:Glossary: DefinationsAppendix C: Composite Materials

2542x_appa

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