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)
Form and dimensions of cross sections
other quantities involved and case no Formula for K in y frac14TL
KGFormula for shear stress
5 Solid triangular section (equilaterial)K frac14
a4ffiffiffi3
p
80tmax frac14
20T
a3at midpoint of each side
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
peth82 lt a lt 120THORN
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
For a frac14 90 Q frac14 01604b3 frac14 00567c3
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
K frac14 2Cr4 where C varies withh
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
r3where B varies with
h
ras follows For two flat sides where
04h
r4 06
B frac14 06366 thorn 25303h
r 11157
h
r
2
thorn 49568h
r
3
85886h
r
4
thorn 69849h
r
5
For four flat sides where 04h
r4 0293
B frac14 06366 thorn 26298h
r 56147
h
r
2
thorn 30853h
r
3
(Data from Refs 12 and 13)
SEC107]
Torsion
403
TABLE 101 Formulas for torsional deformation and stress (Continued)
TABLE 101 Formulas for torsional deformation and stress (Continued)
Form and dimensions of cross sections
other quantities involved and case no Formula for K in y frac14TL
KGFormula for shear stress
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
eth1 n2THORNeth1 n4THORNl2
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
FormulasforStressandStrain
[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)
TABLE 101 Formulas for torsional deformation and stress (Continued )
Form and dimensions of cross sections
other quantities involved and case no Formula for K in y frac14TL
KGFormula for shear stress
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
A frac14area of section
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
FormulasforStressandStrain
[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
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 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
TABLE 101 Formulas for torsional deformation and stress (Continued)
TABLE 101 Formulas for torsional deformation and stress (Continued )
Form and dimensions of cross sections
other quantities involved and case no Formula for K in y frac14TL
KGFormula for shear stress
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
where K1 frac14 ab3 1
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
(Data from Refs 12 and 13)
408
FormulasforStressandStrain
[CHAP10
28 Shaft with one keyway K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
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
r3where B varies with
b
ras follows For 024
b
r4 05
B frac14 K1 thorn K2
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
(Data from Refs 12 and 13)
29 Shaft with two keyways K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 034a=b4 15
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
r3where B varies with
b
ras follows For 024
b
r4 05
B frac14 K1 thorn K2
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
(Data from Refs 12 and 13)
SEC107]
Torsion
409
TABLE 101 Formulas for torsional deformation and stress (Continued)
TABLE 101 Formulas for torsional deformation and stress (Continued)
Form and dimensions of cross sections
other quantities involved and case no Formula for K in y frac14TL
KGFormula for shear stress
30 Shaft with four keyways K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 04
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 034a=b4 12
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
r3where B varies with
b
ras follows For 024
b
r4 04
B frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 054a=b4 12
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
(Data from Refs 12 and 13)
31 Shaft with one spline K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 14
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
r3where B varies with
b
ras follows For 04
b
r4 05
B frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 14
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
(Data from Refs 12 and 13)
410
FormulasforStressandStrain
[CHAP10
32 Shaft with two splines K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 14
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
r3where B varies with
b
ras follows For 04
b
r4 05
B frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 14
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
(Data from Refs 12 and 13)
33 Shaft with four splines K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 10
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
r3where B varies with
b
ras follows For 04
b
r4 05
B frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 10
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
(Data from Refs 12 and 13)
SEC107]
Torsion
411
TABLE 101 Formulas for torsional deformation and stress (Continued)
TABLE 101 Formulas for torsional deformation and stress (Continued)
Form and dimensions of cross sections
other quantities involved and case no Formula for K in y frac14TL
KGFormula for shear stress
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
r3where B varies with
a
rover the
range 014a
r4 05 as follows For one groove
B frac14 10259 thorn 11802a
r 27897
a
r
2
thorn 37092a
r
3
For two grooves
B frac14 10055 thorn 15427a
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
(Data from Refs 12 and 13)
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
(Data from Refs 12 and 13)
412
FormulasforStressandStrain
[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
etht1THORNmax frac14 tGy0 at the surface everywhere
3 Hat sectione frac14 b
3h2b thorn 6h2b1 8b31
h3 thorn 6h2b thorn 6h2b1 thorn 8b31 thorn 12hb2
1
K frac14t3
3ethh thorn 2b thorn 2b1THORN
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
Ey00 throughout the thickness at corners A and F
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 )
Cross section reference no Constants Selected maximum values
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
etht1THORNmax frac14 tGy0 at the surface everywhere
5 Twin channel with
flanges outwardK frac14
t3
3eth2b thorn 4b1THORN
Cw frac14tb2
24eth8b3
1 thorn 6h2b1 thorn h2b 12b21hTHORN
ethsxTHORNmax frac14hb
4Ey00 throughout the thickness at points B and C if h gt b1
ethsxTHORNmax frac14hb
4
bb1
2
Ey00 throughout the thickness at points A and D if h lt b1
etht2THORNmax frac14b
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
etht2THORNmax frac14b
4b2
1 hb
4 hb1
Ey000 throughout the thickness at a point midway between corners B and C if
b1 lth
21 thorn
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
2thorn
b
2h
r
etht1THORNmax frac14 tGy0 at the surface everywhere
6 Wide flanged beam
with equal flanges
K frac14 13eth2t3b thorn t3
whTHORN
Cw frac14h2tb3
24
ethsxTHORNmax frac14hb
4Ey00 throughout the thickness at points A and B
etht2THORNmax frac14 hb2
16Ey000 throughout the thickness at a point midway between A and B
etht1THORNmax frac14 tGy0 at the surface everywhere
414
FormulasforStressandStrain
[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
ethsxTHORNmax frac14hb2
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
etht2THORNmax frac141
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
etht2THORNmax frac14hb2
4
b thorn h
2b thorn h
2
Ey000 throughout the thickness at a distancebethb thorn hTHORN
2b thorn hfrom point A
etht1THORNmax frac14 tGy0 at the surface everywhere
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
etht1THORNmax frac14 tGy0 at the surface everywhere
SEC107]
Torsion
415
TABLE 102 Formulas for torsional properties and stresses in thin-walled open cross sections (Continued)
TABLE 102 Formulas for torsional properties and stresses in thin-walled open cross sections (Continued )
Cross section reference no Constants Selected maximum values
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
etht1THORNmax frac14 tGy0 at the surface everywhere
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
etht1THORNmax frac14 tGy0 at the surface everywhere
416
FormulasforStressandStrain
[CHAP10
T
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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
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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--)
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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
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32 cm b = k 2
--shy
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lJ
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- 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
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)
Form and dimensions of cross sections
other quantities involved and case no Formula for K in y frac14TL
KGFormula for shear stress
5 Solid triangular section (equilaterial)K frac14
a4ffiffiffi3
p
80tmax frac14
20T
a3at midpoint of each side
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
peth82 lt a lt 120THORN
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
For a frac14 90 Q frac14 01604b3 frac14 00567c3
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
K frac14 2Cr4 where C varies withh
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
r3where B varies with
h
ras follows For two flat sides where
04h
r4 06
B frac14 06366 thorn 25303h
r 11157
h
r
2
thorn 49568h
r
3
85886h
r
4
thorn 69849h
r
5
For four flat sides where 04h
r4 0293
B frac14 06366 thorn 26298h
r 56147
h
r
2
thorn 30853h
r
3
(Data from Refs 12 and 13)
SEC107]
Torsion
403
TABLE 101 Formulas for torsional deformation and stress (Continued)
TABLE 101 Formulas for torsional deformation and stress (Continued)
Form and dimensions of cross sections
other quantities involved and case no Formula for K in y frac14TL
KGFormula for shear stress
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
eth1 n2THORNeth1 n4THORNl2
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
FormulasforStressandStrain
[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)
TABLE 101 Formulas for torsional deformation and stress (Continued )
Form and dimensions of cross sections
other quantities involved and case no Formula for K in y frac14TL
KGFormula for shear stress
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
A frac14area of section
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
FormulasforStressandStrain
[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
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 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
TABLE 101 Formulas for torsional deformation and stress (Continued)
TABLE 101 Formulas for torsional deformation and stress (Continued )
Form and dimensions of cross sections
other quantities involved and case no Formula for K in y frac14TL
KGFormula for shear stress
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
where K1 frac14 ab3 1
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
(Data from Refs 12 and 13)
408
FormulasforStressandStrain
[CHAP10
28 Shaft with one keyway K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
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
r3where B varies with
b
ras follows For 024
b
r4 05
B frac14 K1 thorn K2
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
(Data from Refs 12 and 13)
29 Shaft with two keyways K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 034a=b4 15
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
r3where B varies with
b
ras follows For 024
b
r4 05
B frac14 K1 thorn K2
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
(Data from Refs 12 and 13)
SEC107]
Torsion
409
TABLE 101 Formulas for torsional deformation and stress (Continued)
TABLE 101 Formulas for torsional deformation and stress (Continued)
Form and dimensions of cross sections
other quantities involved and case no Formula for K in y frac14TL
KGFormula for shear stress
30 Shaft with four keyways K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 04
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 034a=b4 12
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
r3where B varies with
b
ras follows For 024
b
r4 04
B frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 054a=b4 12
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
(Data from Refs 12 and 13)
31 Shaft with one spline K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 14
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
r3where B varies with
b
ras follows For 04
b
r4 05
B frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 14
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
(Data from Refs 12 and 13)
410
FormulasforStressandStrain
[CHAP10
32 Shaft with two splines K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 14
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
r3where B varies with
b
ras follows For 04
b
r4 05
B frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 14
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
(Data from Refs 12 and 13)
33 Shaft with four splines K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 10
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
r3where B varies with
b
ras follows For 04
b
r4 05
B frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 10
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
(Data from Refs 12 and 13)
SEC107]
Torsion
411
TABLE 101 Formulas for torsional deformation and stress (Continued)
TABLE 101 Formulas for torsional deformation and stress (Continued)
Form and dimensions of cross sections
other quantities involved and case no Formula for K in y frac14TL
KGFormula for shear stress
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
r3where B varies with
a
rover the
range 014a
r4 05 as follows For one groove
B frac14 10259 thorn 11802a
r 27897
a
r
2
thorn 37092a
r
3
For two grooves
B frac14 10055 thorn 15427a
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
(Data from Refs 12 and 13)
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
(Data from Refs 12 and 13)
412
FormulasforStressandStrain
[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
etht1THORNmax frac14 tGy0 at the surface everywhere
3 Hat sectione frac14 b
3h2b thorn 6h2b1 8b31
h3 thorn 6h2b thorn 6h2b1 thorn 8b31 thorn 12hb2
1
K frac14t3
3ethh thorn 2b thorn 2b1THORN
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
Ey00 throughout the thickness at corners A and F
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 )
Cross section reference no Constants Selected maximum values
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
etht1THORNmax frac14 tGy0 at the surface everywhere
5 Twin channel with
flanges outwardK frac14
t3
3eth2b thorn 4b1THORN
Cw frac14tb2
24eth8b3
1 thorn 6h2b1 thorn h2b 12b21hTHORN
ethsxTHORNmax frac14hb
4Ey00 throughout the thickness at points B and C if h gt b1
ethsxTHORNmax frac14hb
4
bb1
2
Ey00 throughout the thickness at points A and D if h lt b1
etht2THORNmax frac14b
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
etht2THORNmax frac14b
4b2
1 hb
4 hb1
Ey000 throughout the thickness at a point midway between corners B and C if
b1 lth
21 thorn
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
2thorn
b
2h
r
etht1THORNmax frac14 tGy0 at the surface everywhere
6 Wide flanged beam
with equal flanges
K frac14 13eth2t3b thorn t3
whTHORN
Cw frac14h2tb3
24
ethsxTHORNmax frac14hb
4Ey00 throughout the thickness at points A and B
etht2THORNmax frac14 hb2
16Ey000 throughout the thickness at a point midway between A and B
etht1THORNmax frac14 tGy0 at the surface everywhere
414
FormulasforStressandStrain
[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
ethsxTHORNmax frac14hb2
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
etht2THORNmax frac141
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
etht2THORNmax frac14hb2
4
b thorn h
2b thorn h
2
Ey000 throughout the thickness at a distancebethb thorn hTHORN
2b thorn hfrom point A
etht1THORNmax frac14 tGy0 at the surface everywhere
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
etht1THORNmax frac14 tGy0 at the surface everywhere
SEC107]
Torsion
415
TABLE 102 Formulas for torsional properties and stresses in thin-walled open cross sections (Continued)
TABLE 102 Formulas for torsional properties and stresses in thin-walled open cross sections (Continued )
Cross section reference no Constants Selected maximum values
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
etht1THORNmax frac14 tGy0 at the surface everywhere
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
etht1THORNmax frac14 tGy0 at the surface everywhere
416
FormulasforStressandStrain
[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
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~ 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
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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 stress (Continued)
Form and dimensions of cross sections
other quantities involved and case no Formula for K in y frac14TL
KGFormula for shear stress
5 Solid triangular section (equilaterial)K frac14
a4ffiffiffi3
p
80tmax frac14
20T
a3at midpoint of each side
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
peth82 lt a lt 120THORN
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
For a frac14 90 Q frac14 01604b3 frac14 00567c3
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
K frac14 2Cr4 where C varies withh
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
r3where B varies with
h
ras follows For two flat sides where
04h
r4 06
B frac14 06366 thorn 25303h
r 11157
h
r
2
thorn 49568h
r
3
85886h
r
4
thorn 69849h
r
5
For four flat sides where 04h
r4 0293
B frac14 06366 thorn 26298h
r 56147
h
r
2
thorn 30853h
r
3
(Data from Refs 12 and 13)
SEC107]
Torsion
403
TABLE 101 Formulas for torsional deformation and stress (Continued)
TABLE 101 Formulas for torsional deformation and stress (Continued)
Form and dimensions of cross sections
other quantities involved and case no Formula for K in y frac14TL
KGFormula for shear stress
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
eth1 n2THORNeth1 n4THORNl2
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
FormulasforStressandStrain
[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)
TABLE 101 Formulas for torsional deformation and stress (Continued )
Form and dimensions of cross sections
other quantities involved and case no Formula for K in y frac14TL
KGFormula for shear stress
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
A frac14area of section
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
FormulasforStressandStrain
[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
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 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
TABLE 101 Formulas for torsional deformation and stress (Continued)
TABLE 101 Formulas for torsional deformation and stress (Continued )
Form and dimensions of cross sections
other quantities involved and case no Formula for K in y frac14TL
KGFormula for shear stress
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
where K1 frac14 ab3 1
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
(Data from Refs 12 and 13)
408
FormulasforStressandStrain
[CHAP10
28 Shaft with one keyway K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
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
r3where B varies with
b
ras follows For 024
b
r4 05
B frac14 K1 thorn K2
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
(Data from Refs 12 and 13)
29 Shaft with two keyways K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 034a=b4 15
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
r3where B varies with
b
ras follows For 024
b
r4 05
B frac14 K1 thorn K2
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
(Data from Refs 12 and 13)
SEC107]
Torsion
409
TABLE 101 Formulas for torsional deformation and stress (Continued)
TABLE 101 Formulas for torsional deformation and stress (Continued)
Form and dimensions of cross sections
other quantities involved and case no Formula for K in y frac14TL
KGFormula for shear stress
30 Shaft with four keyways K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 04
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 034a=b4 12
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
r3where B varies with
b
ras follows For 024
b
r4 04
B frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 054a=b4 12
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
(Data from Refs 12 and 13)
31 Shaft with one spline K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 14
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
r3where B varies with
b
ras follows For 04
b
r4 05
B frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 14
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
(Data from Refs 12 and 13)
410
FormulasforStressandStrain
[CHAP10
32 Shaft with two splines K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 14
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
r3where B varies with
b
ras follows For 04
b
r4 05
B frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 14
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
(Data from Refs 12 and 13)
33 Shaft with four splines K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 10
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
r3where B varies with
b
ras follows For 04
b
r4 05
B frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 10
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
(Data from Refs 12 and 13)
SEC107]
Torsion
411
TABLE 101 Formulas for torsional deformation and stress (Continued)
TABLE 101 Formulas for torsional deformation and stress (Continued)
Form and dimensions of cross sections
other quantities involved and case no Formula for K in y frac14TL
KGFormula for shear stress
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
r3where B varies with
a
rover the
range 014a
r4 05 as follows For one groove
B frac14 10259 thorn 11802a
r 27897
a
r
2
thorn 37092a
r
3
For two grooves
B frac14 10055 thorn 15427a
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
(Data from Refs 12 and 13)
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
(Data from Refs 12 and 13)
412
FormulasforStressandStrain
[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
etht1THORNmax frac14 tGy0 at the surface everywhere
3 Hat sectione frac14 b
3h2b thorn 6h2b1 8b31
h3 thorn 6h2b thorn 6h2b1 thorn 8b31 thorn 12hb2
1
K frac14t3
3ethh thorn 2b thorn 2b1THORN
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
Ey00 throughout the thickness at corners A and F
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 )
Cross section reference no Constants Selected maximum values
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
etht1THORNmax frac14 tGy0 at the surface everywhere
5 Twin channel with
flanges outwardK frac14
t3
3eth2b thorn 4b1THORN
Cw frac14tb2
24eth8b3
1 thorn 6h2b1 thorn h2b 12b21hTHORN
ethsxTHORNmax frac14hb
4Ey00 throughout the thickness at points B and C if h gt b1
ethsxTHORNmax frac14hb
4
bb1
2
Ey00 throughout the thickness at points A and D if h lt b1
etht2THORNmax frac14b
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
etht2THORNmax frac14b
4b2
1 hb
4 hb1
Ey000 throughout the thickness at a point midway between corners B and C if
b1 lth
21 thorn
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
2thorn
b
2h
r
etht1THORNmax frac14 tGy0 at the surface everywhere
6 Wide flanged beam
with equal flanges
K frac14 13eth2t3b thorn t3
whTHORN
Cw frac14h2tb3
24
ethsxTHORNmax frac14hb
4Ey00 throughout the thickness at points A and B
etht2THORNmax frac14 hb2
16Ey000 throughout the thickness at a point midway between A and B
etht1THORNmax frac14 tGy0 at the surface everywhere
414
FormulasforStressandStrain
[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
ethsxTHORNmax frac14hb2
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
etht2THORNmax frac141
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
etht2THORNmax frac14hb2
4
b thorn h
2b thorn h
2
Ey000 throughout the thickness at a distancebethb thorn hTHORN
2b thorn hfrom point A
etht1THORNmax frac14 tGy0 at the surface everywhere
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
etht1THORNmax frac14 tGy0 at the surface everywhere
SEC107]
Torsion
415
TABLE 102 Formulas for torsional properties and stresses in thin-walled open cross sections (Continued)
TABLE 102 Formulas for torsional properties and stresses in thin-walled open cross sections (Continued )
Cross section reference no Constants Selected maximum values
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
etht1THORNmax frac14 tGy0 at the surface everywhere
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
etht1THORNmax frac14 tGy0 at the surface everywhere
416
FormulasforStressandStrain
[CHAP10
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tgt U0 -codo es ~ b35 Om ~ B1- 5 erA) (S q1 1 CTA) 5f Crt g CYIJ_ (I)
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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 ~
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
K frac14 2Cr4 where C varies withh
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
r3where B varies with
h
ras follows For two flat sides where
04h
r4 06
B frac14 06366 thorn 25303h
r 11157
h
r
2
thorn 49568h
r
3
85886h
r
4
thorn 69849h
r
5
For four flat sides where 04h
r4 0293
B frac14 06366 thorn 26298h
r 56147
h
r
2
thorn 30853h
r
3
(Data from Refs 12 and 13)
SEC107]
Torsion
403
TABLE 101 Formulas for torsional deformation and stress (Continued)
TABLE 101 Formulas for torsional deformation and stress (Continued)
Form and dimensions of cross sections
other quantities involved and case no Formula for K in y frac14TL
KGFormula for shear stress
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
eth1 n2THORNeth1 n4THORNl2
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
FormulasforStressandStrain
[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)
TABLE 101 Formulas for torsional deformation and stress (Continued )
Form and dimensions of cross sections
other quantities involved and case no Formula for K in y frac14TL
KGFormula for shear stress
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
A frac14area of section
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
FormulasforStressandStrain
[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
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 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
TABLE 101 Formulas for torsional deformation and stress (Continued)
TABLE 101 Formulas for torsional deformation and stress (Continued )
Form and dimensions of cross sections
other quantities involved and case no Formula for K in y frac14TL
KGFormula for shear stress
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
where K1 frac14 ab3 1
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
(Data from Refs 12 and 13)
408
FormulasforStressandStrain
[CHAP10
28 Shaft with one keyway K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
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
r3where B varies with
b
ras follows For 024
b
r4 05
B frac14 K1 thorn K2
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
(Data from Refs 12 and 13)
29 Shaft with two keyways K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 034a=b4 15
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
r3where B varies with
b
ras follows For 024
b
r4 05
B frac14 K1 thorn K2
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
(Data from Refs 12 and 13)
SEC107]
Torsion
409
TABLE 101 Formulas for torsional deformation and stress (Continued)
TABLE 101 Formulas for torsional deformation and stress (Continued)
Form and dimensions of cross sections
other quantities involved and case no Formula for K in y frac14TL
KGFormula for shear stress
30 Shaft with four keyways K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 04
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 034a=b4 12
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
r3where B varies with
b
ras follows For 024
b
r4 04
B frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 054a=b4 12
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
(Data from Refs 12 and 13)
31 Shaft with one spline K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 14
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
r3where B varies with
b
ras follows For 04
b
r4 05
B frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 14
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
(Data from Refs 12 and 13)
410
FormulasforStressandStrain
[CHAP10
32 Shaft with two splines K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 14
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
r3where B varies with
b
ras follows For 04
b
r4 05
B frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 14
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
(Data from Refs 12 and 13)
33 Shaft with four splines K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 10
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
r3where B varies with
b
ras follows For 04
b
r4 05
B frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 10
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
(Data from Refs 12 and 13)
SEC107]
Torsion
411
TABLE 101 Formulas for torsional deformation and stress (Continued)
TABLE 101 Formulas for torsional deformation and stress (Continued)
Form and dimensions of cross sections
other quantities involved and case no Formula for K in y frac14TL
KGFormula for shear stress
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
r3where B varies with
a
rover the
range 014a
r4 05 as follows For one groove
B frac14 10259 thorn 11802a
r 27897
a
r
2
thorn 37092a
r
3
For two grooves
B frac14 10055 thorn 15427a
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
(Data from Refs 12 and 13)
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
(Data from Refs 12 and 13)
412
FormulasforStressandStrain
[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
etht1THORNmax frac14 tGy0 at the surface everywhere
3 Hat sectione frac14 b
3h2b thorn 6h2b1 8b31
h3 thorn 6h2b thorn 6h2b1 thorn 8b31 thorn 12hb2
1
K frac14t3
3ethh thorn 2b thorn 2b1THORN
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
Ey00 throughout the thickness at corners A and F
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 )
Cross section reference no Constants Selected maximum values
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
etht1THORNmax frac14 tGy0 at the surface everywhere
5 Twin channel with
flanges outwardK frac14
t3
3eth2b thorn 4b1THORN
Cw frac14tb2
24eth8b3
1 thorn 6h2b1 thorn h2b 12b21hTHORN
ethsxTHORNmax frac14hb
4Ey00 throughout the thickness at points B and C if h gt b1
ethsxTHORNmax frac14hb
4
bb1
2
Ey00 throughout the thickness at points A and D if h lt b1
etht2THORNmax frac14b
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
etht2THORNmax frac14b
4b2
1 hb
4 hb1
Ey000 throughout the thickness at a point midway between corners B and C if
b1 lth
21 thorn
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
2thorn
b
2h
r
etht1THORNmax frac14 tGy0 at the surface everywhere
6 Wide flanged beam
with equal flanges
K frac14 13eth2t3b thorn t3
whTHORN
Cw frac14h2tb3
24
ethsxTHORNmax frac14hb
4Ey00 throughout the thickness at points A and B
etht2THORNmax frac14 hb2
16Ey000 throughout the thickness at a point midway between A and B
etht1THORNmax frac14 tGy0 at the surface everywhere
414
FormulasforStressandStrain
[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
ethsxTHORNmax frac14hb2
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
etht2THORNmax frac141
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
etht2THORNmax frac14hb2
4
b thorn h
2b thorn h
2
Ey000 throughout the thickness at a distancebethb thorn hTHORN
2b thorn hfrom point A
etht1THORNmax frac14 tGy0 at the surface everywhere
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
etht1THORNmax frac14 tGy0 at the surface everywhere
SEC107]
Torsion
415
TABLE 102 Formulas for torsional properties and stresses in thin-walled open cross sections (Continued)
TABLE 102 Formulas for torsional properties and stresses in thin-walled open cross sections (Continued )
Cross section reference no Constants Selected maximum values
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
etht1THORNmax frac14 tGy0 at the surface everywhere
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
etht1THORNmax frac14 tGy0 at the surface everywhere
416
FormulasforStressandStrain
[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 stress (Continued)
Form and dimensions of cross sections
other quantities involved and case no Formula for K in y frac14TL
KGFormula for shear stress
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
eth1 n2THORNeth1 n4THORNl2
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
FormulasforStressandStrain
[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)
TABLE 101 Formulas for torsional deformation and stress (Continued )
Form and dimensions of cross sections
other quantities involved and case no Formula for K in y frac14TL
KGFormula for shear stress
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
A frac14area of section
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
FormulasforStressandStrain
[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
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 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
TABLE 101 Formulas for torsional deformation and stress (Continued)
TABLE 101 Formulas for torsional deformation and stress (Continued )
Form and dimensions of cross sections
other quantities involved and case no Formula for K in y frac14TL
KGFormula for shear stress
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
where K1 frac14 ab3 1
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
(Data from Refs 12 and 13)
408
FormulasforStressandStrain
[CHAP10
28 Shaft with one keyway K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
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
r3where B varies with
b
ras follows For 024
b
r4 05
B frac14 K1 thorn K2
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
(Data from Refs 12 and 13)
29 Shaft with two keyways K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 034a=b4 15
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
r3where B varies with
b
ras follows For 024
b
r4 05
B frac14 K1 thorn K2
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
(Data from Refs 12 and 13)
SEC107]
Torsion
409
TABLE 101 Formulas for torsional deformation and stress (Continued)
TABLE 101 Formulas for torsional deformation and stress (Continued)
Form and dimensions of cross sections
other quantities involved and case no Formula for K in y frac14TL
KGFormula for shear stress
30 Shaft with four keyways K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 04
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 034a=b4 12
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
r3where B varies with
b
ras follows For 024
b
r4 04
B frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 054a=b4 12
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
(Data from Refs 12 and 13)
31 Shaft with one spline K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 14
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
r3where B varies with
b
ras follows For 04
b
r4 05
B frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 14
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
(Data from Refs 12 and 13)
410
FormulasforStressandStrain
[CHAP10
32 Shaft with two splines K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 14
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
r3where B varies with
b
ras follows For 04
b
r4 05
B frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 14
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
(Data from Refs 12 and 13)
33 Shaft with four splines K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 10
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
r3where B varies with
b
ras follows For 04
b
r4 05
B frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 10
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
(Data from Refs 12 and 13)
SEC107]
Torsion
411
TABLE 101 Formulas for torsional deformation and stress (Continued)
TABLE 101 Formulas for torsional deformation and stress (Continued)
Form and dimensions of cross sections
other quantities involved and case no Formula for K in y frac14TL
KGFormula for shear stress
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
r3where B varies with
a
rover the
range 014a
r4 05 as follows For one groove
B frac14 10259 thorn 11802a
r 27897
a
r
2
thorn 37092a
r
3
For two grooves
B frac14 10055 thorn 15427a
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
(Data from Refs 12 and 13)
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
(Data from Refs 12 and 13)
412
FormulasforStressandStrain
[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
etht1THORNmax frac14 tGy0 at the surface everywhere
3 Hat sectione frac14 b
3h2b thorn 6h2b1 8b31
h3 thorn 6h2b thorn 6h2b1 thorn 8b31 thorn 12hb2
1
K frac14t3
3ethh thorn 2b thorn 2b1THORN
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
Ey00 throughout the thickness at corners A and F
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 )
Cross section reference no Constants Selected maximum values
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
etht1THORNmax frac14 tGy0 at the surface everywhere
5 Twin channel with
flanges outwardK frac14
t3
3eth2b thorn 4b1THORN
Cw frac14tb2
24eth8b3
1 thorn 6h2b1 thorn h2b 12b21hTHORN
ethsxTHORNmax frac14hb
4Ey00 throughout the thickness at points B and C if h gt b1
ethsxTHORNmax frac14hb
4
bb1
2
Ey00 throughout the thickness at points A and D if h lt b1
etht2THORNmax frac14b
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
etht2THORNmax frac14b
4b2
1 hb
4 hb1
Ey000 throughout the thickness at a point midway between corners B and C if
b1 lth
21 thorn
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
2thorn
b
2h
r
etht1THORNmax frac14 tGy0 at the surface everywhere
6 Wide flanged beam
with equal flanges
K frac14 13eth2t3b thorn t3
whTHORN
Cw frac14h2tb3
24
ethsxTHORNmax frac14hb
4Ey00 throughout the thickness at points A and B
etht2THORNmax frac14 hb2
16Ey000 throughout the thickness at a point midway between A and B
etht1THORNmax frac14 tGy0 at the surface everywhere
414
FormulasforStressandStrain
[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
ethsxTHORNmax frac14hb2
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
etht2THORNmax frac141
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
etht2THORNmax frac14hb2
4
b thorn h
2b thorn h
2
Ey000 throughout the thickness at a distancebethb thorn hTHORN
2b thorn hfrom point A
etht1THORNmax frac14 tGy0 at the surface everywhere
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
etht1THORNmax frac14 tGy0 at the surface everywhere
SEC107]
Torsion
415
TABLE 102 Formulas for torsional properties and stresses in thin-walled open cross sections (Continued)
TABLE 102 Formulas for torsional properties and stresses in thin-walled open cross sections (Continued )
Cross section reference no Constants Selected maximum values
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
etht1THORNmax frac14 tGy0 at the surface everywhere
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
etht1THORNmax frac14 tGy0 at the surface everywhere
416
FormulasforStressandStrain
[CHAP10
T
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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
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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)
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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
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r aI~ vo e ~ ~ - fl - (11-S00 (Cj-cYt) (200(Y)
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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
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KG l 5Go~5bc~4(1G0 12)Ol~ tltCjfcM2)
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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 ~
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)
TABLE 101 Formulas for torsional deformation and stress (Continued )
Form and dimensions of cross sections
other quantities involved and case no Formula for K in y frac14TL
KGFormula for shear stress
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
A frac14area of section
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
FormulasforStressandStrain
[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
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 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
TABLE 101 Formulas for torsional deformation and stress (Continued)
TABLE 101 Formulas for torsional deformation and stress (Continued )
Form and dimensions of cross sections
other quantities involved and case no Formula for K in y frac14TL
KGFormula for shear stress
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
where K1 frac14 ab3 1
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
(Data from Refs 12 and 13)
408
FormulasforStressandStrain
[CHAP10
28 Shaft with one keyway K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
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
r3where B varies with
b
ras follows For 024
b
r4 05
B frac14 K1 thorn K2
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
(Data from Refs 12 and 13)
29 Shaft with two keyways K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 034a=b4 15
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
r3where B varies with
b
ras follows For 024
b
r4 05
B frac14 K1 thorn K2
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
(Data from Refs 12 and 13)
SEC107]
Torsion
409
TABLE 101 Formulas for torsional deformation and stress (Continued)
TABLE 101 Formulas for torsional deformation and stress (Continued)
Form and dimensions of cross sections
other quantities involved and case no Formula for K in y frac14TL
KGFormula for shear stress
30 Shaft with four keyways K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 04
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 034a=b4 12
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
r3where B varies with
b
ras follows For 024
b
r4 04
B frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 054a=b4 12
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
(Data from Refs 12 and 13)
31 Shaft with one spline K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 14
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
r3where B varies with
b
ras follows For 04
b
r4 05
B frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 14
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
(Data from Refs 12 and 13)
410
FormulasforStressandStrain
[CHAP10
32 Shaft with two splines K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 14
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
r3where B varies with
b
ras follows For 04
b
r4 05
B frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 14
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
(Data from Refs 12 and 13)
33 Shaft with four splines K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 10
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
r3where B varies with
b
ras follows For 04
b
r4 05
B frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 10
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
(Data from Refs 12 and 13)
SEC107]
Torsion
411
TABLE 101 Formulas for torsional deformation and stress (Continued)
TABLE 101 Formulas for torsional deformation and stress (Continued)
Form and dimensions of cross sections
other quantities involved and case no Formula for K in y frac14TL
KGFormula for shear stress
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
r3where B varies with
a
rover the
range 014a
r4 05 as follows For one groove
B frac14 10259 thorn 11802a
r 27897
a
r
2
thorn 37092a
r
3
For two grooves
B frac14 10055 thorn 15427a
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
(Data from Refs 12 and 13)
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
(Data from Refs 12 and 13)
412
FormulasforStressandStrain
[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
etht1THORNmax frac14 tGy0 at the surface everywhere
3 Hat sectione frac14 b
3h2b thorn 6h2b1 8b31
h3 thorn 6h2b thorn 6h2b1 thorn 8b31 thorn 12hb2
1
K frac14t3
3ethh thorn 2b thorn 2b1THORN
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
Ey00 throughout the thickness at corners A and F
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 )
Cross section reference no Constants Selected maximum values
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
etht1THORNmax frac14 tGy0 at the surface everywhere
5 Twin channel with
flanges outwardK frac14
t3
3eth2b thorn 4b1THORN
Cw frac14tb2
24eth8b3
1 thorn 6h2b1 thorn h2b 12b21hTHORN
ethsxTHORNmax frac14hb
4Ey00 throughout the thickness at points B and C if h gt b1
ethsxTHORNmax frac14hb
4
bb1
2
Ey00 throughout the thickness at points A and D if h lt b1
etht2THORNmax frac14b
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
etht2THORNmax frac14b
4b2
1 hb
4 hb1
Ey000 throughout the thickness at a point midway between corners B and C if
b1 lth
21 thorn
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
2thorn
b
2h
r
etht1THORNmax frac14 tGy0 at the surface everywhere
6 Wide flanged beam
with equal flanges
K frac14 13eth2t3b thorn t3
whTHORN
Cw frac14h2tb3
24
ethsxTHORNmax frac14hb
4Ey00 throughout the thickness at points A and B
etht2THORNmax frac14 hb2
16Ey000 throughout the thickness at a point midway between A and B
etht1THORNmax frac14 tGy0 at the surface everywhere
414
FormulasforStressandStrain
[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
ethsxTHORNmax frac14hb2
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
etht2THORNmax frac141
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
etht2THORNmax frac14hb2
4
b thorn h
2b thorn h
2
Ey000 throughout the thickness at a distancebethb thorn hTHORN
2b thorn hfrom point A
etht1THORNmax frac14 tGy0 at the surface everywhere
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
etht1THORNmax frac14 tGy0 at the surface everywhere
SEC107]
Torsion
415
TABLE 102 Formulas for torsional properties and stresses in thin-walled open cross sections (Continued)
TABLE 102 Formulas for torsional properties and stresses in thin-walled open cross sections (Continued )
Cross section reference no Constants Selected maximum values
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
etht1THORNmax frac14 tGy0 at the surface everywhere
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
etht1THORNmax frac14 tGy0 at the surface everywhere
416
FormulasforStressandStrain
[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 stress (Continued )
Form and dimensions of cross sections
other quantities involved and case no Formula for K in y frac14TL
KGFormula for shear stress
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
A frac14area of section
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
FormulasforStressandStrain
[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
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 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
TABLE 101 Formulas for torsional deformation and stress (Continued)
TABLE 101 Formulas for torsional deformation and stress (Continued )
Form and dimensions of cross sections
other quantities involved and case no Formula for K in y frac14TL
KGFormula for shear stress
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
where K1 frac14 ab3 1
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
(Data from Refs 12 and 13)
408
FormulasforStressandStrain
[CHAP10
28 Shaft with one keyway K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
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
r3where B varies with
b
ras follows For 024
b
r4 05
B frac14 K1 thorn K2
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
(Data from Refs 12 and 13)
29 Shaft with two keyways K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 034a=b4 15
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
r3where B varies with
b
ras follows For 024
b
r4 05
B frac14 K1 thorn K2
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
(Data from Refs 12 and 13)
SEC107]
Torsion
409
TABLE 101 Formulas for torsional deformation and stress (Continued)
TABLE 101 Formulas for torsional deformation and stress (Continued)
Form and dimensions of cross sections
other quantities involved and case no Formula for K in y frac14TL
KGFormula for shear stress
30 Shaft with four keyways K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 04
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 034a=b4 12
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
r3where B varies with
b
ras follows For 024
b
r4 04
B frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 054a=b4 12
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
(Data from Refs 12 and 13)
31 Shaft with one spline K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 14
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
r3where B varies with
b
ras follows For 04
b
r4 05
B frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 14
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
(Data from Refs 12 and 13)
410
FormulasforStressandStrain
[CHAP10
32 Shaft with two splines K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 14
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
r3where B varies with
b
ras follows For 04
b
r4 05
B frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 14
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
(Data from Refs 12 and 13)
33 Shaft with four splines K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 10
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
r3where B varies with
b
ras follows For 04
b
r4 05
B frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 10
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
(Data from Refs 12 and 13)
SEC107]
Torsion
411
TABLE 101 Formulas for torsional deformation and stress (Continued)
TABLE 101 Formulas for torsional deformation and stress (Continued)
Form and dimensions of cross sections
other quantities involved and case no Formula for K in y frac14TL
KGFormula for shear stress
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
r3where B varies with
a
rover the
range 014a
r4 05 as follows For one groove
B frac14 10259 thorn 11802a
r 27897
a
r
2
thorn 37092a
r
3
For two grooves
B frac14 10055 thorn 15427a
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
(Data from Refs 12 and 13)
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
(Data from Refs 12 and 13)
412
FormulasforStressandStrain
[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
etht1THORNmax frac14 tGy0 at the surface everywhere
3 Hat sectione frac14 b
3h2b thorn 6h2b1 8b31
h3 thorn 6h2b thorn 6h2b1 thorn 8b31 thorn 12hb2
1
K frac14t3
3ethh thorn 2b thorn 2b1THORN
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
Ey00 throughout the thickness at corners A and F
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 )
Cross section reference no Constants Selected maximum values
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
etht1THORNmax frac14 tGy0 at the surface everywhere
5 Twin channel with
flanges outwardK frac14
t3
3eth2b thorn 4b1THORN
Cw frac14tb2
24eth8b3
1 thorn 6h2b1 thorn h2b 12b21hTHORN
ethsxTHORNmax frac14hb
4Ey00 throughout the thickness at points B and C if h gt b1
ethsxTHORNmax frac14hb
4
bb1
2
Ey00 throughout the thickness at points A and D if h lt b1
etht2THORNmax frac14b
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
etht2THORNmax frac14b
4b2
1 hb
4 hb1
Ey000 throughout the thickness at a point midway between corners B and C if
b1 lth
21 thorn
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
2thorn
b
2h
r
etht1THORNmax frac14 tGy0 at the surface everywhere
6 Wide flanged beam
with equal flanges
K frac14 13eth2t3b thorn t3
whTHORN
Cw frac14h2tb3
24
ethsxTHORNmax frac14hb
4Ey00 throughout the thickness at points A and B
etht2THORNmax frac14 hb2
16Ey000 throughout the thickness at a point midway between A and B
etht1THORNmax frac14 tGy0 at the surface everywhere
414
FormulasforStressandStrain
[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
ethsxTHORNmax frac14hb2
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
etht2THORNmax frac141
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
etht2THORNmax frac14hb2
4
b thorn h
2b thorn h
2
Ey000 throughout the thickness at a distancebethb thorn hTHORN
2b thorn hfrom point A
etht1THORNmax frac14 tGy0 at the surface everywhere
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
etht1THORNmax frac14 tGy0 at the surface everywhere
SEC107]
Torsion
415
TABLE 102 Formulas for torsional properties and stresses in thin-walled open cross sections (Continued)
TABLE 102 Formulas for torsional properties and stresses in thin-walled open cross sections (Continued )
Cross section reference no Constants Selected maximum values
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
etht1THORNmax frac14 tGy0 at the surface everywhere
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
etht1THORNmax frac14 tGy0 at the surface everywhere
416
FormulasforStressandStrain
[CHAP10
T
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~ =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 ~
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
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 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
TABLE 101 Formulas for torsional deformation and stress (Continued)
TABLE 101 Formulas for torsional deformation and stress (Continued )
Form and dimensions of cross sections
other quantities involved and case no Formula for K in y frac14TL
KGFormula for shear stress
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
where K1 frac14 ab3 1
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
(Data from Refs 12 and 13)
408
FormulasforStressandStrain
[CHAP10
28 Shaft with one keyway K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
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
r3where B varies with
b
ras follows For 024
b
r4 05
B frac14 K1 thorn K2
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
(Data from Refs 12 and 13)
29 Shaft with two keyways K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 034a=b4 15
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
r3where B varies with
b
ras follows For 024
b
r4 05
B frac14 K1 thorn K2
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
(Data from Refs 12 and 13)
SEC107]
Torsion
409
TABLE 101 Formulas for torsional deformation and stress (Continued)
TABLE 101 Formulas for torsional deformation and stress (Continued)
Form and dimensions of cross sections
other quantities involved and case no Formula for K in y frac14TL
KGFormula for shear stress
30 Shaft with four keyways K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 04
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 034a=b4 12
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
r3where B varies with
b
ras follows For 024
b
r4 04
B frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 054a=b4 12
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
(Data from Refs 12 and 13)
31 Shaft with one spline K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 14
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
r3where B varies with
b
ras follows For 04
b
r4 05
B frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 14
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
(Data from Refs 12 and 13)
410
FormulasforStressandStrain
[CHAP10
32 Shaft with two splines K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 14
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
r3where B varies with
b
ras follows For 04
b
r4 05
B frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 14
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
(Data from Refs 12 and 13)
33 Shaft with four splines K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 10
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
r3where B varies with
b
ras follows For 04
b
r4 05
B frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 10
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
(Data from Refs 12 and 13)
SEC107]
Torsion
411
TABLE 101 Formulas for torsional deformation and stress (Continued)
TABLE 101 Formulas for torsional deformation and stress (Continued)
Form and dimensions of cross sections
other quantities involved and case no Formula for K in y frac14TL
KGFormula for shear stress
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
r3where B varies with
a
rover the
range 014a
r4 05 as follows For one groove
B frac14 10259 thorn 11802a
r 27897
a
r
2
thorn 37092a
r
3
For two grooves
B frac14 10055 thorn 15427a
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
(Data from Refs 12 and 13)
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
(Data from Refs 12 and 13)
412
FormulasforStressandStrain
[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
etht1THORNmax frac14 tGy0 at the surface everywhere
3 Hat sectione frac14 b
3h2b thorn 6h2b1 8b31
h3 thorn 6h2b thorn 6h2b1 thorn 8b31 thorn 12hb2
1
K frac14t3
3ethh thorn 2b thorn 2b1THORN
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
Ey00 throughout the thickness at corners A and F
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 )
Cross section reference no Constants Selected maximum values
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
etht1THORNmax frac14 tGy0 at the surface everywhere
5 Twin channel with
flanges outwardK frac14
t3
3eth2b thorn 4b1THORN
Cw frac14tb2
24eth8b3
1 thorn 6h2b1 thorn h2b 12b21hTHORN
ethsxTHORNmax frac14hb
4Ey00 throughout the thickness at points B and C if h gt b1
ethsxTHORNmax frac14hb
4
bb1
2
Ey00 throughout the thickness at points A and D if h lt b1
etht2THORNmax frac14b
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
etht2THORNmax frac14b
4b2
1 hb
4 hb1
Ey000 throughout the thickness at a point midway between corners B and C if
b1 lth
21 thorn
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
2thorn
b
2h
r
etht1THORNmax frac14 tGy0 at the surface everywhere
6 Wide flanged beam
with equal flanges
K frac14 13eth2t3b thorn t3
whTHORN
Cw frac14h2tb3
24
ethsxTHORNmax frac14hb
4Ey00 throughout the thickness at points A and B
etht2THORNmax frac14 hb2
16Ey000 throughout the thickness at a point midway between A and B
etht1THORNmax frac14 tGy0 at the surface everywhere
414
FormulasforStressandStrain
[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
ethsxTHORNmax frac14hb2
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
etht2THORNmax frac141
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
etht2THORNmax frac14hb2
4
b thorn h
2b thorn h
2
Ey000 throughout the thickness at a distancebethb thorn hTHORN
2b thorn hfrom point A
etht1THORNmax frac14 tGy0 at the surface everywhere
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
etht1THORNmax frac14 tGy0 at the surface everywhere
SEC107]
Torsion
415
TABLE 102 Formulas for torsional properties and stresses in thin-walled open cross sections (Continued)
TABLE 102 Formulas for torsional properties and stresses in thin-walled open cross sections (Continued )
Cross section reference no Constants Selected maximum values
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
etht1THORNmax frac14 tGy0 at the surface everywhere
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
etht1THORNmax frac14 tGy0 at the surface everywhere
416
FormulasforStressandStrain
[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 stress (Continued )
Form and dimensions of cross sections
other quantities involved and case no Formula for K in y frac14TL
KGFormula for shear stress
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
where K1 frac14 ab3 1
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
(Data from Refs 12 and 13)
408
FormulasforStressandStrain
[CHAP10
28 Shaft with one keyway K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
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
r3where B varies with
b
ras follows For 024
b
r4 05
B frac14 K1 thorn K2
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
(Data from Refs 12 and 13)
29 Shaft with two keyways K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 034a=b4 15
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
r3where B varies with
b
ras follows For 024
b
r4 05
B frac14 K1 thorn K2
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
(Data from Refs 12 and 13)
SEC107]
Torsion
409
TABLE 101 Formulas for torsional deformation and stress (Continued)
TABLE 101 Formulas for torsional deformation and stress (Continued)
Form and dimensions of cross sections
other quantities involved and case no Formula for K in y frac14TL
KGFormula for shear stress
30 Shaft with four keyways K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 04
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 034a=b4 12
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
r3where B varies with
b
ras follows For 024
b
r4 04
B frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 054a=b4 12
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
(Data from Refs 12 and 13)
31 Shaft with one spline K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 14
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
r3where B varies with
b
ras follows For 04
b
r4 05
B frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 14
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
(Data from Refs 12 and 13)
410
FormulasforStressandStrain
[CHAP10
32 Shaft with two splines K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 14
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
r3where B varies with
b
ras follows For 04
b
r4 05
B frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 14
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
(Data from Refs 12 and 13)
33 Shaft with four splines K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 10
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
r3where B varies with
b
ras follows For 04
b
r4 05
B frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 10
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
(Data from Refs 12 and 13)
SEC107]
Torsion
411
TABLE 101 Formulas for torsional deformation and stress (Continued)
TABLE 101 Formulas for torsional deformation and stress (Continued)
Form and dimensions of cross sections
other quantities involved and case no Formula for K in y frac14TL
KGFormula for shear stress
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
r3where B varies with
a
rover the
range 014a
r4 05 as follows For one groove
B frac14 10259 thorn 11802a
r 27897
a
r
2
thorn 37092a
r
3
For two grooves
B frac14 10055 thorn 15427a
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
(Data from Refs 12 and 13)
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
(Data from Refs 12 and 13)
412
FormulasforStressandStrain
[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
etht1THORNmax frac14 tGy0 at the surface everywhere
3 Hat sectione frac14 b
3h2b thorn 6h2b1 8b31
h3 thorn 6h2b thorn 6h2b1 thorn 8b31 thorn 12hb2
1
K frac14t3
3ethh thorn 2b thorn 2b1THORN
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
Ey00 throughout the thickness at corners A and F
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 )
Cross section reference no Constants Selected maximum values
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
etht1THORNmax frac14 tGy0 at the surface everywhere
5 Twin channel with
flanges outwardK frac14
t3
3eth2b thorn 4b1THORN
Cw frac14tb2
24eth8b3
1 thorn 6h2b1 thorn h2b 12b21hTHORN
ethsxTHORNmax frac14hb
4Ey00 throughout the thickness at points B and C if h gt b1
ethsxTHORNmax frac14hb
4
bb1
2
Ey00 throughout the thickness at points A and D if h lt b1
etht2THORNmax frac14b
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
etht2THORNmax frac14b
4b2
1 hb
4 hb1
Ey000 throughout the thickness at a point midway between corners B and C if
b1 lth
21 thorn
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
2thorn
b
2h
r
etht1THORNmax frac14 tGy0 at the surface everywhere
6 Wide flanged beam
with equal flanges
K frac14 13eth2t3b thorn t3
whTHORN
Cw frac14h2tb3
24
ethsxTHORNmax frac14hb
4Ey00 throughout the thickness at points A and B
etht2THORNmax frac14 hb2
16Ey000 throughout the thickness at a point midway between A and B
etht1THORNmax frac14 tGy0 at the surface everywhere
414
FormulasforStressandStrain
[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
ethsxTHORNmax frac14hb2
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
etht2THORNmax frac141
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
etht2THORNmax frac14hb2
4
b thorn h
2b thorn h
2
Ey000 throughout the thickness at a distancebethb thorn hTHORN
2b thorn hfrom point A
etht1THORNmax frac14 tGy0 at the surface everywhere
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
etht1THORNmax frac14 tGy0 at the surface everywhere
SEC107]
Torsion
415
TABLE 102 Formulas for torsional properties and stresses in thin-walled open cross sections (Continued)
TABLE 102 Formulas for torsional properties and stresses in thin-walled open cross sections (Continued )
Cross section reference no Constants Selected maximum values
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
etht1THORNmax frac14 tGy0 at the surface everywhere
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
etht1THORNmax frac14 tGy0 at the surface everywhere
416
FormulasforStressandStrain
[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 ~
28 Shaft with one keyway K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
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
r3where B varies with
b
ras follows For 024
b
r4 05
B frac14 K1 thorn K2
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
(Data from Refs 12 and 13)
29 Shaft with two keyways K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 034a=b4 15
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
r3where B varies with
b
ras follows For 024
b
r4 05
B frac14 K1 thorn K2
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
(Data from Refs 12 and 13)
SEC107]
Torsion
409
TABLE 101 Formulas for torsional deformation and stress (Continued)
TABLE 101 Formulas for torsional deformation and stress (Continued)
Form and dimensions of cross sections
other quantities involved and case no Formula for K in y frac14TL
KGFormula for shear stress
30 Shaft with four keyways K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 04
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 034a=b4 12
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
r3where B varies with
b
ras follows For 024
b
r4 04
B frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 054a=b4 12
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
(Data from Refs 12 and 13)
31 Shaft with one spline K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 14
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
r3where B varies with
b
ras follows For 04
b
r4 05
B frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 14
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
(Data from Refs 12 and 13)
410
FormulasforStressandStrain
[CHAP10
32 Shaft with two splines K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 14
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
r3where B varies with
b
ras follows For 04
b
r4 05
B frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 14
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
(Data from Refs 12 and 13)
33 Shaft with four splines K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 10
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
r3where B varies with
b
ras follows For 04
b
r4 05
B frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 10
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
(Data from Refs 12 and 13)
SEC107]
Torsion
411
TABLE 101 Formulas for torsional deformation and stress (Continued)
TABLE 101 Formulas for torsional deformation and stress (Continued)
Form and dimensions of cross sections
other quantities involved and case no Formula for K in y frac14TL
KGFormula for shear stress
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
r3where B varies with
a
rover the
range 014a
r4 05 as follows For one groove
B frac14 10259 thorn 11802a
r 27897
a
r
2
thorn 37092a
r
3
For two grooves
B frac14 10055 thorn 15427a
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
(Data from Refs 12 and 13)
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
(Data from Refs 12 and 13)
412
FormulasforStressandStrain
[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
etht1THORNmax frac14 tGy0 at the surface everywhere
3 Hat sectione frac14 b
3h2b thorn 6h2b1 8b31
h3 thorn 6h2b thorn 6h2b1 thorn 8b31 thorn 12hb2
1
K frac14t3
3ethh thorn 2b thorn 2b1THORN
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
Ey00 throughout the thickness at corners A and F
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 )
Cross section reference no Constants Selected maximum values
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
etht1THORNmax frac14 tGy0 at the surface everywhere
5 Twin channel with
flanges outwardK frac14
t3
3eth2b thorn 4b1THORN
Cw frac14tb2
24eth8b3
1 thorn 6h2b1 thorn h2b 12b21hTHORN
ethsxTHORNmax frac14hb
4Ey00 throughout the thickness at points B and C if h gt b1
ethsxTHORNmax frac14hb
4
bb1
2
Ey00 throughout the thickness at points A and D if h lt b1
etht2THORNmax frac14b
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
etht2THORNmax frac14b
4b2
1 hb
4 hb1
Ey000 throughout the thickness at a point midway between corners B and C if
b1 lth
21 thorn
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
2thorn
b
2h
r
etht1THORNmax frac14 tGy0 at the surface everywhere
6 Wide flanged beam
with equal flanges
K frac14 13eth2t3b thorn t3
whTHORN
Cw frac14h2tb3
24
ethsxTHORNmax frac14hb
4Ey00 throughout the thickness at points A and B
etht2THORNmax frac14 hb2
16Ey000 throughout the thickness at a point midway between A and B
etht1THORNmax frac14 tGy0 at the surface everywhere
414
FormulasforStressandStrain
[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
ethsxTHORNmax frac14hb2
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
etht2THORNmax frac141
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
etht2THORNmax frac14hb2
4
b thorn h
2b thorn h
2
Ey000 throughout the thickness at a distancebethb thorn hTHORN
2b thorn hfrom point A
etht1THORNmax frac14 tGy0 at the surface everywhere
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
etht1THORNmax frac14 tGy0 at the surface everywhere
SEC107]
Torsion
415
TABLE 102 Formulas for torsional properties and stresses in thin-walled open cross sections (Continued)
TABLE 102 Formulas for torsional properties and stresses in thin-walled open cross sections (Continued )
Cross section reference no Constants Selected maximum values
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
etht1THORNmax frac14 tGy0 at the surface everywhere
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
etht1THORNmax frac14 tGy0 at the surface everywhere
416
FormulasforStressandStrain
[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
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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 stress (Continued)
Form and dimensions of cross sections
other quantities involved and case no Formula for K in y frac14TL
KGFormula for shear stress
30 Shaft with four keyways K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 04
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 034a=b4 12
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
r3where B varies with
b
ras follows For 024
b
r4 04
B frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 054a=b4 12
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
(Data from Refs 12 and 13)
31 Shaft with one spline K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 14
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
r3where B varies with
b
ras follows For 04
b
r4 05
B frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 14
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
(Data from Refs 12 and 13)
410
FormulasforStressandStrain
[CHAP10
32 Shaft with two splines K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 14
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
r3where B varies with
b
ras follows For 04
b
r4 05
B frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 14
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
(Data from Refs 12 and 13)
33 Shaft with four splines K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 10
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
r3where B varies with
b
ras follows For 04
b
r4 05
B frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 10
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
(Data from Refs 12 and 13)
SEC107]
Torsion
411
TABLE 101 Formulas for torsional deformation and stress (Continued)
TABLE 101 Formulas for torsional deformation and stress (Continued)
Form and dimensions of cross sections
other quantities involved and case no Formula for K in y frac14TL
KGFormula for shear stress
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
r3where B varies with
a
rover the
range 014a
r4 05 as follows For one groove
B frac14 10259 thorn 11802a
r 27897
a
r
2
thorn 37092a
r
3
For two grooves
B frac14 10055 thorn 15427a
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
(Data from Refs 12 and 13)
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
(Data from Refs 12 and 13)
412
FormulasforStressandStrain
[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
etht1THORNmax frac14 tGy0 at the surface everywhere
3 Hat sectione frac14 b
3h2b thorn 6h2b1 8b31
h3 thorn 6h2b thorn 6h2b1 thorn 8b31 thorn 12hb2
1
K frac14t3
3ethh thorn 2b thorn 2b1THORN
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
Ey00 throughout the thickness at corners A and F
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 )
Cross section reference no Constants Selected maximum values
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
etht1THORNmax frac14 tGy0 at the surface everywhere
5 Twin channel with
flanges outwardK frac14
t3
3eth2b thorn 4b1THORN
Cw frac14tb2
24eth8b3
1 thorn 6h2b1 thorn h2b 12b21hTHORN
ethsxTHORNmax frac14hb
4Ey00 throughout the thickness at points B and C if h gt b1
ethsxTHORNmax frac14hb
4
bb1
2
Ey00 throughout the thickness at points A and D if h lt b1
etht2THORNmax frac14b
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
etht2THORNmax frac14b
4b2
1 hb
4 hb1
Ey000 throughout the thickness at a point midway between corners B and C if
b1 lth
21 thorn
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
2thorn
b
2h
r
etht1THORNmax frac14 tGy0 at the surface everywhere
6 Wide flanged beam
with equal flanges
K frac14 13eth2t3b thorn t3
whTHORN
Cw frac14h2tb3
24
ethsxTHORNmax frac14hb
4Ey00 throughout the thickness at points A and B
etht2THORNmax frac14 hb2
16Ey000 throughout the thickness at a point midway between A and B
etht1THORNmax frac14 tGy0 at the surface everywhere
414
FormulasforStressandStrain
[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
ethsxTHORNmax frac14hb2
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
etht2THORNmax frac141
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
etht2THORNmax frac14hb2
4
b thorn h
2b thorn h
2
Ey000 throughout the thickness at a distancebethb thorn hTHORN
2b thorn hfrom point A
etht1THORNmax frac14 tGy0 at the surface everywhere
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
etht1THORNmax frac14 tGy0 at the surface everywhere
SEC107]
Torsion
415
TABLE 102 Formulas for torsional properties and stresses in thin-walled open cross sections (Continued)
TABLE 102 Formulas for torsional properties and stresses in thin-walled open cross sections (Continued )
Cross section reference no Constants Selected maximum values
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
etht1THORNmax frac14 tGy0 at the surface everywhere
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
etht1THORNmax frac14 tGy0 at the surface everywhere
416
FormulasforStressandStrain
[CHAP10
T
1shy ------shyI~---q1-Sq ---~
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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
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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)
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KG ()
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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
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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
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k 5 6~ $ S CM 4 _ ~ )
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Ytltj - 0 k fjf Icyyen) 2 ~ Moi-i CI l 0 ~ t en~ utl tfOduo C~S+icD G0
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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
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pound estue Co mr1-oV+c 1l()C ) ((() es
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pound eSfuc(W cor~Ov-L ~)(~(lIO cc de u60-pound I)(~ r _ Utl)
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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~)
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(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
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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 ~
32 Shaft with two splines K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 14
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
r3where B varies with
b
ras follows For 04
b
r4 05
B frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 14
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
(Data from Refs 12 and 13)
33 Shaft with four splines K frac14 2Cr4 where C varies withb
ras follows
For 04b
r4 05
C frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 10
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
r3where B varies with
b
ras follows For 04
b
r4 05
B frac14 K1 thorn K2
b
rthorn K3
b
r
2
thorn K4
b
r
3
where for 024a=b4 10
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
(Data from Refs 12 and 13)
SEC107]
Torsion
411
TABLE 101 Formulas for torsional deformation and stress (Continued)
TABLE 101 Formulas for torsional deformation and stress (Continued)
Form and dimensions of cross sections
other quantities involved and case no Formula for K in y frac14TL
KGFormula for shear stress
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
r3where B varies with
a
rover the
range 014a
r4 05 as follows For one groove
B frac14 10259 thorn 11802a
r 27897
a
r
2
thorn 37092a
r
3
For two grooves
B frac14 10055 thorn 15427a
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
(Data from Refs 12 and 13)
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
(Data from Refs 12 and 13)
412
FormulasforStressandStrain
[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
etht1THORNmax frac14 tGy0 at the surface everywhere
3 Hat sectione frac14 b
3h2b thorn 6h2b1 8b31
h3 thorn 6h2b thorn 6h2b1 thorn 8b31 thorn 12hb2
1
K frac14t3
3ethh thorn 2b thorn 2b1THORN
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
Ey00 throughout the thickness at corners A and F
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 )
Cross section reference no Constants Selected maximum values
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
etht1THORNmax frac14 tGy0 at the surface everywhere
5 Twin channel with
flanges outwardK frac14
t3
3eth2b thorn 4b1THORN
Cw frac14tb2
24eth8b3
1 thorn 6h2b1 thorn h2b 12b21hTHORN
ethsxTHORNmax frac14hb
4Ey00 throughout the thickness at points B and C if h gt b1
ethsxTHORNmax frac14hb
4
bb1
2
Ey00 throughout the thickness at points A and D if h lt b1
etht2THORNmax frac14b
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
etht2THORNmax frac14b
4b2
1 hb
4 hb1
Ey000 throughout the thickness at a point midway between corners B and C if
b1 lth
21 thorn
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
2thorn
b
2h
r
etht1THORNmax frac14 tGy0 at the surface everywhere
6 Wide flanged beam
with equal flanges
K frac14 13eth2t3b thorn t3
whTHORN
Cw frac14h2tb3
24
ethsxTHORNmax frac14hb
4Ey00 throughout the thickness at points A and B
etht2THORNmax frac14 hb2
16Ey000 throughout the thickness at a point midway between A and B
etht1THORNmax frac14 tGy0 at the surface everywhere
414
FormulasforStressandStrain
[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
ethsxTHORNmax frac14hb2
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
etht2THORNmax frac141
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
etht2THORNmax frac14hb2
4
b thorn h
2b thorn h
2
Ey000 throughout the thickness at a distancebethb thorn hTHORN
2b thorn hfrom point A
etht1THORNmax frac14 tGy0 at the surface everywhere
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
etht1THORNmax frac14 tGy0 at the surface everywhere
SEC107]
Torsion
415
TABLE 102 Formulas for torsional properties and stresses in thin-walled open cross sections (Continued)
TABLE 102 Formulas for torsional properties and stresses in thin-walled open cross sections (Continued )
Cross section reference no Constants Selected maximum values
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
etht1THORNmax frac14 tGy0 at the surface everywhere
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
etht1THORNmax frac14 tGy0 at the surface everywhere
416
FormulasforStressandStrain
[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
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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 stress (Continued)
Form and dimensions of cross sections
other quantities involved and case no Formula for K in y frac14TL
KGFormula for shear stress
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
r3where B varies with
a
rover the
range 014a
r4 05 as follows For one groove
B frac14 10259 thorn 11802a
r 27897
a
r
2
thorn 37092a
r
3
For two grooves
B frac14 10055 thorn 15427a
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
(Data from Refs 12 and 13)
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
(Data from Refs 12 and 13)
412
FormulasforStressandStrain
[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
etht1THORNmax frac14 tGy0 at the surface everywhere
3 Hat sectione frac14 b
3h2b thorn 6h2b1 8b31
h3 thorn 6h2b thorn 6h2b1 thorn 8b31 thorn 12hb2
1
K frac14t3
3ethh thorn 2b thorn 2b1THORN
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
Ey00 throughout the thickness at corners A and F
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 )
Cross section reference no Constants Selected maximum values
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
etht1THORNmax frac14 tGy0 at the surface everywhere
5 Twin channel with
flanges outwardK frac14
t3
3eth2b thorn 4b1THORN
Cw frac14tb2
24eth8b3
1 thorn 6h2b1 thorn h2b 12b21hTHORN
ethsxTHORNmax frac14hb
4Ey00 throughout the thickness at points B and C if h gt b1
ethsxTHORNmax frac14hb
4
bb1
2
Ey00 throughout the thickness at points A and D if h lt b1
etht2THORNmax frac14b
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
etht2THORNmax frac14b
4b2
1 hb
4 hb1
Ey000 throughout the thickness at a point midway between corners B and C if
b1 lth
21 thorn
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
2thorn
b
2h
r
etht1THORNmax frac14 tGy0 at the surface everywhere
6 Wide flanged beam
with equal flanges
K frac14 13eth2t3b thorn t3
whTHORN
Cw frac14h2tb3
24
ethsxTHORNmax frac14hb
4Ey00 throughout the thickness at points A and B
etht2THORNmax frac14 hb2
16Ey000 throughout the thickness at a point midway between A and B
etht1THORNmax frac14 tGy0 at the surface everywhere
414
FormulasforStressandStrain
[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
ethsxTHORNmax frac14hb2
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
etht2THORNmax frac141
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
etht2THORNmax frac14hb2
4
b thorn h
2b thorn h
2
Ey000 throughout the thickness at a distancebethb thorn hTHORN
2b thorn hfrom point A
etht1THORNmax frac14 tGy0 at the surface everywhere
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
etht1THORNmax frac14 tGy0 at the surface everywhere
SEC107]
Torsion
415
TABLE 102 Formulas for torsional properties and stresses in thin-walled open cross sections (Continued)
TABLE 102 Formulas for torsional properties and stresses in thin-walled open cross sections (Continued )
Cross section reference no Constants Selected maximum values
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
etht1THORNmax frac14 tGy0 at the surface everywhere
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
etht1THORNmax frac14 tGy0 at the surface everywhere
416
FormulasforStressandStrain
[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 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
etht1THORNmax frac14 tGy0 at the surface everywhere
3 Hat sectione frac14 b
3h2b thorn 6h2b1 8b31
h3 thorn 6h2b thorn 6h2b1 thorn 8b31 thorn 12hb2
1
K frac14t3
3ethh thorn 2b thorn 2b1THORN
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
Ey00 throughout the thickness at corners A and F
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 )
Cross section reference no Constants Selected maximum values
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
etht1THORNmax frac14 tGy0 at the surface everywhere
5 Twin channel with
flanges outwardK frac14
t3
3eth2b thorn 4b1THORN
Cw frac14tb2
24eth8b3
1 thorn 6h2b1 thorn h2b 12b21hTHORN
ethsxTHORNmax frac14hb
4Ey00 throughout the thickness at points B and C if h gt b1
ethsxTHORNmax frac14hb
4
bb1
2
Ey00 throughout the thickness at points A and D if h lt b1
etht2THORNmax frac14b
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
etht2THORNmax frac14b
4b2
1 hb
4 hb1
Ey000 throughout the thickness at a point midway between corners B and C if
b1 lth
21 thorn
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
2thorn
b
2h
r
etht1THORNmax frac14 tGy0 at the surface everywhere
6 Wide flanged beam
with equal flanges
K frac14 13eth2t3b thorn t3
whTHORN
Cw frac14h2tb3
24
ethsxTHORNmax frac14hb
4Ey00 throughout the thickness at points A and B
etht2THORNmax frac14 hb2
16Ey000 throughout the thickness at a point midway between A and B
etht1THORNmax frac14 tGy0 at the surface everywhere
414
FormulasforStressandStrain
[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
ethsxTHORNmax frac14hb2
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
etht2THORNmax frac141
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
etht2THORNmax frac14hb2
4
b thorn h
2b thorn h
2
Ey000 throughout the thickness at a distancebethb thorn hTHORN
2b thorn hfrom point A
etht1THORNmax frac14 tGy0 at the surface everywhere
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
etht1THORNmax frac14 tGy0 at the surface everywhere
SEC107]
Torsion
415
TABLE 102 Formulas for torsional properties and stresses in thin-walled open cross sections (Continued)
TABLE 102 Formulas for torsional properties and stresses in thin-walled open cross sections (Continued )
Cross section reference no Constants Selected maximum values
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
etht1THORNmax frac14 tGy0 at the surface everywhere
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
etht1THORNmax frac14 tGy0 at the surface everywhere
416
FormulasforStressandStrain
[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 102 Formulas for torsional properties and stresses in thin-walled open cross sections (Continued )
Cross section reference no Constants Selected maximum values
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
etht1THORNmax frac14 tGy0 at the surface everywhere
5 Twin channel with
flanges outwardK frac14
t3
3eth2b thorn 4b1THORN
Cw frac14tb2
24eth8b3
1 thorn 6h2b1 thorn h2b 12b21hTHORN
ethsxTHORNmax frac14hb
4Ey00 throughout the thickness at points B and C if h gt b1
ethsxTHORNmax frac14hb
4
bb1
2
Ey00 throughout the thickness at points A and D if h lt b1
etht2THORNmax frac14b
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
etht2THORNmax frac14b
4b2
1 hb
4 hb1
Ey000 throughout the thickness at a point midway between corners B and C if
b1 lth
21 thorn
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
2thorn
b
2h
r
etht1THORNmax frac14 tGy0 at the surface everywhere
6 Wide flanged beam
with equal flanges
K frac14 13eth2t3b thorn t3
whTHORN
Cw frac14h2tb3
24
ethsxTHORNmax frac14hb
4Ey00 throughout the thickness at points A and B
etht2THORNmax frac14 hb2
16Ey000 throughout the thickness at a point midway between A and B
etht1THORNmax frac14 tGy0 at the surface everywhere
414
FormulasforStressandStrain
[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
ethsxTHORNmax frac14hb2
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
etht2THORNmax frac141
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
etht2THORNmax frac14hb2
4
b thorn h
2b thorn h
2
Ey000 throughout the thickness at a distancebethb thorn hTHORN
2b thorn hfrom point A
etht1THORNmax frac14 tGy0 at the surface everywhere
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
etht1THORNmax frac14 tGy0 at the surface everywhere
SEC107]
Torsion
415
TABLE 102 Formulas for torsional properties and stresses in thin-walled open cross sections (Continued)
TABLE 102 Formulas for torsional properties and stresses in thin-walled open cross sections (Continued )
Cross section reference no Constants Selected maximum values
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
etht1THORNmax frac14 tGy0 at the surface everywhere
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
etht1THORNmax frac14 tGy0 at the surface everywhere
416
FormulasforStressandStrain
[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 ~
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
ethsxTHORNmax frac14hb2
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
etht2THORNmax frac141
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
etht2THORNmax frac14hb2
4
b thorn h
2b thorn h
2
Ey000 throughout the thickness at a distancebethb thorn hTHORN
2b thorn hfrom point A
etht1THORNmax frac14 tGy0 at the surface everywhere
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
etht1THORNmax frac14 tGy0 at the surface everywhere
SEC107]
Torsion
415
TABLE 102 Formulas for torsional properties and stresses in thin-walled open cross sections (Continued)
TABLE 102 Formulas for torsional properties and stresses in thin-walled open cross sections (Continued )
Cross section reference no Constants Selected maximum values
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
etht1THORNmax frac14 tGy0 at the surface everywhere
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
etht1THORNmax frac14 tGy0 at the surface everywhere
416
FormulasforStressandStrain
[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 102 Formulas for torsional properties and stresses in thin-walled open cross sections (Continued )
Cross section reference no Constants Selected maximum values
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
etht1THORNmax frac14 tGy0 at the surface everywhere
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
etht1THORNmax frac14 tGy0 at the surface everywhere
416
FormulasforStressandStrain
[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 ~
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 ~
-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 ~
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 ~
~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 ~
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 ~