Page 69 - Marks Calculation for Machine Design

P1: Sanjay
January 4, 2005
16:18
Brown.cls
Brown˙C02
51
BEAMS
Note that the bending moment (M) is zero at both ends, and increases linearly to a
maximum positive value (Ca/L) where the couple acts. At the point where the couple
acts, that is at a distance (a), there is a discontinuity in the bending moment of magnitude
(C) downward. So from where the couple acts, the bending moment starts at a maxi-
mum negative value (−Cb/L) and increases linearly back to zero. Note that the slopes
of these two increasing values of bending moment are equal, and therefore the lines are
parallel.
If the distance (a) is less than the distance (b), then the maximum bending moment
(M max ) is given by Eq. (2.14a). If the distance (a) is greater than the distance (b), then the
maximum bending moment (M max ) is given by Eq. (2.14b).
Cb
M max = a < b (2.14a)
L
Ca
M max = a > b (2.14b)
L
If the distance (a) is equal to the distance (b), which means are the couple (C) acts at
the midpoint of the beam, then (a) and (b) each is equal to half the length of the beam (L).
Therefore, the bending moment distribution will be symmetrical about the midpoint of the
beam, and the maximum bending moment (M max ) is given by Eq. (2.15)
C L
M max = a = b = (2.15)
2 2
U.S. Customary SI/Metric
Example 2. Calculate the shear force (V ) and Example 2. Calculate the shear force (V ) and
bending moment (M) for a simply-supported bending moment (M) for a simply-supported
beam with a concentrated couple (C) at a dis- beam with a concentrated couple (C) at a dis-
tance (L/6) from the left end of the beam, where tance (L/4) from the left end of the beam, where
C = 15 ft· kip = 15,000 ft · lb F = 20 kN · m = 20,000 N · m
1
1
L = 12 ft, a = 4 ft, b = 8ft L = 4m, a = 1 m, b = 2 m
2 2
solution solution
Step 1. Establish the distance (x) from the left Step 1. Establish the distance (x) from the left
end of the beam, where end of the beam, where
L 12 ft L 4m
x = = = 2ft x = = = 1m
6 6 4 4
Step 2. Determine the shear force (V ) from Step 2. Determine the shear force (V ) from
Fig. 2.27 as Fig. 2.27 as
C 15,000 ft · lb C 20,000 N · m
V = = V = =
L 12 ft L 4m
= 1,250 lb = 5,000 N
Step 3. Determine the bending moment (M) Step 3. Determine the bending moment (M)
from Eq. (2.13a). from Eq. (2.13a).
C 15,000 ft · 1b C 20,000 N · m
M = x = (2ft) M = x = (1m)
L 12 ft L 4m
= (1,250 lb)(2ft) = 2,500 ft · lb = (5,000 N)(1m) = 5,000 N · m