622
11 VARIATIONAL PRINCIPLES IN STABILITY ANALYSIS
lcr /(c3l)
5
3
1
0
-1
1
2
3
4
5
c1/c2
-3
-5
Fig. 11.2. The critical force vs. the ratio of the spring stiffness values
Now let us turn to Eulers concept for finding the critical load in the
main state of

11.2 Variational description of critical loads
645
Before we start proving the Papkovich theorem, we would like to
introduce a couple of definitions and establish one important auxiliary
proposition.
Let a mechanical system be subjected to independent mec

648
11 VARIATIONAL PRINCIPLES IN STABILITY ANALYSIS
A hyperplane is called a base one for area if at least one point of
boundary of area belongs to the hyperplane while area is wholly
contained by one of the half-spaces created by this hyperplane.
Finally

11.2 Variational description of critical loads
637
min r ( z ) = r ( z1+ ) = 1+ ,
where the minimum is searched for among all vectors z that satisfy the
condition z T rG z > 0 .
Similarly,
The lowest (by absolute value) negative value of functional r(z) i

11.4 Stability of equilibrium of an elastic body
667
To change from functional B2 to functional B3, we just omit the term
2C ijkm pr ur ,i v p , j vk ,m in the integrand of B2, and this approach is sometimes
interpreted mechanically as the following condi

11.2 Variational description of critical loads
639
11.2.2 A remark on the effect of constraints on the stability of a
linearized elastic system
The effect of constraints on the stability of an elastic system is not so
easily found out as it may seem from

11.1 Stability of systems with a finite number of degrees of freedom
623
The system of linear homogeneous equations (1.43) with respect to
variations u and v has a nonzero solution only if the following condition
holds:
(c1 c2 )uo + c3l = 0 ,
which gives

656
11 VARIATIONAL PRINCIPLES IN STABILITY ANALYSIS
1
1
i j = (ui , j + u j ,i ) + km uk ,i um, j .
2
2
(3.2)
Kroneckers delta k m is introduced here in order to keep our convention of
summing over repeated indexes placed on different levels. As we can se

632
11 VARIATIONAL PRINCIPLES IN STABILITY ANALYSIS
A remark on a non-invariant critical load with respect to the
choice of generalized coordinates
There is an important question that comes up with regard to the linearized
analysis of stability. This ques

11.3 Geometrically nonlinear problems in elasticity
661
Now let us turn to the geometrically nonlinear equations of equilibrium
for small slopes the squares of which have the same order of smallness as
the elongations and the shears. According to (3.17),

642
11 VARIATIONAL PRINCIPLES IN STABILITY ANALYSIS
11.2.3 Papkovich theorem of convexity of the stability area
The case when a structure is subjected to one fixed external action is rather
an exception than a rule. Nearly always an engineer deals with a

11.1 Stability of systems with a finite number of degrees of freedom
631
Recall that the linearized formulation of the stability problem implies
the check of stability of an equilibrium of a linear system, not just any
equilibrium. Engineers know well tha

652
11 VARIATIONAL PRINCIPLES IN STABILITY ANALYSIS
The equilibrium state of the system is found from the condition that the L
function of m variables from (2.20) should be stationary with the
additional conditions (2.21). The linear formulation of the pr

668
11 VARIATIONAL PRINCIPLES IN STABILITY ANALYSIS
A remark on a mechanical interpretation of particular terms in
the stability functional
As can be easily seen, the integral
E(v) =
vi , j + v j ,i vk ,m + vm,k
1 ijkm
1
C vi , j vk ,m d = C ijkm
2
2
2

11.2 Variational description of critical loads
bij = P1
647
2 gm
2 g1
,
+ . + Pm
qi q j
qi q j
and the values of bij are linear homogeneous functions of the components
of the unit load vector, P .
Coefficients a ijk depend only on energy E and do not de

11.2 Variational description of critical loads
b = ro1a
641
where a = |[a1,an]|T .
In these designations, the constraint equation (2.9) can be treated as a
condition of E-orthogonality between the constraint vector b and vector z:
bT ro z = (b, z )E = 0 .

11.3 Geometrically nonlinear problems in elasticity
659
infinitesimal volume is extracted from an elastic body and the equilibrium
equations in projections onto the respective axes are composed for it in its
deformed state [12]. However, we can use a diff

11.1 Stability of systems with a finite number of degrees of freedom
c1 + c2
H=
0
621
0
,
c1u1 c2u2
+ c3
l
l
and now it is clear why the condition of positive definiteness of the
Hessian matrix, H, hence the condition of stability of the main state,

11.1 Stability of systems with a finite number of degrees of freedom
629
parameter . At the beginning the equilibrium remains stable, but as the
load continues to grow we have to cross the lower boundary of the
darkened area and thus make the equilibrium

11.4 Stability of equilibrium of an elastic body
665
where ij denote the components of the rotation tensor for the original
state of equilibrium, and ij denote the components of the rotation tensor
defined by the displacement vector v. In other words,
ij

638
11 VARIATIONAL PRINCIPLES IN STABILITY ANALYSIS
( z , rG z ) = 0 .
(2.8)
However, the equality (2.8) cannot be interpreted in terms of orthogonality
of vectors z and z because matrix rG does not generate any metric.
As all (p + q) eigenvectors are lin

644
11 VARIATIONAL PRINCIPLES IN STABILITY ANALYSIS
p2
6
3
-3
6
3
-3
9
12 15
p1
-6
-9
Fig. 11.7. An area of equilibrium stability
The property of convexity of the area is not extrinsic to this problem.
It turns out that the following very important theore

11.2 Variational description of critical loads
635
coordinates from the state of equilibrium which permit sign alteration of
the second variation of the systems full potential energy.
The eigenvector q is said to define a mode of buckling (loss of stabili

628
11 VARIATIONAL PRINCIPLES IN STABILITY ANALYSIS
and this looks suspicious. Intuition says the system should not lose its
stability at all when 20, so there is a contradiction with (1.52). But this
is just a first impression which can be deceptive.
The

11.2 Variational description of critical loads
651
expressions of the strain energy and the work of external forces, i.e. in a
way based on the equations of equilibrium only. The equations of
equilibrium must be composed for the deformed state of the syst

11.3 Geometrically nonlinear problems in elasticity
657
elongations and the shears are small. It means all components of the |[ij]|
tensor are much less than one by absolute value. In other words, for all
combinations of indexes the following estimate mus

634
11 VARIATIONAL PRINCIPLES IN STABILITY ANALYSIS
with a finite number of DOFs has been described in sufficient detail in
many books and textbooks such as [1], [18], [2]. Therefore we can omit
demonstrations of its techniques in application to the above

660
11 VARIATIONAL PRINCIPLES IN STABILITY ANALYSIS
The condition of L = 0 is a principle of virtual displacements in its
mechanical interpretation. The expressions of strains ij, as well as those of
their variations ij, will depend on how the geometrical

11.2 Variational description of critical loads
rij11 =
2L
(qo1 , q2 ) ,
qi q j
i, j = 1, n .
653
(2.27)
The linearized formulation of this stability problem gives
r11 = ro11 rG11 .
(2.28)
It is easy to notice from the general formula (1.27) of the compon

11.1 Stability of systems with a finite number of degrees of freedom
L = c1
v
u
v
+ c2
+ c3 v = c1 + c2
2
2
2
2
2
2
2
2
633
vu 2
2
l + c u v .
3
2
2
v2
Calculation of second derivatives of the full potential strain energy in
the state of equilibrium give

11.4 Stability of equilibrium of an elastic body
663
11.4 Stability of equilibrium of an elastic body
Suppose we know the state of equilibrium for our mechanical system. Let
the displacements of the system in this state be described by vector u. By
formal

640
11 VARIATIONAL PRINCIPLES IN STABILITY ANALYSIS
Finally, it is easy to see that the v = 0 constraint is also a stabilizing one
because the system with it cannot lose its stability at all under any kinds of
external actions.
We would like to note, retu

664
11 VARIATIONAL PRINCIPLES IN STABILITY ANALYSIS
But C(e + f) = where is a stress vector in the equilibrium state of
interest. Apparently, this vector depends on the load intensity
(nonlinearly in a general case), so
= () .
(4.4)
The variation of the

(
j
Xi = 0
i j = C i jkm k m
1
1
(ui , j + u j ,i ) + km uk ,i um, j
2
2
),
ui = ui
(i j + ki m j uk , m ) n j = p i
i j =
,i jj ki m j uk , m
| ij |< 1
i j n j +
ui = ui
1 ki m j
( k m i j ) k j nm = p i
2
i j = C i jkm k m
1
i j = ei j + km ki m j
2
1

630
11 VARIATIONAL PRINCIPLES IN STABILITY ANALYSIS
E = c1
v2
2
u2
2 + l 2
2
l 2 2 2l 2
.
+ c2
+ c3
= c1
+ c2
+ c3
2
2
2
2
2
2
At the same time, the force potential is represented by an expansion with
the accuracy up to second-order terms, which gives
l