4.1. Torsión de elementos no circularesmateriales.azc.uam.mx/gjl/Clases/SOLIDOSII/S9.pdf · Tema...

Tema 4 Torsioacuten en barras y en tubos no circulares

41 Torsioacuten de elementos no circulares

Denotando con L la longitud de la barra con a el lado maacutes ancho y con b el lado maacutes angosto de

su seccioacuten transversal y con T la magnitud de los momentos torsionantes de los pares aplicados

a la barra de la Figura el esfuerzo cortante maacuteximo que ocurre a lo largo de la liacutenea centran de

la cara maacutes ancha de la barra es igual a

τ =T

c1ab2(41)

El aacutengulo de giro se calcula como

θ =TL

c2ab2G(42)

Los coeficientes c1 y c2 dependen de la razoacuten ab dados en la tabla estas ecuaciones son vaacutelidas

dentro del rango elaacutestico

ab c1 c2

10 0208 01406

12 0219 01661

15 0231 01958

20 0246 0229

25 0258 0249

30 0267 0263

40 0282 0281

50 0291 0291

100 0312 0312

infin 0333 0333

En la tabla anterior los coeficientes c1 y c2 son iguales para la razoacuten ab gt 5 Para tales valores

c1 = c2 =1

3(1minus 0630ba) (43)

42 Torsioacuten de elementos huecos de pares delgada

El esfuerzo cortante τ en cualquier punto de un elemento hueco de paredes delgadas se determina

con la siguiente expresioacuten

τ =T

2ta(44)

Donde T es la magnitud de los momentos torsionantes t el espesor del elemento y a es el aacuterea

bordeada por la liacutenea central Para calcular el aacutengulo θ utilice la tabla anexa

6

TABLE 101 Formulas for torsional deformation and stressGENERAL FORMULAS y frac14 TL=KG and t frac14 T=Q where y frac14 angle of twist (radians) T frac14 twisting moment (force-length) L frac14 length t frac14unit shear stress (force per unit area) G frac14 modulus of

rigidity (force per unit area) K (length to the fourth) and Q (length cubed) are functions of the cross section

Form and dimensions of cross sections

other quantities involved and case no Formula for K in y frac14TL

KGFormula for shear stress

1 Solid circular section K frac14 12pr4

tmax frac142T

pr3at boundary

2 Solid elliptical section K frac14pa3b3

a2 thorn b2tmax frac14

2T

pab2at ends of minor axis

3 Solid square section K frac14 225a4

tmax frac140601T

a3at midpoint of each side

4 Solid rectangular section K frac14 ab3 16

3 336

b

a1

b4

12a4

for a5 b tmax frac14

3T

8ab21 thorn 06095

b

athorn 08865

b

a

2

18023b

a

3

thorn 09100b

a

4

at the midpoint of each longer side for a5 b

107 Tables

SEC107]

Torsion

401

TABLE 101 Formulas for torsional deformation and stress (Continued)




5 Solid triangular section (equilaterial)K frac14

a4ffiffiffi3

p

80tmax frac14

20T


6 Isosceles triangle

(Note See also Ref 21 for graphs of stress

magnitudes and locations and stiffness

factors)

For 23lt a=b lt

ffiffiffi3

peth39 lt a lt 82THORN

K frac14a3b3

15a2 thorn 20b2

approximate formula which is exact at a frac14 60

where K frac14 002165c4

Forffiffiffi3

plt a=b lt 2

ffiffiffi3


K frac14 00915b4 a

b 08592

approximate formula which is exact at

a frac14 90 where K frac14 00261c4 (errors lt 4) (Ref 20)

For 39 lt a lt 120

Q frac14K

bfrac120200 thorn 0309a=b 00418etha=bTHORN2

approximate formula which is exact at a frac14 60 and a frac14 90

For a frac14 60 Q frac14 00768b3 frac14 00500c3


tmax at center of longest side

7 Circular segmental section

[Note h frac14 reth1 cos aTHORN

K frac14 2Cr4 where C varies withh

ras follows

For 04h

r4 10

C frac14 07854 00333h

r 26183

h

r

2

thorn 41595h

r

3

30769h

r

4

thorn09299h

r

5

tmax frac14TB

r3where B varies with

h

r

as follows For 04h

r4 10

B frac14 06366 thorn 17598h

r 54897

h

r

2

thorn14062h

r

3

14510h

r

4

thorn 6434h

r

5

(Data from Refs 12 and 13)

402

FormulasforStressandStrain

[CHAP10

8 Circular sector

(Note See also Ref 21)

K frac14 Cr4 where C varies withap

as follows

For 014ap4 20

C frac14 00034 00697apthorn 05825

ap

2

02950ap

3

thorn 00874ap

4

00111ap

5

tmax frac14T

Br3on a radial boundary B varies

withap

as follows For 014ap4 10

B frac14 00117 02137apthorn 22475

ap

2

46709ap

3

thorn 51764ap

4

22000ap

5

ethData from Ref 17)

9 Circular shaft with opposite sides

flattened

(Note h frac14 r wTHORN


ras follows

For two flat sides where 04h

r408

C frac14 07854 04053h

r 35810

h

r

2

thorn 52708h

r

3

20772h

r

4

For four flat sides where

04h

r4 0293

C frac14 07854 07000h

r 77982

h

r

2

thorn 14578h

r

3

tmax frac14TB


h

ras follows For two flat sides where

04h

r4 06


r 11157

h

r

2

thorn 49568h

r

3

85886h

r

4

thorn 69849h

r

5

For four flat sides where 04h

r4 0293


r 56147

h

r

2

thorn 30853h

r

3


SEC107]

Torsion

403






10 Hollow concentric circular section K frac14 12pethr4

0 r4i THORN tmax frac14

2Tro

pethr4o r4

i THORNat outer boundary

11 Eccentric hollow circular sectionK frac14

pethD4 d4THORN

32C

where

C frac14 1 thorn16n2

eth1 n2THORNeth1 n4THORNl2

thorn384n4

eth1 n2THORN2eth1 n4THORN

4l4

tmax frac1416TDF

pethD4 d4THORN

F frac14 1 thorn4n2

1 n2lthorn

32n2


thorn48n2eth1 thorn 2n2 thorn 3n4 thorn 2n6THORN

eth1 n2THORNeth1 n4THORNeth1 n6THORNl3

thorn64n2eth2 thorn 12n2 thorn 19n4 thorn 28n6 thorn 18n8 thorn 14n10 thorn 3n12THORN

eth1 n2THORNeth1 n4THORNeth1 n6THORNeth1 n8THORNl4 (Ref 10)

12 Hollow elliptical section outer and

inner boundaries similar ellipsesK frac14

pa3b3

a2 thorn b2eth1 q4THORN

where

q frac14ao

afrac14

bo

b

(Note The wall thickness is not constant)

tmax frac142T

pab2eth1 q4THORNat ends of minor axis on outer surface

13 Hollow thin-walled section of uniform

thickness U frac14 length of elliptical

median boundary shown dashed

U frac14 petha thorn b tTHORN 1 thorn 0258etha bTHORN2

etha thorn b tTHORN2

ethapproximatelyTHORN

K frac144p2tfrac12etha 1

2tTHORN2ethb 1

2tTHORN2

Utaverage frac14

T

2ptetha 12tTHORNethb 1

2tTHORN

(stress is nearly uniform if t is small)

404


[CHAP10

14 Any thin tube of uniform thickness

U frac14 length of median boundary

A frac14mean of areas enclosed by outer

and inner boundaries or (approximate)

area within median boundary

K frac144A2t

Utaverage frac14

T

2tA(stress is nearly uniform if t is small)

15 Any thin tube U and A as for

case 14 t frac14 thickness at any pointK frac14

4A2ETHdU=t

taverage on any thickness AB frac14T

2tAethtmaxwhere t is a minimum)

16 Hollow rectangle thin-walled

(Note For thick-walled hollow rectangles

see Refs 16 and 25 Reference 25

illustrates how to extend the work

presented to cases with more than

one enclosed region)

K frac142tt1etha tTHORN2ethb t1THORN

2

at thorn bt1 t2 t21 taverage frac14

T

2tetha tTHORNethb t1THORNnear midlength of short sides

T

2t1etha tTHORNethb t1THORNnear midlength of long sides

8gtgtgtltgtgtgt

(There will be higher stresses at inner corners unless fillets of fairly large radius

are provided)

SEC107]

Torsion

405


TABLE 101 Formulas for torsional deformation and stress (Continued )




17 Thin circular open tube of uniform

thickness r frac14 mean radius

K frac14 23prt3

tmax frac14T eth6pr thorn 18tTHORN

4p2r2t2

along both edges remote from ends (this assumes t is small comopared with mean

radius)

18 Any thin open tube of uniform

thickness U frac14 length of median line

shown dashed

K frac141

3Ut3 tmax frac14

T eth3U thorn 18tTHORN

U2t2

along both edges remote from ends (this assumes t small compared wtih least

radius of curvature of median line otherwise use the formulas given for cases

19ndash26)

19 Any elongated section with axis of

symmetry OX U frac14 length A frac14 area of

section Ix frac14moment of inertia about

axis of symmetry

K frac144Ix

1 thorn 16Ix=AU2

20 Any elongated section or thin open tube

dU frac14 elementary length along median

line t frac14 thickness normal to median line

A frac14area of section

K frac14F

3 thorn 4F=AU2where F frac14

ethU

0

t3dU

21 Any solid fairly compact section

without reentrant angles J frac14polar

moment of inertia about centroid axis


K frac14A4

40J

For all solid sections of irregular form (cases 19ndash26 inclusive) the maximum shear

stress occurs at or very near one of the points where the largest inscribed circle

touches the boundary and of these at the one where the curvature of the

boundary is algebraically least (Convexity represents positive and concavity

negative curvature of the boundary) At a point where the curvature is positive

(boundary of section straight or convex) this maximum stress is given approxi-

mately by

tmax frac14 GyL

C or tmax frac14T

KC

where

C frac14D

1 thornp2D4

16A2

1 thorn 015p2D4

16A2

D

2r

Dfrac14diameter of largest inscribed circle

rfrac14 radius of curvature of boundary at the point (positive for this case)

Afrac14 area of the section

Unless at some point on the boundary there is a sharp reentant angle causing

high local stress

406


[CHAP10

22 Trapezoid K frac14 112

bethm thorn nTHORNethm2 thorn n2THORN VLm4 Vsn4

where VL frac14 010504 010s thorn 00848s2

006746s3 thorn 00515s4

Vs frac14 010504 thorn 010s thorn 00848s2

thorn 006746s3 thorn 00515s4

s frac14m n

b

(Ref 11)

23 T-section flange thickness uniform

For definitions of rD t and t1 see

case 26

K frac14 K1 thorn K2 thorn aD4

where K1 frac14 ab3 1

3 021

b

a1

b4

12a4

K2 frac14 cd3 1

3 0105

d

c1

d4

192c4

a frac14t

t1

015 thorn 010r

b

D frac14ethb thorn rTHORN2 thorn rd thorn d2=4

eth2r thorn bTHORN

for d lt 2ethb thorn rTHORN

24 L-section b5d For definitions of r and

D see case 26



3 021

b

a1

b4

12a4

K2 frac14 cd3 1

3 0105

d

c1

d4

192c4

a frac14d

b007 thorn 0076

r

b

D frac14 2frac12d thorn b thorn 3r

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2eth2r thorn bTHORNeth2r thorn d

p

for b lt 2ethd thorn rTHORN

At a point where the curvature is negative (boundary of section concave or

reentrant) this maximum stress is given approximately by

tmax frac14 GyL

C or tmax frac14T

KC

where C frac14D

1 thornp2D4

16A2

1 thorn 0118 ln 1 D

2r

0238

D

2r

tanh

2fp

and DA and r have the same meaning as before and f frac14 a positive angle through

which a tangent to the boundary rotates in turning or traveling around the

reentrant portion measured in radians (here r is negative)

The preceding formulas should also be used for cases 17 and 18 when t is

relatively large compared with radius of median line

SEC107]

Torsion

407






25 U- or Z-section K frac14 sum of Krsquos of constituent L-sections computed

as for case 24

26 I-section flange thickness uniform

r frac14fillet radius D frac14diameter largest

inscribed circle t frac14 b if b lt d t frac14 d

if d lt b t1 frac14 b if b gt d t1 frac14 d if d gt b

K frac14 2K1 thorn K2 thorn 2aD4


3 021

b

a1

b4

12a4

K2 frac14 1

3cd3

a frac14t

t1

015 thorn 01r

b

Use expression for D from case 23

27 Split hollow shaft K frac14 2Cr4o where C varies with

ri

ro

as follows

For 024ri

ro

4 06

C frac14 K1 thorn K2

ri

ro

thorn K3

ri

ro

2

thorn K4

ri

ro

3

where for 014h=ri 4 10

K1 frac14 04427 thorn 00064h

ri

00201h

ri

2

K2 frac14 08071 04047h

ri

thorn 01051h

ri

2

K3 frac14 00469 thorn 12063h

ri

03538h

ri

2

K4 frac14 05023 09618h

ri

thorn 03639h

ri

2

At M t frac14TB

r3o

where B varies withri

ro

as follows

For 024ri

ro

406

B frac14 K1 thorn K2

ri

ro

thorn K3

ri

ro

2

thorn K4

ri

ro

3

where fore 014h=ri 4 10

K1 frac14 20014 01400h

ri

03231h

ri

3

K2 frac14 29047 thorn 30069h

ri

thorn 40500h

ri

2

K3 frac14 15721 65077h

ri

12496h

ri

2

K4 frac14 29553 thorn 41115h

ri

thorn 18845h

ri

2


408


[CHAP10

28 Shaft with one keyway K frac14 2Cr4 where C varies withb

ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3

where for 034a=b4 15

K1 frac14 07854

K2 frac14 00848 thorn 01234a

b 00847

a

b

2

K3 frac14 04318 22000a

bthorn 07633

a

b

2

K4 frac14 00780 thorn 20618a

b 05234

a

b

2

At M t frac14TB


b

ras follows For 024

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3

where for 054a=b415

K1 frac14 11690 03168a

bthorn 00490

a

b

2

K2 frac14 043490 15096a

bthorn 08677

a

b

2

K3 frac14 11830 thorn 42764a

b 17024

a

b

2

K4 frac14 08812 02627a

b 01897

a

b

2


29 Shaft with two keyways K frac14 2Cr4 where C varies withb

ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00795 thorn 01286a

b 01169

a

b

2

K3 frac14 14126 38589a

bthorn 13292

a

b

2

K4 frac14 07098 thorn 41936a

b 11053

a

b

2

At M t frac14TB


b

ras follows For 024

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3

where for 054a=b415

K1 frac14 12512 05406a

bthorn 00387

a

b

2

K2 frac14 09385 thorn 23450a

bthorn 03256

a

b

2

K3 frac14 72650 15338a

bthorn 31138

a

b

2

K4 frac14 11152 thorn 33710a

b 10007

a

b

2


SEC107]

Torsion

409






30 Shaft with four keyways K frac14 2Cr4 where C varies withb

ras follows

For 04b

r4 04


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 01496 thorn 02773a

b 02110

a

b

2

K3 frac14 29138 82354a

bthorn 25782

a

b

2

K4 frac14 22991 thorn 12097a

b 22838

a

b

2

At M t frac14TB


b

ras follows For 024

b

r4 04


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 10434 thorn 10449a

b 02977

a

b

2

K2 frac14 00958 98401a

bthorn 16847

a

b

2

K3 frac14 15749 69650a

bthorn 14222

a

b

2

K4 frac14 35878 thorn 88696a

b 47545

a

b

2


31 Shaft with one spline K frac14 2Cr4 where C varies withb

ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00264 01187a

bthorn 00868

a

b

2

K3 frac14 02017 thorn 09019a

b 04947

a

b

2

K4 frac14 02911 14875a

bthorn 20651

a

b

2

At M t frac14TB


b

ras follows For 04

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 06366

K2 frac14 00023 thorn 00168a

bthorn 00093

a

b

2

K3 frac14 00052 thorn 00225a

b 03300

a

b

2

K4 frac14 00984 04936a

bthorn 02179

a

b

2


410


[CHAP10

32 Shaft with two splines K frac14 2Cr4 where C varies withb

ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00204 01307a

bthorn 01157

a

b

2

K3 frac14 02075 thorn 11544a

b 05937

a

b

2

K4 frac14 03608 22582a

bthorn 37336

a

b

2

At M t frac14TB


b

ras follows For 04

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 06366

K2 frac14 00069 00229a

bthorn 00637

a

b

2

K3 frac14 00675 thorn 03996a

b 10514

a

b

2

K4 frac14 03582 18324a

bthorn 15393

a

b

2


33 Shaft with four splines K frac14 2Cr4 where C varies withb

ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00595 03397a

bthorn 03239

a

b

2

K3 frac14 06008 thorn 31396a

b 20693

a

b

2

K4 frac14 10869 62451a

bthorn 94190

a

b

2

At M t frac14TB


b

ras follows For 04

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 06366

K2 frac14 00114 00789a

bthorn 01767

a

b

2

K3 frac14 01207 thorn 10291a

b 23589

a

b

2

K4 frac14 05132 34300a

bthorn 40226

a

b

2


SEC107]

Torsion

411






34 Pinned shaft with one two or four

grooves

K frac14 2Cr4 where C varies witha

rover the range

04a

r4 05 as follows For one groove

C frac14 07854 00225a

r 14154

a

r

2

thorn 09167a

r

3

For two grooves

C frac14 07854 00147a

r 30649

a

r

2

thorn 25453a

r

3

For four grooves

C frac14 07854 00409a

r 62371

a

r

2

thorn 72538a

r

3

At M t frac14TB


a

rover the

range 014a


B frac14 10259 thorn 11802a

r 27897

a

r

2

thorn 37092a

r

3

For two grooves


r 29501

a

r

2

thorn 70534a

r

3

For four grooves

B frac14 12135 29697a

rthorn 33713

a

r

2

99506a

r

3

thorn 13049a

r

4


35 Cross shaft K frac14 2Cs4 where C varies withr

sover the

range 04r

s4 09 as follows

C frac14 11266 03210r

sthorn 31519

r

s

2

14347r

s

3

thorn 15223r

s

4

47767r

s

5

At M t frac14BM T

s3where BM varies with

r

sover the range 04

r

s4 05 as follows

BM frac14 06010 thorn 01059r

s 09180

r

s

2

thorn 37335r

s

3

28686r

s

4

At N t frac14BN T

s3where BN varies with

r

sover the range 034

r

s4 09 as follows

BN frac14 03281 thorn 91405r

s 42520

r

s

2

thorn 10904r

s

3

13395r

s

4

thorn 66054r

s

5

(Note BN gt BM for r=s gt 032THORN


412


[CHAP10

TABLE 102 Formulas for torsional properties and stresses in thin-walled open cross sectionsNOTATION Point 0 indicates the shear center e frac14 distance from a reference to the shear center K frac14 torsional stiffness constant (length to the fourth power) Cw frac14warping constant (length to the

sixth power) t1 frac14 shear stress due to torsional rigidity of the cross section (force per unit area) t2 frac14 shear stress due to warping rigidity of the cross section (force per unit area) sx frac14 bending stress

due to warping rigidity of the cross section (force per unit area) E frac14modulus of elasticity of the material (force per unit area) and G frac14modulus of rigidity (shear modulus) of the material (force per

unit area)

The appropriate values of y0 y00 and y000 are found in Table 103 for the loading and boundary restraints desired

Cross section reference no Constants Selected maximum values

1 Channele frac14

3b2

h thorn 6b

K frac14t3

3ethh thorn 2bTHORN

Cw frac14h2b3t

12

2h thorn 3b

h thorn 6b

ethsxTHORNmax frac14hb

2

h thorn 3b

h thorn 6bEy00 throughout the thickness at corners A and D

etht2THORNmax frac14hb2

4

h thorn 3b

h thorn 6b

2

Ey000 throughout the thickness at a distance bh thorn 3b

h thorn 6bfrom corners A and D

etht1THORNmax frac14 tGy0 at the surface everywhere

2 C-sectione frac14 b

3h2b thorn 6h2b1 8b31

h3 thorn 6h2b thorn 6h2b1 thorn 8b31 12hb2

1

K frac14t3

3ethh thorn 2b thorn 2b1THORN

Cw frac14 th2b2

2b1 thorn

b

3 e

2eb1

bthorn

2b21

h

thornh2e2

2b thorn b1 thorn

h

6

2b21

h

thorn

2b31

3ethb thorn eTHORN2

ethsxTHORNmax frac14h

2ethb eTHORN thorn b1ethb thorn eTHORN

Ey00 throughout the thickness at corners A and F

etht2THORNmax frac14h

4ethb eTHORNeth2b1 thorn b eTHORN thorn

b21

2ethb thorn eTHORN

Ey000 throughout the thickness on the top and bottom flanges at a

distance e from corners C and D


3 Hat sectione frac14 b


h3 thorn 6h2b thorn 6h2b1 thorn 8b31 thorn 12hb2

1

K frac14t3


Cw frac14 th2b2

2b1 thorn

b

3 e

2eb1

b

2b21

h

thornh2e2

2b thorn b1 thorn

h

6thorn

2b21

h

thorn

2b31

3ethb thorn eTHORN2

sx frac14h

2ethb eTHORN b1ethb thorn eTHORN


sx frac14h

2ethb eTHORNEy00 throughout the thickness at corners B and E

t2 frac14h2ethb eTHORN2

8ethb thorn eTHORNthorn

b21

2ethb thorn eTHORN

hb1

2ethb eTHORN

Ey000 throughout the thickness at a distance

hethb eTHORN

2ethb thorn eTHORN

from corner B toward corner A

t2 frac14b2

1

2ethb thorn eTHORN

hb1

2ethb eTHORN

h

4ethb eTHORN2

Ey000 throughout the thickness at a distance e

from corner C toward corner B

t1 frac14 tGy0 at the surface everywhere

SEC107]

Torsion

413

TABLE 102 Formulas for torsional properties and stresses in thin-walled open cross sections (Continued )


4 Twin channel with

flanges inwardK frac14

t3

3eth2b thorn 4b1THORN

Cw frac14tb2

24eth8b3

1 thorn 6h2b1 thorn h2b thorn 12b21hTHORN

ethsxTHORNmax frac14b

2b1 thorn

h

2

Ey00 throughout the thickness at points A and D

etht2THORNmax frac14b

16eth4b2

1 thorn 4b1h thorn hbTHORNEy000 throughout the thickness midway between corners B and C


5 Twin channel with

flanges outwardK frac14

t3


Cw frac14tb2

24eth8b3

1 thorn 6h2b1 thorn h2b 12b21hTHORN


4Ey00 throughout the thickness at points B and C if h gt b1


4

bb1

2

Ey00 throughout the thickness at points A and D if h lt b1


4

h

2 b1

2

Ey000 throughout the thickness at a distanceh

2from corner B toward point A if

b1 gth

21 thorn

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

2thorn

b

2h

r


4b2

1 hb

4 hb1

Ey000 throughout the thickness at a point midway between corners B and C if

b1 lth

21 thorn


2thorn

b

2h

r


6 Wide flanged beam

with equal flanges

K frac14 13eth2t3b thorn t3

whTHORN

Cw frac14h2tb3

24


4Ey00 throughout the thickness at points A and B

etht2THORNmax frac14 hb2

16Ey000 throughout the thickness at a point midway between A and B


414


[CHAP10

7 Wide flanged beam

with unequal flangese frac14

t1b31h

t1b31 thorn t2b3

2

K frac14 13etht3

1b1 thorn t32b2 thorn t3

whTHORN

Cw frac14h2t1t2b3

1b32

12etht1b31 thorn t2b3

2THORN

ethsxTHORNmax frac14hb1

2

t2b32

t1b31 thorn t2b3

2

Ey00 throughout the thickness at points A and B if t2b22 gt t1b2

1


2

t1b31

t1b31 thorn t2b3

2

Ey00 throughout the thickness at points C and D if t2b22 lt t1b2

1

etht2THORNmax frac141

8

ht2b32b2

1

t1b31 thorn t2b3

2

Ey000 throughout the thickness at a point midway between A and B if t2b2 gt t1b1


8

ht1b31b2

2

t1b31 thorn t2b3

2

Ey000 throughout the thickness at a point midway between C and D if t2b2 lt t1b1

etht1THORNmax frac14 tmaxGy0 at the surface on the thickest portion

8 Z-sectionK frac14

t3

3eth2b thorn hTHORN

Cw frac14th2b3

12

b thorn 2h

2b thorn h

ethsxTHORNmax frac14

hb

2

b thorn h

2b thorn hEy00 throughout the thickness at points A and D


4

b thorn h

2b thorn h

2

Ey000 throughout the thickness at a distancebethb thorn hTHORN

2b thorn hfrom point A


9 Segment of a circular

tube

(Note If t=r is small a can

be larger than p to

evaluate constants for

the case when the

walls overlap)

e frac14 2rsin a a cos aa sin a cos a

K frac14 23t3ra

Cw frac142tr5

3a3 6

ethsin a a cos aTHORN2

a sin a cos a

ethsxTHORNmax frac14 ethr2a re sin aTHORNEy00 throughout the thickness at points A and B

etht2THORNmax frac14 r2 eeth1 cos aTHORN ra2

2

Ey000 throughout the thickness at midlength


SEC107]

Torsion

415

TABLE 102 Formulas for torsional properties and stresses in thin-walled open cross sections (Continued)



10 e frac14 0707ab2 3a 2b

2a3 etha bTHORN3

K frac14 23t3etha thorn bTHORN

Cw frac14ta4b3

6

4a thorn 3b

2a3 etha bTHORN3

ethsxTHORNmax frac14a2b

2

2a2 thorn 3ab b2

2a3 etha bTHORN3Ey00 throughout the thickness at points A and E

t2 frac14a2b2

4

a2 2ab b2

2a3 etha bTHORN3Ey000 throughout the thickness at point C


11 K frac14 13eth4t3b thorn t3

waTHORN

Cw frac14a2b3t

3cos2 a

(Note Expressions are equally valid for thorn and a)

ethsxTHORNmax frac14ab

2cos aEy00 throughout the thickness at points A and C

etht2THORNmax frac14ab2

4cos aEy000 throughout the thickness at point B


416


[CHAP10

T

1shy ------shyI~---q1-Sq ---~

hJelaquotplo Uf) +ubo cuQdrocb de auVYlnio ef--r-ucJura d~ secoOl G ~ S em I lO b cVVl slt sor(e-~ Q uY elt FLXf 20 ton OYC1 n 1-- e

T - 12 fS ilaquoo f -m )c--tc-r m t c C~fxflD Lor -to n 1 e en cadct una

ce o~ cuo+ro pared~ dlt- dlcha tv bo ~upoticl)do ~ ~) UY esre~or UflfF0rfYC d~ OdYo6UY ~ h) dos poredes dc- QSOSCrrl

~ Q~ a-h-cs de o sOlti cm

So J L o~ 0) i)O(~C de epLscr Uf~ CCftrc ~~---~o If UY --~~ 6r~Q ~t IY -oca p or 0 ~ (ca

tgt U0 -codo es ~ b35 Om ~ B1- 5 erA) (S q1 1 CTA) 5f Crt g CYIJ_ (I)

pu~amp~o que el es )cpoundOr cS constOYlt c

e estuerzo cadofl--c ~ LOeb PQnd es bull

r-= T l 1-5 00 K91= -em

2-l Q( tl (0-06(111) (~j- qi ~Cfl2)

1 r=-S~13gt~- 1 -6-)

b) I u~ coo CSlk r J (Jr 0 b ~ f t ci (o cleJimi tudo por u Yt-o

1---q~S2 PI) i co do e~ ~ qcs L

Qm ~Jqb5~) lb~~IX~ - t) g 31 b (IY) L -l~)

I eSfgtettO ~V 0 coros cc es~SQ( b es r -L -= 1 + SOO V ~ - (f(Fshy

Zl QfY 1 (0 0~OVHs g3l1 b1IY(2)

~ =-nl bbb ~ eM

t I e~fuC((o cn os ltafQS de ~~r t1- esmiddot ~I- -= ji ~o lt~( -lto-___yv

2 hCYY 2 (tgt5a~c1)(Samp3~bcwf)

Y- tib3qo~ ~2

-JCf pb Dc protemo CH)~CYOI de~c(m fie e Q ~~u 0 centJ que ~ rClI letS poundCCCoYes l e ~ +C(C Uytu lot~11ud L2-YY)

Yv6dUO etci~~1 co ~ 100000 ~FtfmL ~ reCJdo~ ce OS~OY J 0 gtL(5

S()ULO~ ~ II ~CjuO ~c cletmiddotrMto con ~C1 ~~CltO( -~= ll

KG ()

(omoJ0 fI etc Ilt cs ) ( Cl coY So~0 -tcshy 2-hb (o - t1Y-Lb - -tS~shy

Q t2~ b-t -~ - t~

j ~ cs c mo dUo ~c ( a LO( --0 Y -c

G =-~ - 1-DOOOO Kc~ G 2bO I 2 L S 04 I ~ _ l3) 1(+v) - 2 ( + 6 ) 4 s) QY2

4

COYl porcd de CSfcsor

LCl co f S toY +- c K cs I

tlt= tmiddotI-(Q_)2 tb-tt-- 2 (a o~06)tlIOlb_ shy ~t~ l lCs s - Of ~o 9Y Qt i-bt- - 2 t2 0amp (O YQ6)- 635 (0YOb) - 2 (D4Db)lshy

ll1)

r aI~ vo e ~ ~ - fl - (11-S00 (Cj-cYt) (200(Y)

kG (1- ~ fl-i 4 tM~ (2 (0 - - )- o-~-~----221 ~ ~A()(

bJ TIJ 0 co~ par cd de C P CSDf JQ( 10-bc t O ~o5 LM L =-- O SO~ cY

Lo coYs~01tc k ~s

(- rz~2~ LO-~lY(h t~ = Uolto~)lO~oS)iIOb - QCQltltlb 3S - D30S)Z

0+2 +- b- - t~- -t ~ 10 (0 SO~) -- b 3S (cgt~o s) - (0 S02)2 - (O~3(h)2

k 5 6~ $ S CM 4 _ ~ )

1 df~0 smiddot

~ - n - (21- slt)O k-~ r -QY) (200 c~) _

KG l 5Go~5bc~4(1G0 12)Ol~ tltCjfcM2)

1 Jern po_ De--~ ml ese c mOVYU f -t-o rl) C)( tYO c e ~ or S 01 qlC

puc~c 0Pc(jr~c j c 6Vl~JO de ~((O c~ os COuffiICLS de Cone( cto qu c p Jc0e f co pgtr tal 0 vi ~s F-IJCzo ~ fYD d c

Ytltj - 0 k fjf Icyyen) 2 ~ Moi-i CI l 0 ~ t en~ utl tfOduo C~S+icD G0

1 ~ ~ 2 21 3 Sq Lj 3 b k~ Ft 2 ~ ( a lc u c 0 ( d c ( 0 ~SO( f O CfY t f

l~ So colvYV ra lt --ic(ci as siCJvic~+cs proPlcddUcf CjeOr1-C~ fCdS

Q) S6do oc=-40clY j be IO em ~ Lc~ 250 cVV

b) Sdhdo ~(AcVV ) 6( ~lScY j LL=~ eM

c) ~Ueca ct- 4Dc m) blt 40 C YV ) ef pcsor t= Scyy ) l- 250c ty

So ut ( Q (6du~o de rI ~ de 2 U L ex +o~+c es

C ~--L 2 23 oS q Lj)t iltjfc fyenL

2~ - v) 2(--042)l 2S0

I 1

Q) C()L fVYa cuodfnda I

pound estue Co mr1-oV+c 1l()C ) ((() es

t060lt

D TTh~ =- [) b6 T If) 2q

q1 1

e-spc JociD 1 de 0 cc (z)

I = iM~-lt ~ ~ )

060

I vo or de a e-s

0 - Qc - 20 C1t ~ L~) 2

SlJ S u ~ (YdD (4) cn (3)

T - (La tlt9Ecm 1) (~a c~3 o foO

[ IYlOYrcft-o m()l( 1YO qu Sd p=gtr-h Cl

co LI M ta e sshy

f ) 3 ~ S ~ ~ kSpound - c rfI J --- ~)

~udd-u~cY1co On ~ hz) cf) (IS)

1lt 32c (Ilt SC) l~ - 3 ~ (OlQ) ( - il lo q)~n SJs+ Ittl ~( n do (H)) ~ Y (tL)

~ ~ ilo bH o~~ K~f-() (lSOcm)

~S S~22b(Yl~)(qLI133 cug ~flYl)

c) CdUrlA 0 cuadra6a ~u cLO

pound eSfuc(W cor~Ov-L ~)(~(lIO cc de u60-pound I)(~ r _ Utl)

ltLq-t) lb--)

C OrYIO cs cuodt-oda Q=b CI es~e (ClW C-Ck b ce

r es fucreo cor-to f-tc cs ~)(~ T ~ 1 r-)(~ it la-t~2 _ ( C)

2-t (q-t)

(~dmiddot ~IJ~ C~~ os J()~r~ cc fx~ I t- ~ a C~ ( Q) = Uo (~( 02) (2)( 5ch) (40crI - SUllY

if =- 2 L SOO tlt~f - C-M I - 00)

f ~ro ~ detc(M ltJ be ~abu ~ ~ TL l2

KG

60nJc ( ~ =- ~ (q--tL - l(S)L (LO-sl l 2 4 3fS crlt~ ---l2~)

1 q~ - 21c- 2 (4D)lS)-L SYshy

~~middoth1U~et)OO L1o~CU)j us propc6od~ el l)

~ -= (12 SoO 1lt~~-cYV (1~CIV )

(114 ~1-Cc(Vl) (122330q~V~t(Jl~)

P- $ 4Q Xr rltXl ~ - O 0 ~l 0 I

pound ~ro sc d1-cfmfd de -ta6as cent -=- TL _ (amp)

K~

doYdc d tlOf de K c~ k~ ~ C2S 04 ~()

Sus --1 --J~crlCO ~) C( 8-) N =- 2 2 S (OcI)- lt =- ~O 000 ( iV1 -~)

lt(Jst~u~cndOCSt1) ~o~ ProP ccO-d~s CI(b)

cent (1 ~ )dH tltCjs -ernJ ( 1 $0lt kfI~ _

(3(OOOOctvl) Ut2) n)~OQ~ (3fcm2)

~=- L 003 )lt 03 rod ~ =o 0 ss 1-L~ ~

b) 001 - =cc-hfltJo- 016

E es~uc-czo co-r--o tc m~~f1D es de -Ct~CJgt ~2q--1

r~ t1 [1 +O (OIlS - O ~HS ~y-- V02 l)3t o ~ 06~)1-(9) QSteJOr~O de (q)

T ~ Sg lJ(j ___ 1 ___ _(6)

3 [H-QbOq~ O~~bS(y- L~On lty--QqD()(~~ ~ lo~ VoOfLS de Q ~b ~o

Cl Clc 2-shy

b4c-M ~

32 cm b = k 2

--shy

LSCfr)i

- lScrt -

lJ

~ =lLS~ _ 039 (lt) q 32C1V) - - U-lt

SV$ +- v ~cfld() U ) ) ~2) ~ c va Of ce cor-ton~c c ( (0

1 ~ (3211) (1 ~c( It) t(~amp em) - __ 3 [I ~ - - - -()~ (Os~)1()bQqS t~3q) -0 ~lt6b s(oslt)~_ gDlmiddot- COYl)3+

IT 103 bl+Os4~ ~9F-tM I -l)

r l ~ro ~c (L fer m 10 de +u 6~os ~ - n (4)

kG

~dc K t ~ ( - Q bS [~ _ ~ 1b hq t _~)l -LJs )

- 1 12Q ~

TABLE 101 Formulas for torsional deformation and stressGENERAL FORMULAS y frac14 TL=KG and t frac14 T=Q where y frac14 angle of twist (radians) T frac14 twisting moment (force-length) L frac14 length t frac14unit shear stress (force per unit area) G frac14 modulus of

rigidity (force per unit area) K (length to the fourth) and Q (length cubed) are functions of the cross section




1 Solid circular section K frac14 12pr4

tmax frac142T

pr3at boundary

2 Solid elliptical section K frac14pa3b3

a2 thorn b2tmax frac14

2T

pab2at ends of minor axis

3 Solid square section K frac14 225a4

tmax frac140601T


4 Solid rectangular section K frac14 ab3 16

3 336

b

a1

b4

12a4

for a5 b tmax frac14

3T

8ab21 thorn 06095

b

athorn 08865

b

a

2

18023b

a

3

thorn 09100b

a

4

at the midpoint of each longer side for a5 b

107 Tables

SEC107]

Torsion

401






a4ffiffiffi3

p

80tmax frac14

20T





factors)

For 23lt a=b lt

ffiffiffi3


K frac14a3b3

15a2 thorn 20b2



Forffiffiffi3

plt a=b lt 2

ffiffiffi3


K frac14 00915b4 a

b 08592



For 39 lt a lt 120

Q frac14K









ras follows

For 04h

r4 10

C frac14 07854 00333h

r 26183

h

r

2

thorn 41595h

r

3

30769h

r

4

thorn09299h

r

5

tmax frac14TB


h

r

as follows For 04h

r4 10


r 54897

h

r

2

thorn14062h

r

3

14510h

r

4

thorn 6434h

r

5


402


[CHAP10

8 Circular sector



as follows

For 014ap4 20

C frac14 00034 00697apthorn 05825

ap

2

02950ap

3

thorn 00874ap

4

00111ap

5

tmax frac14T


withap


B frac14 00117 02137apthorn 22475

ap

2

46709ap

3

thorn 51764ap

4

22000ap

5



flattened



ras follows


r408

C frac14 07854 04053h

r 35810

h

r

2

thorn 52708h

r

3

20772h

r

4


04h

r4 0293

C frac14 07854 07000h

r 77982

h

r

2

thorn 14578h

r

3

tmax frac14TB


h


04h

r4 06


r 11157

h

r

2

thorn 49568h

r

3

85886h

r

4

thorn 69849h

r

5


r4 0293


r 56147

h

r

2

thorn 30853h

r

3


SEC107]

Torsion

403








2Tro

pethr4o r4



pethD4 d4THORN

32C

where



thorn384n4


4l4

tmax frac1416TDF

pethD4 d4THORN

F frac14 1 thorn4n2

1 n2lthorn

32n2








pa3b3


where

q frac14ao

afrac14

bo

b


tmax frac142T









2tTHORN2ethb 1

2tTHORN2

Utaverage frac14

T


2tTHORN


404


[CHAP10






K frac144A2t

Utaverage frac14

T




4A2ETHdU=t










2


T


T


8gtgtgtltgtgtgt


are provided)

SEC107]

Torsion

405








K frac14 23prt3


4p2r2t2


radius)



shown dashed

K frac141

3Ut3 tmax frac14


U2t2



19ndash26)




axis of symmetry

K frac144Ix

1 thorn 16Ix=AU2





K frac14F


ethU

0

t3dU





K frac14A4

40J







mately by

tmax frac14 GyL

C or tmax frac14T

KC

where

C frac14D

1 thornp2D4

16A2

1 thorn 015p2D4

16A2

D

2r





high local stress

406


[CHAP10




006746s3 thorn 00515s4



s frac14m n

b

(Ref 11)



case 26



3 021

b

a1

b4

12a4

K2 frac14 cd3 1

3 0105

d

c1

d4

192c4

a frac14t

t1

015 thorn 010r

b


eth2r thorn bTHORN



D see case 26



3 021

b

a1

b4

12a4

K2 frac14 cd3 1

3 0105

d

c1

d4

192c4

a frac14d

b007 thorn 0076

r

b



p




tmax frac14 GyL

C or tmax frac14T

KC

where C frac14D

1 thornp2D4

16A2

1 thorn 0118 ln 1 D

2r

0238

D

2r

tanh

2fp






SEC107]

Torsion

407







as for case 24







3 021

b

a1

b4

12a4

K2 frac14 1

3cd3

a frac14t

t1

015 thorn 01r

b



ri

ro

as follows

For 024ri

ro

4 06


ri

ro

thorn K3

ri

ro

2

thorn K4

ri

ro

3


K1 frac14 04427 thorn 00064h

ri

00201h

ri

2

K2 frac14 08071 04047h

ri

thorn 01051h

ri

2

K3 frac14 00469 thorn 12063h

ri

03538h

ri

2

K4 frac14 05023 09618h

ri

thorn 03639h

ri

2

At M t frac14TB

r3o


ro

as follows

For 024ri

ro

406


ri

ro

thorn K3

ri

ro

2

thorn K4

ri

ro

3


K1 frac14 20014 01400h

ri

03231h

ri

3

K2 frac14 29047 thorn 30069h

ri

thorn 40500h

ri

2

K3 frac14 15721 65077h

ri

12496h

ri

2

K4 frac14 29553 thorn 41115h

ri

thorn 18845h

ri

2


408


[CHAP10


ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00848 thorn 01234a

b 00847

a

b

2

K3 frac14 04318 22000a

bthorn 07633

a

b

2

K4 frac14 00780 thorn 20618a

b 05234

a

b

2

At M t frac14TB


b

ras follows For 024

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3

where for 054a=b415

K1 frac14 11690 03168a

bthorn 00490

a

b

2

K2 frac14 043490 15096a

bthorn 08677

a

b

2

K3 frac14 11830 thorn 42764a

b 17024

a

b

2

K4 frac14 08812 02627a

b 01897

a

b

2



ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00795 thorn 01286a

b 01169

a

b

2

K3 frac14 14126 38589a

bthorn 13292

a

b

2

K4 frac14 07098 thorn 41936a

b 11053

a

b

2

At M t frac14TB


b

ras follows For 024

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3

where for 054a=b415

K1 frac14 12512 05406a

bthorn 00387

a

b

2

K2 frac14 09385 thorn 23450a

bthorn 03256

a

b

2

K3 frac14 72650 15338a

bthorn 31138

a

b

2

K4 frac14 11152 thorn 33710a

b 10007

a

b

2


SEC107]

Torsion

409







ras follows

For 04b

r4 04


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 01496 thorn 02773a

b 02110

a

b

2

K3 frac14 29138 82354a

bthorn 25782

a

b

2

K4 frac14 22991 thorn 12097a

b 22838

a

b

2

At M t frac14TB


b

ras follows For 024

b

r4 04


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 10434 thorn 10449a

b 02977

a

b

2

K2 frac14 00958 98401a

bthorn 16847

a

b

2

K3 frac14 15749 69650a

bthorn 14222

a

b

2

K4 frac14 35878 thorn 88696a

b 47545

a

b

2



ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00264 01187a

bthorn 00868

a

b

2

K3 frac14 02017 thorn 09019a

b 04947

a

b

2

K4 frac14 02911 14875a

bthorn 20651

a

b

2

At M t frac14TB


b

ras follows For 04

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 06366

K2 frac14 00023 thorn 00168a

bthorn 00093

a

b

2

K3 frac14 00052 thorn 00225a

b 03300

a

b

2

K4 frac14 00984 04936a

bthorn 02179

a

b

2


410


[CHAP10


ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00204 01307a

bthorn 01157

a

b

2

K3 frac14 02075 thorn 11544a

b 05937

a

b

2

K4 frac14 03608 22582a

bthorn 37336

a

b

2

At M t frac14TB


b

ras follows For 04

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 06366

K2 frac14 00069 00229a

bthorn 00637

a

b

2

K3 frac14 00675 thorn 03996a

b 10514

a

b

2

K4 frac14 03582 18324a

bthorn 15393

a

b

2



ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00595 03397a

bthorn 03239

a

b

2

K3 frac14 06008 thorn 31396a

b 20693

a

b

2

K4 frac14 10869 62451a

bthorn 94190

a

b

2

At M t frac14TB


b

ras follows For 04

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 06366

K2 frac14 00114 00789a

bthorn 01767

a

b

2

K3 frac14 01207 thorn 10291a

b 23589

a

b

2

K4 frac14 05132 34300a

bthorn 40226

a

b

2


SEC107]

Torsion

411







grooves


rover the range

04a


C frac14 07854 00225a

r 14154

a

r

2

thorn 09167a

r

3

For two grooves

C frac14 07854 00147a

r 30649

a

r

2

thorn 25453a

r

3

For four grooves

C frac14 07854 00409a

r 62371

a

r

2

thorn 72538a

r

3

At M t frac14TB


a

rover the

range 014a



r 27897

a

r

2

thorn 37092a

r

3

For two grooves


r 29501

a

r

2

thorn 70534a

r

3

For four grooves

B frac14 12135 29697a

rthorn 33713

a

r

2

99506a

r

3

thorn 13049a

r

4



sover the

range 04r

s4 09 as follows

C frac14 11266 03210r

sthorn 31519

r

s

2

14347r

s

3

thorn 15223r

s

4

47767r

s

5

At M t frac14BM T


r

sover the range 04

r

s4 05 as follows


s 09180

r

s

2

thorn 37335r

s

3

28686r

s

4

At N t frac14BN T


r

sover the range 034

r

s4 09 as follows


s 42520

r

s

2

thorn 10904r

s

3

13395r

s

4

thorn 66054r

s

5



412


[CHAP10




unit area)



1 Channele frac14

3b2

h thorn 6b

K frac14t3

3ethh thorn 2bTHORN

Cw frac14h2b3t

12

2h thorn 3b

h thorn 6b


2

h thorn 3b



4

h thorn 3b

h thorn 6b

2







1

K frac14t3


Cw frac14 th2b2

2b1 thorn

b

3 e

2eb1

bthorn

2b21

h

thornh2e2

2b thorn b1 thorn

h

6

2b21

h

thorn

2b31

3ethb thorn eTHORN2






b21

2ethb thorn eTHORN







1

K frac14t3


Cw frac14 th2b2

2b1 thorn

b

3 e

2eb1

b

2b21

h

thornh2e2

2b thorn b1 thorn

h

6thorn

2b21

h

thorn

2b31

3ethb thorn eTHORN2

sx frac14h



sx frac14h




b21

2ethb thorn eTHORN

hb1

2ethb eTHORN


hethb eTHORN

2ethb thorn eTHORN


t2 frac14b2

1

2ethb thorn eTHORN

hb1

2ethb eTHORN

h

4ethb eTHORN2




SEC107]

Torsion

413



4 Twin channel with


t3


Cw frac14tb2

24eth8b3



2b1 thorn

h

2



16eth4b2



5 Twin channel with


t3


Cw frac14tb2

24eth8b3





4

bb1

2



4

h

2 b1

2



b1 gth

21 thorn


2thorn

b

2h

r


4b2

1 hb

4 hb1


b1 lth

21 thorn


2thorn

b

2h

r


6 Wide flanged beam

with equal flanges


whTHORN

Cw frac14h2tb3

24






414


[CHAP10

7 Wide flanged beam


t1b31h

t1b31 thorn t2b3

2

K frac14 13etht3


whTHORN

Cw frac14h2t1t2b3

1b32


2THORN


2

t2b32

t1b31 thorn t2b3

2


1


2

t1b31

t1b31 thorn t2b3

2


1


8

ht2b32b2

1

t1b31 thorn t2b3

2



8

ht1b31b2

2

t1b31 thorn t2b3

2



8 Z-sectionK frac14

t3

3eth2b thorn hTHORN

Cw frac14th2b3

12

b thorn 2h

2b thorn h


hb

2

b thorn h



4

b thorn h

2b thorn h

2





tube


be larger than p to


the case when the

walls overlap)


K frac14 23t3ra

Cw frac142tr5

3a3 6


a sin a cos a



2



SEC107]

Torsion

415




10 e frac14 0707ab2 3a 2b

2a3 etha bTHORN3


Cw frac14ta4b3

6

4a thorn 3b

2a3 etha bTHORN3


2

2a2 thorn 3ab b2


t2 frac14a2b2

4

a2 2ab b2




waTHORN

Cw frac14a2b3t

3cos2 a







416


[CHAP10

T

1shy ------shyI~---q1-Sq ---~









r-= T l 1-5 00 K91= -em

2-l Q( tl (0-06(111) (~j- qi ~Cfl2)

1 r=-S~13gt~- 1 -6-)






~ =-nl bbb ~ eM



Y- tib3qo~ ~2




KG ()


Q t2~ b-t -~ - t~



4




ll1)


kG (1- ~ fl-i 4 tM~ (2 (0 - - )- o-~-~----221 ~ ~A()(


Lo coYs~01tc k ~s



k 5 6~ $ S CM 4 _ ~ )

1 df~0 smiddot

~ - n - (21- slt)O k-~ r -QY) (200 c~) _












2~ - v) 2(--042)l 2S0

I 1



t060lt

D TTh~ =- [) b6 T If) 2q

q1 1


I = iM~-lt ~ ~ )

060

I vo or de a e-s

0 - Qc - 20 C1t ~ L~) 2




co LI M ta e sshy

f ) 3 ~ S ~ ~ kSpound - c rfI J --- ~)







ltLq-t) lb--)



2-t (q-t)


if =- 2 L SOO tlt~f - C-M I - 00)


KG


1 q~ - 21c- 2 (4D)lS)-L SYshy


~ -= (12 SoO 1lt~~-cYV (1~CIV )

(114 ~1-Cc(Vl) (122330q~V~t(Jl~)

P- $ 4Q Xr rltXl ~ - O 0 ~l 0 I


K~






~=- L 003 )lt 03 rod ~ =o 0 ss 1-L~ ~

b) 001 - =cc-hfltJo- 016



T ~ Sg lJ(j ___ 1 ___ _(6)


Cl Clc 2-shy

b4c-M ~

32 cm b = k 2

--shy

LSCfr)i

- lScrt -

lJ

~ =lLS~ _ 039 (lt) q 32C1V) - - U-lt





kG


- 1 12Q ~






a4ffiffiffi3

p

80tmax frac14

20T





factors)

For 23lt a=b lt

ffiffiffi3


K frac14a3b3

15a2 thorn 20b2



Forffiffiffi3

plt a=b lt 2

ffiffiffi3


K frac14 00915b4 a

b 08592



For 39 lt a lt 120

Q frac14K









ras follows

For 04h

r4 10

C frac14 07854 00333h

r 26183

h

r

2

thorn 41595h

r

3

30769h

r

4

thorn09299h

r

5

tmax frac14TB


h

r

as follows For 04h

r4 10


r 54897

h

r

2

thorn14062h

r

3

14510h

r

4

thorn 6434h

r

5


402


[CHAP10

8 Circular sector



as follows

For 014ap4 20

C frac14 00034 00697apthorn 05825

ap

2

02950ap

3

thorn 00874ap

4

00111ap

5

tmax frac14T


withap


B frac14 00117 02137apthorn 22475

ap

2

46709ap

3

thorn 51764ap

4

22000ap

5



flattened



ras follows


r408

C frac14 07854 04053h

r 35810

h

r

2

thorn 52708h

r

3

20772h

r

4


04h

r4 0293

C frac14 07854 07000h

r 77982

h

r

2

thorn 14578h

r

3

tmax frac14TB


h


04h

r4 06


r 11157

h

r

2

thorn 49568h

r

3

85886h

r

4

thorn 69849h

r

5


r4 0293


r 56147

h

r

2

thorn 30853h

r

3


SEC107]

Torsion

403








2Tro

pethr4o r4



pethD4 d4THORN

32C

where



thorn384n4


4l4

tmax frac1416TDF

pethD4 d4THORN

F frac14 1 thorn4n2

1 n2lthorn

32n2








pa3b3


where

q frac14ao

afrac14

bo

b


tmax frac142T









2tTHORN2ethb 1

2tTHORN2

Utaverage frac14

T


2tTHORN


404


[CHAP10






K frac144A2t

Utaverage frac14

T




4A2ETHdU=t










2


T


T


8gtgtgtltgtgtgt


are provided)

SEC107]

Torsion

405








K frac14 23prt3


4p2r2t2


radius)



shown dashed

K frac141

3Ut3 tmax frac14


U2t2



19ndash26)




axis of symmetry

K frac144Ix

1 thorn 16Ix=AU2





K frac14F


ethU

0

t3dU





K frac14A4

40J







mately by

tmax frac14 GyL

C or tmax frac14T

KC

where

C frac14D

1 thornp2D4

16A2

1 thorn 015p2D4

16A2

D

2r





high local stress

406


[CHAP10




006746s3 thorn 00515s4



s frac14m n

b

(Ref 11)



case 26



3 021

b

a1

b4

12a4

K2 frac14 cd3 1

3 0105

d

c1

d4

192c4

a frac14t

t1

015 thorn 010r

b


eth2r thorn bTHORN



D see case 26



3 021

b

a1

b4

12a4

K2 frac14 cd3 1

3 0105

d

c1

d4

192c4

a frac14d

b007 thorn 0076

r

b



p




tmax frac14 GyL

C or tmax frac14T

KC

where C frac14D

1 thornp2D4

16A2

1 thorn 0118 ln 1 D

2r

0238

D

2r

tanh

2fp






SEC107]

Torsion

407







as for case 24







3 021

b

a1

b4

12a4

K2 frac14 1

3cd3

a frac14t

t1

015 thorn 01r

b



ri

ro

as follows

For 024ri

ro

4 06


ri

ro

thorn K3

ri

ro

2

thorn K4

ri

ro

3


K1 frac14 04427 thorn 00064h

ri

00201h

ri

2

K2 frac14 08071 04047h

ri

thorn 01051h

ri

2

K3 frac14 00469 thorn 12063h

ri

03538h

ri

2

K4 frac14 05023 09618h

ri

thorn 03639h

ri

2

At M t frac14TB

r3o


ro

as follows

For 024ri

ro

406


ri

ro

thorn K3

ri

ro

2

thorn K4

ri

ro

3


K1 frac14 20014 01400h

ri

03231h

ri

3

K2 frac14 29047 thorn 30069h

ri

thorn 40500h

ri

2

K3 frac14 15721 65077h

ri

12496h

ri

2

K4 frac14 29553 thorn 41115h

ri

thorn 18845h

ri

2


408


[CHAP10


ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00848 thorn 01234a

b 00847

a

b

2

K3 frac14 04318 22000a

bthorn 07633

a

b

2

K4 frac14 00780 thorn 20618a

b 05234

a

b

2

At M t frac14TB


b

ras follows For 024

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3

where for 054a=b415

K1 frac14 11690 03168a

bthorn 00490

a

b

2

K2 frac14 043490 15096a

bthorn 08677

a

b

2

K3 frac14 11830 thorn 42764a

b 17024

a

b

2

K4 frac14 08812 02627a

b 01897

a

b

2



ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00795 thorn 01286a

b 01169

a

b

2

K3 frac14 14126 38589a

bthorn 13292

a

b

2

K4 frac14 07098 thorn 41936a

b 11053

a

b

2

At M t frac14TB


b

ras follows For 024

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3

where for 054a=b415

K1 frac14 12512 05406a

bthorn 00387

a

b

2

K2 frac14 09385 thorn 23450a

bthorn 03256

a

b

2

K3 frac14 72650 15338a

bthorn 31138

a

b

2

K4 frac14 11152 thorn 33710a

b 10007

a

b

2


SEC107]

Torsion

409







ras follows

For 04b

r4 04


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 01496 thorn 02773a

b 02110

a

b

2

K3 frac14 29138 82354a

bthorn 25782

a

b

2

K4 frac14 22991 thorn 12097a

b 22838

a

b

2

At M t frac14TB


b

ras follows For 024

b

r4 04


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 10434 thorn 10449a

b 02977

a

b

2

K2 frac14 00958 98401a

bthorn 16847

a

b

2

K3 frac14 15749 69650a

bthorn 14222

a

b

2

K4 frac14 35878 thorn 88696a

b 47545

a

b

2



ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00264 01187a

bthorn 00868

a

b

2

K3 frac14 02017 thorn 09019a

b 04947

a

b

2

K4 frac14 02911 14875a

bthorn 20651

a

b

2

At M t frac14TB


b

ras follows For 04

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 06366

K2 frac14 00023 thorn 00168a

bthorn 00093

a

b

2

K3 frac14 00052 thorn 00225a

b 03300

a

b

2

K4 frac14 00984 04936a

bthorn 02179

a

b

2


410


[CHAP10


ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00204 01307a

bthorn 01157

a

b

2

K3 frac14 02075 thorn 11544a

b 05937

a

b

2

K4 frac14 03608 22582a

bthorn 37336

a

b

2

At M t frac14TB


b

ras follows For 04

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 06366

K2 frac14 00069 00229a

bthorn 00637

a

b

2

K3 frac14 00675 thorn 03996a

b 10514

a

b

2

K4 frac14 03582 18324a

bthorn 15393

a

b

2



ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00595 03397a

bthorn 03239

a

b

2

K3 frac14 06008 thorn 31396a

b 20693

a

b

2

K4 frac14 10869 62451a

bthorn 94190

a

b

2

At M t frac14TB


b

ras follows For 04

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 06366

K2 frac14 00114 00789a

bthorn 01767

a

b

2

K3 frac14 01207 thorn 10291a

b 23589

a

b

2

K4 frac14 05132 34300a

bthorn 40226

a

b

2


SEC107]

Torsion

411







grooves


rover the range

04a


C frac14 07854 00225a

r 14154

a

r

2

thorn 09167a

r

3

For two grooves

C frac14 07854 00147a

r 30649

a

r

2

thorn 25453a

r

3

For four grooves

C frac14 07854 00409a

r 62371

a

r

2

thorn 72538a

r

3

At M t frac14TB


a

rover the

range 014a



r 27897

a

r

2

thorn 37092a

r

3

For two grooves


r 29501

a

r

2

thorn 70534a

r

3

For four grooves

B frac14 12135 29697a

rthorn 33713

a

r

2

99506a

r

3

thorn 13049a

r

4



sover the

range 04r

s4 09 as follows

C frac14 11266 03210r

sthorn 31519

r

s

2

14347r

s

3

thorn 15223r

s

4

47767r

s

5

At M t frac14BM T


r

sover the range 04

r

s4 05 as follows


s 09180

r

s

2

thorn 37335r

s

3

28686r

s

4

At N t frac14BN T


r

sover the range 034

r

s4 09 as follows


s 42520

r

s

2

thorn 10904r

s

3

13395r

s

4

thorn 66054r

s

5



412


[CHAP10




unit area)



1 Channele frac14

3b2

h thorn 6b

K frac14t3

3ethh thorn 2bTHORN

Cw frac14h2b3t

12

2h thorn 3b

h thorn 6b


2

h thorn 3b



4

h thorn 3b

h thorn 6b

2







1

K frac14t3


Cw frac14 th2b2

2b1 thorn

b

3 e

2eb1

bthorn

2b21

h

thornh2e2

2b thorn b1 thorn

h

6

2b21

h

thorn

2b31

3ethb thorn eTHORN2






b21

2ethb thorn eTHORN







1

K frac14t3


Cw frac14 th2b2

2b1 thorn

b

3 e

2eb1

b

2b21

h

thornh2e2

2b thorn b1 thorn

h

6thorn

2b21

h

thorn

2b31

3ethb thorn eTHORN2

sx frac14h



sx frac14h




b21

2ethb thorn eTHORN

hb1

2ethb eTHORN


hethb eTHORN

2ethb thorn eTHORN


t2 frac14b2

1

2ethb thorn eTHORN

hb1

2ethb eTHORN

h

4ethb eTHORN2




SEC107]

Torsion

413



4 Twin channel with


t3


Cw frac14tb2

24eth8b3



2b1 thorn

h

2



16eth4b2



5 Twin channel with


t3


Cw frac14tb2

24eth8b3





4

bb1

2



4

h

2 b1

2



b1 gth

21 thorn


2thorn

b

2h

r


4b2

1 hb

4 hb1


b1 lth

21 thorn


2thorn

b

2h

r


6 Wide flanged beam

with equal flanges


whTHORN

Cw frac14h2tb3

24






414


[CHAP10

7 Wide flanged beam


t1b31h

t1b31 thorn t2b3

2

K frac14 13etht3


whTHORN

Cw frac14h2t1t2b3

1b32


2THORN


2

t2b32

t1b31 thorn t2b3

2


1


2

t1b31

t1b31 thorn t2b3

2


1


8

ht2b32b2

1

t1b31 thorn t2b3

2



8

ht1b31b2

2

t1b31 thorn t2b3

2



8 Z-sectionK frac14

t3

3eth2b thorn hTHORN

Cw frac14th2b3

12

b thorn 2h

2b thorn h


hb

2

b thorn h



4

b thorn h

2b thorn h

2





tube


be larger than p to


the case when the

walls overlap)


K frac14 23t3ra

Cw frac142tr5

3a3 6


a sin a cos a



2



SEC107]

Torsion

415




10 e frac14 0707ab2 3a 2b

2a3 etha bTHORN3


Cw frac14ta4b3

6

4a thorn 3b

2a3 etha bTHORN3


2

2a2 thorn 3ab b2


t2 frac14a2b2

4

a2 2ab b2




waTHORN

Cw frac14a2b3t

3cos2 a







416


[CHAP10

T

1shy ------shyI~---q1-Sq ---~









r-= T l 1-5 00 K91= -em

2-l Q( tl (0-06(111) (~j- qi ~Cfl2)

1 r=-S~13gt~- 1 -6-)






~ =-nl bbb ~ eM



Y- tib3qo~ ~2




KG ()


Q t2~ b-t -~ - t~



4




ll1)


kG (1- ~ fl-i 4 tM~ (2 (0 - - )- o-~-~----221 ~ ~A()(


Lo coYs~01tc k ~s



k 5 6~ $ S CM 4 _ ~ )

1 df~0 smiddot

~ - n - (21- slt)O k-~ r -QY) (200 c~) _












2~ - v) 2(--042)l 2S0

I 1



t060lt

D TTh~ =- [) b6 T If) 2q

q1 1


I = iM~-lt ~ ~ )

060

I vo or de a e-s

0 - Qc - 20 C1t ~ L~) 2




co LI M ta e sshy

f ) 3 ~ S ~ ~ kSpound - c rfI J --- ~)







ltLq-t) lb--)



2-t (q-t)


if =- 2 L SOO tlt~f - C-M I - 00)


KG


1 q~ - 21c- 2 (4D)lS)-L SYshy


~ -= (12 SoO 1lt~~-cYV (1~CIV )

(114 ~1-Cc(Vl) (122330q~V~t(Jl~)

P- $ 4Q Xr rltXl ~ - O 0 ~l 0 I


K~






~=- L 003 )lt 03 rod ~ =o 0 ss 1-L~ ~

b) 001 - =cc-hfltJo- 016



T ~ Sg lJ(j ___ 1 ___ _(6)


Cl Clc 2-shy

b4c-M ~

32 cm b = k 2

--shy

LSCfr)i

- lScrt -

lJ

~ =lLS~ _ 039 (lt) q 32C1V) - - U-lt





kG


- 1 12Q ~

8 Circular sector



as follows

For 014ap4 20

C frac14 00034 00697apthorn 05825

ap

2

02950ap

3

thorn 00874ap

4

00111ap

5

tmax frac14T


withap


B frac14 00117 02137apthorn 22475

ap

2

46709ap

3

thorn 51764ap

4

22000ap

5



flattened



ras follows


r408

C frac14 07854 04053h

r 35810

h

r

2

thorn 52708h

r

3

20772h

r

4


04h

r4 0293

C frac14 07854 07000h

r 77982

h

r

2

thorn 14578h

r

3

tmax frac14TB


h


04h

r4 06


r 11157

h

r

2

thorn 49568h

r

3

85886h

r

4

thorn 69849h

r

5


r4 0293


r 56147

h

r

2

thorn 30853h

r

3


SEC107]

Torsion

403








2Tro

pethr4o r4



pethD4 d4THORN

32C

where



thorn384n4


4l4

tmax frac1416TDF

pethD4 d4THORN

F frac14 1 thorn4n2

1 n2lthorn

32n2








pa3b3


where

q frac14ao

afrac14

bo

b


tmax frac142T









2tTHORN2ethb 1

2tTHORN2

Utaverage frac14

T


2tTHORN


404


[CHAP10






K frac144A2t

Utaverage frac14

T




4A2ETHdU=t










2


T


T


8gtgtgtltgtgtgt


are provided)

SEC107]

Torsion

405








K frac14 23prt3


4p2r2t2


radius)



shown dashed

K frac141

3Ut3 tmax frac14


U2t2



19ndash26)




axis of symmetry

K frac144Ix

1 thorn 16Ix=AU2





K frac14F


ethU

0

t3dU





K frac14A4

40J







mately by

tmax frac14 GyL

C or tmax frac14T

KC

where

C frac14D

1 thornp2D4

16A2

1 thorn 015p2D4

16A2

D

2r





high local stress

406


[CHAP10




006746s3 thorn 00515s4



s frac14m n

b

(Ref 11)



case 26



3 021

b

a1

b4

12a4

K2 frac14 cd3 1

3 0105

d

c1

d4

192c4

a frac14t

t1

015 thorn 010r

b


eth2r thorn bTHORN



D see case 26



3 021

b

a1

b4

12a4

K2 frac14 cd3 1

3 0105

d

c1

d4

192c4

a frac14d

b007 thorn 0076

r

b



p




tmax frac14 GyL

C or tmax frac14T

KC

where C frac14D

1 thornp2D4

16A2

1 thorn 0118 ln 1 D

2r

0238

D

2r

tanh

2fp






SEC107]

Torsion

407







as for case 24







3 021

b

a1

b4

12a4

K2 frac14 1

3cd3

a frac14t

t1

015 thorn 01r

b



ri

ro

as follows

For 024ri

ro

4 06


ri

ro

thorn K3

ri

ro

2

thorn K4

ri

ro

3


K1 frac14 04427 thorn 00064h

ri

00201h

ri

2

K2 frac14 08071 04047h

ri

thorn 01051h

ri

2

K3 frac14 00469 thorn 12063h

ri

03538h

ri

2

K4 frac14 05023 09618h

ri

thorn 03639h

ri

2

At M t frac14TB

r3o


ro

as follows

For 024ri

ro

406


ri

ro

thorn K3

ri

ro

2

thorn K4

ri

ro

3


K1 frac14 20014 01400h

ri

03231h

ri

3

K2 frac14 29047 thorn 30069h

ri

thorn 40500h

ri

2

K3 frac14 15721 65077h

ri

12496h

ri

2

K4 frac14 29553 thorn 41115h

ri

thorn 18845h

ri

2


408


[CHAP10


ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00848 thorn 01234a

b 00847

a

b

2

K3 frac14 04318 22000a

bthorn 07633

a

b

2

K4 frac14 00780 thorn 20618a

b 05234

a

b

2

At M t frac14TB


b

ras follows For 024

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3

where for 054a=b415

K1 frac14 11690 03168a

bthorn 00490

a

b

2

K2 frac14 043490 15096a

bthorn 08677

a

b

2

K3 frac14 11830 thorn 42764a

b 17024

a

b

2

K4 frac14 08812 02627a

b 01897

a

b

2



ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00795 thorn 01286a

b 01169

a

b

2

K3 frac14 14126 38589a

bthorn 13292

a

b

2

K4 frac14 07098 thorn 41936a

b 11053

a

b

2

At M t frac14TB


b

ras follows For 024

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3

where for 054a=b415

K1 frac14 12512 05406a

bthorn 00387

a

b

2

K2 frac14 09385 thorn 23450a

bthorn 03256

a

b

2

K3 frac14 72650 15338a

bthorn 31138

a

b

2

K4 frac14 11152 thorn 33710a

b 10007

a

b

2


SEC107]

Torsion

409







ras follows

For 04b

r4 04


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 01496 thorn 02773a

b 02110

a

b

2

K3 frac14 29138 82354a

bthorn 25782

a

b

2

K4 frac14 22991 thorn 12097a

b 22838

a

b

2

At M t frac14TB


b

ras follows For 024

b

r4 04


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 10434 thorn 10449a

b 02977

a

b

2

K2 frac14 00958 98401a

bthorn 16847

a

b

2

K3 frac14 15749 69650a

bthorn 14222

a

b

2

K4 frac14 35878 thorn 88696a

b 47545

a

b

2



ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00264 01187a

bthorn 00868

a

b

2

K3 frac14 02017 thorn 09019a

b 04947

a

b

2

K4 frac14 02911 14875a

bthorn 20651

a

b

2

At M t frac14TB


b

ras follows For 04

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 06366

K2 frac14 00023 thorn 00168a

bthorn 00093

a

b

2

K3 frac14 00052 thorn 00225a

b 03300

a

b

2

K4 frac14 00984 04936a

bthorn 02179

a

b

2


410


[CHAP10


ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00204 01307a

bthorn 01157

a

b

2

K3 frac14 02075 thorn 11544a

b 05937

a

b

2

K4 frac14 03608 22582a

bthorn 37336

a

b

2

At M t frac14TB


b

ras follows For 04

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 06366

K2 frac14 00069 00229a

bthorn 00637

a

b

2

K3 frac14 00675 thorn 03996a

b 10514

a

b

2

K4 frac14 03582 18324a

bthorn 15393

a

b

2



ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00595 03397a

bthorn 03239

a

b

2

K3 frac14 06008 thorn 31396a

b 20693

a

b

2

K4 frac14 10869 62451a

bthorn 94190

a

b

2

At M t frac14TB


b

ras follows For 04

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 06366

K2 frac14 00114 00789a

bthorn 01767

a

b

2

K3 frac14 01207 thorn 10291a

b 23589

a

b

2

K4 frac14 05132 34300a

bthorn 40226

a

b

2


SEC107]

Torsion

411







grooves


rover the range

04a


C frac14 07854 00225a

r 14154

a

r

2

thorn 09167a

r

3

For two grooves

C frac14 07854 00147a

r 30649

a

r

2

thorn 25453a

r

3

For four grooves

C frac14 07854 00409a

r 62371

a

r

2

thorn 72538a

r

3

At M t frac14TB


a

rover the

range 014a



r 27897

a

r

2

thorn 37092a

r

3

For two grooves


r 29501

a

r

2

thorn 70534a

r

3

For four grooves

B frac14 12135 29697a

rthorn 33713

a

r

2

99506a

r

3

thorn 13049a

r

4



sover the

range 04r

s4 09 as follows

C frac14 11266 03210r

sthorn 31519

r

s

2

14347r

s

3

thorn 15223r

s

4

47767r

s

5

At M t frac14BM T


r

sover the range 04

r

s4 05 as follows


s 09180

r

s

2

thorn 37335r

s

3

28686r

s

4

At N t frac14BN T


r

sover the range 034

r

s4 09 as follows


s 42520

r

s

2

thorn 10904r

s

3

13395r

s

4

thorn 66054r

s

5



412


[CHAP10




unit area)



1 Channele frac14

3b2

h thorn 6b

K frac14t3

3ethh thorn 2bTHORN

Cw frac14h2b3t

12

2h thorn 3b

h thorn 6b


2

h thorn 3b



4

h thorn 3b

h thorn 6b

2







1

K frac14t3


Cw frac14 th2b2

2b1 thorn

b

3 e

2eb1

bthorn

2b21

h

thornh2e2

2b thorn b1 thorn

h

6

2b21

h

thorn

2b31

3ethb thorn eTHORN2






b21

2ethb thorn eTHORN







1

K frac14t3


Cw frac14 th2b2

2b1 thorn

b

3 e

2eb1

b

2b21

h

thornh2e2

2b thorn b1 thorn

h

6thorn

2b21

h

thorn

2b31

3ethb thorn eTHORN2

sx frac14h



sx frac14h




b21

2ethb thorn eTHORN

hb1

2ethb eTHORN


hethb eTHORN

2ethb thorn eTHORN


t2 frac14b2

1

2ethb thorn eTHORN

hb1

2ethb eTHORN

h

4ethb eTHORN2




SEC107]

Torsion

413



4 Twin channel with


t3


Cw frac14tb2

24eth8b3



2b1 thorn

h

2



16eth4b2



5 Twin channel with


t3


Cw frac14tb2

24eth8b3





4

bb1

2



4

h

2 b1

2



b1 gth

21 thorn


2thorn

b

2h

r


4b2

1 hb

4 hb1


b1 lth

21 thorn


2thorn

b

2h

r


6 Wide flanged beam

with equal flanges


whTHORN

Cw frac14h2tb3

24






414


[CHAP10

7 Wide flanged beam


t1b31h

t1b31 thorn t2b3

2

K frac14 13etht3


whTHORN

Cw frac14h2t1t2b3

1b32


2THORN


2

t2b32

t1b31 thorn t2b3

2


1


2

t1b31

t1b31 thorn t2b3

2


1


8

ht2b32b2

1

t1b31 thorn t2b3

2



8

ht1b31b2

2

t1b31 thorn t2b3

2



8 Z-sectionK frac14

t3

3eth2b thorn hTHORN

Cw frac14th2b3

12

b thorn 2h

2b thorn h


hb

2

b thorn h



4

b thorn h

2b thorn h

2





tube


be larger than p to


the case when the

walls overlap)


K frac14 23t3ra

Cw frac142tr5

3a3 6


a sin a cos a



2



SEC107]

Torsion

415




10 e frac14 0707ab2 3a 2b

2a3 etha bTHORN3


Cw frac14ta4b3

6

4a thorn 3b

2a3 etha bTHORN3


2

2a2 thorn 3ab b2


t2 frac14a2b2

4

a2 2ab b2




waTHORN

Cw frac14a2b3t

3cos2 a







416


[CHAP10

T

1shy ------shyI~---q1-Sq ---~









r-= T l 1-5 00 K91= -em

2-l Q( tl (0-06(111) (~j- qi ~Cfl2)

1 r=-S~13gt~- 1 -6-)






~ =-nl bbb ~ eM



Y- tib3qo~ ~2




KG ()


Q t2~ b-t -~ - t~



4




ll1)


kG (1- ~ fl-i 4 tM~ (2 (0 - - )- o-~-~----221 ~ ~A()(


Lo coYs~01tc k ~s



k 5 6~ $ S CM 4 _ ~ )

1 df~0 smiddot

~ - n - (21- slt)O k-~ r -QY) (200 c~) _












2~ - v) 2(--042)l 2S0

I 1



t060lt

D TTh~ =- [) b6 T If) 2q

q1 1


I = iM~-lt ~ ~ )

060

I vo or de a e-s

0 - Qc - 20 C1t ~ L~) 2




co LI M ta e sshy

f ) 3 ~ S ~ ~ kSpound - c rfI J --- ~)







ltLq-t) lb--)



2-t (q-t)


if =- 2 L SOO tlt~f - C-M I - 00)


KG


1 q~ - 21c- 2 (4D)lS)-L SYshy


~ -= (12 SoO 1lt~~-cYV (1~CIV )

(114 ~1-Cc(Vl) (122330q~V~t(Jl~)

P- $ 4Q Xr rltXl ~ - O 0 ~l 0 I


K~






~=- L 003 )lt 03 rod ~ =o 0 ss 1-L~ ~

b) 001 - =cc-hfltJo- 016



T ~ Sg lJ(j ___ 1 ___ _(6)


Cl Clc 2-shy

b4c-M ~

32 cm b = k 2

--shy

LSCfr)i

- lScrt -

lJ

~ =lLS~ _ 039 (lt) q 32C1V) - - U-lt





kG


- 1 12Q ~







2Tro

pethr4o r4



pethD4 d4THORN

32C

where



thorn384n4


4l4

tmax frac1416TDF

pethD4 d4THORN

F frac14 1 thorn4n2

1 n2lthorn

32n2








pa3b3


where

q frac14ao

afrac14

bo

b


tmax frac142T









2tTHORN2ethb 1

2tTHORN2

Utaverage frac14

T


2tTHORN


404


[CHAP10






K frac144A2t

Utaverage frac14

T




4A2ETHdU=t










2


T


T


8gtgtgtltgtgtgt


are provided)

SEC107]

Torsion

405








K frac14 23prt3


4p2r2t2


radius)



shown dashed

K frac141

3Ut3 tmax frac14


U2t2



19ndash26)




axis of symmetry

K frac144Ix

1 thorn 16Ix=AU2





K frac14F


ethU

0

t3dU





K frac14A4

40J







mately by

tmax frac14 GyL

C or tmax frac14T

KC

where

C frac14D

1 thornp2D4

16A2

1 thorn 015p2D4

16A2

D

2r





high local stress

406


[CHAP10




006746s3 thorn 00515s4



s frac14m n

b

(Ref 11)



case 26



3 021

b

a1

b4

12a4

K2 frac14 cd3 1

3 0105

d

c1

d4

192c4

a frac14t

t1

015 thorn 010r

b


eth2r thorn bTHORN



D see case 26



3 021

b

a1

b4

12a4

K2 frac14 cd3 1

3 0105

d

c1

d4

192c4

a frac14d

b007 thorn 0076

r

b



p




tmax frac14 GyL

C or tmax frac14T

KC

where C frac14D

1 thornp2D4

16A2

1 thorn 0118 ln 1 D

2r

0238

D

2r

tanh

2fp






SEC107]

Torsion

407







as for case 24







3 021

b

a1

b4

12a4

K2 frac14 1

3cd3

a frac14t

t1

015 thorn 01r

b



ri

ro

as follows

For 024ri

ro

4 06


ri

ro

thorn K3

ri

ro

2

thorn K4

ri

ro

3


K1 frac14 04427 thorn 00064h

ri

00201h

ri

2

K2 frac14 08071 04047h

ri

thorn 01051h

ri

2

K3 frac14 00469 thorn 12063h

ri

03538h

ri

2

K4 frac14 05023 09618h

ri

thorn 03639h

ri

2

At M t frac14TB

r3o


ro

as follows

For 024ri

ro

406


ri

ro

thorn K3

ri

ro

2

thorn K4

ri

ro

3


K1 frac14 20014 01400h

ri

03231h

ri

3

K2 frac14 29047 thorn 30069h

ri

thorn 40500h

ri

2

K3 frac14 15721 65077h

ri

12496h

ri

2

K4 frac14 29553 thorn 41115h

ri

thorn 18845h

ri

2


408


[CHAP10


ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00848 thorn 01234a

b 00847

a

b

2

K3 frac14 04318 22000a

bthorn 07633

a

b

2

K4 frac14 00780 thorn 20618a

b 05234

a

b

2

At M t frac14TB


b

ras follows For 024

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3

where for 054a=b415

K1 frac14 11690 03168a

bthorn 00490

a

b

2

K2 frac14 043490 15096a

bthorn 08677

a

b

2

K3 frac14 11830 thorn 42764a

b 17024

a

b

2

K4 frac14 08812 02627a

b 01897

a

b

2



ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00795 thorn 01286a

b 01169

a

b

2

K3 frac14 14126 38589a

bthorn 13292

a

b

2

K4 frac14 07098 thorn 41936a

b 11053

a

b

2

At M t frac14TB


b

ras follows For 024

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3

where for 054a=b415

K1 frac14 12512 05406a

bthorn 00387

a

b

2

K2 frac14 09385 thorn 23450a

bthorn 03256

a

b

2

K3 frac14 72650 15338a

bthorn 31138

a

b

2

K4 frac14 11152 thorn 33710a

b 10007

a

b

2


SEC107]

Torsion

409







ras follows

For 04b

r4 04


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 01496 thorn 02773a

b 02110

a

b

2

K3 frac14 29138 82354a

bthorn 25782

a

b

2

K4 frac14 22991 thorn 12097a

b 22838

a

b

2

At M t frac14TB


b

ras follows For 024

b

r4 04


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 10434 thorn 10449a

b 02977

a

b

2

K2 frac14 00958 98401a

bthorn 16847

a

b

2

K3 frac14 15749 69650a

bthorn 14222

a

b

2

K4 frac14 35878 thorn 88696a

b 47545

a

b

2



ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00264 01187a

bthorn 00868

a

b

2

K3 frac14 02017 thorn 09019a

b 04947

a

b

2

K4 frac14 02911 14875a

bthorn 20651

a

b

2

At M t frac14TB


b

ras follows For 04

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 06366

K2 frac14 00023 thorn 00168a

bthorn 00093

a

b

2

K3 frac14 00052 thorn 00225a

b 03300

a

b

2

K4 frac14 00984 04936a

bthorn 02179

a

b

2


410


[CHAP10


ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00204 01307a

bthorn 01157

a

b

2

K3 frac14 02075 thorn 11544a

b 05937

a

b

2

K4 frac14 03608 22582a

bthorn 37336

a

b

2

At M t frac14TB


b

ras follows For 04

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 06366

K2 frac14 00069 00229a

bthorn 00637

a

b

2

K3 frac14 00675 thorn 03996a

b 10514

a

b

2

K4 frac14 03582 18324a

bthorn 15393

a

b

2



ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00595 03397a

bthorn 03239

a

b

2

K3 frac14 06008 thorn 31396a

b 20693

a

b

2

K4 frac14 10869 62451a

bthorn 94190

a

b

2

At M t frac14TB


b

ras follows For 04

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 06366

K2 frac14 00114 00789a

bthorn 01767

a

b

2

K3 frac14 01207 thorn 10291a

b 23589

a

b

2

K4 frac14 05132 34300a

bthorn 40226

a

b

2


SEC107]

Torsion

411







grooves


rover the range

04a


C frac14 07854 00225a

r 14154

a

r

2

thorn 09167a

r

3

For two grooves

C frac14 07854 00147a

r 30649

a

r

2

thorn 25453a

r

3

For four grooves

C frac14 07854 00409a

r 62371

a

r

2

thorn 72538a

r

3

At M t frac14TB


a

rover the

range 014a



r 27897

a

r

2

thorn 37092a

r

3

For two grooves


r 29501

a

r

2

thorn 70534a

r

3

For four grooves

B frac14 12135 29697a

rthorn 33713

a

r

2

99506a

r

3

thorn 13049a

r

4



sover the

range 04r

s4 09 as follows

C frac14 11266 03210r

sthorn 31519

r

s

2

14347r

s

3

thorn 15223r

s

4

47767r

s

5

At M t frac14BM T


r

sover the range 04

r

s4 05 as follows


s 09180

r

s

2

thorn 37335r

s

3

28686r

s

4

At N t frac14BN T


r

sover the range 034

r

s4 09 as follows


s 42520

r

s

2

thorn 10904r

s

3

13395r

s

4

thorn 66054r

s

5



412


[CHAP10




unit area)



1 Channele frac14

3b2

h thorn 6b

K frac14t3

3ethh thorn 2bTHORN

Cw frac14h2b3t

12

2h thorn 3b

h thorn 6b


2

h thorn 3b



4

h thorn 3b

h thorn 6b

2







1

K frac14t3


Cw frac14 th2b2

2b1 thorn

b

3 e

2eb1

bthorn

2b21

h

thornh2e2

2b thorn b1 thorn

h

6

2b21

h

thorn

2b31

3ethb thorn eTHORN2






b21

2ethb thorn eTHORN







1

K frac14t3


Cw frac14 th2b2

2b1 thorn

b

3 e

2eb1

b

2b21

h

thornh2e2

2b thorn b1 thorn

h

6thorn

2b21

h

thorn

2b31

3ethb thorn eTHORN2

sx frac14h



sx frac14h




b21

2ethb thorn eTHORN

hb1

2ethb eTHORN


hethb eTHORN

2ethb thorn eTHORN


t2 frac14b2

1

2ethb thorn eTHORN

hb1

2ethb eTHORN

h

4ethb eTHORN2




SEC107]

Torsion

413



4 Twin channel with


t3


Cw frac14tb2

24eth8b3



2b1 thorn

h

2



16eth4b2



5 Twin channel with


t3


Cw frac14tb2

24eth8b3





4

bb1

2



4

h

2 b1

2



b1 gth

21 thorn


2thorn

b

2h

r


4b2

1 hb

4 hb1


b1 lth

21 thorn


2thorn

b

2h

r


6 Wide flanged beam

with equal flanges


whTHORN

Cw frac14h2tb3

24






414


[CHAP10

7 Wide flanged beam


t1b31h

t1b31 thorn t2b3

2

K frac14 13etht3


whTHORN

Cw frac14h2t1t2b3

1b32


2THORN


2

t2b32

t1b31 thorn t2b3

2


1


2

t1b31

t1b31 thorn t2b3

2


1


8

ht2b32b2

1

t1b31 thorn t2b3

2



8

ht1b31b2

2

t1b31 thorn t2b3

2



8 Z-sectionK frac14

t3

3eth2b thorn hTHORN

Cw frac14th2b3

12

b thorn 2h

2b thorn h


hb

2

b thorn h



4

b thorn h

2b thorn h

2





tube


be larger than p to


the case when the

walls overlap)


K frac14 23t3ra

Cw frac142tr5

3a3 6


a sin a cos a



2



SEC107]

Torsion

415




10 e frac14 0707ab2 3a 2b

2a3 etha bTHORN3


Cw frac14ta4b3

6

4a thorn 3b

2a3 etha bTHORN3


2

2a2 thorn 3ab b2


t2 frac14a2b2

4

a2 2ab b2




waTHORN

Cw frac14a2b3t

3cos2 a







416


[CHAP10

T

1shy ------shyI~---q1-Sq ---~









r-= T l 1-5 00 K91= -em

2-l Q( tl (0-06(111) (~j- qi ~Cfl2)

1 r=-S~13gt~- 1 -6-)






~ =-nl bbb ~ eM



Y- tib3qo~ ~2




KG ()


Q t2~ b-t -~ - t~



4




ll1)


kG (1- ~ fl-i 4 tM~ (2 (0 - - )- o-~-~----221 ~ ~A()(


Lo coYs~01tc k ~s



k 5 6~ $ S CM 4 _ ~ )

1 df~0 smiddot

~ - n - (21- slt)O k-~ r -QY) (200 c~) _












2~ - v) 2(--042)l 2S0

I 1



t060lt

D TTh~ =- [) b6 T If) 2q

q1 1


I = iM~-lt ~ ~ )

060

I vo or de a e-s

0 - Qc - 20 C1t ~ L~) 2




co LI M ta e sshy

f ) 3 ~ S ~ ~ kSpound - c rfI J --- ~)







ltLq-t) lb--)



2-t (q-t)


if =- 2 L SOO tlt~f - C-M I - 00)


KG


1 q~ - 21c- 2 (4D)lS)-L SYshy


~ -= (12 SoO 1lt~~-cYV (1~CIV )

(114 ~1-Cc(Vl) (122330q~V~t(Jl~)

P- $ 4Q Xr rltXl ~ - O 0 ~l 0 I


K~






~=- L 003 )lt 03 rod ~ =o 0 ss 1-L~ ~

b) 001 - =cc-hfltJo- 016



T ~ Sg lJ(j ___ 1 ___ _(6)


Cl Clc 2-shy

b4c-M ~

32 cm b = k 2

--shy

LSCfr)i

- lScrt -

lJ

~ =lLS~ _ 039 (lt) q 32C1V) - - U-lt





kG


- 1 12Q ~






K frac144A2t

Utaverage frac14

T




4A2ETHdU=t










2


T


T


8gtgtgtltgtgtgt


are provided)

SEC107]

Torsion

405








K frac14 23prt3


4p2r2t2


radius)



shown dashed

K frac141

3Ut3 tmax frac14


U2t2



19ndash26)




axis of symmetry

K frac144Ix

1 thorn 16Ix=AU2





K frac14F


ethU

0

t3dU





K frac14A4

40J







mately by

tmax frac14 GyL

C or tmax frac14T

KC

where

C frac14D

1 thornp2D4

16A2

1 thorn 015p2D4

16A2

D

2r





high local stress

406


[CHAP10




006746s3 thorn 00515s4



s frac14m n

b

(Ref 11)



case 26



3 021

b

a1

b4

12a4

K2 frac14 cd3 1

3 0105

d

c1

d4

192c4

a frac14t

t1

015 thorn 010r

b


eth2r thorn bTHORN



D see case 26



3 021

b

a1

b4

12a4

K2 frac14 cd3 1

3 0105

d

c1

d4

192c4

a frac14d

b007 thorn 0076

r

b



p




tmax frac14 GyL

C or tmax frac14T

KC

where C frac14D

1 thornp2D4

16A2

1 thorn 0118 ln 1 D

2r

0238

D

2r

tanh

2fp






SEC107]

Torsion

407







as for case 24







3 021

b

a1

b4

12a4

K2 frac14 1

3cd3

a frac14t

t1

015 thorn 01r

b



ri

ro

as follows

For 024ri

ro

4 06


ri

ro

thorn K3

ri

ro

2

thorn K4

ri

ro

3


K1 frac14 04427 thorn 00064h

ri

00201h

ri

2

K2 frac14 08071 04047h

ri

thorn 01051h

ri

2

K3 frac14 00469 thorn 12063h

ri

03538h

ri

2

K4 frac14 05023 09618h

ri

thorn 03639h

ri

2

At M t frac14TB

r3o


ro

as follows

For 024ri

ro

406


ri

ro

thorn K3

ri

ro

2

thorn K4

ri

ro

3


K1 frac14 20014 01400h

ri

03231h

ri

3

K2 frac14 29047 thorn 30069h

ri

thorn 40500h

ri

2

K3 frac14 15721 65077h

ri

12496h

ri

2

K4 frac14 29553 thorn 41115h

ri

thorn 18845h

ri

2


408


[CHAP10


ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00848 thorn 01234a

b 00847

a

b

2

K3 frac14 04318 22000a

bthorn 07633

a

b

2

K4 frac14 00780 thorn 20618a

b 05234

a

b

2

At M t frac14TB


b

ras follows For 024

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3

where for 054a=b415

K1 frac14 11690 03168a

bthorn 00490

a

b

2

K2 frac14 043490 15096a

bthorn 08677

a

b

2

K3 frac14 11830 thorn 42764a

b 17024

a

b

2

K4 frac14 08812 02627a

b 01897

a

b

2



ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00795 thorn 01286a

b 01169

a

b

2

K3 frac14 14126 38589a

bthorn 13292

a

b

2

K4 frac14 07098 thorn 41936a

b 11053

a

b

2

At M t frac14TB


b

ras follows For 024

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3

where for 054a=b415

K1 frac14 12512 05406a

bthorn 00387

a

b

2

K2 frac14 09385 thorn 23450a

bthorn 03256

a

b

2

K3 frac14 72650 15338a

bthorn 31138

a

b

2

K4 frac14 11152 thorn 33710a

b 10007

a

b

2


SEC107]

Torsion

409







ras follows

For 04b

r4 04


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 01496 thorn 02773a

b 02110

a

b

2

K3 frac14 29138 82354a

bthorn 25782

a

b

2

K4 frac14 22991 thorn 12097a

b 22838

a

b

2

At M t frac14TB


b

ras follows For 024

b

r4 04


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 10434 thorn 10449a

b 02977

a

b

2

K2 frac14 00958 98401a

bthorn 16847

a

b

2

K3 frac14 15749 69650a

bthorn 14222

a

b

2

K4 frac14 35878 thorn 88696a

b 47545

a

b

2



ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00264 01187a

bthorn 00868

a

b

2

K3 frac14 02017 thorn 09019a

b 04947

a

b

2

K4 frac14 02911 14875a

bthorn 20651

a

b

2

At M t frac14TB


b

ras follows For 04

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 06366

K2 frac14 00023 thorn 00168a

bthorn 00093

a

b

2

K3 frac14 00052 thorn 00225a

b 03300

a

b

2

K4 frac14 00984 04936a

bthorn 02179

a

b

2


410


[CHAP10


ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00204 01307a

bthorn 01157

a

b

2

K3 frac14 02075 thorn 11544a

b 05937

a

b

2

K4 frac14 03608 22582a

bthorn 37336

a

b

2

At M t frac14TB


b

ras follows For 04

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 06366

K2 frac14 00069 00229a

bthorn 00637

a

b

2

K3 frac14 00675 thorn 03996a

b 10514

a

b

2

K4 frac14 03582 18324a

bthorn 15393

a

b

2



ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00595 03397a

bthorn 03239

a

b

2

K3 frac14 06008 thorn 31396a

b 20693

a

b

2

K4 frac14 10869 62451a

bthorn 94190

a

b

2

At M t frac14TB


b

ras follows For 04

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 06366

K2 frac14 00114 00789a

bthorn 01767

a

b

2

K3 frac14 01207 thorn 10291a

b 23589

a

b

2

K4 frac14 05132 34300a

bthorn 40226

a

b

2


SEC107]

Torsion

411







grooves


rover the range

04a


C frac14 07854 00225a

r 14154

a

r

2

thorn 09167a

r

3

For two grooves

C frac14 07854 00147a

r 30649

a

r

2

thorn 25453a

r

3

For four grooves

C frac14 07854 00409a

r 62371

a

r

2

thorn 72538a

r

3

At M t frac14TB


a

rover the

range 014a



r 27897

a

r

2

thorn 37092a

r

3

For two grooves


r 29501

a

r

2

thorn 70534a

r

3

For four grooves

B frac14 12135 29697a

rthorn 33713

a

r

2

99506a

r

3

thorn 13049a

r

4



sover the

range 04r

s4 09 as follows

C frac14 11266 03210r

sthorn 31519

r

s

2

14347r

s

3

thorn 15223r

s

4

47767r

s

5

At M t frac14BM T


r

sover the range 04

r

s4 05 as follows


s 09180

r

s

2

thorn 37335r

s

3

28686r

s

4

At N t frac14BN T


r

sover the range 034

r

s4 09 as follows


s 42520

r

s

2

thorn 10904r

s

3

13395r

s

4

thorn 66054r

s

5



412


[CHAP10




unit area)



1 Channele frac14

3b2

h thorn 6b

K frac14t3

3ethh thorn 2bTHORN

Cw frac14h2b3t

12

2h thorn 3b

h thorn 6b


2

h thorn 3b



4

h thorn 3b

h thorn 6b

2







1

K frac14t3


Cw frac14 th2b2

2b1 thorn

b

3 e

2eb1

bthorn

2b21

h

thornh2e2

2b thorn b1 thorn

h

6

2b21

h

thorn

2b31

3ethb thorn eTHORN2






b21

2ethb thorn eTHORN







1

K frac14t3


Cw frac14 th2b2

2b1 thorn

b

3 e

2eb1

b

2b21

h

thornh2e2

2b thorn b1 thorn

h

6thorn

2b21

h

thorn

2b31

3ethb thorn eTHORN2

sx frac14h



sx frac14h




b21

2ethb thorn eTHORN

hb1

2ethb eTHORN


hethb eTHORN

2ethb thorn eTHORN


t2 frac14b2

1

2ethb thorn eTHORN

hb1

2ethb eTHORN

h

4ethb eTHORN2




SEC107]

Torsion

413



4 Twin channel with


t3


Cw frac14tb2

24eth8b3



2b1 thorn

h

2



16eth4b2



5 Twin channel with


t3


Cw frac14tb2

24eth8b3





4

bb1

2



4

h

2 b1

2



b1 gth

21 thorn


2thorn

b

2h

r


4b2

1 hb

4 hb1


b1 lth

21 thorn


2thorn

b

2h

r


6 Wide flanged beam

with equal flanges


whTHORN

Cw frac14h2tb3

24






414


[CHAP10

7 Wide flanged beam


t1b31h

t1b31 thorn t2b3

2

K frac14 13etht3


whTHORN

Cw frac14h2t1t2b3

1b32


2THORN


2

t2b32

t1b31 thorn t2b3

2


1


2

t1b31

t1b31 thorn t2b3

2


1


8

ht2b32b2

1

t1b31 thorn t2b3

2



8

ht1b31b2

2

t1b31 thorn t2b3

2



8 Z-sectionK frac14

t3

3eth2b thorn hTHORN

Cw frac14th2b3

12

b thorn 2h

2b thorn h


hb

2

b thorn h



4

b thorn h

2b thorn h

2





tube


be larger than p to


the case when the

walls overlap)


K frac14 23t3ra

Cw frac142tr5

3a3 6


a sin a cos a



2



SEC107]

Torsion

415




10 e frac14 0707ab2 3a 2b

2a3 etha bTHORN3


Cw frac14ta4b3

6

4a thorn 3b

2a3 etha bTHORN3


2

2a2 thorn 3ab b2


t2 frac14a2b2

4

a2 2ab b2




waTHORN

Cw frac14a2b3t

3cos2 a







416


[CHAP10

T

1shy ------shyI~---q1-Sq ---~









r-= T l 1-5 00 K91= -em

2-l Q( tl (0-06(111) (~j- qi ~Cfl2)

1 r=-S~13gt~- 1 -6-)






~ =-nl bbb ~ eM



Y- tib3qo~ ~2




KG ()


Q t2~ b-t -~ - t~



4




ll1)


kG (1- ~ fl-i 4 tM~ (2 (0 - - )- o-~-~----221 ~ ~A()(


Lo coYs~01tc k ~s



k 5 6~ $ S CM 4 _ ~ )

1 df~0 smiddot

~ - n - (21- slt)O k-~ r -QY) (200 c~) _












2~ - v) 2(--042)l 2S0

I 1



t060lt

D TTh~ =- [) b6 T If) 2q

q1 1


I = iM~-lt ~ ~ )

060

I vo or de a e-s

0 - Qc - 20 C1t ~ L~) 2




co LI M ta e sshy

f ) 3 ~ S ~ ~ kSpound - c rfI J --- ~)







ltLq-t) lb--)



2-t (q-t)


if =- 2 L SOO tlt~f - C-M I - 00)


KG


1 q~ - 21c- 2 (4D)lS)-L SYshy


~ -= (12 SoO 1lt~~-cYV (1~CIV )

(114 ~1-Cc(Vl) (122330q~V~t(Jl~)

P- $ 4Q Xr rltXl ~ - O 0 ~l 0 I


K~






~=- L 003 )lt 03 rod ~ =o 0 ss 1-L~ ~

b) 001 - =cc-hfltJo- 016



T ~ Sg lJ(j ___ 1 ___ _(6)


Cl Clc 2-shy

b4c-M ~

32 cm b = k 2

--shy

LSCfr)i

- lScrt -

lJ

~ =lLS~ _ 039 (lt) q 32C1V) - - U-lt





kG


- 1 12Q ~







K frac14 23prt3


4p2r2t2


radius)



shown dashed

K frac141

3Ut3 tmax frac14


U2t2



19ndash26)




axis of symmetry

K frac144Ix

1 thorn 16Ix=AU2





K frac14F


ethU

0

t3dU





K frac14A4

40J







mately by

tmax frac14 GyL

C or tmax frac14T

KC

where

C frac14D

1 thornp2D4

16A2

1 thorn 015p2D4

16A2

D

2r





high local stress

406


[CHAP10




006746s3 thorn 00515s4



s frac14m n

b

(Ref 11)



case 26



3 021

b

a1

b4

12a4

K2 frac14 cd3 1

3 0105

d

c1

d4

192c4

a frac14t

t1

015 thorn 010r

b


eth2r thorn bTHORN



D see case 26



3 021

b

a1

b4

12a4

K2 frac14 cd3 1

3 0105

d

c1

d4

192c4

a frac14d

b007 thorn 0076

r

b



p




tmax frac14 GyL

C or tmax frac14T

KC

where C frac14D

1 thornp2D4

16A2

1 thorn 0118 ln 1 D

2r

0238

D

2r

tanh

2fp






SEC107]

Torsion

407







as for case 24







3 021

b

a1

b4

12a4

K2 frac14 1

3cd3

a frac14t

t1

015 thorn 01r

b



ri

ro

as follows

For 024ri

ro

4 06


ri

ro

thorn K3

ri

ro

2

thorn K4

ri

ro

3


K1 frac14 04427 thorn 00064h

ri

00201h

ri

2

K2 frac14 08071 04047h

ri

thorn 01051h

ri

2

K3 frac14 00469 thorn 12063h

ri

03538h

ri

2

K4 frac14 05023 09618h

ri

thorn 03639h

ri

2

At M t frac14TB

r3o


ro

as follows

For 024ri

ro

406


ri

ro

thorn K3

ri

ro

2

thorn K4

ri

ro

3


K1 frac14 20014 01400h

ri

03231h

ri

3

K2 frac14 29047 thorn 30069h

ri

thorn 40500h

ri

2

K3 frac14 15721 65077h

ri

12496h

ri

2

K4 frac14 29553 thorn 41115h

ri

thorn 18845h

ri

2


408


[CHAP10


ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00848 thorn 01234a

b 00847

a

b

2

K3 frac14 04318 22000a

bthorn 07633

a

b

2

K4 frac14 00780 thorn 20618a

b 05234

a

b

2

At M t frac14TB


b

ras follows For 024

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3

where for 054a=b415

K1 frac14 11690 03168a

bthorn 00490

a

b

2

K2 frac14 043490 15096a

bthorn 08677

a

b

2

K3 frac14 11830 thorn 42764a

b 17024

a

b

2

K4 frac14 08812 02627a

b 01897

a

b

2



ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00795 thorn 01286a

b 01169

a

b

2

K3 frac14 14126 38589a

bthorn 13292

a

b

2

K4 frac14 07098 thorn 41936a

b 11053

a

b

2

At M t frac14TB


b

ras follows For 024

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3

where for 054a=b415

K1 frac14 12512 05406a

bthorn 00387

a

b

2

K2 frac14 09385 thorn 23450a

bthorn 03256

a

b

2

K3 frac14 72650 15338a

bthorn 31138

a

b

2

K4 frac14 11152 thorn 33710a

b 10007

a

b

2


SEC107]

Torsion

409







ras follows

For 04b

r4 04


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 01496 thorn 02773a

b 02110

a

b

2

K3 frac14 29138 82354a

bthorn 25782

a

b

2

K4 frac14 22991 thorn 12097a

b 22838

a

b

2

At M t frac14TB


b

ras follows For 024

b

r4 04


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 10434 thorn 10449a

b 02977

a

b

2

K2 frac14 00958 98401a

bthorn 16847

a

b

2

K3 frac14 15749 69650a

bthorn 14222

a

b

2

K4 frac14 35878 thorn 88696a

b 47545

a

b

2



ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00264 01187a

bthorn 00868

a

b

2

K3 frac14 02017 thorn 09019a

b 04947

a

b

2

K4 frac14 02911 14875a

bthorn 20651

a

b

2

At M t frac14TB


b

ras follows For 04

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 06366

K2 frac14 00023 thorn 00168a

bthorn 00093

a

b

2

K3 frac14 00052 thorn 00225a

b 03300

a

b

2

K4 frac14 00984 04936a

bthorn 02179

a

b

2


410


[CHAP10


ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00204 01307a

bthorn 01157

a

b

2

K3 frac14 02075 thorn 11544a

b 05937

a

b

2

K4 frac14 03608 22582a

bthorn 37336

a

b

2

At M t frac14TB


b

ras follows For 04

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 06366

K2 frac14 00069 00229a

bthorn 00637

a

b

2

K3 frac14 00675 thorn 03996a

b 10514

a

b

2

K4 frac14 03582 18324a

bthorn 15393

a

b

2



ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00595 03397a

bthorn 03239

a

b

2

K3 frac14 06008 thorn 31396a

b 20693

a

b

2

K4 frac14 10869 62451a

bthorn 94190

a

b

2

At M t frac14TB


b

ras follows For 04

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 06366

K2 frac14 00114 00789a

bthorn 01767

a

b

2

K3 frac14 01207 thorn 10291a

b 23589

a

b

2

K4 frac14 05132 34300a

bthorn 40226

a

b

2


SEC107]

Torsion

411







grooves


rover the range

04a


C frac14 07854 00225a

r 14154

a

r

2

thorn 09167a

r

3

For two grooves

C frac14 07854 00147a

r 30649

a

r

2

thorn 25453a

r

3

For four grooves

C frac14 07854 00409a

r 62371

a

r

2

thorn 72538a

r

3

At M t frac14TB


a

rover the

range 014a



r 27897

a

r

2

thorn 37092a

r

3

For two grooves


r 29501

a

r

2

thorn 70534a

r

3

For four grooves

B frac14 12135 29697a

rthorn 33713

a

r

2

99506a

r

3

thorn 13049a

r

4



sover the

range 04r

s4 09 as follows

C frac14 11266 03210r

sthorn 31519

r

s

2

14347r

s

3

thorn 15223r

s

4

47767r

s

5

At M t frac14BM T


r

sover the range 04

r

s4 05 as follows


s 09180

r

s

2

thorn 37335r

s

3

28686r

s

4

At N t frac14BN T


r

sover the range 034

r

s4 09 as follows


s 42520

r

s

2

thorn 10904r

s

3

13395r

s

4

thorn 66054r

s

5



412


[CHAP10




unit area)



1 Channele frac14

3b2

h thorn 6b

K frac14t3

3ethh thorn 2bTHORN

Cw frac14h2b3t

12

2h thorn 3b

h thorn 6b


2

h thorn 3b



4

h thorn 3b

h thorn 6b

2







1

K frac14t3


Cw frac14 th2b2

2b1 thorn

b

3 e

2eb1

bthorn

2b21

h

thornh2e2

2b thorn b1 thorn

h

6

2b21

h

thorn

2b31

3ethb thorn eTHORN2






b21

2ethb thorn eTHORN







1

K frac14t3


Cw frac14 th2b2

2b1 thorn

b

3 e

2eb1

b

2b21

h

thornh2e2

2b thorn b1 thorn

h

6thorn

2b21

h

thorn

2b31

3ethb thorn eTHORN2

sx frac14h



sx frac14h




b21

2ethb thorn eTHORN

hb1

2ethb eTHORN


hethb eTHORN

2ethb thorn eTHORN


t2 frac14b2

1

2ethb thorn eTHORN

hb1

2ethb eTHORN

h

4ethb eTHORN2




SEC107]

Torsion

413



4 Twin channel with


t3


Cw frac14tb2

24eth8b3



2b1 thorn

h

2



16eth4b2



5 Twin channel with


t3


Cw frac14tb2

24eth8b3





4

bb1

2



4

h

2 b1

2



b1 gth

21 thorn


2thorn

b

2h

r


4b2

1 hb

4 hb1


b1 lth

21 thorn


2thorn

b

2h

r


6 Wide flanged beam

with equal flanges


whTHORN

Cw frac14h2tb3

24






414


[CHAP10

7 Wide flanged beam


t1b31h

t1b31 thorn t2b3

2

K frac14 13etht3


whTHORN

Cw frac14h2t1t2b3

1b32


2THORN


2

t2b32

t1b31 thorn t2b3

2


1


2

t1b31

t1b31 thorn t2b3

2


1


8

ht2b32b2

1

t1b31 thorn t2b3

2



8

ht1b31b2

2

t1b31 thorn t2b3

2



8 Z-sectionK frac14

t3

3eth2b thorn hTHORN

Cw frac14th2b3

12

b thorn 2h

2b thorn h


hb

2

b thorn h



4

b thorn h

2b thorn h

2





tube


be larger than p to


the case when the

walls overlap)


K frac14 23t3ra

Cw frac142tr5

3a3 6


a sin a cos a



2



SEC107]

Torsion

415




10 e frac14 0707ab2 3a 2b

2a3 etha bTHORN3


Cw frac14ta4b3

6

4a thorn 3b

2a3 etha bTHORN3


2

2a2 thorn 3ab b2


t2 frac14a2b2

4

a2 2ab b2




waTHORN

Cw frac14a2b3t

3cos2 a







416


[CHAP10

T

1shy ------shyI~---q1-Sq ---~









r-= T l 1-5 00 K91= -em

2-l Q( tl (0-06(111) (~j- qi ~Cfl2)

1 r=-S~13gt~- 1 -6-)






~ =-nl bbb ~ eM



Y- tib3qo~ ~2




KG ()


Q t2~ b-t -~ - t~



4




ll1)


kG (1- ~ fl-i 4 tM~ (2 (0 - - )- o-~-~----221 ~ ~A()(


Lo coYs~01tc k ~s



k 5 6~ $ S CM 4 _ ~ )

1 df~0 smiddot

~ - n - (21- slt)O k-~ r -QY) (200 c~) _












2~ - v) 2(--042)l 2S0

I 1



t060lt

D TTh~ =- [) b6 T If) 2q

q1 1


I = iM~-lt ~ ~ )

060

I vo or de a e-s

0 - Qc - 20 C1t ~ L~) 2




co LI M ta e sshy

f ) 3 ~ S ~ ~ kSpound - c rfI J --- ~)







ltLq-t) lb--)



2-t (q-t)


if =- 2 L SOO tlt~f - C-M I - 00)


KG


1 q~ - 21c- 2 (4D)lS)-L SYshy


~ -= (12 SoO 1lt~~-cYV (1~CIV )

(114 ~1-Cc(Vl) (122330q~V~t(Jl~)

P- $ 4Q Xr rltXl ~ - O 0 ~l 0 I


K~






~=- L 003 )lt 03 rod ~ =o 0 ss 1-L~ ~

b) 001 - =cc-hfltJo- 016



T ~ Sg lJ(j ___ 1 ___ _(6)


Cl Clc 2-shy

b4c-M ~

32 cm b = k 2

--shy

LSCfr)i

- lScrt -

lJ

~ =lLS~ _ 039 (lt) q 32C1V) - - U-lt





kG


- 1 12Q ~




006746s3 thorn 00515s4



s frac14m n

b

(Ref 11)



case 26



3 021

b

a1

b4

12a4

K2 frac14 cd3 1

3 0105

d

c1

d4

192c4

a frac14t

t1

015 thorn 010r

b


eth2r thorn bTHORN



D see case 26



3 021

b

a1

b4

12a4

K2 frac14 cd3 1

3 0105

d

c1

d4

192c4

a frac14d

b007 thorn 0076

r

b



p




tmax frac14 GyL

C or tmax frac14T

KC

where C frac14D

1 thornp2D4

16A2

1 thorn 0118 ln 1 D

2r

0238

D

2r

tanh

2fp






SEC107]

Torsion

407







as for case 24







3 021

b

a1

b4

12a4

K2 frac14 1

3cd3

a frac14t

t1

015 thorn 01r

b



ri

ro

as follows

For 024ri

ro

4 06


ri

ro

thorn K3

ri

ro

2

thorn K4

ri

ro

3


K1 frac14 04427 thorn 00064h

ri

00201h

ri

2

K2 frac14 08071 04047h

ri

thorn 01051h

ri

2

K3 frac14 00469 thorn 12063h

ri

03538h

ri

2

K4 frac14 05023 09618h

ri

thorn 03639h

ri

2

At M t frac14TB

r3o


ro

as follows

For 024ri

ro

406


ri

ro

thorn K3

ri

ro

2

thorn K4

ri

ro

3


K1 frac14 20014 01400h

ri

03231h

ri

3

K2 frac14 29047 thorn 30069h

ri

thorn 40500h

ri

2

K3 frac14 15721 65077h

ri

12496h

ri

2

K4 frac14 29553 thorn 41115h

ri

thorn 18845h

ri

2


408


[CHAP10


ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00848 thorn 01234a

b 00847

a

b

2

K3 frac14 04318 22000a

bthorn 07633

a

b

2

K4 frac14 00780 thorn 20618a

b 05234

a

b

2

At M t frac14TB


b

ras follows For 024

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3

where for 054a=b415

K1 frac14 11690 03168a

bthorn 00490

a

b

2

K2 frac14 043490 15096a

bthorn 08677

a

b

2

K3 frac14 11830 thorn 42764a

b 17024

a

b

2

K4 frac14 08812 02627a

b 01897

a

b

2



ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00795 thorn 01286a

b 01169

a

b

2

K3 frac14 14126 38589a

bthorn 13292

a

b

2

K4 frac14 07098 thorn 41936a

b 11053

a

b

2

At M t frac14TB


b

ras follows For 024

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3

where for 054a=b415

K1 frac14 12512 05406a

bthorn 00387

a

b

2

K2 frac14 09385 thorn 23450a

bthorn 03256

a

b

2

K3 frac14 72650 15338a

bthorn 31138

a

b

2

K4 frac14 11152 thorn 33710a

b 10007

a

b

2


SEC107]

Torsion

409







ras follows

For 04b

r4 04


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 01496 thorn 02773a

b 02110

a

b

2

K3 frac14 29138 82354a

bthorn 25782

a

b

2

K4 frac14 22991 thorn 12097a

b 22838

a

b

2

At M t frac14TB


b

ras follows For 024

b

r4 04


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 10434 thorn 10449a

b 02977

a

b

2

K2 frac14 00958 98401a

bthorn 16847

a

b

2

K3 frac14 15749 69650a

bthorn 14222

a

b

2

K4 frac14 35878 thorn 88696a

b 47545

a

b

2



ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00264 01187a

bthorn 00868

a

b

2

K3 frac14 02017 thorn 09019a

b 04947

a

b

2

K4 frac14 02911 14875a

bthorn 20651

a

b

2

At M t frac14TB


b

ras follows For 04

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 06366

K2 frac14 00023 thorn 00168a

bthorn 00093

a

b

2

K3 frac14 00052 thorn 00225a

b 03300

a

b

2

K4 frac14 00984 04936a

bthorn 02179

a

b

2


410


[CHAP10


ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00204 01307a

bthorn 01157

a

b

2

K3 frac14 02075 thorn 11544a

b 05937

a

b

2

K4 frac14 03608 22582a

bthorn 37336

a

b

2

At M t frac14TB


b

ras follows For 04

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 06366

K2 frac14 00069 00229a

bthorn 00637

a

b

2

K3 frac14 00675 thorn 03996a

b 10514

a

b

2

K4 frac14 03582 18324a

bthorn 15393

a

b

2



ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00595 03397a

bthorn 03239

a

b

2

K3 frac14 06008 thorn 31396a

b 20693

a

b

2

K4 frac14 10869 62451a

bthorn 94190

a

b

2

At M t frac14TB


b

ras follows For 04

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 06366

K2 frac14 00114 00789a

bthorn 01767

a

b

2

K3 frac14 01207 thorn 10291a

b 23589

a

b

2

K4 frac14 05132 34300a

bthorn 40226

a

b

2


SEC107]

Torsion

411







grooves


rover the range

04a


C frac14 07854 00225a

r 14154

a

r

2

thorn 09167a

r

3

For two grooves

C frac14 07854 00147a

r 30649

a

r

2

thorn 25453a

r

3

For four grooves

C frac14 07854 00409a

r 62371

a

r

2

thorn 72538a

r

3

At M t frac14TB


a

rover the

range 014a



r 27897

a

r

2

thorn 37092a

r

3

For two grooves


r 29501

a

r

2

thorn 70534a

r

3

For four grooves

B frac14 12135 29697a

rthorn 33713

a

r

2

99506a

r

3

thorn 13049a

r

4



sover the

range 04r

s4 09 as follows

C frac14 11266 03210r

sthorn 31519

r

s

2

14347r

s

3

thorn 15223r

s

4

47767r

s

5

At M t frac14BM T


r

sover the range 04

r

s4 05 as follows


s 09180

r

s

2

thorn 37335r

s

3

28686r

s

4

At N t frac14BN T


r

sover the range 034

r

s4 09 as follows


s 42520

r

s

2

thorn 10904r

s

3

13395r

s

4

thorn 66054r

s

5



412


[CHAP10




unit area)



1 Channele frac14

3b2

h thorn 6b

K frac14t3

3ethh thorn 2bTHORN

Cw frac14h2b3t

12

2h thorn 3b

h thorn 6b


2

h thorn 3b



4

h thorn 3b

h thorn 6b

2







1

K frac14t3


Cw frac14 th2b2

2b1 thorn

b

3 e

2eb1

bthorn

2b21

h

thornh2e2

2b thorn b1 thorn

h

6

2b21

h

thorn

2b31

3ethb thorn eTHORN2






b21

2ethb thorn eTHORN







1

K frac14t3


Cw frac14 th2b2

2b1 thorn

b

3 e

2eb1

b

2b21

h

thornh2e2

2b thorn b1 thorn

h

6thorn

2b21

h

thorn

2b31

3ethb thorn eTHORN2

sx frac14h



sx frac14h




b21

2ethb thorn eTHORN

hb1

2ethb eTHORN


hethb eTHORN

2ethb thorn eTHORN


t2 frac14b2

1

2ethb thorn eTHORN

hb1

2ethb eTHORN

h

4ethb eTHORN2




SEC107]

Torsion

413



4 Twin channel with


t3


Cw frac14tb2

24eth8b3



2b1 thorn

h

2



16eth4b2



5 Twin channel with


t3


Cw frac14tb2

24eth8b3





4

bb1

2



4

h

2 b1

2



b1 gth

21 thorn


2thorn

b

2h

r


4b2

1 hb

4 hb1


b1 lth

21 thorn


2thorn

b

2h

r


6 Wide flanged beam

with equal flanges


whTHORN

Cw frac14h2tb3

24






414


[CHAP10

7 Wide flanged beam


t1b31h

t1b31 thorn t2b3

2

K frac14 13etht3


whTHORN

Cw frac14h2t1t2b3

1b32


2THORN


2

t2b32

t1b31 thorn t2b3

2


1


2

t1b31

t1b31 thorn t2b3

2


1


8

ht2b32b2

1

t1b31 thorn t2b3

2



8

ht1b31b2

2

t1b31 thorn t2b3

2



8 Z-sectionK frac14

t3

3eth2b thorn hTHORN

Cw frac14th2b3

12

b thorn 2h

2b thorn h


hb

2

b thorn h



4

b thorn h

2b thorn h

2





tube


be larger than p to


the case when the

walls overlap)


K frac14 23t3ra

Cw frac142tr5

3a3 6


a sin a cos a



2



SEC107]

Torsion

415




10 e frac14 0707ab2 3a 2b

2a3 etha bTHORN3


Cw frac14ta4b3

6

4a thorn 3b

2a3 etha bTHORN3


2

2a2 thorn 3ab b2


t2 frac14a2b2

4

a2 2ab b2




waTHORN

Cw frac14a2b3t

3cos2 a







416


[CHAP10

T

1shy ------shyI~---q1-Sq ---~









r-= T l 1-5 00 K91= -em

2-l Q( tl (0-06(111) (~j- qi ~Cfl2)

1 r=-S~13gt~- 1 -6-)






~ =-nl bbb ~ eM



Y- tib3qo~ ~2




KG ()


Q t2~ b-t -~ - t~



4




ll1)


kG (1- ~ fl-i 4 tM~ (2 (0 - - )- o-~-~----221 ~ ~A()(


Lo coYs~01tc k ~s



k 5 6~ $ S CM 4 _ ~ )

1 df~0 smiddot

~ - n - (21- slt)O k-~ r -QY) (200 c~) _












2~ - v) 2(--042)l 2S0

I 1



t060lt

D TTh~ =- [) b6 T If) 2q

q1 1


I = iM~-lt ~ ~ )

060

I vo or de a e-s

0 - Qc - 20 C1t ~ L~) 2




co LI M ta e sshy

f ) 3 ~ S ~ ~ kSpound - c rfI J --- ~)







ltLq-t) lb--)



2-t (q-t)


if =- 2 L SOO tlt~f - C-M I - 00)


KG


1 q~ - 21c- 2 (4D)lS)-L SYshy


~ -= (12 SoO 1lt~~-cYV (1~CIV )

(114 ~1-Cc(Vl) (122330q~V~t(Jl~)

P- $ 4Q Xr rltXl ~ - O 0 ~l 0 I


K~






~=- L 003 )lt 03 rod ~ =o 0 ss 1-L~ ~

b) 001 - =cc-hfltJo- 016



T ~ Sg lJ(j ___ 1 ___ _(6)


Cl Clc 2-shy

b4c-M ~

32 cm b = k 2

--shy

LSCfr)i

- lScrt -

lJ

~ =lLS~ _ 039 (lt) q 32C1V) - - U-lt





kG


- 1 12Q ~






as for case 24







3 021

b

a1

b4

12a4

K2 frac14 1

3cd3

a frac14t

t1

015 thorn 01r

b



ri

ro

as follows

For 024ri

ro

4 06


ri

ro

thorn K3

ri

ro

2

thorn K4

ri

ro

3


K1 frac14 04427 thorn 00064h

ri

00201h

ri

2

K2 frac14 08071 04047h

ri

thorn 01051h

ri

2

K3 frac14 00469 thorn 12063h

ri

03538h

ri

2

K4 frac14 05023 09618h

ri

thorn 03639h

ri

2

At M t frac14TB

r3o


ro

as follows

For 024ri

ro

406


ri

ro

thorn K3

ri

ro

2

thorn K4

ri

ro

3


K1 frac14 20014 01400h

ri

03231h

ri

3

K2 frac14 29047 thorn 30069h

ri

thorn 40500h

ri

2

K3 frac14 15721 65077h

ri

12496h

ri

2

K4 frac14 29553 thorn 41115h

ri

thorn 18845h

ri

2


408


[CHAP10


ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00848 thorn 01234a

b 00847

a

b

2

K3 frac14 04318 22000a

bthorn 07633

a

b

2

K4 frac14 00780 thorn 20618a

b 05234

a

b

2

At M t frac14TB


b

ras follows For 024

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3

where for 054a=b415

K1 frac14 11690 03168a

bthorn 00490

a

b

2

K2 frac14 043490 15096a

bthorn 08677

a

b

2

K3 frac14 11830 thorn 42764a

b 17024

a

b

2

K4 frac14 08812 02627a

b 01897

a

b

2



ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00795 thorn 01286a

b 01169

a

b

2

K3 frac14 14126 38589a

bthorn 13292

a

b

2

K4 frac14 07098 thorn 41936a

b 11053

a

b

2

At M t frac14TB


b

ras follows For 024

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3

where for 054a=b415

K1 frac14 12512 05406a

bthorn 00387

a

b

2

K2 frac14 09385 thorn 23450a

bthorn 03256

a

b

2

K3 frac14 72650 15338a

bthorn 31138

a

b

2

K4 frac14 11152 thorn 33710a

b 10007

a

b

2


SEC107]

Torsion

409







ras follows

For 04b

r4 04


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 01496 thorn 02773a

b 02110

a

b

2

K3 frac14 29138 82354a

bthorn 25782

a

b

2

K4 frac14 22991 thorn 12097a

b 22838

a

b

2

At M t frac14TB


b

ras follows For 024

b

r4 04


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 10434 thorn 10449a

b 02977

a

b

2

K2 frac14 00958 98401a

bthorn 16847

a

b

2

K3 frac14 15749 69650a

bthorn 14222

a

b

2

K4 frac14 35878 thorn 88696a

b 47545

a

b

2



ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00264 01187a

bthorn 00868

a

b

2

K3 frac14 02017 thorn 09019a

b 04947

a

b

2

K4 frac14 02911 14875a

bthorn 20651

a

b

2

At M t frac14TB


b

ras follows For 04

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 06366

K2 frac14 00023 thorn 00168a

bthorn 00093

a

b

2

K3 frac14 00052 thorn 00225a

b 03300

a

b

2

K4 frac14 00984 04936a

bthorn 02179

a

b

2


410


[CHAP10


ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00204 01307a

bthorn 01157

a

b

2

K3 frac14 02075 thorn 11544a

b 05937

a

b

2

K4 frac14 03608 22582a

bthorn 37336

a

b

2

At M t frac14TB


b

ras follows For 04

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 06366

K2 frac14 00069 00229a

bthorn 00637

a

b

2

K3 frac14 00675 thorn 03996a

b 10514

a

b

2

K4 frac14 03582 18324a

bthorn 15393

a

b

2



ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00595 03397a

bthorn 03239

a

b

2

K3 frac14 06008 thorn 31396a

b 20693

a

b

2

K4 frac14 10869 62451a

bthorn 94190

a

b

2

At M t frac14TB


b

ras follows For 04

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 06366

K2 frac14 00114 00789a

bthorn 01767

a

b

2

K3 frac14 01207 thorn 10291a

b 23589

a

b

2

K4 frac14 05132 34300a

bthorn 40226

a

b

2


SEC107]

Torsion

411







grooves


rover the range

04a


C frac14 07854 00225a

r 14154

a

r

2

thorn 09167a

r

3

For two grooves

C frac14 07854 00147a

r 30649

a

r

2

thorn 25453a

r

3

For four grooves

C frac14 07854 00409a

r 62371

a

r

2

thorn 72538a

r

3

At M t frac14TB


a

rover the

range 014a



r 27897

a

r

2

thorn 37092a

r

3

For two grooves


r 29501

a

r

2

thorn 70534a

r

3

For four grooves

B frac14 12135 29697a

rthorn 33713

a

r

2

99506a

r

3

thorn 13049a

r

4



sover the

range 04r

s4 09 as follows

C frac14 11266 03210r

sthorn 31519

r

s

2

14347r

s

3

thorn 15223r

s

4

47767r

s

5

At M t frac14BM T


r

sover the range 04

r

s4 05 as follows


s 09180

r

s

2

thorn 37335r

s

3

28686r

s

4

At N t frac14BN T


r

sover the range 034

r

s4 09 as follows


s 42520

r

s

2

thorn 10904r

s

3

13395r

s

4

thorn 66054r

s

5



412


[CHAP10




unit area)



1 Channele frac14

3b2

h thorn 6b

K frac14t3

3ethh thorn 2bTHORN

Cw frac14h2b3t

12

2h thorn 3b

h thorn 6b


2

h thorn 3b



4

h thorn 3b

h thorn 6b

2







1

K frac14t3


Cw frac14 th2b2

2b1 thorn

b

3 e

2eb1

bthorn

2b21

h

thornh2e2

2b thorn b1 thorn

h

6

2b21

h

thorn

2b31

3ethb thorn eTHORN2






b21

2ethb thorn eTHORN







1

K frac14t3


Cw frac14 th2b2

2b1 thorn

b

3 e

2eb1

b

2b21

h

thornh2e2

2b thorn b1 thorn

h

6thorn

2b21

h

thorn

2b31

3ethb thorn eTHORN2

sx frac14h



sx frac14h




b21

2ethb thorn eTHORN

hb1

2ethb eTHORN


hethb eTHORN

2ethb thorn eTHORN


t2 frac14b2

1

2ethb thorn eTHORN

hb1

2ethb eTHORN

h

4ethb eTHORN2




SEC107]

Torsion

413



4 Twin channel with


t3


Cw frac14tb2

24eth8b3



2b1 thorn

h

2



16eth4b2



5 Twin channel with


t3


Cw frac14tb2

24eth8b3





4

bb1

2



4

h

2 b1

2



b1 gth

21 thorn


2thorn

b

2h

r


4b2

1 hb

4 hb1


b1 lth

21 thorn


2thorn

b

2h

r


6 Wide flanged beam

with equal flanges


whTHORN

Cw frac14h2tb3

24






414


[CHAP10

7 Wide flanged beam


t1b31h

t1b31 thorn t2b3

2

K frac14 13etht3


whTHORN

Cw frac14h2t1t2b3

1b32


2THORN


2

t2b32

t1b31 thorn t2b3

2


1


2

t1b31

t1b31 thorn t2b3

2


1


8

ht2b32b2

1

t1b31 thorn t2b3

2



8

ht1b31b2

2

t1b31 thorn t2b3

2



8 Z-sectionK frac14

t3

3eth2b thorn hTHORN

Cw frac14th2b3

12

b thorn 2h

2b thorn h


hb

2

b thorn h



4

b thorn h

2b thorn h

2





tube


be larger than p to


the case when the

walls overlap)


K frac14 23t3ra

Cw frac142tr5

3a3 6


a sin a cos a



2



SEC107]

Torsion

415




10 e frac14 0707ab2 3a 2b

2a3 etha bTHORN3


Cw frac14ta4b3

6

4a thorn 3b

2a3 etha bTHORN3


2

2a2 thorn 3ab b2


t2 frac14a2b2

4

a2 2ab b2




waTHORN

Cw frac14a2b3t

3cos2 a







416


[CHAP10

T

1shy ------shyI~---q1-Sq ---~









r-= T l 1-5 00 K91= -em

2-l Q( tl (0-06(111) (~j- qi ~Cfl2)

1 r=-S~13gt~- 1 -6-)






~ =-nl bbb ~ eM



Y- tib3qo~ ~2




KG ()


Q t2~ b-t -~ - t~



4




ll1)


kG (1- ~ fl-i 4 tM~ (2 (0 - - )- o-~-~----221 ~ ~A()(


Lo coYs~01tc k ~s



k 5 6~ $ S CM 4 _ ~ )

1 df~0 smiddot

~ - n - (21- slt)O k-~ r -QY) (200 c~) _












2~ - v) 2(--042)l 2S0

I 1



t060lt

D TTh~ =- [) b6 T If) 2q

q1 1


I = iM~-lt ~ ~ )

060

I vo or de a e-s

0 - Qc - 20 C1t ~ L~) 2




co LI M ta e sshy

f ) 3 ~ S ~ ~ kSpound - c rfI J --- ~)







ltLq-t) lb--)



2-t (q-t)


if =- 2 L SOO tlt~f - C-M I - 00)


KG


1 q~ - 21c- 2 (4D)lS)-L SYshy


~ -= (12 SoO 1lt~~-cYV (1~CIV )

(114 ~1-Cc(Vl) (122330q~V~t(Jl~)

P- $ 4Q Xr rltXl ~ - O 0 ~l 0 I


K~






~=- L 003 )lt 03 rod ~ =o 0 ss 1-L~ ~

b) 001 - =cc-hfltJo- 016



T ~ Sg lJ(j ___ 1 ___ _(6)


Cl Clc 2-shy

b4c-M ~

32 cm b = k 2

--shy

LSCfr)i

- lScrt -

lJ

~ =lLS~ _ 039 (lt) q 32C1V) - - U-lt





kG


- 1 12Q ~


ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00848 thorn 01234a

b 00847

a

b

2

K3 frac14 04318 22000a

bthorn 07633

a

b

2

K4 frac14 00780 thorn 20618a

b 05234

a

b

2

At M t frac14TB


b

ras follows For 024

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3

where for 054a=b415

K1 frac14 11690 03168a

bthorn 00490

a

b

2

K2 frac14 043490 15096a

bthorn 08677

a

b

2

K3 frac14 11830 thorn 42764a

b 17024

a

b

2

K4 frac14 08812 02627a

b 01897

a

b

2



ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00795 thorn 01286a

b 01169

a

b

2

K3 frac14 14126 38589a

bthorn 13292

a

b

2

K4 frac14 07098 thorn 41936a

b 11053

a

b

2

At M t frac14TB


b

ras follows For 024

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3

where for 054a=b415

K1 frac14 12512 05406a

bthorn 00387

a

b

2

K2 frac14 09385 thorn 23450a

bthorn 03256

a

b

2

K3 frac14 72650 15338a

bthorn 31138

a

b

2

K4 frac14 11152 thorn 33710a

b 10007

a

b

2


SEC107]

Torsion

409







ras follows

For 04b

r4 04


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 01496 thorn 02773a

b 02110

a

b

2

K3 frac14 29138 82354a

bthorn 25782

a

b

2

K4 frac14 22991 thorn 12097a

b 22838

a

b

2

At M t frac14TB


b

ras follows For 024

b

r4 04


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 10434 thorn 10449a

b 02977

a

b

2

K2 frac14 00958 98401a

bthorn 16847

a

b

2

K3 frac14 15749 69650a

bthorn 14222

a

b

2

K4 frac14 35878 thorn 88696a

b 47545

a

b

2



ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00264 01187a

bthorn 00868

a

b

2

K3 frac14 02017 thorn 09019a

b 04947

a

b

2

K4 frac14 02911 14875a

bthorn 20651

a

b

2

At M t frac14TB


b

ras follows For 04

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 06366

K2 frac14 00023 thorn 00168a

bthorn 00093

a

b

2

K3 frac14 00052 thorn 00225a

b 03300

a

b

2

K4 frac14 00984 04936a

bthorn 02179

a

b

2


410


[CHAP10


ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00204 01307a

bthorn 01157

a

b

2

K3 frac14 02075 thorn 11544a

b 05937

a

b

2

K4 frac14 03608 22582a

bthorn 37336

a

b

2

At M t frac14TB


b

ras follows For 04

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 06366

K2 frac14 00069 00229a

bthorn 00637

a

b

2

K3 frac14 00675 thorn 03996a

b 10514

a

b

2

K4 frac14 03582 18324a

bthorn 15393

a

b

2



ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00595 03397a

bthorn 03239

a

b

2

K3 frac14 06008 thorn 31396a

b 20693

a

b

2

K4 frac14 10869 62451a

bthorn 94190

a

b

2

At M t frac14TB


b

ras follows For 04

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 06366

K2 frac14 00114 00789a

bthorn 01767

a

b

2

K3 frac14 01207 thorn 10291a

b 23589

a

b

2

K4 frac14 05132 34300a

bthorn 40226

a

b

2


SEC107]

Torsion

411







grooves


rover the range

04a


C frac14 07854 00225a

r 14154

a

r

2

thorn 09167a

r

3

For two grooves

C frac14 07854 00147a

r 30649

a

r

2

thorn 25453a

r

3

For four grooves

C frac14 07854 00409a

r 62371

a

r

2

thorn 72538a

r

3

At M t frac14TB


a

rover the

range 014a



r 27897

a

r

2

thorn 37092a

r

3

For two grooves


r 29501

a

r

2

thorn 70534a

r

3

For four grooves

B frac14 12135 29697a

rthorn 33713

a

r

2

99506a

r

3

thorn 13049a

r

4



sover the

range 04r

s4 09 as follows

C frac14 11266 03210r

sthorn 31519

r

s

2

14347r

s

3

thorn 15223r

s

4

47767r

s

5

At M t frac14BM T


r

sover the range 04

r

s4 05 as follows


s 09180

r

s

2

thorn 37335r

s

3

28686r

s

4

At N t frac14BN T


r

sover the range 034

r

s4 09 as follows


s 42520

r

s

2

thorn 10904r

s

3

13395r

s

4

thorn 66054r

s

5



412


[CHAP10




unit area)



1 Channele frac14

3b2

h thorn 6b

K frac14t3

3ethh thorn 2bTHORN

Cw frac14h2b3t

12

2h thorn 3b

h thorn 6b


2

h thorn 3b



4

h thorn 3b

h thorn 6b

2







1

K frac14t3


Cw frac14 th2b2

2b1 thorn

b

3 e

2eb1

bthorn

2b21

h

thornh2e2

2b thorn b1 thorn

h

6

2b21

h

thorn

2b31

3ethb thorn eTHORN2






b21

2ethb thorn eTHORN







1

K frac14t3


Cw frac14 th2b2

2b1 thorn

b

3 e

2eb1

b

2b21

h

thornh2e2

2b thorn b1 thorn

h

6thorn

2b21

h

thorn

2b31

3ethb thorn eTHORN2

sx frac14h



sx frac14h




b21

2ethb thorn eTHORN

hb1

2ethb eTHORN


hethb eTHORN

2ethb thorn eTHORN


t2 frac14b2

1

2ethb thorn eTHORN

hb1

2ethb eTHORN

h

4ethb eTHORN2




SEC107]

Torsion

413



4 Twin channel with


t3


Cw frac14tb2

24eth8b3



2b1 thorn

h

2



16eth4b2



5 Twin channel with


t3


Cw frac14tb2

24eth8b3





4

bb1

2



4

h

2 b1

2



b1 gth

21 thorn


2thorn

b

2h

r


4b2

1 hb

4 hb1


b1 lth

21 thorn


2thorn

b

2h

r


6 Wide flanged beam

with equal flanges


whTHORN

Cw frac14h2tb3

24






414


[CHAP10

7 Wide flanged beam


t1b31h

t1b31 thorn t2b3

2

K frac14 13etht3


whTHORN

Cw frac14h2t1t2b3

1b32


2THORN


2

t2b32

t1b31 thorn t2b3

2


1


2

t1b31

t1b31 thorn t2b3

2


1


8

ht2b32b2

1

t1b31 thorn t2b3

2



8

ht1b31b2

2

t1b31 thorn t2b3

2



8 Z-sectionK frac14

t3

3eth2b thorn hTHORN

Cw frac14th2b3

12

b thorn 2h

2b thorn h


hb

2

b thorn h



4

b thorn h

2b thorn h

2





tube


be larger than p to


the case when the

walls overlap)


K frac14 23t3ra

Cw frac142tr5

3a3 6


a sin a cos a



2



SEC107]

Torsion

415




10 e frac14 0707ab2 3a 2b

2a3 etha bTHORN3


Cw frac14ta4b3

6

4a thorn 3b

2a3 etha bTHORN3


2

2a2 thorn 3ab b2


t2 frac14a2b2

4

a2 2ab b2




waTHORN

Cw frac14a2b3t

3cos2 a







416


[CHAP10

T

1shy ------shyI~---q1-Sq ---~









r-= T l 1-5 00 K91= -em

2-l Q( tl (0-06(111) (~j- qi ~Cfl2)

1 r=-S~13gt~- 1 -6-)






~ =-nl bbb ~ eM



Y- tib3qo~ ~2




KG ()


Q t2~ b-t -~ - t~



4




ll1)


kG (1- ~ fl-i 4 tM~ (2 (0 - - )- o-~-~----221 ~ ~A()(


Lo coYs~01tc k ~s



k 5 6~ $ S CM 4 _ ~ )

1 df~0 smiddot

~ - n - (21- slt)O k-~ r -QY) (200 c~) _












2~ - v) 2(--042)l 2S0

I 1



t060lt

D TTh~ =- [) b6 T If) 2q

q1 1


I = iM~-lt ~ ~ )

060

I vo or de a e-s

0 - Qc - 20 C1t ~ L~) 2




co LI M ta e sshy

f ) 3 ~ S ~ ~ kSpound - c rfI J --- ~)







ltLq-t) lb--)



2-t (q-t)


if =- 2 L SOO tlt~f - C-M I - 00)


KG


1 q~ - 21c- 2 (4D)lS)-L SYshy


~ -= (12 SoO 1lt~~-cYV (1~CIV )

(114 ~1-Cc(Vl) (122330q~V~t(Jl~)

P- $ 4Q Xr rltXl ~ - O 0 ~l 0 I


K~






~=- L 003 )lt 03 rod ~ =o 0 ss 1-L~ ~

b) 001 - =cc-hfltJo- 016



T ~ Sg lJ(j ___ 1 ___ _(6)


Cl Clc 2-shy

b4c-M ~

32 cm b = k 2

--shy

LSCfr)i

- lScrt -

lJ

~ =lLS~ _ 039 (lt) q 32C1V) - - U-lt





kG


- 1 12Q ~






ras follows

For 04b

r4 04


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 01496 thorn 02773a

b 02110

a

b

2

K3 frac14 29138 82354a

bthorn 25782

a

b

2

K4 frac14 22991 thorn 12097a

b 22838

a

b

2

At M t frac14TB


b

ras follows For 024

b

r4 04


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 10434 thorn 10449a

b 02977

a

b

2

K2 frac14 00958 98401a

bthorn 16847

a

b

2

K3 frac14 15749 69650a

bthorn 14222

a

b

2

K4 frac14 35878 thorn 88696a

b 47545

a

b

2



ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00264 01187a

bthorn 00868

a

b

2

K3 frac14 02017 thorn 09019a

b 04947

a

b

2

K4 frac14 02911 14875a

bthorn 20651

a

b

2

At M t frac14TB


b

ras follows For 04

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 06366

K2 frac14 00023 thorn 00168a

bthorn 00093

a

b

2

K3 frac14 00052 thorn 00225a

b 03300

a

b

2

K4 frac14 00984 04936a

bthorn 02179

a

b

2


410


[CHAP10


ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00204 01307a

bthorn 01157

a

b

2

K3 frac14 02075 thorn 11544a

b 05937

a

b

2

K4 frac14 03608 22582a

bthorn 37336

a

b

2

At M t frac14TB


b

ras follows For 04

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 06366

K2 frac14 00069 00229a

bthorn 00637

a

b

2

K3 frac14 00675 thorn 03996a

b 10514

a

b

2

K4 frac14 03582 18324a

bthorn 15393

a

b

2



ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00595 03397a

bthorn 03239

a

b

2

K3 frac14 06008 thorn 31396a

b 20693

a

b

2

K4 frac14 10869 62451a

bthorn 94190

a

b

2

At M t frac14TB


b

ras follows For 04

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 06366

K2 frac14 00114 00789a

bthorn 01767

a

b

2

K3 frac14 01207 thorn 10291a

b 23589

a

b

2

K4 frac14 05132 34300a

bthorn 40226

a

b

2


SEC107]

Torsion

411







grooves


rover the range

04a


C frac14 07854 00225a

r 14154

a

r

2

thorn 09167a

r

3

For two grooves

C frac14 07854 00147a

r 30649

a

r

2

thorn 25453a

r

3

For four grooves

C frac14 07854 00409a

r 62371

a

r

2

thorn 72538a

r

3

At M t frac14TB


a

rover the

range 014a



r 27897

a

r

2

thorn 37092a

r

3

For two grooves


r 29501

a

r

2

thorn 70534a

r

3

For four grooves

B frac14 12135 29697a

rthorn 33713

a

r

2

99506a

r

3

thorn 13049a

r

4



sover the

range 04r

s4 09 as follows

C frac14 11266 03210r

sthorn 31519

r

s

2

14347r

s

3

thorn 15223r

s

4

47767r

s

5

At M t frac14BM T


r

sover the range 04

r

s4 05 as follows


s 09180

r

s

2

thorn 37335r

s

3

28686r

s

4

At N t frac14BN T


r

sover the range 034

r

s4 09 as follows


s 42520

r

s

2

thorn 10904r

s

3

13395r

s

4

thorn 66054r

s

5



412


[CHAP10




unit area)



1 Channele frac14

3b2

h thorn 6b

K frac14t3

3ethh thorn 2bTHORN

Cw frac14h2b3t

12

2h thorn 3b

h thorn 6b


2

h thorn 3b



4

h thorn 3b

h thorn 6b

2







1

K frac14t3


Cw frac14 th2b2

2b1 thorn

b

3 e

2eb1

bthorn

2b21

h

thornh2e2

2b thorn b1 thorn

h

6

2b21

h

thorn

2b31

3ethb thorn eTHORN2






b21

2ethb thorn eTHORN







1

K frac14t3


Cw frac14 th2b2

2b1 thorn

b

3 e

2eb1

b

2b21

h

thornh2e2

2b thorn b1 thorn

h

6thorn

2b21

h

thorn

2b31

3ethb thorn eTHORN2

sx frac14h



sx frac14h




b21

2ethb thorn eTHORN

hb1

2ethb eTHORN


hethb eTHORN

2ethb thorn eTHORN


t2 frac14b2

1

2ethb thorn eTHORN

hb1

2ethb eTHORN

h

4ethb eTHORN2




SEC107]

Torsion

413



4 Twin channel with


t3


Cw frac14tb2

24eth8b3



2b1 thorn

h

2



16eth4b2



5 Twin channel with


t3


Cw frac14tb2

24eth8b3





4

bb1

2



4

h

2 b1

2



b1 gth

21 thorn


2thorn

b

2h

r


4b2

1 hb

4 hb1


b1 lth

21 thorn


2thorn

b

2h

r


6 Wide flanged beam

with equal flanges


whTHORN

Cw frac14h2tb3

24






414


[CHAP10

7 Wide flanged beam


t1b31h

t1b31 thorn t2b3

2

K frac14 13etht3


whTHORN

Cw frac14h2t1t2b3

1b32


2THORN


2

t2b32

t1b31 thorn t2b3

2


1


2

t1b31

t1b31 thorn t2b3

2


1


8

ht2b32b2

1

t1b31 thorn t2b3

2



8

ht1b31b2

2

t1b31 thorn t2b3

2



8 Z-sectionK frac14

t3

3eth2b thorn hTHORN

Cw frac14th2b3

12

b thorn 2h

2b thorn h


hb

2

b thorn h



4

b thorn h

2b thorn h

2





tube


be larger than p to


the case when the

walls overlap)


K frac14 23t3ra

Cw frac142tr5

3a3 6


a sin a cos a



2



SEC107]

Torsion

415




10 e frac14 0707ab2 3a 2b

2a3 etha bTHORN3


Cw frac14ta4b3

6

4a thorn 3b

2a3 etha bTHORN3


2

2a2 thorn 3ab b2


t2 frac14a2b2

4

a2 2ab b2




waTHORN

Cw frac14a2b3t

3cos2 a







416


[CHAP10

T

1shy ------shyI~---q1-Sq ---~









r-= T l 1-5 00 K91= -em

2-l Q( tl (0-06(111) (~j- qi ~Cfl2)

1 r=-S~13gt~- 1 -6-)






~ =-nl bbb ~ eM



Y- tib3qo~ ~2




KG ()


Q t2~ b-t -~ - t~



4




ll1)


kG (1- ~ fl-i 4 tM~ (2 (0 - - )- o-~-~----221 ~ ~A()(


Lo coYs~01tc k ~s



k 5 6~ $ S CM 4 _ ~ )

1 df~0 smiddot

~ - n - (21- slt)O k-~ r -QY) (200 c~) _












2~ - v) 2(--042)l 2S0

I 1



t060lt

D TTh~ =- [) b6 T If) 2q

q1 1


I = iM~-lt ~ ~ )

060

I vo or de a e-s

0 - Qc - 20 C1t ~ L~) 2




co LI M ta e sshy

f ) 3 ~ S ~ ~ kSpound - c rfI J --- ~)







ltLq-t) lb--)



2-t (q-t)


if =- 2 L SOO tlt~f - C-M I - 00)


KG


1 q~ - 21c- 2 (4D)lS)-L SYshy


~ -= (12 SoO 1lt~~-cYV (1~CIV )

(114 ~1-Cc(Vl) (122330q~V~t(Jl~)

P- $ 4Q Xr rltXl ~ - O 0 ~l 0 I


K~






~=- L 003 )lt 03 rod ~ =o 0 ss 1-L~ ~

b) 001 - =cc-hfltJo- 016



T ~ Sg lJ(j ___ 1 ___ _(6)


Cl Clc 2-shy

b4c-M ~

32 cm b = k 2

--shy

LSCfr)i

- lScrt -

lJ

~ =lLS~ _ 039 (lt) q 32C1V) - - U-lt





kG


- 1 12Q ~


ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00204 01307a

bthorn 01157

a

b

2

K3 frac14 02075 thorn 11544a

b 05937

a

b

2

K4 frac14 03608 22582a

bthorn 37336

a

b

2

At M t frac14TB


b

ras follows For 04

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 06366

K2 frac14 00069 00229a

bthorn 00637

a

b

2

K3 frac14 00675 thorn 03996a

b 10514

a

b

2

K4 frac14 03582 18324a

bthorn 15393

a

b

2



ras follows

For 04b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 07854

K2 frac14 00595 03397a

bthorn 03239

a

b

2

K3 frac14 06008 thorn 31396a

b 20693

a

b

2

K4 frac14 10869 62451a

bthorn 94190

a

b

2

At M t frac14TB


b

ras follows For 04

b

r4 05


b

rthorn K3

b

r

2

thorn K4

b

r

3


K1 frac14 06366

K2 frac14 00114 00789a

bthorn 01767

a

b

2

K3 frac14 01207 thorn 10291a

b 23589

a

b

2

K4 frac14 05132 34300a

bthorn 40226

a

b

2


SEC107]

Torsion

411







grooves


rover the range

04a


C frac14 07854 00225a

r 14154

a

r

2

thorn 09167a

r

3

For two grooves

C frac14 07854 00147a

r 30649

a

r

2

thorn 25453a

r

3

For four grooves

C frac14 07854 00409a

r 62371

a

r

2

thorn 72538a

r

3

At M t frac14TB


a

rover the

range 014a



r 27897

a

r

2

thorn 37092a

r

3

For two grooves


r 29501

a

r

2

thorn 70534a

r

3

For four grooves

B frac14 12135 29697a

rthorn 33713

a

r

2

99506a

r

3

thorn 13049a

r

4



sover the

range 04r

s4 09 as follows

C frac14 11266 03210r

sthorn 31519

r

s

2

14347r

s

3

thorn 15223r

s

4

47767r

s

5

At M t frac14BM T


r

sover the range 04

r

s4 05 as follows


s 09180

r

s

2

thorn 37335r

s

3

28686r

s

4

At N t frac14BN T


r

sover the range 034

r

s4 09 as follows


s 42520

r

s

2

thorn 10904r

s

3

13395r

s

4

thorn 66054r

s

5



412


[CHAP10




unit area)



1 Channele frac14

3b2

h thorn 6b

K frac14t3

3ethh thorn 2bTHORN

Cw frac14h2b3t

12

2h thorn 3b

h thorn 6b


2

h thorn 3b



4

h thorn 3b

h thorn 6b

2







1

K frac14t3


Cw frac14 th2b2

2b1 thorn

b

3 e

2eb1

bthorn

2b21

h

thornh2e2

2b thorn b1 thorn

h

6

2b21

h

thorn

2b31

3ethb thorn eTHORN2






b21

2ethb thorn eTHORN







1

K frac14t3


Cw frac14 th2b2

2b1 thorn

b

3 e

2eb1

b

2b21

h

thornh2e2

2b thorn b1 thorn

h

6thorn

2b21

h

thorn

2b31

3ethb thorn eTHORN2

sx frac14h



sx frac14h




b21

2ethb thorn eTHORN

hb1

2ethb eTHORN


hethb eTHORN

2ethb thorn eTHORN


t2 frac14b2

1

2ethb thorn eTHORN

hb1

2ethb eTHORN

h

4ethb eTHORN2




SEC107]

Torsion

413



4 Twin channel with


t3


Cw frac14tb2

24eth8b3



2b1 thorn

h

2



16eth4b2



5 Twin channel with


t3


Cw frac14tb2

24eth8b3





4

bb1

2



4

h

2 b1

2



b1 gth

21 thorn


2thorn

b

2h

r


4b2

1 hb

4 hb1


b1 lth

21 thorn


2thorn

b

2h

r


6 Wide flanged beam

with equal flanges


whTHORN

Cw frac14h2tb3

24






414


[CHAP10

7 Wide flanged beam


t1b31h

t1b31 thorn t2b3

2

K frac14 13etht3


whTHORN

Cw frac14h2t1t2b3

1b32


2THORN


2

t2b32

t1b31 thorn t2b3

2


1


2

t1b31

t1b31 thorn t2b3

2


1


8

ht2b32b2

1

t1b31 thorn t2b3

2



8

ht1b31b2

2

t1b31 thorn t2b3

2



8 Z-sectionK frac14

t3

3eth2b thorn hTHORN

Cw frac14th2b3

12

b thorn 2h

2b thorn h


hb

2

b thorn h



4

b thorn h

2b thorn h

2





tube


be larger than p to


the case when the

walls overlap)


K frac14 23t3ra

Cw frac142tr5

3a3 6


a sin a cos a



2



SEC107]

Torsion

415




10 e frac14 0707ab2 3a 2b

2a3 etha bTHORN3


Cw frac14ta4b3

6

4a thorn 3b

2a3 etha bTHORN3


2

2a2 thorn 3ab b2


t2 frac14a2b2

4

a2 2ab b2




waTHORN

Cw frac14a2b3t

3cos2 a







416


[CHAP10

T

1shy ------shyI~---q1-Sq ---~









r-= T l 1-5 00 K91= -em

2-l Q( tl (0-06(111) (~j- qi ~Cfl2)

1 r=-S~13gt~- 1 -6-)






~ =-nl bbb ~ eM



Y- tib3qo~ ~2




KG ()


Q t2~ b-t -~ - t~



4




ll1)


kG (1- ~ fl-i 4 tM~ (2 (0 - - )- o-~-~----221 ~ ~A()(


Lo coYs~01tc k ~s



k 5 6~ $ S CM 4 _ ~ )

1 df~0 smiddot

~ - n - (21- slt)O k-~ r -QY) (200 c~) _












2~ - v) 2(--042)l 2S0

I 1



t060lt

D TTh~ =- [) b6 T If) 2q

q1 1


I = iM~-lt ~ ~ )

060

I vo or de a e-s

0 - Qc - 20 C1t ~ L~) 2




co LI M ta e sshy

f ) 3 ~ S ~ ~ kSpound - c rfI J --- ~)







ltLq-t) lb--)



2-t (q-t)


if =- 2 L SOO tlt~f - C-M I - 00)


KG


1 q~ - 21c- 2 (4D)lS)-L SYshy


~ -= (12 SoO 1lt~~-cYV (1~CIV )

(114 ~1-Cc(Vl) (122330q~V~t(Jl~)

P- $ 4Q Xr rltXl ~ - O 0 ~l 0 I


K~






~=- L 003 )lt 03 rod ~ =o 0 ss 1-L~ ~

b) 001 - =cc-hfltJo- 016



T ~ Sg lJ(j ___ 1 ___ _(6)


Cl Clc 2-shy

b4c-M ~

32 cm b = k 2

--shy

LSCfr)i

- lScrt -

lJ

~ =lLS~ _ 039 (lt) q 32C1V) - - U-lt





kG


- 1 12Q ~






grooves


rover the range

04a


C frac14 07854 00225a

r 14154

a

r

2

thorn 09167a

r

3

For two grooves

C frac14 07854 00147a

r 30649

a

r

2

thorn 25453a

r

3

For four grooves

C frac14 07854 00409a

r 62371

a

r

2

thorn 72538a

r

3

At M t frac14TB


a

rover the

range 014a



r 27897

a

r

2

thorn 37092a

r

3

For two grooves


r 29501

a

r

2

thorn 70534a

r

3

For four grooves

B frac14 12135 29697a

rthorn 33713

a

r

2

99506a

r

3

thorn 13049a

r

4



sover the

range 04r

s4 09 as follows

C frac14 11266 03210r

sthorn 31519

r

s

2

14347r

s

3

thorn 15223r

s

4

47767r

s

5

At M t frac14BM T


r

sover the range 04

r

s4 05 as follows


s 09180

r

s

2

thorn 37335r

s

3

28686r

s

4

At N t frac14BN T


r

sover the range 034

r

s4 09 as follows


s 42520

r

s

2

thorn 10904r

s

3

13395r

s

4

thorn 66054r

s

5



412


[CHAP10




unit area)



1 Channele frac14

3b2

h thorn 6b

K frac14t3

3ethh thorn 2bTHORN

Cw frac14h2b3t

12

2h thorn 3b

h thorn 6b


2

h thorn 3b



4

h thorn 3b

h thorn 6b

2







1

K frac14t3


Cw frac14 th2b2

2b1 thorn

b

3 e

2eb1

bthorn

2b21

h

thornh2e2

2b thorn b1 thorn

h

6

2b21

h

thorn

2b31

3ethb thorn eTHORN2






b21

2ethb thorn eTHORN







1

K frac14t3


Cw frac14 th2b2

2b1 thorn

b

3 e

2eb1

b

2b21

h

thornh2e2

2b thorn b1 thorn

h

6thorn

2b21

h

thorn

2b31

3ethb thorn eTHORN2

sx frac14h



sx frac14h




b21

2ethb thorn eTHORN

hb1

2ethb eTHORN


hethb eTHORN

2ethb thorn eTHORN


t2 frac14b2

1

2ethb thorn eTHORN

hb1

2ethb eTHORN

h

4ethb eTHORN2




SEC107]

Torsion

413



4 Twin channel with


t3


Cw frac14tb2

24eth8b3



2b1 thorn

h

2



16eth4b2



5 Twin channel with


t3


Cw frac14tb2

24eth8b3





4

bb1

2



4

h

2 b1

2



b1 gth

21 thorn


2thorn

b

2h

r


4b2

1 hb

4 hb1


b1 lth

21 thorn


2thorn

b

2h

r


6 Wide flanged beam

with equal flanges


whTHORN

Cw frac14h2tb3

24






414


[CHAP10

7 Wide flanged beam


t1b31h

t1b31 thorn t2b3

2

K frac14 13etht3


whTHORN

Cw frac14h2t1t2b3

1b32


2THORN


2

t2b32

t1b31 thorn t2b3

2


1


2

t1b31

t1b31 thorn t2b3

2


1


8

ht2b32b2

1

t1b31 thorn t2b3

2



8

ht1b31b2

2

t1b31 thorn t2b3

2



8 Z-sectionK frac14

t3

3eth2b thorn hTHORN

Cw frac14th2b3

12

b thorn 2h

2b thorn h


hb

2

b thorn h



4

b thorn h

2b thorn h

2





tube


be larger than p to


the case when the

walls overlap)


K frac14 23t3ra

Cw frac142tr5

3a3 6


a sin a cos a



2



SEC107]

Torsion

415




10 e frac14 0707ab2 3a 2b

2a3 etha bTHORN3


Cw frac14ta4b3

6

4a thorn 3b

2a3 etha bTHORN3


2

2a2 thorn 3ab b2


t2 frac14a2b2

4

a2 2ab b2




waTHORN

Cw frac14a2b3t

3cos2 a







416


[CHAP10

T

1shy ------shyI~---q1-Sq ---~









r-= T l 1-5 00 K91= -em

2-l Q( tl (0-06(111) (~j- qi ~Cfl2)

1 r=-S~13gt~- 1 -6-)






~ =-nl bbb ~ eM



Y- tib3qo~ ~2




KG ()


Q t2~ b-t -~ - t~



4




ll1)


kG (1- ~ fl-i 4 tM~ (2 (0 - - )- o-~-~----221 ~ ~A()(


Lo coYs~01tc k ~s



k 5 6~ $ S CM 4 _ ~ )

1 df~0 smiddot

~ - n - (21- slt)O k-~ r -QY) (200 c~) _












2~ - v) 2(--042)l 2S0

I 1



t060lt

D TTh~ =- [) b6 T If) 2q

q1 1


I = iM~-lt ~ ~ )

060

I vo or de a e-s

0 - Qc - 20 C1t ~ L~) 2




co LI M ta e sshy

f ) 3 ~ S ~ ~ kSpound - c rfI J --- ~)







ltLq-t) lb--)



2-t (q-t)


if =- 2 L SOO tlt~f - C-M I - 00)


KG


1 q~ - 21c- 2 (4D)lS)-L SYshy


~ -= (12 SoO 1lt~~-cYV (1~CIV )

(114 ~1-Cc(Vl) (122330q~V~t(Jl~)

P- $ 4Q Xr rltXl ~ - O 0 ~l 0 I


K~






~=- L 003 )lt 03 rod ~ =o 0 ss 1-L~ ~

b) 001 - =cc-hfltJo- 016



T ~ Sg lJ(j ___ 1 ___ _(6)


Cl Clc 2-shy

b4c-M ~

32 cm b = k 2

--shy

LSCfr)i

- lScrt -

lJ

~ =lLS~ _ 039 (lt) q 32C1V) - - U-lt





kG


- 1 12Q ~




unit area)



1 Channele frac14

3b2

h thorn 6b

K frac14t3

3ethh thorn 2bTHORN

Cw frac14h2b3t

12

2h thorn 3b

h thorn 6b


2

h thorn 3b



4

h thorn 3b

h thorn 6b

2







1

K frac14t3


Cw frac14 th2b2

2b1 thorn

b

3 e

2eb1

bthorn

2b21

h

thornh2e2

2b thorn b1 thorn

h

6

2b21

h

thorn

2b31

3ethb thorn eTHORN2






b21

2ethb thorn eTHORN







1

K frac14t3


Cw frac14 th2b2

2b1 thorn

b

3 e

2eb1

b

2b21

h

thornh2e2

2b thorn b1 thorn

h

6thorn

2b21

h

thorn

2b31

3ethb thorn eTHORN2

sx frac14h



sx frac14h




b21

2ethb thorn eTHORN

hb1

2ethb eTHORN


hethb eTHORN

2ethb thorn eTHORN


t2 frac14b2

1

2ethb thorn eTHORN

hb1

2ethb eTHORN

h

4ethb eTHORN2




SEC107]

Torsion

413



4 Twin channel with


t3


Cw frac14tb2

24eth8b3



2b1 thorn

h

2



16eth4b2



5 Twin channel with


t3


Cw frac14tb2

24eth8b3





4

bb1

2



4

h

2 b1

2



b1 gth

21 thorn


2thorn

b

2h

r


4b2

1 hb

4 hb1


b1 lth

21 thorn


2thorn

b

2h

r


6 Wide flanged beam

with equal flanges


whTHORN

Cw frac14h2tb3

24






414


[CHAP10

7 Wide flanged beam


t1b31h

t1b31 thorn t2b3

2

K frac14 13etht3


whTHORN

Cw frac14h2t1t2b3

1b32


2THORN


2

t2b32

t1b31 thorn t2b3

2


1


2

t1b31

t1b31 thorn t2b3

2


1


8

ht2b32b2

1

t1b31 thorn t2b3

2



8

ht1b31b2

2

t1b31 thorn t2b3

2



8 Z-sectionK frac14

t3

3eth2b thorn hTHORN

Cw frac14th2b3

12

b thorn 2h

2b thorn h


hb

2

b thorn h



4

b thorn h

2b thorn h

2





tube


be larger than p to


the case when the

walls overlap)


K frac14 23t3ra

Cw frac142tr5

3a3 6


a sin a cos a



2



SEC107]

Torsion

415




10 e frac14 0707ab2 3a 2b

2a3 etha bTHORN3


Cw frac14ta4b3

6

4a thorn 3b

2a3 etha bTHORN3


2

2a2 thorn 3ab b2


t2 frac14a2b2

4

a2 2ab b2




waTHORN

Cw frac14a2b3t

3cos2 a







416


[CHAP10

T

1shy ------shyI~---q1-Sq ---~









r-= T l 1-5 00 K91= -em

2-l Q( tl (0-06(111) (~j- qi ~Cfl2)

1 r=-S~13gt~- 1 -6-)






~ =-nl bbb ~ eM



Y- tib3qo~ ~2




KG ()


Q t2~ b-t -~ - t~



4




ll1)


kG (1- ~ fl-i 4 tM~ (2 (0 - - )- o-~-~----221 ~ ~A()(


Lo coYs~01tc k ~s



k 5 6~ $ S CM 4 _ ~ )

1 df~0 smiddot

~ - n - (21- slt)O k-~ r -QY) (200 c~) _












2~ - v) 2(--042)l 2S0

I 1



t060lt

D TTh~ =- [) b6 T If) 2q

q1 1


I = iM~-lt ~ ~ )

060

I vo or de a e-s

0 - Qc - 20 C1t ~ L~) 2




co LI M ta e sshy

f ) 3 ~ S ~ ~ kSpound - c rfI J --- ~)







ltLq-t) lb--)



2-t (q-t)


if =- 2 L SOO tlt~f - C-M I - 00)


KG


1 q~ - 21c- 2 (4D)lS)-L SYshy


~ -= (12 SoO 1lt~~-cYV (1~CIV )

(114 ~1-Cc(Vl) (122330q~V~t(Jl~)

P- $ 4Q Xr rltXl ~ - O 0 ~l 0 I


K~






~=- L 003 )lt 03 rod ~ =o 0 ss 1-L~ ~

b) 001 - =cc-hfltJo- 016



T ~ Sg lJ(j ___ 1 ___ _(6)


Cl Clc 2-shy

b4c-M ~

32 cm b = k 2

--shy

LSCfr)i

- lScrt -

lJ

~ =lLS~ _ 039 (lt) q 32C1V) - - U-lt





kG


- 1 12Q ~



4 Twin channel with


t3


Cw frac14tb2

24eth8b3



2b1 thorn

h

2



16eth4b2



5 Twin channel with


t3


Cw frac14tb2

24eth8b3





4

bb1

2



4

h

2 b1

2



b1 gth

21 thorn


2thorn

b

2h

r


4b2

1 hb

4 hb1


b1 lth

21 thorn


2thorn

b

2h

r


6 Wide flanged beam

with equal flanges


whTHORN

Cw frac14h2tb3

24






414


[CHAP10

7 Wide flanged beam


t1b31h

t1b31 thorn t2b3

2

K frac14 13etht3


whTHORN

Cw frac14h2t1t2b3

1b32


2THORN


2

t2b32

t1b31 thorn t2b3

2


1


2

t1b31

t1b31 thorn t2b3

2


1


8

ht2b32b2

1

t1b31 thorn t2b3

2



8

ht1b31b2

2

t1b31 thorn t2b3

2



8 Z-sectionK frac14

t3

3eth2b thorn hTHORN

Cw frac14th2b3

12

b thorn 2h

2b thorn h


hb

2

b thorn h



4

b thorn h

2b thorn h

2





tube


be larger than p to


the case when the

walls overlap)


K frac14 23t3ra

Cw frac142tr5

3a3 6


a sin a cos a



2



SEC107]

Torsion

415




10 e frac14 0707ab2 3a 2b

2a3 etha bTHORN3


Cw frac14ta4b3

6

4a thorn 3b

2a3 etha bTHORN3


2

2a2 thorn 3ab b2


t2 frac14a2b2

4

a2 2ab b2




waTHORN

Cw frac14a2b3t

3cos2 a







416


[CHAP10

T

1shy ------shyI~---q1-Sq ---~









r-= T l 1-5 00 K91= -em

2-l Q( tl (0-06(111) (~j- qi ~Cfl2)

1 r=-S~13gt~- 1 -6-)






~ =-nl bbb ~ eM



Y- tib3qo~ ~2




KG ()


Q t2~ b-t -~ - t~



4




ll1)


kG (1- ~ fl-i 4 tM~ (2 (0 - - )- o-~-~----221 ~ ~A()(


Lo coYs~01tc k ~s



k 5 6~ $ S CM 4 _ ~ )

1 df~0 smiddot

~ - n - (21- slt)O k-~ r -QY) (200 c~) _












2~ - v) 2(--042)l 2S0

I 1



t060lt

D TTh~ =- [) b6 T If) 2q

q1 1


I = iM~-lt ~ ~ )

060

I vo or de a e-s

0 - Qc - 20 C1t ~ L~) 2




co LI M ta e sshy

f ) 3 ~ S ~ ~ kSpound - c rfI J --- ~)







ltLq-t) lb--)



2-t (q-t)


if =- 2 L SOO tlt~f - C-M I - 00)


KG


1 q~ - 21c- 2 (4D)lS)-L SYshy


~ -= (12 SoO 1lt~~-cYV (1~CIV )

(114 ~1-Cc(Vl) (122330q~V~t(Jl~)

P- $ 4Q Xr rltXl ~ - O 0 ~l 0 I


K~






~=- L 003 )lt 03 rod ~ =o 0 ss 1-L~ ~

b) 001 - =cc-hfltJo- 016



T ~ Sg lJ(j ___ 1 ___ _(6)


Cl Clc 2-shy

b4c-M ~

32 cm b = k 2

--shy

LSCfr)i

- lScrt -

lJ

~ =lLS~ _ 039 (lt) q 32C1V) - - U-lt





kG


- 1 12Q ~

7 Wide flanged beam


t1b31h

t1b31 thorn t2b3

2

K frac14 13etht3


whTHORN

Cw frac14h2t1t2b3

1b32


2THORN


2

t2b32

t1b31 thorn t2b3

2


1


2

t1b31

t1b31 thorn t2b3

2


1


8

ht2b32b2

1

t1b31 thorn t2b3

2



8

ht1b31b2

2

t1b31 thorn t2b3

2



8 Z-sectionK frac14

t3

3eth2b thorn hTHORN

Cw frac14th2b3

12

b thorn 2h

2b thorn h


hb

2

b thorn h



4

b thorn h

2b thorn h

2





tube


be larger than p to


the case when the

walls overlap)


K frac14 23t3ra

Cw frac142tr5

3a3 6


a sin a cos a



2



SEC107]

Torsion

415




10 e frac14 0707ab2 3a 2b

2a3 etha bTHORN3


Cw frac14ta4b3

6

4a thorn 3b

2a3 etha bTHORN3


2

2a2 thorn 3ab b2


t2 frac14a2b2

4

a2 2ab b2




waTHORN

Cw frac14a2b3t

3cos2 a







416


[CHAP10

T

1shy ------shyI~---q1-Sq ---~









r-= T l 1-5 00 K91= -em

2-l Q( tl (0-06(111) (~j- qi ~Cfl2)

1 r=-S~13gt~- 1 -6-)






~ =-nl bbb ~ eM



Y- tib3qo~ ~2




KG ()


Q t2~ b-t -~ - t~



4




ll1)


kG (1- ~ fl-i 4 tM~ (2 (0 - - )- o-~-~----221 ~ ~A()(


Lo coYs~01tc k ~s



k 5 6~ $ S CM 4 _ ~ )

1 df~0 smiddot

~ - n - (21- slt)O k-~ r -QY) (200 c~) _












2~ - v) 2(--042)l 2S0

I 1



t060lt

D TTh~ =- [) b6 T If) 2q

q1 1


I = iM~-lt ~ ~ )

060

I vo or de a e-s

0 - Qc - 20 C1t ~ L~) 2




co LI M ta e sshy

f ) 3 ~ S ~ ~ kSpound - c rfI J --- ~)







ltLq-t) lb--)



2-t (q-t)


if =- 2 L SOO tlt~f - C-M I - 00)


KG


1 q~ - 21c- 2 (4D)lS)-L SYshy


~ -= (12 SoO 1lt~~-cYV (1~CIV )

(114 ~1-Cc(Vl) (122330q~V~t(Jl~)

P- $ 4Q Xr rltXl ~ - O 0 ~l 0 I


K~






~=- L 003 )lt 03 rod ~ =o 0 ss 1-L~ ~

b) 001 - =cc-hfltJo- 016



T ~ Sg lJ(j ___ 1 ___ _(6)


Cl Clc 2-shy

b4c-M ~

32 cm b = k 2

--shy

LSCfr)i

- lScrt -

lJ

~ =lLS~ _ 039 (lt) q 32C1V) - - U-lt





kG


- 1 12Q ~



10 e frac14 0707ab2 3a 2b

2a3 etha bTHORN3


Cw frac14ta4b3

6

4a thorn 3b

2a3 etha bTHORN3


2

2a2 thorn 3ab b2


t2 frac14a2b2

4

a2 2ab b2




waTHORN

Cw frac14a2b3t

3cos2 a







416


[CHAP10

T

1shy ------shyI~---q1-Sq ---~









r-= T l 1-5 00 K91= -em

2-l Q( tl (0-06(111) (~j- qi ~Cfl2)

1 r=-S~13gt~- 1 -6-)






~ =-nl bbb ~ eM



Y- tib3qo~ ~2




KG ()


Q t2~ b-t -~ - t~



4




ll1)


kG (1- ~ fl-i 4 tM~ (2 (0 - - )- o-~-~----221 ~ ~A()(


Lo coYs~01tc k ~s



k 5 6~ $ S CM 4 _ ~ )

1 df~0 smiddot

~ - n - (21- slt)O k-~ r -QY) (200 c~) _












2~ - v) 2(--042)l 2S0

I 1



t060lt

D TTh~ =- [) b6 T If) 2q

q1 1


I = iM~-lt ~ ~ )

060

I vo or de a e-s

0 - Qc - 20 C1t ~ L~) 2




co LI M ta e sshy

f ) 3 ~ S ~ ~ kSpound - c rfI J --- ~)







ltLq-t) lb--)



2-t (q-t)


if =- 2 L SOO tlt~f - C-M I - 00)


KG


1 q~ - 21c- 2 (4D)lS)-L SYshy


~ -= (12 SoO 1lt~~-cYV (1~CIV )

(114 ~1-Cc(Vl) (122330q~V~t(Jl~)

P- $ 4Q Xr rltXl ~ - O 0 ~l 0 I


K~






~=- L 003 )lt 03 rod ~ =o 0 ss 1-L~ ~

b) 001 - =cc-hfltJo- 016



T ~ Sg lJ(j ___ 1 ___ _(6)


Cl Clc 2-shy

b4c-M ~

32 cm b = k 2

--shy

LSCfr)i

- lScrt -

lJ

~ =lLS~ _ 039 (lt) q 32C1V) - - U-lt





kG


- 1 12Q ~

T

1shy ------shyI~---q1-Sq ---~









r-= T l 1-5 00 K91= -em

2-l Q( tl (0-06(111) (~j- qi ~Cfl2)

1 r=-S~13gt~- 1 -6-)






~ =-nl bbb ~ eM



Y- tib3qo~ ~2




KG ()


Q t2~ b-t -~ - t~



4




ll1)


kG (1- ~ fl-i 4 tM~ (2 (0 - - )- o-~-~----221 ~ ~A()(


Lo coYs~01tc k ~s



k 5 6~ $ S CM 4 _ ~ )

1 df~0 smiddot

~ - n - (21- slt)O k-~ r -QY) (200 c~) _












2~ - v) 2(--042)l 2S0

I 1



t060lt

D TTh~ =- [) b6 T If) 2q

q1 1


I = iM~-lt ~ ~ )

060

I vo or de a e-s

0 - Qc - 20 C1t ~ L~) 2




co LI M ta e sshy

f ) 3 ~ S ~ ~ kSpound - c rfI J --- ~)







ltLq-t) lb--)



2-t (q-t)


if =- 2 L SOO tlt~f - C-M I - 00)


KG


1 q~ - 21c- 2 (4D)lS)-L SYshy


~ -= (12 SoO 1lt~~-cYV (1~CIV )

(114 ~1-Cc(Vl) (122330q~V~t(Jl~)

P- $ 4Q Xr rltXl ~ - O 0 ~l 0 I


K~






~=- L 003 )lt 03 rod ~ =o 0 ss 1-L~ ~

b) 001 - =cc-hfltJo- 016



T ~ Sg lJ(j ___ 1 ___ _(6)


Cl Clc 2-shy

b4c-M ~

32 cm b = k 2

--shy

LSCfr)i

- lScrt -

lJ

~ =lLS~ _ 039 (lt) q 32C1V) - - U-lt





kG


- 1 12Q ~




KG ()


Q t2~ b-t -~ - t~



4




ll1)


kG (1- ~ fl-i 4 tM~ (2 (0 - - )- o-~-~----221 ~ ~A()(


Lo coYs~01tc k ~s



k 5 6~ $ S CM 4 _ ~ )

1 df~0 smiddot

~ - n - (21- slt)O k-~ r -QY) (200 c~) _












2~ - v) 2(--042)l 2S0

I 1



t060lt

D TTh~ =- [) b6 T If) 2q

q1 1


I = iM~-lt ~ ~ )

060

I vo or de a e-s

0 - Qc - 20 C1t ~ L~) 2




co LI M ta e sshy

f ) 3 ~ S ~ ~ kSpound - c rfI J --- ~)







ltLq-t) lb--)



2-t (q-t)


if =- 2 L SOO tlt~f - C-M I - 00)


KG


1 q~ - 21c- 2 (4D)lS)-L SYshy


~ -= (12 SoO 1lt~~-cYV (1~CIV )

(114 ~1-Cc(Vl) (122330q~V~t(Jl~)

P- $ 4Q Xr rltXl ~ - O 0 ~l 0 I


K~






~=- L 003 )lt 03 rod ~ =o 0 ss 1-L~ ~

b) 001 - =cc-hfltJo- 016



T ~ Sg lJ(j ___ 1 ___ _(6)


Cl Clc 2-shy

b4c-M ~

32 cm b = k 2

--shy

LSCfr)i

- lScrt -

lJ

~ =lLS~ _ 039 (lt) q 32C1V) - - U-lt





kG


- 1 12Q ~











2~ - v) 2(--042)l 2S0

I 1



t060lt

D TTh~ =- [) b6 T If) 2q

q1 1


I = iM~-lt ~ ~ )

060

I vo or de a e-s

0 - Qc - 20 C1t ~ L~) 2




co LI M ta e sshy

f ) 3 ~ S ~ ~ kSpound - c rfI J --- ~)







ltLq-t) lb--)



2-t (q-t)


if =- 2 L SOO tlt~f - C-M I - 00)


KG


1 q~ - 21c- 2 (4D)lS)-L SYshy


~ -= (12 SoO 1lt~~-cYV (1~CIV )

(114 ~1-Cc(Vl) (122330q~V~t(Jl~)

P- $ 4Q Xr rltXl ~ - O 0 ~l 0 I


K~






~=- L 003 )lt 03 rod ~ =o 0 ss 1-L~ ~

b) 001 - =cc-hfltJo- 016



T ~ Sg lJ(j ___ 1 ___ _(6)


Cl Clc 2-shy

b4c-M ~

32 cm b = k 2

--shy

LSCfr)i

- lScrt -

lJ

~ =lLS~ _ 039 (lt) q 32C1V) - - U-lt





kG


- 1 12Q ~







ltLq-t) lb--)



2-t (q-t)


if =- 2 L SOO tlt~f - C-M I - 00)


KG


1 q~ - 21c- 2 (4D)lS)-L SYshy


~ -= (12 SoO 1lt~~-cYV (1~CIV )

(114 ~1-Cc(Vl) (122330q~V~t(Jl~)

P- $ 4Q Xr rltXl ~ - O 0 ~l 0 I


K~






~=- L 003 )lt 03 rod ~ =o 0 ss 1-L~ ~

b) 001 - =cc-hfltJo- 016



T ~ Sg lJ(j ___ 1 ___ _(6)


Cl Clc 2-shy

b4c-M ~

32 cm b = k 2

--shy

LSCfr)i

- lScrt -

lJ

~ =lLS~ _ 039 (lt) q 32C1V) - - U-lt





kG


- 1 12Q ~


K~






~=- L 003 )lt 03 rod ~ =o 0 ss 1-L~ ~

b) 001 - =cc-hfltJo- 016



T ~ Sg lJ(j ___ 1 ___ _(6)


Cl Clc 2-shy

b4c-M ~

32 cm b = k 2

--shy

LSCfr)i

- lScrt -

lJ

~ =lLS~ _ 039 (lt) q 32C1V) - - U-lt





kG


- 1 12Q ~

4.1. Torsión de elementos no circularesmateriales.azc.uam.mx/gjl/Clases/SOLIDOSII/S9.pdf · Tema...

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Transcript of 4.1. Torsión de elementos no circularesmateriales.azc.uam.mx/gjl/Clases/SOLIDOSII/S9.pdf · Tema...