Civil Engineering Reference
In-Depth Information
3
4
7
3
5
BE Region I
2
2
6
1
1
Figure 16.1
Cantilever beam: discretisation into finite and boundary elements
In the above,
^`
0
c
t
is a vector containing tractions, if all interface displacements are
zero, and
K
BE
is the pseudo “stiffness matrix” of the BE region.
For the example problem we have
t
u
-
½
-
½
x
1
x
1
°
°
°
°
t
u
°
°
°
°
y
1
y
1
°
°
°
°
°
°
°
°
t
u
x
2
x
2
^`
^`
t
;
u
®
¾
®
¾
(16.2)
c
c
t
u
°
°
°
°
y
2
y
2
°
°
°
°
t
u
°
°
°
°
x
3
x
3
°
°
°
°
°
t
°
°
u
°
¯
¿
¯
¿
y
3
y
3
For the finite element region we can write a relationship between interface
displacements and interface nodal forces as
^` ^`
^`
c
F
F
K
u
(16.3)
FE
c
co
where
^`
0
F
is the force vector at the interface when all interface displacements are zero
and
K
FE
the condensed stiffness matrix of the finite element region which involves only
the interface nodes. In equations (16.1) and (16.3) we have already implicitly assumed
that compatibility conditions are satisfied (i.e., displacements of the BE and FE regions
are the same at nodes 1-3). Figure 16.2 shows the forces that exist at the interface. For
the BE region these are boundary stresses, whereas for the FE region these are nodal
point forces.
Search WWH ::
Custom Search