Civil Engineering Reference
In-Depth Information
10
Finite Element Method (FEM)
10.1
Introduction
Stability analysis procedures for tunnels, slopes and dam foundations must account for
inhomogeneous and anisotropic ground conditions, arbitrary geometries and complex
boundary conditions. Furthermore, the computation of general three-dimensional stress-
strain states must be possible. Also, the simulation of support measures and of different
stages and sequences of construction must be possible. In particular, an adequate consider-
ation of the interaction of rock mass and support is essential for design and construction.
The best way to solve such complex problems is to use numerical analysis procedures. These
are either based on continuum mechanics concepts such as the fi nite element method (FEM)
or based on discontinuum mechanics concepts such as the distinct element method (DEM).
FEM has proven to be especially powerful and, in recent years, has become a useful tool
in the fi eld of rock engineering. The basic principle of this method and the way it is ap-
plied by WBI to rock engineering problems will be described later. A more detailed treat-
ment of this subject can be found, for example, in Wittke (1990) and Wittke (2000b).
10.2
The Principle of FEM
FEM is based on the subdivision of a selected continuum section - the so-called “com-
putation section” - into individual elements of fi nite size, the so-called “fi nite elements”.
In the two-dimensional case each element has the shape of a polygon with nodes at the
corners and in some cases also along the lines connecting the corners or within the element.
In case of a stress-strain analysis, the nodal displacements are calculated. The displace-
ments of other points within the element are evaluated by interpolation functions. With
the aid of the interpolation functions the computation of displacements at each point
of the element is reduced to the determination of the nodal displacements.
The principle of the method will be described subsequently using the simple example of
a plane triangular element with three nodes (Fig. 10.1).
The displacement components in the x and y directions are denoted as u and v, respectively.
It is assumed that the displacement components are linear dependent on the coordinates:
(10.1)
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