Page 64 - Marks Calculation for Machine Design

P1: Sanjay
16:18
January 4, 2005
Brown.cls
Brown˙C02
STRENGTH OF MACHINES
46
If the distance (a) is greater than the distance (b), then the maximum deflection ( max )
caused by this loading configuration is given by Eq. (2.10),
Fb a (L + b) 3/2 a(L + b)
max = at x = (2.10)
3 EIL 3 3
located at a point to the left of where the force (F) acts. It is clear that the distance (x) for
the maximum deflection is not the place where the force (F) acts.
If the distance (a) is less than the distance (b), then consider a mirror image of the beam
where the distances (a) and (b) swap values.
The deflection ( a ) at the point where the force (F) acts, the distance (a), is given by
Eq. (2.11),
2 2
Fa b
a = at x = a (2.11)
3 EIL
where it is not entirely obvious that the deflection ( a ) is less than the maximum deflection
( max ).
U.S. Customary SI/Metric
Example 4. Calculate the deflection ( ) of Example 4. Calculate the deflection ( ) of
a simply-supported beam with a concentrated a simply-supported beam with a concentrated
force (F) at an intermediate point a distance force (F) at an intermediate point a distance
(x) of (3L/8), where (x) of (3L/8), where
F = 10 kip = 10,000 lb F = 45 kN = 45,000 N
L = 8 ft, a = 6 ft, b = 2ft L = 3m, a = 2m, b = 1m
9
2
2
6
E = 30 × 10 lb/in. (steel) E = 207 × 10 N/m (steel)
I = 4in 4 I = 201 cm 4
solution solution
Step 1. Determine the distance (x). Step 1. Determine the distance (x).
3L 3 (8ft) 3L 3 (3m)
x = = = 3ft x = = = 1.125 m
8 8 8 8
Step 2. Calculate the stiffness (EI). Step 2. Calculate the stiffness (EI).
6
2
4
4
2
9
EI = (30 × 10 lb/in )(4in ) EI = (207 × 10 N/m )(201 cm )
2
8
1.2 × 10 lb · in × 1ft 2 1m 4
= ×
144 in 2 (100 cm) 4
5
5
= 8.33 × 10 lb · ft 2 = 4.16 × 10 N · m 2
Step 3. Determine the deflection ( ) from Step 3. Determine the deflection ( ) from
Eq. (2.9a). Eq. (2.9a).
Fbx 2 2 2 Fbx 2 2 2
= (L − b − x ) = (L − b − x )
6 (EI) L 6 (EI) L
(10,000 lb)(2ft)(3ft) (45,000 N)(1m)(1.125 m)
= =
2
5
5
2
6(8.33 × 10 lb · ft )(8ft) 6 (4.16 × 10 N · m )(3m)
2
2
2
2
2
2
× [(8ft) − (2ft) − (3ft) ] ×[(3m) − (1m) − (1.125 m) ]