Information Technology Reference
In-Depth Information
Brebbia, C.A., Telles, J.C.F. and Wrobel, L.C., 1984,
Boundary Element Techniques
, Springer-Verlag,
Berlin.
Gipson, G.S., 1987,
Boundary Element Fundamentals
, Computational Mechanics Publications, South-
hampton, UK.
Haberman, R., 1987,
Elementary Applied Partial Differential Equations
, Prentice-Hall, Englewood Cliffs,
New Jersey.
Stern, M., 1989, Static analysis of beams, plates and shells, in Beskos, D.E., Ed.,
Boundary Element
Methods in Structural Analysis
, ASCE, New York.
Problems
9.1 Obtain the stiffness and transfer matrices from Eq. (9.11). Check your results with the
matrices of Chapter 4.
9.2 Use the same procedure as that for beams to derive the boundary element formulation
for bars with axial loading.
Hint:
The fundamental solution is
1
2
|
u
∗
=
x
−
ξ
|
9.3 The governing differential equ
at
ion for the small t
ra
nsverse motion of a tight string
with applied distributed load
p
z
is
d
2
w/
dx
2
=−
p
z
/
N
, where
N
is the tensile ax-
w(
)
=
w(
)
=
.
ial force. The boundary conditions are
Begin with the extended
Galerkin's method formula. Integrate by parts to obtain the boundary integral equa-
tion. Find the fundamental solution by solving
d
2
0
L
0
w/
dx
2
=−
δ(ξ
,x
)/
N
w
∗
=
1
Answer:
The fundamental solution is
2
|
x
−
ξ
|
9.4 Begin with the equation
d
2
u
dx
2
/
+
u
+
x
=
0, with boundary conditions
u
(
0
)
=
1
and
u
Obtain the boundary integral equation from the extended Galerkin's
formula. Find the exact boundary values
q
0
=
(
1
)
=
1
.
du
/
dx
|
x
=
0
and
q
1
=
du
/
dx
|
x
=
1
.
The fundamental solution is
u
∗
=
Hint:
A
cos
x
+
B
sin
x
.
Answer:
q
0
=
1
/
sin 1
−
1,
q
1
=
cos 1
/
sin 1
−
1
9.5 Use the boundary element method to calculate the response at
2 of a bar of
lengt
h
L
with both ends constrained. The bar is loaded by a linearly varying axial
load
p
x
=
ξ
=
L
/
p
0
(
1
−
x
/
L
).
p
0
L
2
Answer:
u
L
/
2
=−
/(
16
EA
)
9.6 Verify that Eq. (9.22) satisfies Eq. (9.21).
Hint:
Substitute Eq. (9.22) into the left-hand side of Eq. (9.21). Then investigate
x
=
ξ
and
x
=
ξ.
9.7 Use four constant elements for a 1
1 square cross section of a bar under torsion. Form
the linear boundary element equations by hand calculations. Solve for the torsional
constant.
Hint:
The values of the warping function at the nodes are not unique, so the value
at one of the nodes can be set to zero at the outset. The exact result for the torsional
constant is 0.1406.
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