Tema 4: Torsiуn en barras y en tubos no circulares

Tema 4: Torsión en barras y en tubos no circulares
4.1.
Torsión de elementos no circulares
Denotando con L la longitud de la barra, con a el lado más ancho y con b el lado más angosto de
su sección transversal y con T la magnitud de los momentos torsionantes de los pares aplicados
a la barra de la Figura, el esfuerzo cortante máximo, que ocurre a lo largo de la línea centran de
la cara más ancha de la barra, es igual a:
T
c1 ab2
(4.1)
TL
c2 ab2 G
(4.2)
τ=
El ángulo de giro se calcula como:
θ=
Los coeficientes c1 y c2 dependen de la razón a/b, dados en la tabla, estas ecuaciones son válidas
dentro del rango elástico.
a/b
c1
c2
1.0
0.208
0.1406
1.2
0.219
0.1661
1.5
0.231
0.1958
2.0
0.246
0.229
2.5
0.258
0.249
3.0
0.267
0.263
4.0
0.282
0.281
5.0
0.291
0.291
10.0
0.312
0.312
∞
0.333
0.333
En la tabla anterior, los coeficientes c1 y c2 son iguales para la razón a/b > 5. Para tales valores:
c1 = c2 =
4.2.
1
(1 − 0,630b/a)
3
(4.3)
Torsión 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 expresión
τ=
T
2tã
(4.4)
Donde T es la magnitud de los momentos torsionantes, t el espesor del elemento y ã es el área
bordeada por la línea central. Para calcular el ángulo θ utilice la tabla anexa.
6
SEC.
10.7
Tables
Formulas for torsional deformation and stress
GENERAL FORMULAS: y ¼ TL=KG and t ¼ T =Q, where y ¼ angle of twist (radians); T ¼ twisting moment (force-length); L ¼ length, t ¼ unit shear stress (force per unit area); G ¼ modulus of
10.7]
TABLE 10.1
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 ¼
1. Solid circular section
K¼
1
4
2 pr
2. Solid elliptical section
K¼
pa3 b3
a2 þ b2
3. Solid square section
K ¼ 2:25a4
4. Solid rectangular section
K ¼ ab3
TL
KG
tmax ¼
2T
pr3
tmax ¼
2T
pab2
tmax ¼
0:601T
a3
tmax ¼
at boundary
at ends of minor axis
at midpoint of each side
"
2
3
4 #
3T
b
b
b
b
1 þ 0:6095 þ 0:8865
1:8023
þ 0:9100
2
8ab
a
a
a
a
at the midpoint of each longer side for a 5 b
Torsion
16
b
b4
for a 5 b
3:36
1
3
a
12a4
Formula for shear stress
401
402
TABLE 10.1
Formulas for torsional deformation and stress (Continued)
Form and dimensions of cross sections,
Formula for K in y ¼
5. Solid triangular section (equilaterial)
pffiffiffi
a4 3
K¼
80
6. Isosceles triangle
For
2
3
K¼
a3 b3
15a2 þ 20b2
(Note: See also Ref. 21 for graphs of stress
magnitudes and locations and stiffness
factors)
TL
KG
Formula for shear stress
tmax ¼
< a=b <
pffiffiffi
3
ð39 < a < 82 Þ
at midpoint of each side
For 39 < a < 120
Q¼
approximate formula which is exact at a ¼ 60
where K ¼ 0:02165c4 .
pffiffiffi
pffiffiffi
For 3 < a=b
< a < 120 Þ
a < 2 3 ð82
K ¼ 0:0915b4 0:8592
b
20T
a3
K
b½0:200 þ 0:309a=b 0:0418ða=bÞ2 approximate formula which is exact at a ¼ 60 and a ¼ 90
For a ¼ 60
For a ¼ 90
Formulas for Stress and Strain
other quantities involved, and case no.
Q ¼ 0:0768b3 ¼ 0:0500c3
Q ¼ 0:1604b3 ¼ 0:0567c3
tmax at center of longest side
approximate formula which is exact at
a ¼ 90 where K ¼ 0:0261c4 (errors < 4%) (Ref. 20)
7. Circular segmental section
K ¼ 2Cr4 where C varies with
For 0 4
h
as follows.
r
h
4 1:0:
r
2
h
h
C ¼ 0:7854 0:0333 2:6183
r
r
[Note: h ¼ rð1 cos aÞ
3
4
5
h
h
h
þ 4:1595
3:0769
þ0:9299
r
r
r
tmax ¼
TB
h
where B varies with
r3
r
as follows. For 0 4
h
4 1:0 :
r
2
h
h
B ¼ 0:6366 þ 1:7598 5:4897
r
r
3
4
5
h
h
h
þ14:062
14:510
þ 6:434
r
r
r
[CHAP. 10
(Data from Refs. 12 and 13)
SEC.
TABLE 10.1
Formulas for torsional deformation and stress (Continued)
10.7]
8. Circular sector
K ¼ Cr4 where C varies with
For 0:1 4
a
as follows:
p
a
4 2:0:
p
with
a2
a
C ¼ 0:0034 0:0697 þ 0:5825
p
p
0:2950
(Note: See also Ref. 21)
a 3
p
tmax ¼
þ 0:0874
a 4
p
0:0111
T
on a radial boundary. B varies
Br3
a
a
as follows. For 0:1 4 4 1:0:
p
p
a2
a
B ¼ 0:0117 0:2137 þ 2:2475
p
p
a5
a3
a4
a5
4:6709
þ 5:1764
2:2000
p
p
p
p
ðData from Ref. 17)
9. Circular shaft with opposite sides
flattened
h
as follows:
K ¼ 2Cr where C varies with
r
4
TB
h
as follows. For two flat sides where
where B varies with
r3
r
h
4 0:6:
r
h
4 0:8:
r
2
h
h
C ¼ 0:7854 0:4053 3:5810
r
r
3
4
h
h
þ 5:2708
2:0772
r
r
2
3
h
h
h
B ¼ 0:6366 þ 2:5303 11:157
þ 49:568
r
r
r
For four flat sides where
h
0 4 4 0:293 :
r
For four flat sides where 0 4
For two flat sides where 0 4
(Note: h ¼ r wÞ
tmax ¼
C ¼ 0:7854 0:7000
4
5
h
h
85:886
þ 69:849
r
r
h
4 0:293:
r
2
3
h
h
h
þ 30:853
B ¼ 0:6366 þ 2:6298 5:6147
r
r
r
(Data from Refs. 12 and 13)
Torsion
2
3
h
h
h
þ 14:578
7:7982
r
r
r
04
403
TABLE 10.1
Formulas for torsional deformation and stress (Continued)
other quantities involved, and case no.
11. Eccentric hollow circular section
TL
KG
K ¼ 12 pðr40 r4i Þ
K¼
pðD4 d4 Þ
32C
where
Formula for shear stress
tmax ¼
2Tro
at outer boundary
pðr4o r4i Þ
tmax ¼
16TDF
pðD4 d4 Þ
F ¼1þ
16n2
384n4
l2 þ
l4
C ¼ 1þ
ð1 n2 Þð1 n4 Þ
ð1 n2 Þ2 ð1 n4 Þ4
4n2
32n2
48n2 ð1 þ 2n2 þ 3n4 þ 2n6 Þ 3
lþ
l2 þ
l
ð1 n2 Þð1 n4 Þð1 n6 Þ
1 n2
ð1 n2 Þð1 n4 Þ
þ
12. Hollow elliptical section, outer and
inner boundaries similar ellipses
K¼
pa3 b3
ð1 q4 Þ
a2 þ b2
tmax ¼
64n2 ð2 þ 12n2 þ 19n4 þ 28n6 þ 18n8 þ 14n10 þ 3n12 Þ 4
l
ð1 n2 Þð1 n4 Þð1 n6 Þð1 n8 Þ
2T
pab2 ð1 q4 Þ
(Ref. 10)
Formulas for Stress and Strain
10. Hollow concentric circular section
Formula for K in y ¼
404
Form and dimensions of cross sections,
at ends of minor axis on outer surface
where
q¼
ao bo
¼
a
b
(Note: The wall thickness is not constant)
13. Hollow, thin-walled section of uniform
thickness; U ¼ length of elliptical
median boundary, shown dashed:
"
U ¼ pða þ b tÞ 1 þ 0:258
4p2 t½ða 12 tÞ2 ðb 12 tÞ2 U
taverage ¼
T
2ptða 12 tÞðb 12 tÞ
(stress is nearly uniform if t is small)
#
ða bÞ2
2
ða þ b tÞ
[CHAP. 10
ðapproximatelyÞ
K¼
SEC.
TABLE 10.1
Formulas for torsional deformation and stress (Continued)
15. Any thin tube. U and A as for
case 14; t ¼ thickness at any point
16. Hollow rectangle, thin-walled
K¼
4A2 t
U
K ¼Ð
K¼
4A2
dU =t
2tt1 ða tÞ2 ðb t1 Þ2
at þ bt1 t2 t21
taverage ¼
T
2tA
10.7]
14. Any thin tube of uniform thickness;
U ¼ length of median boundary;
A ¼ mean of areas enclosed by outer
and inner boundaries, or (approximate)
area within median boundary
(stress is nearly uniform if t is small)
taverage on any thickness AB ¼
taverage ¼
T
2tA
ðtmax where t is a minimum)
8
>
>
>
<
T
2tða tÞðb t1 Þ
near midlength of short sides
>
>
>
:
T
2t1 ða tÞðb t1 Þ
near midlength of long sides
(There will be higher stresses at inner corners unless fillets of fairly large radius
are provided)
Torsion
(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.)
405
TABLE 10.1
Formulas for torsional deformation and stress (Continued )
Form and dimensions of cross sections,
17. Thin circular open tube of uniform
thickness; r ¼ mean radius
Formula for K in y ¼
K ¼ 23 prt3
TL
KG
Formula for shear stress
tmax ¼
T ð6pr þ 1:8tÞ
4p2 r2 t2
18. Any thin open tube of uniform
thickness; U ¼ length of median line,
shown dashed
K¼
19. Any elongated section with axis of
symmetry OX; U ¼ length, A ¼ area of
section, Ix ¼ moment of inertia about
axis of symmetry
K¼
1 3
Ut
3
tmax ¼
T ð3U þ 1:8tÞ
U 2 t2
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
19–26)
4Ix
1 þ 16Ix =AU 2
For all solid sections of irregular form (cases 19–26 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 approximately by
tmax ¼ G
20. Any elongated section or thin open tube;
dU ¼ elementary length along median
line, t ¼ thickness normal to median line,
A ¼ area of section
F
3 þ 4F=AU 2
where F ¼
0
t3 dU
A4
40J
or tmax ¼
T
C
K
where
C¼
K¼
y
C
L
2 4
D
p D
D
1 þ 0:15
16A2 2r
p2 D4
1þ
16A2
D ¼ diameter of largest inscribed circle
r ¼ radius of curvature of boundary at the point (positive for this case)
A ¼ area of the section
*Unless at some point on the boundary there is a sharp reentant angle, causing
high local stress.
[CHAP. 10
21. Any solid, fairly compact section
without reentrant angles, J ¼ polar
moment of inertia about centroid axis,
A ¼ area of section
K¼
ðU
Formulas for Stress and Strain
along both edges remote from ends (this assumes t is small comopared with mean
radius)
406
other quantities involved, and case no.
SEC.
TABLE 10.1
Formulas for torsional deformation and stress (Continued)
1
K ¼ 12
bðm þ nÞðm2 þ n2 Þ VL m4 Vs n4
where VL ¼ 0:10504 0:10s þ 0:0848s2
0:06746s3 þ 0:0515s4
Vs ¼ 0:10504 þ 0:10s þ 0:0848s2
þ 0:06746s3 þ 0:0515s4
mn
s¼
b
(Ref. 11)
23. T-section, flange thickness uniform.
For definitions of r; D; t; and t1 , see
case 26.
K ¼ K1 þ K2 þ aD4
1
b
b4
0:21
1
12a4
3
a
1
d
d4
1
K2 ¼ cd3 0:105
3
c
192c4
t
r
a¼
0:15 þ 0:10
t1
b
where K1 ¼ ab3
10.7]
22. Trapezoid
At a point where the curvature is negative (boundary of section concave or
reentrant), this maximum stress is given approximately by
T
C
K
D
D
D
2f
0:238
tanh
where C ¼
1 þ 0:118 ln 1 2r
2r
p
p2 D4
1þ
16A2
y
tmax ¼ G C
L
or tmax ¼
and D; A, and r have the same meaning as before and f ¼ 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.
ðb þ rÞ2 þ rd þ d2 =4
ð2r þ bÞ
for d < 2ðb þ rÞ
D¼
24. L-section; b 5 d. For definitions of r and
D, see case 26.
Torsion
K ¼ K1 þ K2 þ aD4
1
b
b4
where K1 ¼ ab3 0:21
1
12a4
3
a
1
d
d4
1
K2 ¼ cd3 0:105
192c4
3
c
d
r
a¼
0:07 þ 0:076
b
b
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
D ¼ 2½d þ b þ 3r 2ð2r þ bÞð2r þ d
for b < 2ðd þ rÞ
407
TABLE 10.1
Formulas for torsional deformation and stress (Continued )
other quantities involved, and case no.
Formula for K in y ¼
TL
KG
K ¼ sum of K’s of constituent L-sections computed
as for case 24
26. I-section, flange thickness uniform;
r ¼ fillet radius, D ¼ diameter largest
inscribed circle, t ¼ b if b < d; t ¼ d
if d < b; t1 ¼ b if b > d; t1 ¼ d if d > b
K ¼ 2K1 þ K2 þ 2aD4
Formulas for Stress and Strain
25. U- or Z-section
Formula for shear stress
1
b
b4
0:21
1
4
3
a
12a
K2 ¼ 13 cd3
t
r
a¼
0:15 þ 0:1
t1
b
where K1 ¼ ab3
Use expression for D from case 23
27. Split hollow shaft
K ¼ 2Cr4o where C varies with
For 0:2 4
ri
as follows:
ro
ri
4 0:6:
ro
C ¼ K1 þ K2
2
3
ri
r
r
þ K3 i þ K4 i
ro
ro
ro
where for 0:1 4 h=ri 4 1:0,
K1
K2
K4
At M ; t ¼
TB
r
where B varies with i as follows.
r3o
ro
For 0:2 4
ri
4 0:6:
ro
B ¼ K1 þ K2
2
3
ri
r
r
þ K3 i þ K4 i
ro
ro
ro
where fore 0:1 4 h=ri 4 1:0,
3
h
h
0:3231
ri
ri
2
h
h
K2 ¼ 2:9047 þ 3:0069 þ 4:0500
ri
ri
2
h
h
K3 ¼ 15:721 6:5077 12:496
ri
ri
2
h
h
K4 ¼ 29:553 þ 4:1115 þ 18:845
ri
ri
K1 ¼
2:0014 0:1400
(Data from Refs. 12 and 13)
[CHAP. 10
K3
2
h
h
¼ 0:4427 þ 0:0064 0:0201
ri
ri
2
h
h
¼ 0:8071 0:4047 þ 0:1051
ri
ri
2
h
h
¼ 0:0469 þ 1:2063 0:3538
ri
ri
2
h
h
¼ 0:5023 0:9618 þ 0:3639
ri
ri
408
Form and dimensions of cross sections,
SEC.
TABLE 10.1
Formulas for torsional deformation and stress (Continued)
K ¼ 2Cr4 where C varies with
For 0 4
b
as follows.
r
b
4 0:5:
r
C ¼ K1 þ K2
At M ; t ¼
TB
b
b
where B varies with as follows. For 0:2 4 4 0:5 :
r3
r
r
B ¼ K1 þ K2
2
3
b
b
b
þ K4
þ K3
r
r
r
10.7]
28. Shaft with one keyway
2
3
b
b
b
þ K3
þ K4
r
r
r
where for 0:5 4 a=b 4 1:5;
a2
a
1:1690 0:3168 þ 0:0490
b
b
a2
a
K2 ¼ 0:43490 1:5096 þ 0:8677
b
b
a2
a
K3 ¼ 1:1830 þ 4:2764 1:7024
b
b
a2
a
K4 ¼ 0:8812 0:2627 0:1897
b
b
K1 ¼
where for 0:3 4 a=b 4 1:5;
K1 ¼
0:7854
a2
a
K2 ¼ 0:0848 þ 0:1234 0:0847
b
b
a2
a
K3 ¼ 0:4318 2:2000 þ 0:7633
b
b
a2
a
K4 ¼ 0:0780 þ 2:0618 0:5234
b
b
29. Shaft with two keyways
K ¼ 2Cr4 where C varies with
For 0 4
b
as follows.
r
b
4 0:5:
r
C ¼ K1 þ K2
2
3
b
b
b
þ K3
þ K4
r
r
r
where for 0:3 4 a=b 4 1:5;
K1 ¼
At M ; t ¼
TB
b
b
where B varies with
as follows. For 0:2 4 4 0:5 :
r3
r
r
B ¼ K1 þ K2
2
3
b
b
b
þ K4
þ K3
r
r
r
where for 0:5 4 a=b 4 1:5;
a2
a
1:2512 0:5406 þ 0:0387
b
b
a
a2
K2 ¼ 0:9385 þ 2:3450 þ 0:3256
b
b
a2
a
K3 ¼ 7:2650 15:338 þ 3:1138
b
b
a2
a
K4 ¼ 11:152 þ 33:710 10:007
b
b
K1 ¼
Torsion
0:7854
a2
a
K2 ¼ 0:0795 þ 0:1286 0:1169
b
b
a2
a
K3 ¼ 1:4126 3:8589 þ 1:3292
b
b
a2
a
K4 ¼ 0:7098 þ 4:1936 1:1053
b
b
(Data from Refs. 12 and 13)
(Data from Refs. 12 and 13)
409
Formulas for torsional deformation and stress (Continued)
Form and dimensions of cross sections,
30. Shaft with four keyways
Formula for K in y ¼
K ¼ 2Cr4 where C varies with
For 0 4
TL
KG
b
as follows.
r
b
4 0:4:
r
C ¼ K1 þ K2
2
3
b
b
b
þ K4
þ K3
r
r
r
0:7854
a2
a
K2 ¼ 0:1496 þ 0:2773 0:2110
b
b
a2
a
K3 ¼ 2:9138 8:2354 þ 2:5782
b
b
a2
a
K4 ¼ 2:2991 þ 12:097 2:2838
b
b
For 0 4
b
as follows.
r
b
4 0:5:
r
C ¼ K1 þ K2
2
3
b
b
b
þ K4
þ K3
r
r
r
where for 0:2 4 a=b 4 1:4;
K1 ¼
0:7854
2
3
b
b
b
þ K3
þ K4
r
r
r
where for 0:5 4 a=b 4 1:2;
a2
a
1:0434 þ 1:0449 0:2977
b
b
a2
a
K2 ¼ 0:0958 9:8401 þ 1:6847
b
b
a2
a
K3 ¼ 15:749 6:9650 þ 14:222
b
b
a2
a
K4 ¼ 35:878 þ 88:696 47:545
b
b
(Data from Refs. 12 and 13)
At M ; t ¼
TB
b
b
where B varies with
as follows. For 0 4 4 0:5;
r3
r
r
B ¼ K1 þ K2
2
3
b
b
b
þ K3
þ K4
r
r
r
where for 0:2 4 a=b 4 1:4;
K1 ¼
0:6366
a2
a
K2 ¼ 0:0023 þ 0:0168 þ 0:0093
b
b
a2
a
K3 ¼ 0:0052 þ 0:0225 0:3300
b
b
a2
a
K4 ¼ 0:0984 0:4936 þ 0:2179
b
b
(Data from Refs. 12 and 13)
[CHAP. 10
a2
a
0:0264 0:1187 þ 0:0868
b
b
a2
a
K3 ¼ 0:2017 þ 0:9019 0:4947
b
b
a2
a
K4 ¼ 0:2911 1:4875 þ 2:0651
b
b
K2 ¼
TB
b
b
where B varies with
as follows. For 0:2 4 4 0:4;
r3
r
r
K1 ¼
K1 ¼
K ¼ 2Cr4 where C varies with
At M ; t ¼
B ¼ K1 þ K2
where for 0:3 4 a=b 4 1:2;
31. Shaft with one spline
Formula for shear stress
Formulas for Stress and Strain
other quantities involved, and case no.
410
TABLE 10.1
SEC.
TABLE 10.1
Formulas for torsional deformation and stress (Continued)
K ¼ 2Cr4 where C varies with
For 0 4
b
as follows.
r
b
4 0:5:
r
C ¼ K1 þ K2
At M ; t ¼
TB
b
b
as follows. For 0 4 4 0:5;
where B varies with
r3
r
r
B ¼ K1 þ K2
2
3
b
b
b
þ K4
þ K3
r
r
r
2
3
b
b
b
þ K4
þ K3
r
r
r
where for 0:2 4 a=b 4 1:4;
K1 ¼
where for 0:2 4 a=b 4 1:4;
10.7]
32. Shaft with two splines
0:6366
a2
a
0:0069 0:0229 þ 0:0637
b
b
a
a2
K3 ¼ 0:0675 þ 0:3996 1:0514
b
b
a2
a
K4 ¼ 0:3582 1:8324 þ 1:5393
b
b
K2 ¼
K1 ¼
0:7854
a2
a
0:0204 0:1307 þ 0:1157
b
b
a
a2
K3 ¼ 0:2075 þ 1:1544 0:5937
b
b
a2
a
K4 ¼ 0:3608 2:2582 þ 3:7336
b
b
K2 ¼
33. Shaft with four splines
K ¼ 2Cr4 where C varies with
For 0 4
b
as follows.
r
b
4 0:5:
r
C ¼ K1 þ K2
2
3
b
b
b
þ K3
þ K4
r
r
r
where for 0:2 4 a=b 4 1:0;
(Data from Refs. 12 and 13)
At M ; t ¼
TB
b
b
where B varies with
as follows. For 0 4 4 0:5;
r3
r
r
B ¼ K1 þ K2
2
3
b
b
b
þ K4
þ K3
r
r
r
where for 0:2 4 a=b 4 1:0;
K1 ¼
0:6366
a2
a
0:0114 0:0789 þ 0:1767
b
b
a2
a
K3 ¼ 0:1207 þ 1:0291 2:3589
b
b
a2
a
K4 ¼ 0:5132 3:4300 þ 4:0226
b
b
K2 ¼
K1 ¼
0:7854
(Data from Refs. 12 and 13)
Torsion
a2
a
0:0595 0:3397 þ 0:3239
b
b
a2
a
K3 ¼ 0:6008 þ 3:1396 2:0693
b
b
a2
a
K4 ¼ 1:0869 6:2451 þ 9:4190
b
b
K2 ¼
411
412
TABLE 10.1
Formulas for torsional deformation and stress (Continued)
other quantities involved, and case no.
34. Pinned shaft with one, two, or four
grooves
Formula for K in y ¼
K ¼ 2Cr4 where C varies with
04
TL
KG
a
over the range
r
a
4 0:5 as follows. For one groove:
r
Formula for shear stress
At M ; t ¼
TB
a
over the
where B varies with
r3
r
range 0:1 4
a
4 0:5 as follows. For one groove:
r
a2
a3
a
þ 0:9167
C ¼ 0:7854 0:0225 1:4154
r
r
r
a2
a3
a
B ¼ 1:0259 þ 1:1802 2:7897
þ 3:7092
r
r
r
For two grooves:
For two grooves:
a2
a3
a
C ¼ 0:7854 0:0147 3:0649
þ 2:5453
r
r
r
a2
a3
a
þ 7:0534
B ¼ 1:0055 þ 1:5427 2:9501
r
r
r
For four grooves:
For four grooves:
a2
a3
a
C ¼ 0:7854 0:0409 6:2371
þ 7:2538
r
r
r
a2
a3
a4
a
B ¼ 1:2135 2:9697 þ 33:713
99:506
þ 130:49
r
r
r
r
Formulas for Stress and Strain
Form and dimensions of cross sections,
(Data from Refs. 12 and 13)
35. Cross shaft
r
K ¼ 2Cs where C varies with over the
s
4
range 0 4
r
4 0:9 as follows:
s
r2
r3
r
C ¼ 1:1266 0:3210 þ 3:1519
14:347
s
s
s
r4
r5
þ 15:223
4:7767
s
s
At M ; t ¼
BM T
r
r
where BM varies with over the range 0 4 4 0:5 as follows:
s3
s
s
r2
r3
r4
r
þ 3:7335
2:8686
BM ¼ 0:6010 þ 0:1059 0:9180
s
s
s
s
At N; t ¼
BN T
r
r
where BN varies with over the range 0:3 4 4 0:9 as follows:
s3
s
s
r2
r3
r4
r5
r
þ 109:04
133:95
þ 66:054
BN ¼ 0:3281 þ 9:1405 42:520
s
s
s
s
s
(Data from Refs. 12 and 13)
[CHAP. 10
(Note: BN > BM for r=s > 0:32Þ
TABLE 10.2
Formulas for torsional properties and stresses in thin-walled open cross sections
sixth power); t1 ¼ shear stress due to torsional rigidity of the cross section (force per unit area); t2 ¼ shear stress due to warping rigidity of the cross section (force per unit area); sx ¼ bending stress
unit area)
The appropriate values of y0 ; y00 , and y000 are found in Table 10.3 for the loading and boundary restraints desired
Cross section, reference no.
1. Channel
Constants
3b
h þ 6b
K¼
t3
ðh þ 2bÞ
3
Cw ¼
2. C-section
e¼b
h2 b3 t 2h þ 3b
12 h þ 6b
h3
3h2 b þ 6h2 b1 8b31
þ 6h2 b þ 6h2 b1 þ 8b31 12hb21
t3
ðh þ 2b þ 2b1 Þ
3
2 2
h b
b
2eb1 2b21
Cw ¼ t
b1 þ e þ
2
b
h
3
K¼
þ
3. Hat section
Selected maximum values
2
e¼
e¼b
ðsx Þmax ¼
hb h þ 3b 00
Ey throughout the thickness at corners A and D
2 h þ 6b
ðt2 Þmax ¼
2
hb2 h þ 3b
h þ 3b
from corners A and D
Ey000 throughout the thickness at a distance b
h þ 6b
4 h þ 6b
ðt1 Þmax ¼ tGy0 at the surface everywhere
ðsx Þmax ¼
h
ðb eÞ þ b1 ðb þ eÞ Ey00 throughout the thickness at corners A and F
2
ðt2 Þmax ¼
h
b2
ðb eÞð2b1 þ b eÞ þ 1 ðb þ eÞ Ey000 throughout the thickness on the top and bottom flanges at a
4
2
distance e from corners C and D
ðt1 Þmax ¼ tGy0 at the surface everywhere
h2 e2
h 2b2
2b3
b þ b1 þ 1 þ 1 ðb þ eÞ2
2
h
3
6
3h2 b þ 6h2 b1 8b31
h3 þ 6h2 b þ 6h2 b1 þ 8b31 þ 12hb21
h
ðb eÞ b1 ðb þ eÞ Ey00 throughout the thickness at corners A and F
2
h
ðb eÞEy00 throughout the thickness at corners B and E
2
"
#
h2 ðb eÞ2 b21
hb
hðb eÞ
þ ðb þ eÞ 1 ðb eÞ Ey000 throughout the thickness at a distance
t2 ¼
2ðb þ eÞ
8ðb þ eÞ
2
2
sx ¼
from corner B toward corner A
2
b
hb
h
t2 ¼ 1 ðb þ eÞ 1 ðb eÞ ðb eÞ2 Ey000 throughout the thickness at a distance e
4
2
2
from corner C toward corner B
413
h2 e2
h 2b2
2b3
b þ b1 þ þ 1 þ 1 ðb þ eÞ2 6
2
h
3
sx ¼
Torsion
t3
ðh þ 2b þ 2b1 Þ
3
2 2
h b
b
2eb1 2b21
b1 þ e Cw ¼ t
3
2
b
h
K¼
þ
10.7]
due to warping rigidity of the cross section (force per unit area); E ¼ modulus of elasticity of the material (force per unit area); and G ¼ modulus of rigidity (shear modulus) of the material (force per
SEC.
NOTATION: Point 0 indicates the shear center. e ¼ distance from a reference to the shear center; K ¼ torsional stiffness constant (length to the fourth power); Cw ¼ warping constant (length to the
t1 ¼ tGy0 at the surface everywhere
TABLE 10.2
Formulas for torsional properties and stresses in thin-walled open cross sections (Continued )
4. Twin channel with
flanges inward
Constants
K¼
t3
ð2b þ 4b1 Þ
3
tb2
ð8b31 þ 6h2 b1 þ h2 b þ 12b21 hÞ
24
ðsx Þmax
b
h
b þ Ey00 throughout the thickness at points A and D
¼
2 1 2
ðt2 Þmax ¼
b
ð4b21 þ 4b1 h þ hbÞEy000 throughout the thickness midway between corners B and C
16
ðt1 Þmax ¼ tGy0 at the surface everywhere
5. Twin channel with
flanges outward
K¼
t3
ð2b þ 4b1 Þ
3
Cw ¼
tb2
ð8b31 þ 6h2 b1 þ h2 b 12b21 hÞ
24
hb 00
Ey throughout the thickness at points B and C if h > b1
4
hb bb1
¼
Ey00 throughout the thickness at points A and D if h < b1
2
4
ðsx Þmax ¼
ðsx Þmax
ðt2 Þmax ¼
ðt2 Þmax ¼
2
b h
h
b1 Ey000 throughout the thickness at a distance
from corner B toward point A if
4 2
2
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!
h
1
b
b1 >
1þ
þ
2
2 2h
Formulas for Stress and Strain
Cw ¼
Selected maximum values
414
Cross section, reference no.
b 2 hb
b hb1 Ey000 throughout the thickness at a point midway between corners B and C if
4 1
4
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!
h
1
b
b1 <
1þ
þ
2
2 2h
ðt1 Þmax ¼ tGy0 at the surface everywhere
6. Wide flanged beam
with equal flanges
K ¼ 13 ð2t3 b þ t3w hÞ
Cw ¼
h2 tb3
24
ðsx Þmax ¼
hb 00
Ey throughout the thickness at points A and B
4
hb2
Ey000 throughout the thickness at a point midway between A and B
16
ðt1 Þmax ¼ tGy0 at the surface everywhere
[CHAP. 10
ðt2 Þmax ¼ TABLE 10.2
Formulas for torsional properties and stresses in thin-walled open cross sections (Continued
t1 b31 h
t1 b31 þ t2 b32
K ¼ 13 ðt31 b1 þ t32 b2 þ t3w hÞ
Cw ¼
h2 t1 t2 b31 b32
12ðt1 b31 þ t2 b32 Þ
ðsx Þmax ¼
hb1
t2 b32
Ey00 throughout the thickness at points A and B if t2 b22 > t1 b21
2 t1 b31 þ t2 b32
ðsx Þmax ¼
hb2
t1 b31
Ey00 throughout the thickness at points C and D if t2 b22 < t1 b21
2 t1 b31 þ t2 b32
ðt2 Þmax ¼
1 ht2 b32 b21
Ey000 throughout the thickness at a point midway between A and B if t2 b2 > t1 b1
8 t1 b31 þ t2 b32
ðt2 Þmax ¼
1 ht1 b31 b22
Ey000 throughout the thickness at a point midway between C and D if t2 b2 < t1 b1
8 t1 b31 þ t2 b32
10.7]
e¼
SEC.
7. Wide flanged beam
with unequal flanges
)
ðt1 Þmax ¼ tmax Gy0 at the surface on the thickest portion
8. Z-section
t3
ð2b þ hÞ
3
th2 b3 b þ 2h
Cw ¼
2b þ h
12
K¼
ðsx Þmax ¼
hb b þ h
Ey00 throughout the thickness at points A and D
2 2b þ h
ðt2 Þmax ¼
2
hb2 b þ h
bðb þ hÞ
from point A
Ey000 throughout the thickness at a distance
4
2b þ h
2b þ h
ðt1 Þmax ¼ tGy0 at the surface everywhere
9. Segment of a circular
tube
e ¼ 2r
sin a a cos a
a sin a cos a
K ¼ 23 t3 ra
Cw ¼
ra2
Ey000 throughout the thickness at midlength
ðt2 Þmax ¼ r2 eð1 cos aÞ 2
ðt1 Þmax ¼ tGy0 at the surface everywhere
Torsion
415
(Note: If t=r is small, a can
be larger than p to
evaluate constants for
the case when the
walls overlap)
"
#
2tr5 3
ðsin a a cos aÞ2
a 6
3
a sin a cos a
ðsx Þmax ¼ ðr2 a re sin aÞEy00 throughout the thickness at points A and B
416
TABLE 10.2
Formulas for torsional properties and stresses in thin-walled open cross sections (Continued )
10.
Constants
e ¼ 0:707ab2
3a 2b
2a3 ða bÞ3
K ¼ 23 t3 ða þ bÞ
Selected maximum values
ðsx Þmax ¼
t2 ¼
4 3
Cw ¼
11.
ta b
4a þ 3b
6 2a3 ða bÞ3
K ¼ 13 ð4t3 b þ t3w aÞ
a2 b 2a2 þ 3ab b2 00
Ey throughout the thickness at points A and E
2 2a3 ða bÞ3
a2 b2 a2 2ab b2
Ey000 throughout the thickness at point C
4 2a3 ða bÞ3
ðt1 Þmax ¼ tGy0 at the surface everywhere
ðsx Þmax ¼
ab
cos aEy00 throughout the thickness at points A and C
2
ðt2 Þmax ¼
ab2
cos aEy000 throughout the thickness at point B
4
Formulas for Stress and Strain
Cross section, reference no.
2 3
Cw ¼
a b t
cos2 a
3
(Note: Expressions are equally valid for þ and a)
ðt1 Þmax ¼ tGy0 at the surface everywhere
[CHAP. 10
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