Page 183 - Marks Calculation for Machine Design

P1: Shibu
January 4, 2005
14:25
Brown˙C04
Brown.cls
COMBINED LOADINGS
The thermal stress due to a temperature drop ( T ) is given by Eq. (4.6) where the thermal
strain (ε T ) is multiplied by the modulus of elasticity (E) 165
σ thermal = Eε T = Eα( T ) (4.2)
and (α) is the coefficient of thermal expansion of the pipe.
Combining these two normal stresses, both of which are constant over the cross section
of the pipe, gives the single stress (σ xx ) shown in Eq. (4.3),
L

σ xx = σ axial + σ thermal = Eε axial + Eε T = E + α( T ) (4.3)
L
where
L L installed − L o
= (4.4)
L L o
Stress Elements. The general stress element shown in Fig. 4.2 becomes the uniaxial stress
element shown in Fig. 4.15, where the normal stress (σ xx ) is given by Eq. (4.3) and both
the normal stress (σ yy ) and the shear stress (τ xy ) are zero.
s yy 0
t xy
0
t xy ∆L
s xx s xx s = E L + a(∆T)
xx
→
s xx
t xy
0
t xy
s yy 0
FIGURE 4.15 Stress element for axial and thermal loads.

U.S. Customary SI/Metric
Example 4. Determine the maximum stress Example 4. Determine the maximum stress
(σ xx ) due to a combination of axial and ther- (σ xx ) due to a combination of axial and ther-
mal loads like those for the machine element in mal loads like those for the machine element in
Fig. 4.14, where Fig. 4.14, where
L o = 3 ft (1/32 of an inch too short) L o = 1m
L installed = 3.0026 ft L installed = 1.0008 m
◦
T =−80 F T =−45 C
◦
◦
◦
α = 6.5 × 10 −6 in/in · F (steel) α = 12 × 10 −6 cm/cm · C(steel)
9
6
2
2
E = 30 × 10 lb/in (steel) E = 207 × 10 N/m (steel)
solution solution
Step 1. Calculate the axial strain (ε axial ) using Step 1. Calculate the axial strain (ε axial ) using
Eq. (4.8) as Eq. (4.8) as
L L installed − L o L L installed − L o
ε axial = = ε axial = =
L L o L L o
(3.0026 ft) − (3ft) (1.0008 m) − (1m)
= =
(3ft) (1m)
0.0026 ft 0.0008 m
= = 0.00087 = = 0.0008
3ft 1m