64
2 BASIC VARIATIONAL PRINCIPLES
easy to see that the force constraints do not violate the superposition
principle either.
It is often necessary to allow for force constraints in the structural
design practice. For example, one may have to analyze a stru
58
2 BASIC VARIATIONAL PRINCIPLES
F = Fs + FK + FR K ,
(3.13)
where Fs is a certain fixed field from Ps , FK K , FR K RK. Substituting
(3.13) to (3.3) and taking into account the equalities E(FR K) = 0 and
<FK, FR K > = 0 will give
K(F) = E(FK) k(FK) + 2<
76
2 BASIC VARIATIONAL PRINCIPLES
2
1
W3
W1
W2
Fig. 2.5. To the validation of the St.-Venant principle
Let us use the designation 2 for an area between the boundaries 1 and
2, and 3 for an area between the boundaries 2 and . Next, we can
denote interactio
2.3 Castigliano variational principle
E(,u) = (C 1, ) + (Ku, u) ,
k() = (Eu H, Eu u )
55
(3.2)
where
K(,u) = E(,u) k() .
(3.3)
Let F = cfw_, , u be a true stress-and-strain field. The following
statement holds:
K = K(F) K(Fs)
(3.4)
1
for any field Fs = cf
44
2 BASIC VARIATIONAL PRINCIPLES
strain compatibility conditions does not require them to be differentiable
even once.
At this point the following note should be made. One of substantial
differences (though not the only one!) of one-dimensional problems
52
2 BASIC VARIATIONAL PRINCIPLES
s(u + u) s(u) = s(u) .
Substituting all this in (2.8) will yield
L(u + u) L(u) = cfw_2<F, F > s(u) + E(u) .
(2.9)
But the braced expression from (2.9) can be expanded as follows:
2<F, F > s(u) =
= (CAu, Au) + (Ku, u) ( X
2.1 Energy space
( , ) = ( , Au) = (AT , u) + (H, Huu) .
43
(1.31)
Applying (1.30) will annihilate both scalar products in the right part of
(1.31).
Now let (1.29) be true at any satisfying the conditions (1.30).
Multiplying the first of the equations in
2.1 Energy space
47
2.1.8 Lagrangian energy space
Now we are going to construct an important linear set for further
consideration, L, by removing all fields of R o from the linear set P ko. This
can be done by using the scalar product introduced earlier o
50
2 BASIC VARIATIONAL PRINCIPLES
2.2 Lagrange variational principle
2.2.1 Conservative external forces
Up to this point, we did not discuss the nature of external forces X
distributed over the volume of a body and contour forces p which can
depend (both
2.1 Energy space
49
Multiplying both parts of the equality (1.46) scalarly (in the energy
metric) by an arbitrary field Fk L and taking into account the theorem of
field orthogonality (1.4.21) will give
< Fk, Fk1 Fk2> = 0 .
(1.47)
As the selected field Fk
68
2 BASIC VARIATIONAL PRINCIPLES
It may seem this reciprocity of kinematical and force constraints makes
the separate consideration of force constraints unnecessary because force
constraints can always be transformed into kinematical ones. This is
actual
2.1 Energy space
X (u
o
+ r) d +
v p (u
o
+ r) d = 0 .
45
(1.35)
Now using the known property of mixed product of three vectors, a, b
and c [3], in the form a (bc) = b (ac), and seeing that the
components of the vectors uo and are constant throughout the
60
2 BASIC VARIATIONAL PRINCIPLES
with the Ritz method where either the Lagrange functional or the
Castigliano functional is used, respectively, enables one to approach the
exact solution of a problem from both sides.
Lets sum up the results obtained here
2.2 Lagrange variational principle
53
According to the condition (2.12), the right part in (2.14) must be
nonnegative at any, even infinitesimal, value of . But this can be true
only if the factor at raised to power one is zero:
(CAu1, Auk) + (Ku1, uk) (
56
2 BASIC VARIATIONAL PRINCIPLES
The field of variations F is a homogeneously statically admissible field,
therefore this field can be represented as F = Fso with a certain field
FsoPso and an arbitrary numerical parameter . The result is that (3.7) will
2.4 Sensitivity of the strain energy to modifications of a system
(v, u*) = 0 .
63
(4.8)
To illustrate this with an example, we will present a solution of the
Oxford problem formulated as follows10.
Two series of two consecutive experiments are dedicated
54
2 BASIC VARIATIONAL PRINCIPLES
By varying the Lagrange functional in the form (2.16) on the
Lagrangian energy space L and using the same reasoning as before, we
can find its minimum point FL from L . As a result, we have
u = uk + uL, up to an inessenti
2.4 Sensitivity of the strain energy to modifications of a system
61
2.4.2 Remarks on the effect of additional constraints
(kinematical and force)
Kinematical constraints
What is usually called a constraint in structural mechanics is a limitation
imposed
2.3 Castigliano variational principle
57
Now we can introduce the linear set K as a truncation of the set Pso at
the expense of all fields from RK and represent Pso as a direct sum
Pso = K RK
(3.10)
where the orthogonality is understood in the sense of th
2.4 Sensitivity of the strain energy to modifications of a system
67
Introducing the force constraint makes the system statically determinate
immediately because now the equilibrium equations are enough to
determine the reactions. The calculation gives
R1
48
2 BASIC VARIATIONAL PRINCIPLES
Comparing (1.13) and (1.45) gives <F, F> = E(F) wherefrom the
positivity of the energy E on L yields the scalar product on the set L
produced by this bilinear functional. This scalar product is called an
energy (Lagrangia
70
2 BASIC VARIATIONAL PRINCIPLES
should keep in mind that first we glue the unstressed areas and only then
apply external loads to the area as a part of the merged area + .
We leave the proof of this fact to the reader for an exercise.
2.4.4 Modification
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2 BASIC VARIATIONAL PRINCIPLES
2.4.6 Second theorem of the strain energy minimum
Consider two independent actions on the system, one of which being
purely kinematic (no active external forces) and the other purely force (no
given nonzero displacements)
2.4 Sensitivity of the strain energy to modifications of a system
71
When the stiffness properties of an elastic body and/or its surrounding
elastic medium are increased, the strain energy of the system, E* , cannot
rise under a purely force load and cann
62
2 BASIC VARIATIONAL PRINCIPLES
(v, uko) = 0 ,
uko P ko .
(4.4)
This definition of a constraint plane (4.4) includes a vector, v, which we
call a constraint vector.
As a result, the linear set Pko of homogeneously kinematically
admissible displacements
2.4 Sensitivity of the strain energy to modifications of a system
Type of
perturbation
65
Table 2.2
External actions applied to a system
Force and kinematic
Force only
Kinematic only
k = 0
s = 0
Kinematical
constraints
L* L *
K* K
E* E *
E* E *
Force
cons
72
2 BASIC VARIATIONAL PRINCIPLES
eigenvalues of it be nonnegative. We use direct calculation to expand the
characteristic determinant of the matrix C and find out that all its
eigenvalues i (i = 1, 6) are roots of the polynomial
( )3 (2 )2 (3 + 2 ) = 0 .
2.4 Sensitivity of the strain energy to modifications of a system
73
E * = < F , F > = < tF, tF > = t2< F, F > = t2E* ,
=(t X , u ) + (Ep t p , Ep u ) = (t X , tu) + (Ept p , Eptu) = t2s* ,
s*
= (Eu H , Eu t u ) = (Eu tp, Eu t u ) = t2k* .
k*
Now lets a
2.4 Sensitivity of the strain energy to modifications of a system
69
are nothing but one of possible forms of kinematic constraints12 .
As the sum L1* + L2* (just as K1* + K2*) is the value of the respective
functional on the system consisting of separate