Page 72 - Marks Calculation for Machine Design

P1: Sanjay
January 4, 2005
Brown˙C02
Brown.cls
54
U.S. Customary 16:18 STRENGTH OF MACHINES SI/Metric
Step 3. Determine the deflection ( ) from Step 3. Determine the deflection ( ) from
Eq. (2.16a). Eq. (2.16a).
Cx 2 2 2 Cx 2 2 2
= [6 aL − x − 3 a − 2 L ] = [6 aL − x − 3 a − 2 L ]
6 (EI) L 6 (EI) L
(15,000 ft · lb)(2ft) (20,000 N · m)(1m)
= 2 = 5 2
6
6(1.43 × 10 lb · ft )(12 ft) 6 (5.54 × 10 N · m )(4m)
2
2
2
2
2
2
2
2
×[(6(4)(12) − (2) − 3(4) − 2 (12) ) ft ] ×[(6(1.5)(4)−(1) −3 (1.5) −2 (4) ) ft ]
2
2
(30,000 lb · ft ) (20,000 N · m )
= 3 = 7 3
8
(1.03 × 10 lb · ft ) (1.33 × 10 N · m )
2
2
×[(288 − 4 − 48 − 288) ft ] ×[(36 − 1 − 6.75 − 32) ft ]

1 −3 1 2
2
= 2.9 × 10 −4 × [−52 ft ] = 1.5 × 10 ft × [−3.75ft ]
ft
12 in 100 cm
=−0.015 ft × =−0.18 in ↓ =−0.0056 m × =−0.56 cm ↓
ft m
= 0.18 in ↑ = 0.56 cm ↑
Example 5. Calculate the deflection ( a ) at Example 5. Calculate the deflection ( a ) at
the location (a) where the couple (C) acts, the location (a) where the couple (C) acts,
where where
C = 15 ft · kip = 15,000 ft · lb C = 20 kN · m = 20,000 N · m
L = 12 ft, a = 4 ft, b = 8ft L = 4m, a = 1 1 / 2 m, b = 2 1 / 2 m
6
5
EI = 1.43 × 10 lb · ft 2 EI = 5.54 × 10 N · m 2
solution solution
Calculate the deflection ( a ) where the force Calculate the deflection ( a ) where the force
(F) acts from Eq. (2.17). (F) acts from Eq. (2.17).
Cab Cab
a = (2 a − L) a = (2 a − L)
3 (EI) L 3 (EI) L
(15,000 ft · lb)(4ft)(8ft) (20,000 N · m)(1.5m)(2.5m)
= 2 =
2
5
6
3 (1.43 × 10 lb · ft )(12 ft) 3 (5.54 × 10 N · m )(4m)
×[2 (4ft) − 12 ft] ×[2 (1.5m) − 4m]
4
5
4.8 × 10 lb · ft 3 7.5 × 10 N · m 3
= [8 ft − 12 ft] = 6 3 [3 m − 4m]
7
5.15 × 10 lb · ft 3 6.65 × 10 N · m
= (9.32 × 10 −3 )[−4 ft] = (1.13 × 10 −2 )[−1m]
12 in 100 cm
=−0.037 ft × =−0.45 in ↓ =−0.0113 ft × m =−1.13 cm ↓
ft
= 0.45 in ↑ = 1.13 cm ↑
Notice that the deflection ( a ) came out as a negative value, which means it is above the
axis of the beam where the couple (C) acts. As stated earlier, if the couple is located at the
midpoint of the beam, then the deflection at this point is zero.