Page 115 - Marks Calculation for Machine Design

P1: Sanjay
16:18
January 4, 2005
Brown˙C02
Brown.cls
U.S. Customary BEAMS SI/Metric 97
or at the midpoint of the beam, given by or at the midpoint of the beam, given by
Eq. (2.57), Eq. (2.57).
wL 2 wL 2
a
a
M = 1 − 4 M = 1 − 4
max @ 8 L max @ 8 L
midpoint midpoint
(15 lb/ft)(12 ft) 2 2ft (225 N/m)(4m) 2 0.6m
= 1 − 4 = 1 − 4
8 12 ft 8 4m
2
2
(15 lb/ft)(144 ft ) (225 N/m)(16 m ) 3
1
= 1 − 4 = 1 − 4
8 6 8 20
2,160 ft · lb 4 3,600 N · m 3
= 1 − = 1 −
8 6 8 5

1 2
= (270 ft · lb) = (450 N · m)
3 5
= 90 ft · lb = 180 N · m
Note that for these relative values of the overhang (a) and the length of the beam (L),
the bending moment at the supports is less than the bending moment at the midpoint of the
beam. As said earlier, if the overhang (a) is one-fourth the length of the beam (L), that is
(a = L/4), then the maximum deflection at the midpoint will be zero, and the maximum
2
bending moment at the supports will have a magnitude of (wL /32).
Deflection. For this loading configuration, the deflection along the beam has the shape
shown in Fig. 2.78.
w
A D
B C
a a
L
FIGURE 2.78 Beam deflection diagram.
However, formal equations for the deflection of either the overhangs or between the
supports are not available. Seems odd, but even Marks’ Standard Handbook for Mechanical
Engineers does not include deflection equations for this beam configuration. The author
would greatly appreciate any information regarding where these equations might be found.
This complete the first of the two sections focused on simply-supported beams. In the
next section we will present several important cantilevered beams with common loading
configurations.

2.3 CANTILEVERED BEAMS

As stated earlier, cantilevered beams like the one shown in Fig. 2.79, have a special type
of support at one end, as shown on the left at point A in the figure. The other end of the
beam can be free as shown in the right at point B, or can have a roller or pin type support
at the other end as shown in Figs. 2.80 and 2.81, respectively.