Page 58 - Marks Calculation for Machine Design

P1: Sanjay
January 4, 2005
16:18
Brown.cls
Brown˙C02
STRENGTH OF MACHINES
40
located at the midpoint (L/2) of the beam directly under the concentrated force (F). As will
be shown in the following examples, the value for the deflection ( ) at any location along
the beam for this loading configuration will be downward. However, although many loading
configurations produce deflections that are always downward, still others have deflections
that are always upward, and still others where the deflection is both upward and downward,
depending on the location along the length of the beam.
U.S. Customary SI/Metric
Example 4. Calculate the deflection ( ) of a Example 4. Calculate the deflection ( ) of a
beam with a concentrated force (F) at its mid- beam with a concentrated force (F) at its mid-
point a distance (L/4) from the left end of the point a distance (L/4) from the left end of the
beam, where beam, where
F = 12 kip = 12,000 lb F = 55 kN = 55,000 N
L = 6ft L = 2m
2
9
2
6
E = 30 × 10 lb/in (steel) E = 207 × 10 N/m (steel)
I = 2.25 in 4 I = 88 cm 4
solution solution
Step 1. Determine the distance (x). Step 1. Determine the distance (x).
L 6ft L 2m
x = = = 1.5ft x = = = 0.5m
4 4 4 4
Step 2. Calculate the stiffness (EI). Step 2. Calculate the stiffness (EI).
6
9
4
2
2
4
EI = (30 × 10 lb/in )(2.25 in ) EI = (207 × 10 N/m )(88 cm )
7
2
4
2
= 6.75 × 10 lb · in × 1ft /144 in 2 ×1m /(100 cm) 4
5
5
= 4.69 × 10 lb · ft 2 = 1.82 × 10 N · m 2
Step 3. Determine the deflection ( ) from Step 3. Determine the deflection ( ) from
Eq. (2.4). Eq. (2.4).
Fx 2 2 Fx 2 2
= (3L − 4x ) = (3L − 4x )
48 (EI) 48 (EI)
(12,000 lb)(1.5ft) (55,000 N)(0.5m)
= =
2
2
5
5
48 (4.69 × 10 lb · ft ) 48 (1.82 × 10 N · m )
2
2
2
2
×[3 (6ft) − 4(1.5ft) ] ×[3(2m) − 4 (0.5m) ]
(18,000 lb · ft) (27,500 N · m)
= 2 = 6 2
7
(2.25 × 10 lb · ft ) (8.74 × 10 N · m )
2
2
2
2
×[(108 ft ) − (9ft )] ×[(12 m ) − (1m )]

1 −3 1 2
2
= 8.00 × 10 −4 × [99 ft ] = 3.15 × 10 × [11 m ]
ft m
12 in 100 cm
= 0.079 ft × = 0.95 in ↓ = 0.0346 m × = 3.46 cm ↓
ft m
Example 5. Calculate the maximum deflec- Example 5. Calculate the maximum deflec-
tion ( max ) and its location for the beam con- tion ( max ) and its location for the beam con-
figuration in Example 4, where figuration in Example 4, where
F = 12 kip = 12,000 lb F = 55 kN = 55,000 N
L = 6ft L = 2m
5
5
EI = 4.69 × 10 lb · ft 2 EI = 1.82 × 10 N · m 2