Information Technology Reference
In-Depth Information
7.4
Variational Methods
The trial function methods can also be approached from the viewpoint of a variational
principle. Recall that in Chapter 2 the integral (global) forms of the differential equations
represented the three fundamental relationships of solid mechanics—kinematics, material
law, and equilibrium. It is not surprising that the integrals of the variational principles
can be used directly in the same fashion as weighted-residual integrals. In fact, use of
these integrals as the basis of the approximation is referred to as a
direct variational method.
The most popular direct variational method, the so-called Ritz's
11
method, employs the
integral relations of the principle of stationary potential energy, or, more generally, those of
the principle of virtual work. It will be shown in this section that Galerkin's method can be
considered to be either a standard weighted residual method or to be based on a variational
principle.
As with the previous methods, the discretization of the variational integrals leads to a
system of algebraic equations. The unknowns in the trial solution are to be obtained such
that a variational functional is made stationary. This procedure leads to a best approximation
of certain characteristics, e.g., equilibrium or boundary conditions, of the problem.
7.4.1
Ritz's Method
One of the most frequently used approximate methods in mechanics is the
method of Ritz
.
The Ritz method can be used in conjunction with the principle of virtual work which re-
quires that
This leads to a solution which satisfies approximately the conditions
of equilibrium and the static boundary conditions. With a known material law,
the chosen
approximate displacements must satisfy the kinematic conditions and displacement boundary con-
ditions, i.e., they must be kinematically admissible.
Thus, at the outset, it is required that the
assumed displacements
δ
W
=
0
.
u
satisfy
=
D
u
in
V
u
=
u
on
S
u
(7.42)
and then the free parameters
u
are determined such that the conditions of equilibrium
and the static boundary conditions are approximated as closely as possible. It should be
apparent that the Ritz method is closely related to the displacement method.
From Chapter 2, we know that for a continuum the virtual work can be expressed as
T
σ
dV
u
T
p
V
u
T
p
dS
−
δ
W
=−
δ
W
i
−
δ
W
e
=
V
δ
−
V
δ
dV
−
S
p
δ
=
0
(7.43)
For a beam, with no shear deformation, from Chapter 2, Example 2.7 or Chapter 4,
Eq. (4.52),
L
0
δw
EI
L
w
dx
]
0
=
−
δ
W
=
−
p
z
δw
dx
−
[
V
δw
+
M
δθ
0
(7.44)
0
on
S
p
11
Walter Ritz (1878-1909) was born in Switzerland, son of the artist Raphael Ritz. He studied in Zurich, Switzerland,
and G ottingen, Germany, where he obtained his doctorate in 1902. He then worked in Leyden, Paris, and T ubingen.
In 1908, he returned to G ottingen and remained there until his untimely death.
Search WWH ::
Custom Search