8
1 BASIC VARIATIONAL PRINCIPLES
The relation (2.5) implies the idempotency of the boundary condition
extraction operators:
Ep Ep = Ep ,
Eu Eu = Eu .
(2.6)
It should be obvious that the operators are defined on the whole
boundary , but they may have diffe

1 BASIC VARIATIONAL PRINCIPLES OF STATICS
AND GEOMETRY IN STRUCTURAL MECHANICS
We have the right as well as are obliged to subject all our
definitions to critical analysis from the standpoint of their
application and revise them (fundamentally, if need be

1.4 The general principle of statics and geometry
17
Now we use the basic integral identity in the form (2.17) in application
to the elementary body; lets transform the integral into the form
d B 12 =
Au
i
2i
1id u 2i AT1id = Au 2i 1id + K u1i u 2i d .

CONTENTS
B.
XXVII
Tangential stresses in the bending of bars .
.1 Tangential stresses in the bending of straight bars .
.2 Tangential stresses in the bending of curvilinear bars .
Small-curvature bars .
Medium-curvature bars .
Big-curvature bars .
Refe

CONTENTS
9.3.5 Examples of application of the method of two functionals .
Example 1 .
Example 2 .
Example 3 .
References .
10 VARIATIONAL PRINCIPLES IN SPECTRAL PROBLEMS
10.1 Basic concepts. Termilogy .
10.2 The spectral problem as a variational proble

12
1 BASIC VARIATIONAL PRINCIPLES
field F. It should be emphasized that the elements of a field are not
supposed to relate to one another anyhow in the most general case.
Now, lets introduce more notions and definitions which will be useful
for further pr

XX
CONTENTS
Kinematical constraints .
Force constraints .
2.4.3 Build-up of a system .
2.4.4 Modification of stiffness properties of a system .
2.4.5 Perturbation of external actions .
2.4.6 Second theorem of the strain energy minimum .
2.4.7 St.-Venant

CONTENTS
1 BASIC VARIATIONAL PRINCIPLES OF STATIC
AND GEOMETRY IN STRUCTURAL MECHANICS
Preliminaries .
1.1.1 Formally conjugate differential operators .
1.2 Basic integral identity .
1.3 Various types of stress and strain fields .
1.4 The general principl

10
1 BASIC VARIATIONAL PRINCIPLES
Ep = I , Eu = p ,
Ep = , Eu = I u .
(2.11)
When (2.10) and (2.11) hold, the boundary conditions (2.4) can be
written as
p
H p = 0
Huu u = 0
u
(static boundary conditions)
(2.12-a)
(kinematic boundary conditions) (2.12-b)

XXVI CONTENTS
11.1.2 Linear analysis and a linearized formulation of the
stability problem .
11.1.3 Example 1 .
11.1.4 Example 2 paradoxes in the stability analysis .
A remark on a non-invariant critical load to the choise of
generalized coordinates .
11

2
1 BASIC VARIATIONAL PRINCIPLES
Here and further we use a common rule: the same indices on different
levels are used for summation.
The one-dimensional version of the area is an interval, [x1,x2], over
which an independent variable, x, can vary, so that

22
1 BASIC VARIATIONAL PRINCIPLES
1.4.3 Theorem of field orthogonality
We can derive the following statement from the formula (4.13) as a
particular case:
The virtual work of internal stresses of a homogeneously statically
admissible state (field) on the

24
1 BASIC VARIATIONAL PRINCIPLES
1.4.4 Integral identity by Papkovich
P.F. Papkovich [6] derived an integral identity in the theory of elasticity,
which he called a general expression of the work of external forces.
Papkovich used this identity to derive

1.4 The general principle of statics and geometry
15
created by an elastic medium in which the deformable solid is placed. The
forces of this kind are reactions of the elastic medium sometimes called a
response of an elastic foundation.
As one can see fro

References
27
and laws (which we call supplementary) can be derived from the basic
ones by means of various known transformations such as: the Lagrange
transformation, the Legendre transformation, the Friedrichs transformation.
More details on the use of

XXII
CONTENTS
5.2 Static-geometric analogy in the theory of plates.
5.2.1 A stress function vector in the theory of plates .
Physical meaning of the stress function in plane stress .
Physical meaning of the stress function in plate bending .
5.2.2 A sta

1.4 The general principle of statics and geometry
21
It is necessary and sufficient for a certain state 2 of a linearly deformable
mechanical system to be kinematically admissible that the sum of the
virtual work, A12 , of all external forces of any homog

1.1 Preliminaries
3
them as symbolic differentiation operations, although in mathematics [3]
an operator is a bigger notion than a simple differentiation expression.
The differential operators A and B are called formally conjugate
(sometimes Lagrange-conj

2 BASIC VARIATIONAL PRINCIPLES OF
STRUCTURAL MECHANICS
The history of mechanics and physics is a history of attempts to
explain things that happen around us in the world, using a small
number of laws and universal principles. The most successful
and fruit

4
1 BASIC VARIATIONAL PRINCIPLES
a1 + a2
,
Aa =
a1
a2
b b2
1
AT b =
b1 b
2
(1.9)
where the stroke means the differentiation with respect to x. The scalar
products of our interest can be represented now as
l
a
a
(Aa, b) = a1 + 2 b1

34
2 BASIC VARIATIONAL PRINCIPLES
so that it depends only on the final state of the system (recall that the initial
state is zero), then the quantities in the right part of (1.11) should be total
differentials. Calculus of functions of multiple real varia

2.1 Energy space
35
Now, let us give another derivation of the Clapeyron theorem, also to
demonstrate an application of the Papkovich identity. To do this, assume
that all four states of a system are the same in (1.4.23), which gives us the
right to ident

2.1 Energy space
31
expression of the virtual work, B21 , of the internal forces of the state 2 of
the same system on the displacements of the state 1. We have
B12 = (1, 2) (Ku1, u2) ,
B21 = (2, 1) (Ku2, u1) .
(1.6)
Now we use (1.1) and take into account

20
1 BASIC VARIATIONAL PRINCIPLES
is the virtual work of the internal forces which appear in the bars under
external forces. As a result, the virtual work of all internal forces in the
system can be expressed as S01 where is a virtual (homogeneously
kinem

XXIV CONTENTS
8 PARTICULAR CLASSES OF PROBLEMS IN STRUCTURAL
MECHANICS part 5
459
8.1 Compound-profile thin-walled bars .
8.1.1 Pure torsion of a compound-profile thin-walled bars .
8.1.2 A general behavior of a compound-profile thin-walled bar .
Average

28
1 BASIC VARIATIONAL PRINCIPLES
11. Slivker VI (1982) Ritz method in problems of elasticity, based on sequentially
minimizing two functionals (in Russian). Trans. Acad. Sci. USSR, Mech. of
Solids 2: 5764
12. Washizu (1982) Variational methods in elastic

2.1 Energy space
33
2.1.3 Energy of strain. Clapeyron theorem
Consider a true field, F1 = cfw_, , u, which conforms to given external
actions, V = cfw_ X , p , u , applied to an elastic mechanical system. In the
vicinity of the true field F1 , we consider

32
2 BASIC VARIATIONAL PRINCIPLES
The virtual work, A12 , of the external forces of state 1 of a system on actual
displacements of state 2 of the same system is equal to the virtual work,
A21 , of the external forces of state 2 on the actual displacements

1.4 The general principle of statics and geometry
23
All three framed statements above, taken together, will be called a
theorem of field orthogonality. Further on, after we introduce the notion of
an energy metrics, the fundamental relationship (4.19) wi

1.2 Basic integral identity
D( ) =
5
( )
x1 1 xk k
, | = 1+k .
With one dimension (k = 1) we can use the integration by parts, while
with two or three dimensions we can use the GaussOstrogradsky formula
to check that these rules really produce the conjug

CONTENTS
XXIII
6.2.7 Basic variational principles in the theory of open-profile
thin-walled bars .
6.2.8 A remark on non-warped cross-sections in the open-profile
thin-walled bars .
6.3 Allowing for shearing in open-profile thin-walled bars .
6.3.1 Basic

1.2 Basic integral identity
9
Fig. 1.1. Mixed boundary conditions on a piece of the boundary
of the area
If p and u are specified in a local coordinate system, and if the
components of the boundary force vector, p, and of the boundary
displacement vecto

2.1 Energy space
39
homogeneous equation with the geometry operator to have an infinite
number of linearly independent solutions, this would mean mechanically
that the system was internally unstable so that the instability could not be
eliminated by a fin

1.4 The general principle of statics and geometry
13
A field Fk = cfw_k, k, uk will be called homogeneously kinematically
admissible if the displacements uk and the strains k satisfy the geometric
equations inside the area and the homogeneous kinematic bo