Unformatted text preview: 09Ch09.qxd 9/27/08 1:35 PM Page 755 755 SECTION 9.7 Nonprismatic Beams Problem 9.7-2 The cantilever beam ACB shown in the figure supports a uniform q load of intensity q throughout its length. The beam has moments of inertia I2 and I1
in parts AC and CB, respectively.
(a) Using the method of superposition, determine the deflection dB at the free
end due to the uniform load.
(b) Determine the ratio r of the deflection dB to the deflection d1 at the free end
of a prismatic cantilever with moment of inertia I1 carrying the same load.
(c) Plot a graph of the deflection ratio r versus the ratio I2/I1 of the moments of
inertia. (Let I2 /I1 vary from 1 to 5.) A C
2 Solution 9.7-2 Cantilever beam (nonprismatic)
Use the method of superposition (3) Total deflection at point B (a) DEFLECTION dB AT THE FREE END dB (dB)1 + (dB)2 (1) Part CB of the beam:
(b) PRISMATIC BEAM d1
(dB)1 q L4
8EI1 2 qL4
128EI1 Ratio: r dB
d1 (c) GRAPH OF RATIO
(2) Part AC of the beam: 4 dC q(L/2)
a a qL
3EI2 2 qL
384EI2 3 uC q(L/2)
48EI2 (dB)2 L
dC + uC a b
I1 r 1
I2 ; ; ...
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