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

The terms L, A, and E are constant for each particular member. Since the partial
2
derivative ∂F / ∂ P = 2 F( ∂F/ ∂P)
, Castigliano’s second theorem for trusses can be written as
777
CALCuLATINg dEFLECTIONS
OF TRuSSES by CASTIgLIANO’S
THEOREM
where
∆=
∑
⎛∂F
⎞ FL
⎜ ⎟ (17.39)
⎝∂P
⎠ AE
D = displacement of the truss joint
P = external force applied to the truss joint in the direction of D and expressed as a variable
F = internal axial force in a member caused by both the force P and the loads on the truss
L = length of the member
A = cross-sectional area of the member
E = elastic modulus of the member
To determine the partial derivative ∂F/∂P, the external force P must be treated as a
variable, not a specific numeric quantity. Consequently, each internal axial force F must be
expressed as a function of P.
If the deflection is required at a joint at which 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 joint. We obtain the joint deflection by
first differentiating the strain energy with respect to the dummy load and then taking the
limit as the dummy load approaches zero.
procedure for Analysis
The following procedure is recommended when Castigliano’s second theorem is applied to
calculate truss deflections:
1. Load P Expressed as a Variable: If an external load acts on the truss at the joint
where deflections are to be calculated and in the direction of the desired deflection,
then designate that load as the variable P. As a result, subsequent calculations will be
performed in terms of the variable P rather than in terms of the actual numeric value
known for this particular external load. Otherwise, apply a fictitious load (a dummy
load) in the direction of the desired deflection at the particular joint. Designate this
dummy load as P.
2. member Forces F in Terms of P: Develop expressions for the internal axial force
F created in each truss member by the actual external loads and the variable load P. It
is likely that the expression for the internal force in a particular member will include
both a numeric value and a function in terms of P. Assume that tensile forces are positive
and compressive forces are negative.
3. partial Derivatives for Each Truss member: Differentiate the expressions for
the truss-member forces F with respect to P to compute ∂F/ ∂P.
4. Substitute Numeric Value for P: Substitute the actual numeric value for load P
into the expressions for F and ∂F/ ∂P
for each truss member. If a dummy load has been
used for P, its numeric value is zero.
5. Summation: Perform the summation indicated by Equation (17.39) to calculate the
desired joint deflection. A positive answer indicates that the deflection acts in the same
direction as P, a negative answer in the opposite direction.
The use of Castigliano’s theorem to compute truss joint deflections is illustrated in
Examples 17.16 and 17.17.