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w
∗
(ξ
d
d
ξ
w
∗
(ξ
where
Equations (9.9) and (9.10) are the expressions for the responses
along the beam in terms of the Green's function and the variables on the boundary. These
are fundamental relationships on which the boundary element formulation is based. In the
boundary element formulation, the unknown variables on the boundary are found first and
then the responses along the beam are calculated. In Eqs. (9.9) and (9.10) the point
,x
)
=
,x
).
ξ
can be
at any location along the beam. If the boundary variables are to be calculated,
has to be
moved to the boundary so that Eqs. (9.9) and (9.10) involve the boundary variables only.
When the point
ξ
ξ
is moved to the left end of the beam
(ξ
=
0
)
,
Eqs. (9.9) and (9.10), with
the help of Eq. (9.7), become
L
L
3
12
EI
V
L
L
2
4
EI
M
L
1
2
w
1
2
w
L
2
θ
1
12
EI
x
3
w
=
+
+
+
−
+
p
z
(
x
)
dx
0
0
L
L
0
L
L
2
4
EI
V
L
+
1
2
θ
0
−
1
2
θ
L
−
L
2
EI
M
L
−
1
4
EI
x
2
p
z
(
−
θ
0
=−
x
)
dx
0
i.e.,
L
L
3
12
EI
V
L
L
2
4
EI
M
L
1
2
w
1
2
w
L
2
θ
1
12
EI
x
3
p
z
(
−
+
+
+
−
+
x
)
dx
=
0
0
L
L
0
L
1
2
θ
0
−
1
2
θ
L
−
L
2
4
EI
V
L
+
L
2
EI
M
L
−
1
4
EI
x
2
p
z
(
x
)
dx
=
0
0
Similarly, moving
ξ
to the point
ξ
=
L
leads to
L
0
(
L
3
12
EI
V
0
L
2
4
EI
M
0
1
2
w
L
2
θ
1
2
w
1
12
EI
3
−
−
−
−
−
−
)
p
z
(
)
=
x
L
x
dx
0
0
0
L
L
0
(
L
2
4
EI
V
0
−
1
2
θ
0
+
1
2
θ
L
−
L
2
EI
M
0
+
1
4
EI
2
−
x
−
L
)
p
z
(
x
)
dx
=
0
or in matrix form
L
3
6
EI
L
2
2
EI
V
0
M
0
V
L
M
L
=
0
0
−
−
1
011
w
0
L
2
2
EI
L
EI
0
0
−
0
−
101
θ
0
w
−
1
L
10
L
3
6
EI
L
2
2
EI
−
−
0
0
L
0
−
101
θ
L
L
2
2
EI
L
EI
0
0
P
∗
H
V
=
G
0
x
3
p
z
dx
3
L
0
x
2
p
z
dx
1
6
EI
−
(9.11)
0
3
p
z
dx
(
x
−
L
)
3
L
0
(
−
)
2
x
L
p
z
dx
+
B
Equation (9.11) is a system of linear equations containing boundary variables only. This is
the desired result of the boundary element formulation for the beam problem. The boundary
points for the beam serve as boundary elements and the linear equations of Eq. (9.11)
will be the relationships between the variables on these elements. The boundary elements
consisting of boundary points contrasts with the situation for finite elements for beams
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