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Similarly
q
r
3
dS
r
3
a
3
r
2
a
3
∂ω(ξ )
∂ξ
1
2
+
2
r
2
r
3
a
2
−
3
=
r
2
+
ω
(18)
π
r
4
S
The derivatives are taken with respect to
ξ
i
because the variable for
ω(ξ)
is
ξ
and the
derivatives at point
3
known, the stresses can be
calculated from Chapter 1, Eqs. (1.142) and (1.143) at any point inside the cross-section.
Alternatively, a different approach can be employed to compute the derivatives of
ξ
are of interest. With
∂ω/∂ξ
2
and
∂ω/∂ξ
ω
with
respect to
3
. When they are needed at a specific point inside the cross-section, first
calculate the values of
ξ
2
and
ξ
at four surrounding points. These points can be used as the vertices
of a quadrilateral inside which the values of
ω
can be interpolated using shape functions.
Then the derivatives can be calculated from these interpolation functions. For some cases
the derivatives are calculated more accurately with the interpolation functions than from
Eqs. (17) and (18).
ω
9.2.2 Boundary Element Formulation
An analytical solution of the integral relation of Eq. (9.29) is exceedingly difficult to find.
Consequently, numerical methods are employed. The basic steps involved are
1. Discretize the boundary
S
into elements over which approximate shape functions for
u
and
q,
as appropriate, are defined.
2. Introduce these elements into Eq. (9.29) to obtain a system of linear equations.
3. Impose the boundary conditions, and solve these equations.
4. Find the variables
u
and
q
, as desired, inside the body.
The boundary (surface) of the body can be discretized into elements (boundary elements)
in the form of Figs. 9.4 and 9.5. For a two-dimensional analysis, the elements are normally
straight lines (constant or linear elements) or curves (quadratic or higher order). For a
three-dimensional analysis, the elements are usually quadrilaterals and triangles. Linear
and quadratic elements, which are often chosen to be isoparametric elements, are shown
in Fig. 9.6. The shape functions for these elements are the same as those for an element of
the same order used in a finite element analysis. Thus, the assumed distributions of the
variables in Eq. (9.29) can be written in terms of an element shape function
N
i
as
N
i
v
i
u
=
(9.30)
N
i
q
i
=
q
(9.31)
where
v
i
and
q
i
are the nodal values of
u
and
q
of the
i
th element, with dimensions equal
to the number of nodes in the element. For the constant element, which has one node at
its centroid, the unknown variable throughout the element is assumed to be equal to the
variable at the nodal point. This kind of element has only a single DOF for each element
and, hence, is the simplest element.
Substitute Eqs. (9.30) and (9.31) into Eq. (9.29) to obtain
q
∗
N
i
dS
v
i
u
∗
N
i
dS
q
i
bu
∗
dV
M
M
S
(ξ
)
+
=
−
cu
(9.32)
j
S
i
S
i
V
s
i
=
1
i
=
1
s
=
1
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