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needs to be moved to the boundary to establish a boundary element formu-
lation. Also, the right-hand side of (2) has to be given special attention, since a singularity
develops when
The point
ξ
is on the boundary. Treat the boundary of the cross-section in the same
fashion as in Example 9.2 (Fig. 9.3) and perform the integration.
ξ
π
π
0
σ
r
i
a
i
r
2
0
σ
S
σ
dS
=
2
d
θ
=
d
θ
=
πσ
Thus,
dS
u
∗
∂
r
i
a
i
r
2
r
i
a
i
r
2
S
σ
∂
−
lim
→
S
σ
dS
+
S
σ
=
dS
−
πσ
(3)
n
0
S
−
Then (2) becomes
S
σ
∂
u
∗
∂
2
π
q
=−
πσ
+
dS
(4)
n
After the unknown density
on the boundary is calculated, the warping function on the
boundary as well as inside the cross-section can be obtained from (1). The torsional constant
can be found from Eq. (15) of Example 9.2. The derivatives of
σ
ω
ξ
ξ
with respect to
2
and
3
can be expresed as
u
∗
∂ξ
2
∂ω
∂ξ
2
=
1
2
S
σ
∂
dS
(5)
π
u
∗
∂ξ
∂ω
∂ξ
1
2
S
σ
∂
3
=
dS
(6)
π
3
After these quantities are found, the stresses on the cross-section can be calculated using
Chapter 1, Eqs. (1.142) and (1.143).
9.3 Linear Elasticity
The boundary element formulations for two- and three-dimensional elasticity problems
are given in this section. For the two-dimensional case, the formulation is for plane strain
problems. For plane stress problems, the plane strain relationships with an adjustment in
material constants (Chapter 1, Section 1.3.1) can be employed.
9.3.1 Basic Relations
We begin with the extended Galerkin's formula of Chapter 7, Eq. (7.72)
T
T
T
u
∗
D
T
σ
+
u
∗
p
∗
−
(
p
V
)
dV
+
(
p
−
p
)
dS
−
(
u
−
u
)
dS
=
0
(9.48a)
V
S
p
S
u
or
u
i
dV
u
i
dS
p
i
dS
−
V
(σ
+
p
Vi
)
+
S
p
(
−
p
i
)
−
S
u
(
−
)
=
p
i
u
i
u
i
0
(9.48b)
ij, j
p
have been replaced by the weighting functions
u
∗
and
p
∗
,
as is customary in
boundary element theory. This general relationship is often considered as the fundamental
principle underlying the boundary element method.
when
δ
u
and
δ
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