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Unformatted text preview: 09Ch09.qxd 9/27/08 1:31 PM Page 713 SECTION 9.3 Deflection Formulas Problem 9.3-7 Obtain a formula for the ratio dC /dmax of the deflection at the
midpoint to the maximum deflection for a simple beam supporting a concentrated load P (see figure).
From the formula, plot a graph of dC/dmax versus the ratio a/L that
defines the position of the load (0.5 a/L 1). What conclusion do you
draw from the graph? (Use the formulas of Example 9-3.) 713 P A B
L Solution 9.3-7 Simple beam (concentrated load)
48EI Eq. (9-35): dC (3 13L)(3L2 Eq. (9-34): dmax
dmax 9 13LEI Pb(L2 2 dmax 0.5
0.974 (a Ú b) (3 13L)( L2 + 8aL 4a2) a2)3/2 (3 13L) a 1 + 8 dc
b 4b2) b) 16(2aL a/L b, the ratio b versus from 0.5 to 1.0. (a Ú b) Replace the distance b by the distance a by substituting
L a for b:
dmax GRAPH OF dc/dmax VERSUS b
Because a b2)3/2 2 3/2 16(L (a Ú b) Divide numerator and denominator by L2:
dmax (3 13L) a 1 + 8
16L a 2 16 a 2 a
L 4 a2 4 L2 b 3/2 a
L a2 b
L2 a2 b
L2 3/2 b ;
NOTE: The deflection dc at the midpoint of the beam is
almost as large as the maximum deflection dmax. The
greatest difference is only 2.6% and occurs when the
load reaches the end of the beam (b 1). ALTERNATIVE FORM OF THE RATIO
dmax (3 13)(
L 1 + 8b 16(2 b b 2)3/2 4 b 2) ; ...
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This note was uploaded on 12/22/2011 for the course MEEG 310 taught by Professor Staff during the Fall '11 term at University of Delaware.
- Fall '11