194 6 Deflections of Elastic Structures
Computation of displacementsInitial parameters methodn!F(x − aF)nFictitious beam methodEI EIMf Qf
y = ,q =Strain energyM (^) + ⋅ ⋅ ⋅
(^2) (x)dx
U =
2 EI
Work-strain energy method
W=U
Castigliano theorem
qi =∂U
∂Q
Graph multiplication method
EI M^ p^ Mdx^ = EI
Ωp yc
∫
1
Maxwell-Mohr integral
Dkp =
s
EI ds
M (^) p M k
0
Elastic loads method
EA
Nn⋅ NP ⋅ l
Wn =
Double integration method
EIy′′(x) = ± M (x)
EIy(x) = EIy 0 + EIq 0 x− ±
Fig. 6.31 Two fundamental approaches to calculation of displacements of elastic structures
Group 1 This group contains the double integration method, initial parameters
method, and conjugate beammethod (fictitious beam method). All methods of this
group are based on the differential equation of the elastic curve of the beam.
Double integration of differential equation allows finding the equation of an elas-
tic curve of a beam. Integration procedure should be performed for each specified
problem. This procedure leads to appearance of the constants, which should be
determined from the boundary conditions. The number of the constants is twice
more than the number of portions of a beam. For beams with two or more por-
tions, this method becomes very cumbersome.
Initial parameters method is a modification of double integration method. This
method is effective for deriving equation of elastic curve for uniform beams with
large number of portions and any types of loads. The integrating procedure is
performed once at the deriving of universal equation and therefore, practical ap-
plication of this method reduces only to the algebraic procedures.
Conjugate beam method also presents the modification of the double integration
method and allows computing the linearand angular displacements at specified
section of the beam. This method required constructing the fictitious beam and
computing the fictitious bending moment and shear at specified section. In case
of complex loading, this method leads to the cumbersome computation.