Page 203 - Shigley's Mechanical Engineering Design

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178 Mechanical Engineering Design
3 Since M 1 is the redundant reaction at O, write the equation for the angular
deflection at point O. From Castigliano’s theorem this is

∂U
θ O = (6)
∂M 1
We can apply Eq. (4–31), using the variable x as shown in Fig. 4–16b. However, sim-
pler terms can be found by using a variable ˆx that starts at B and is positive to the left.
With this and the expression for R 2 from Eq. (5) the moment equations are

F M 1 l

x
M = − ˆ x 0 ≤ˆ ≤ (7)
2 l 2

F M 1 l l
x
M = − ˆ x − F x − ≤ˆ ≤ l (8)
ˆ
2 l 2 2
For both equations
∂M ˆ x
=− (9)
∂M 1 l
Substituting Eqs. (7) to (9) in Eq. (6), using the form of Eq. (4–31) where F i = M 1 , gives

∂U 1 l/2 F M 1 ˆ x l F M 1
x
θ O = = − ˆ x − d ˆ + − ˆ x
∂M 1 EI 0 2 l l l/2 2 l
l ˆ x

x
ˆ
− F x − − d ˆ = 0
2 l
Canceling 1/EIl, and combining the first two integrals, simplifies this quite readily to
F M 1 2 l
l l
x
x
− ˆ x d ˆ− F ˆ x − ˆ xd ˆ = 0
2 l 0 l/2 2
Integrating gives
3
2
F M 1 l F 3 l Fl 2 l
3
− − l − + l − = 0
2 l 3 3 2 4 2
which reduces to
3Fl
M 1 = (10)
16
4 Substituting Eq. (10) into (5) results in

11F 5F
R 1 = R 2 = (11)
16 16
which again agrees with beam 11 of Table A–9.