Page 104 - Marks Calculation for Machine Design

P1: Sanjay
January 4, 2005
16:18
Brown.cls
Brown˙C02
STRENGTH OF MACHINES
86
Note that the deflection determined in the previous example was positive or upward. For
distances closer to the left support, the deflection will be negative or downward. The location
of the transition point, meaning the point of zero deflection, is not a simple expression, and
depends on the relative values of the length (L) between the supports and the length of the
overhang (a).
Unless the length of the overhang (a) is very short compared to the length (L) between
the supports, the maximum downward deflection occurs at the tip of the overhang, a distance
(x 1 )isequalto(a)fromtherollersupport.Substituting(a)forthedistance(x 1 )inEq.(2.47b)
gives the tip deflection ( Tip ) as
w 4 3 3
Tip = (3 a + 4 a L − aL ) ↑ (2.48)
24 EI
U.S. Customary SI/Metric
Example 5. Calculate the maximum down- Example 5. Calculate the maximum down-
ward deflection ( Tip ) for the beam configu- ward deflection ( Tip ) for the beam configu-
ration in Example 4, where ration in Example 4, where
w = 113 lb/ft w = 1,540 N/m
L = 3 ft, a = 1ft L = 1m, a = 0.3 m
6
6
EI = 2.5 × 10 lb · ft 2 EI = 1.02 × 10 N · m 2
solution solution
The maximum downward deflection ( Tip ) is The maximum downward deflection ( Tip ) is
given by Eq. (2.48). given by Eq. (2.48).
w 4 3 3 w 4 3 3
Tip = (3 a + 4 a L − aL ) ↑ Tip = (3 a + 4 a L − aL ) ↑
24 EI 24 EI
(113 lb/ft) (1,540 N/m)
= =
2
2
6
6
24 (2.5 × 10 lb · ft ) 24 (1.02 × 10 N · m )
4
3
4
3
×[3(1ft) + 4(1ft) (3ft) ×[3(0.3m) + 4(0.3m) (1m)
3
3
−(1ft)(3ft) ] −(0.3m)(1m) ]
113 lb/ft 1,540 N/m
= =
7
7
6.0 × 10 lb · ft 2 2.45 × 10 N · m 2
4
4
×[(3 + 12 − 27) ft ] ×[(0.0243 + 0.108 − 0.3) m ]

1 1
4
4
= 1.88 × 10 −6 (−12 ft ) = 6.29 × 10 −5 (−0.1677 m )
ft 3 m 3
12 in 100 cm
=−0.000023 ft × ↑ =−0.00001 m × ↑
ft m
= 0.00027 in ↓ = 0.001 cm ↓
2.2.9 Double Overhang: Concentrated
Forces at Free Ends
The simply-supported beam in Fig. 2.65 has double overhangs with concentrated forces,
each of magnitude (F), acting directly downward at the free ends: points A and D. The
distance between the supports is labeled (L), and the length of each overhang is labeled
(a). Therefore, the total length of the beam, measured from the left end, is (L + 2a).