Page 372 - Marks Calculation for Machine Design

P1: Sanjay
January 4, 2005
15:14
Brown.cls
Brown˙C08
APPLICATION TO MACHINES
354
and the polar moment of inertia (J group ) can be determined from the expression
3 3
LH HL 2
J group = 2 + + LHd o (8.75)
12 12
where the factor 2 reflects the fact that there are two welds and the terms in brackets represent
the application of the parallel axis theorem to the rectangular weld shapes.
Notice that the shear stress (τ shear ) acts downward at any point on either of the two welds;
however, the shear stress (τ torsion ) acts perpendicular to the radial distance (r o ). There are
four points, labeled (A), (B), (C), and (D) in Fig. 8.13, where the shear stress (τ torsion ) is
maximum. The maximum shear stress (τ max ) is therefore the geometric sum of these two
separate shear stresses and is found using the law of cosines in the scalene triangle formed
by these three stresses and shown in Fig. 8.14.
L
L/2
B A
a
t
r o t torsion
d o shear
a t
b = 180° – a max
O
FIGURE 8.14 Maximum shear stress diagram.
The angle (α) is calculated as
L
α = tan −1 2 (8.76)
d o
where (L) is the length of the weld and (d o ) is the distance from the centroid of the weld
group to the centerline of the weld. The angle (β) is the supplement of the angle (α) and as
shown in Fig. 8.14 is given by
◦
β = 180 − α (8.77)
Therefore, using the law of cosines on the resulting scalene triangle, the maximum shear
stress (τ max ) is determined from Eq. (8.78) as
2 2 2
τ max = τ shear + τ torsion − 2 (τ shear )(τ torsion ) cos β (8.78)

U.S. Customary SI/Metric
Example 3. For the fillet weld and loading Example 3. For the fillet weld and loading
configuration shown in Figs. 8.12 and 8.13, configuration shown in Figs. 8.12 and 8.13,
determine the maximum shear stress (τ max ), determine the maximum shear stress (τ max ),
where where
P = 3,000 lb P = 13,500 N
◦
H = 0.619 in (0.875 in × cos 45 ) H = 1.4 cm = 0.014 m (2 cm × cos 45 )
◦
L = 4in L = 10 cm = 0.1 m
d o = 1.5 in d o = 4cm = 0.04 m
L o = 1ft = 12 in L o = 30 cm = 0.3 m