Civil Engineering Reference
In-Depth Information
Three-dimensional problems are solved in Chapter 12, in which the displacements in
the
z
-direction are given by
w
.
2.19 Conclusions
When viewed from a finite element standpoint, all static equilibrium problems, whether
involving solids or fluids, take the same form, namely,
[
k
m
]
{
u
} = {
f
}
(2.142)
or
{
φ
}
=
{
}
[
k
c
]
q
(2.143)
For simple uncoupled problems the solid [
k
m
] and fluid [
k
c
] matrices have similar
symmetrical structures, so computer programs to construct them will also be similar. How-
ever, for other problems, for example those described by the Navier-Stokes equations,
the constitutive matrices are unsymmetrical and appropriate alternative software will be
necessary.
In the same way, eigenvalue, propagation and transient problems all involve the mass
matrix
[
m
m
] (or a simple multiple of it). Therefore, coding of these different types of
solutions can be expected to contain sections common to all three problems.
So far, single elements have been considered in the discretisation process, and only
the simplest line and rectangular elements have been described. The next chapter is mainly
devoted to a description of programming strategy, but before this, the finite element concept
is extended to embrace meshes of interlinked elements and elements of general shape.
References
Bathe KJ and Wilson EL 1976
Numerical Methods in Finite Element Analysis
. Prentice-Hall, Engle-
wood Cliffs, N.J.
Berg PN 1962
Calculus of Variations
. McGraw-Hill, London, New York.
Biot MA 1941 General theory of three-dimensional consolidation.
J Appl Phys
12
, 155-164.
Cook RD, Malkus DS and Plesha ME 1989
Concepts and Applications of Finite Element Analysis
,
3rd edn. John Wiley & Sons, Chichester, New York.
Finlayson BA 1972
The Method of Weighted Residuals and Variational Principles
. Academic Press,
New York.
Griffiths DV 1989 Advantages of consistent over lumped methods for analysis of beams on elastic
foundations.
Commun Appl Numer Methods
5
(1), 53-60.
Griffiths DV 1994 Coupled analyses in geomechanics. In
Visco-Plastic Behavior of Geomaterials
(eds. Cristescu ND and Gioda G). Springer-Verlag, Wien, New York, pp. 245-317. Chapter 5.
Griffiths DV and Smith IM 1991
Numerical Methods for Engineers
. Blackwell Scientific Publications
Ltd., Oxford.
Horne MR and Merchant W 1965
The Stability of Frames
. Pergamon Press, Oxford.
Jennings A and McKeown JJ 1992
Matrix Computation
. John Wiley & Sons, Chichester, New York.
Leckie FA and Lindberg GM 1963 The effect of lumped parameters on beam frequencies.
The
Aeronaut Q
14
, 234.
