Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

Castigliano’s second theorem applied to beam deflections is expressed by Equation
(17.40). When the expressions just derived for ∂M/ ∂P
and M are substituted, Equation
(17.40) becomes
D=
∫
0
L
⎛∂M
⎞ M
⎜ ⎟ = ∫ - ⎛ ⎞
⎜-
⎟ = ∫
⎝ ∂P
⎠ EI dx L
x 2
wx
L
3
wx
dx
⎝ 2EI
⎠ 2EI dx
0
0
Now integrate this expression over the beam length L to determine the vertical beam
deflection at A:
4
wL
D A = ↓
8EI
Ans.
Since the result is a positive value, the deflection occurs in the direction assumed for
the dummy load P—that is, downward.
(b) Calculation of Slope: To determine the angular rotation
of the cantilever beam at A, a dummy concentrated
moment M′ will be applied. Because the beam is expected
to slope upward from A, the dummy moment will be applied
counterclockwise in this instance.
Again, draw a free-body diagram around end A of
the beam, placing the origin of the x coordinate system
at A. From the diagram, derive the following equation for the internal bending
moment M:
M
2
wx
=- - M′ 0 ≤ x ≤ L
2
Next, differentiate this equation to obtain
∂M
1
∂ M′ =-
Now substitute M′ = 0 into the bending-moment equation to get
M
2
wx
=-
2
Castigliano’s second theorem applied to beam slopes is expressed by Equation (17.41).
When the expressions just derived for ∂M/ ∂ M′
and M are substituted, Equation (17.41)
becomes
θ =
⎛ ∂M
⎞ M
⎝∂ M′
⎠ EI dx
1 ⎛ 2
wx ⎞
2
wx
⎝ 2EI
⎠ 2EI dx
0
∫ ⎜ ⎟ = ∫ - ⎜-
⎟ dx = ∫
0
L L L
0
Integrate this expression over the beam length L to determine the beam slope at A:
3
wL
θ A = (ccw)
Ans.
6EI
Since the result is a positive value, the angular rotation occurs in the same direction
assumed for the dummy moment—that is, counterclockwise (ccw).
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