Civil Engineering Reference
In-Depth Information
be generated with the help of a CAS and the risk of typographical errors can be virtually
eliminated by outputting the results in Fortran format.
The simplified algebraic expressions that form the stiffness matrix of the 4-node quadri-
lateral element by this method have been isolated, and form the basis of subroutine
stiff4
used in Program 11.5 of this topic. A detailed description of the method is given in Griffiths
(1994b).
The same technique can be applied to other element types (Cardoso 1994) and other
element matrices (e.g. 8-node quadrilaterals, 3D elements, mass, conductivity, etc). For
example, the technique is to be found again in Program 7.3, where the conductivity matrix of
a general 4-node quadrilateral element is computed algebraically using subroutine
seep4
.
A similar approach was used to create subroutine
bee8
used in Programs 6.3, 6.8 and
6.9, which generates an algebraic version of the [
B
] matrix for a general 8-node quadrilateral
element, corresponding to any given local coordinate
(ξ, η)
.
3.3 Local coordinates for triangular elements
Local coordinates for triangles are conveniently described in terms of a right-angled isosce-
les triangle of side length equal to unity as shown in Figure 3.4. This approach is exactly
equivalent to “area coordinates” (Zienkiewicz
et al
., 1971) in which any point within the
triangle can be referenced using local coordinates
(L
1
,L
2
)
. Clearly for a plane region, only
two independent coordinates are necessary. However a third “coordinate”
L
3
given by,
L
3
=
1
−
L
1
−
L
2
(3.9)
can sometimes be included to simplify the algebra.
For example, the shape functions for a 3-noded (“constant strain”) triangular element
(Figure 3.4(b)) take the form
N
1
=
L
1
N
2
=
L
3
(3.10)
N
3
=
L
2
y
L
2
3
(0,1)
2
L
3
=
1
−
L
1
−
L
2
1
3
1
2
(0,0)
x
(1,0)
L
1
(a)
(b)
Figure 3.4
(a) General
triangular element (b) Local coordinate system for triangular
elements
