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

17.13 Calculating Deflections of Beams
by Castigliano’s Theorem
781
CALCuLATINg dEFLECTIONS
OF bEAMS by CASTIgLIANO’S
THEOREM
The strain energy in a flexural member was developed in Section 17.5. The total strain
energy in a beam of length L is given by Equation (17.20) as
U
=
L
∫ 0
2
M
2EI dx
To compute the deflection of a beam, the general expression for strain energy given by
Equation (17.20) can be substituted into Equation (17.37) to obtain
D= ∂
∂P
L
∫ 0
2
M
2EI dx
From the rules of calculus, the integral can be differentiated by differentiating inside
the integral sign. If the elastic modulus E and the moment of inertia I are constant with
respect to the applied load P, then
∂
∂P
M
2EI dx =
⎛∂M
⎝ ⎜ ∂P
L
2
L
2
∫ 0
∫ 0
⎞ 1
⎟
⎠ 2EI dx
Differentiation inside the
integral is permissible when P is
not a function of x.
2
Since the partial derivative ∂M / ∂ P = 2 M( ∂M/ ∂P)
, Castigliano’s second theorem for
beam deflections can be written as
where
D=
L
∫ 0
⎛∂M
⎞ M
⎜ ⎟
⎝ ∂P
⎠ EI dx
(17.40)
D = displacement of a point on the beam
P = external force applied to the beam in the direction of D and expressed as a variable
M = internal bending moment in the beam, expressed as a function of x and caused by
both the force P and the loads on the beam
I = moment of inertia of the beam cross section about the neutral axis
E = elastic modulus of the beam
L = length of the beam
Similarly, Castigliano’s second theorem can be used to compute the rotation angle
(i.e., slope) of a beam from
θ =
L
∫ 0
⎛ ∂M
⎞ M
⎜ ⎟
⎝∂ M′
⎠ EI dx
(17.41)
where θ is the rotation angle (or slope) of the beam at a point and M′ is a concentrated moment
applied to the beam in the direction of θ at the point of interest and expressed as a
variable.
If the deflection is required at a point where there is no external load or if the deflection
is required for a direction that is not aligned with the external load, then a dummy load
must be added in the proper direction at the desired point. Likewise, if the slope is required
at a point where there is no external concentrated moment, then a dummy moment must be
added in the proper direction at the desired point.