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were utilized to compute the coefficients
u
i
of the trial functions. Alternative forms of
Eq. (7.37) can be obtained by using integration by parts or the Gauss integral theorem to
form integrals such as
L
1
WL
2
L
3
WL
4
V
(
−
)
=
+
−
WL
u
Wf
dV
u
dV
u
dS
Wf
dV
(7.38)
V
S
V
in which
L
1
,
L
2
,
L
3
,
and
L
4
are differential operators, and
S
is the boundary. In this alternative
formulation, the differentiation of
u
is usually of lower order than that appearing in
L
,so
that the requirement of the order of continuity on
u
can be “weakened”. Hence, this is called
a
weak formulation
or
weak form
. Such a formulation can be advantageous because it can make
the choice of the trial functions easier. The operators
L
1
and
L
3
on the weighting function
W
involve differentiations; consequently, the continuity requirements on
W
are more severe
than before. Thus,
W
must have
C
r
−
1
continuity where
r
is the highest derivative in
L
1
and
L
3
. This requirement can be met by choosing appropriate
W
, e.g., choosing
W
to be equal
to
N
u
of Eq. (7.9).
To illustrate the fundamentals of a weak formulation, consider a linearly elastic beam
with the governing equation
dx
2
EI
d
2
d
2
w
dx
2
−
p
z
=
0
(7.39)
Cast this into the weighted-residual formulation
W
j
d
2
p
z
dx
dx
2
EI
d
2
w
dx
2
−
=
0
(7.40)
L
Integrate the left-hand integral by parts twice to find
EI
d
2
W
j
dx
2
W
j
p
z
dx
L
0
+
L
0
=
d
2
dx
EI
d
2
EI
d
2
w
dx
2
−
d
w
dx
2
dW
j
dx
w
dx
2
+
W
j
0
(7.41)
L
w
Equation (7.41) represents the weak formulation. The order of
is lowered, and boundary
terms have appeared. In the case of the boundary element method of Chapter 9, integration
by parts is continued until all of the derivatives are switched from
w
to
W
j
,
leaving only
boundary terms involving the unknown
w.
In practice, in the weak formulation,
W
j
is often chosen to have the same physical mean-
ing as
. When trial solutions of the form of Eq. (7.9) are used, Eq. (7.41) becomes a set
of simultaneous linear equations. Because of the boundary terms, these equations tend to
become complicated. Hence, it is often desirable to make
W
j
and the trial solution satisfy
certain boundary conditions to eliminate the boundary terms from Eq. (7.41). Designate
W
j
and its derivative in the boundary terms, i.e.,
W
j
and
dW
j
w
/
dx,
as the
forced
or
essential
dx
EI
d
2
d
w
boundary conditions
of Eq. (7.41) and the remaining factors,
dx
2
=−
V
(shear force)
and
EI
d
2
w
dx
2
=−
M
(bending moment), as the
natural boundary conditions
. In a displacement
formulation forced boundary conditions are the displacement conditions, whereas the nat-
ural boundary conditions are the force or static boundary conditions. These definitions of
forced and natural boundary conditions are the same as in Appendix I.
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