Beam deflection examples
Find v(x) and vmax for each of the following cases:
Answer: EIv(x) = 2.083<x - 0>3 – 2.083<x - 40>3 – 3333x
vmax = -0.02565” (up) at x = 23.09”
and 0.1167” (down) at the free end
Answer: standard solution from text (e.g. Gere and Timoshenko)
v = wx(9L3 – 24Lx2 + 16x3)/384EI 0 < x < L/2
v = wL(8x3 -24lx2 +16 L2x –L3)/384EI L/2 < x < L
Assume max. defl. is in range 0 < x < L/2 and differentiate to get v’(x)
Hence get vmax = 7wL4/768EI at x = 0.416L
Answer: EIv(x) = 3PL<x - 0>2/32 - 11P<x - 0>3/96 + P<x – L/2>3/6
EIv’(x) = 3PL<x – 0> - 11P<x – 0>2/32 + P<x – L/2>2/2
Guess x in range L/2 < x < L, and solve v’(x) = 0 giving x = 0.553L
And hence vmax = 0.00932L