104
Formulas for Stress and Strain
TABLE 6.2
[CHAP. 6
Corrections for the transverse sensitivity of electrical resistance strain
gages
e refers to corrected strain value, whereas e^ refers to the strain read from the strain
indicator. The Kt terms are the
Formulas for circular rings (Continued)
Reference no., loading, and load terms
Formulas for moments, loads, and deformations and some selected numerical values
MC
wR2
f9sp y 6ys3 3s4 8 8c 5s2 c 6k2 3ss p y s2 c 2 2c g
36ps
NA
wRs3
12p
VA 0
p
Note: y 5
2
reference no.
1c. Left end simply supported,
right end fixed
Boundary values
C C C1 Ca4
RA W 2 a3
C2 C3 C1 C4
yA
W C4 Ca3 C3 Ca4
P C2 C3 C1 C4
RB W RA
MB
1d. Left end fixed, right end
fixed
yA 0
yB 0
yB 0
W kl sin ka ka sin kl
k
sin kl kl cos kl
RA W
MA
746
Formulas for Stress and Strain
[CHAP. 16
and a tangential tensile inertia stress
st
1 do2
3 nR2 1 3nr2
8 g
16:2-2
The maximum radial stress and maximum tangential stress are
equal, occur at the center, and are
sr max st max
1 do2
3 nR2
8 g
16:2-3
TABLE 9.4
Formulas for curved beams of compact cross section loaded normal to the plane of curvature (Continued )
C5 sin f C2 cos f1 cosf y Ca3 cos2 f Ca6 sin f cos f
C5 sin f C2 cos f1 cos f C3 cos2 f C6 sin f cos f
TA WR
YA
MA 0 yA 0 cA 0
TB 0 yB 0 YB
TABLE 8.1
Shear, moment, slope, and deection formulas for elastic straight beams (Continued)
Transverse shear V RA
6. Uniform temperature
variation from top to bottom from a to l
Slope y yA
200
Bending moment M MA RA x
MA x RA x2 g
T2 T1 hx ai
EI
t
2EI
TABLE 11.2
Formulas for at circular plates of constant thickness (Continued )
yb 0;
yb 0;
Qb qa
2e. Outer edge fixed, inner edge
free
Mra 0
C3 L17 C9 L11
C2 C9 C3 C8
C2 L17 C8 L11
C2 C9 C3 C8
ya Mrb
a
a2
qa3
L
C Qb C6
D
D 14
D 5
Qa Qb
b
q 2
a r2o
a 2a
M
SEC.
9.2]
Curved Beams
275
c2 =Rt 0
b0
0.1
0.297
0.2
0.580
0.3
0.836
0.4
1.056
0.5
1.238
0.6
1.382
0.8
1.577
c2 =Rt 1
b 1:677
1.2
1.721
1.4
1.732
1.5
1.732
2
1.707
3
1.671
4
1.680
5
1.700
Derivations of expressions for b0 =b and for b are also found in Re
94
Formulas for Stress and Strain
Figure 6.4
[CHAP. 6
(a) Stresses in the x and y directions. (b) Principal stresses.
Using the equations given in Table 6.1 at the end of the chapter,
eA eC 200 1000
1678:3 m
1 0:285
1n
q
1
eA eC 2 2eB eA eC 2
1n
q
1
200
700
Formulas for Stress and Strain
[CHAP. 14
several empirical formulas (Ref. 24). Under certain conditions, the
bearing stress between teeth may become important (especially as this
stress affects wear), and this stress may be calculated by the formula
f
APP.
B]
Glossary: Denitions
823
Shear lag: Because of shear strain, the longitudinal tensile or
compressive bending stresses in wide beam flanges decrease with
the distance from the web(s), and this stress reduction is called
shear lag.
Simply-supported:
570
Formulas for Stress and Strain
[CHAP. 13
The superimposed stresses at the joint are, therefore,
s1 4800 1200 0 3600 lb=in2
s01 0 0 33;800 33;800 lb=in2
s2 4800 37;200 18;200 23;800 lb=in2
s02 0 1940 9220 11;160 lb=in2
The maximum stress is a tensile m
TABLE 13.4
Formulas for discontinuity stresses and deformations at the junctions of shells and plates (Continued )
LTA2
For axial tension, E1 E2 , n1 n2 0:3, t1 t2 , R1 sin f1 R2 sin f2 , and for R=t > 5.
R21 1 n1
2t21 sin f1
R21 E1 1 n2
2E2 t1 t2 sin
SEC.
16.6]
Dynamic and Temperature Stresses
761
mum stress is the tangential stress at the ends of the ellipse and is
st DT gE=1 b=a , where a is the major and b the minor semiaxis of the ellipse (Ref. 7).
12. If the disk of case 10 is heated symmetricall
Reaction and deformation formulas for circular arches (Continued )
4. Left end fixed, right
end pinned
340
TABLE 9.3
Since dHA 0 and cA 0,
AMM LPH AHM LPM
AHH AMM A2HM
and
MA AHH LPM AHM LPH
R
AHH AMM A2HM
Use load terms given above for cases 1a1s
General
56
Formulas for Stress and Strain
[CHAP. 3
much higher allowable stress than others. This fact is often recognized
in design; for example, the allowable stress for wooden airplane spars
varies according to the form factor and the proportion of the stress
SEC.
TABLE 9.2
Formulas for circular rings (Continued)
MA
wR2
(
"
2p y2 c 6s k2 2p y s
pp y2
p y3
3
#)
"
NA
Max M occurs at an angular position x1
where x1 > y; x1 > 108:6
; and x1 is found from
x1 y s cos x1 3s 2p 2y yc sin x1 0
Max M MC
(
wR2
2y2 c 6
TABLE 13.4
Formulas for discontinuity stresses and deformations at the junctions of shells and plates (Continued )
2b. Axial load P
LTA1
For axial tension, R1 R2 , E1 E2 , n1 n2 0:3, and for R=t > 5.
DRA
LTAC
LTB1
LTB2
LTBC
Pn1
K
;
2pE1 t1 DRA
cA
Pn1
K
Formulas for torsional deformation and stress (Continued)
Form and dimensions of cross sections,
30. Shaft with four keyways
Formula for K in y
K 2Cr4 where C varies with
For 0 4
TL
KG
b
as follows.
r
b
4 0:4:
r
C K1 K2
2
3
b
b
b
K4
K3
r
r
r
0:7854
a
SEC.
13.5]
Shells of Revolution; Pressure Vessels; Pipes
585
Summing the results from case 7c with correction terms and from case 7b
produces DRA 576:38106 , cA 674:84106 , s1 54:945, s2 98:792,
s01 1227:8, and s02 269:5.
The junction moves radially outwa
SEC.
2.3]
Stress and Strain: Important Relationships
Figure 2.14
31
Plane stress maximum shear stress.
A counterclockwise rotation of 45 of the normal in the 3 direction about axis 2
is represented by
8 09 2
38 9
cos 45 0
sin 45 >
>
<x >
<1>
=
=
6
7
y0
E1C08
11/10/2009
13:19:41
Page 185
Multiple Choice Quiz
Biodegradable_plastic. Plastic. Available at: http:/
en.wikipedia.org/wiki/Plastic.
[22] Green Plastics. Available at: http:/www.greenplastics.com/reference.
185
[23] Young, R. J., and Lovell, P. Int
E1C36
11/10/2009
13:30:52
Page 857
Section 36.1/Microsystem Products
857
and data applications. A CD disk is molded of polycarbonate (Section 8.2), which has ideal
optical and mechanical properties for the application. The disk is 120 mm in diameter and
1
E1C34
11/10/2009
15:58:57
Page 809
Section 34.3/Lithography
809
The diameter of the processable area of the wafer will be slightly less than the outside
diameter of the wafer. The actual number of chips on the wafer may be different from the
value given b
Case no., form of vessel
Manner of loading
4f. Uniform rotation, o rad=s,
about central axis
dm mass density
5. Toroidal shell
5a. Uniform internal or
external pressure, q
force=unit area
Formulas
For y 4 90
wR2
2t
wR2
R
2 cos2 y 2
s2
2t
R1
wR22 sin y
R
Tables
SEC.
11.14
Numerical values for functions used in Table 11.2
Numerical values for the plate coefficients F; C; L, and G for values of b=r; b=a; ro =a, and ro =r, respectively, from 0.05 to 1.0. Poissons ratio is 0.30.
The table headings are given f
SEC.
15.2]
Elastic Stability
711
effect of lateral restraint of the tensile flange of a beam under lateral
buckling; example calculations are presented. Massey and McGuire
(Ref. 54) present graphs of buckling coefficients for both stepped and
tapered cant
SEC.
12.3]
Columns and Other Compression Members
535
resist the transverse shear due to initial obliquity and that consequent
upon such bending as may occur under load. The amount of this shear
is conjectural since the obliquity is accidental and indeterm