Civil Engineering Reference
In-Depth Information
1
Preliminaries
A journey of a thousand miles
begins with a single step
Lao-tzu,
Chinese philosopher
1.1
INTRODUCTION
Nearly all physical phenomena occurring in nature can be described by differential
equations and boundary conditions. In the solution of these
boundary value problems
we aim to determine a response to given boundary conditions. For example we may be
interested in determining the response of the rock mass due to the excavation of a tunnel,
or the response of a structure to dynamic excitations of its foundations (caused by an
earthquake). Analytical solutions of boundary value problems, i.e. solutions that satisfy
both the differential equations (DE) and the boundary conditions (BCs), can only be
obtained for few problems with very simple boundary conditions. For example,
analytical solutions exist for the excavation of a circular tunnel in a homogeneous rock
mass, not really a realistic scenario for practical tunnelling. To be able to solve real life
problems, the engineer must revert to approximate solutions. Two approaches can be
taken: instead of satisfying both the DE and the BCs, one can attempt to satisfy only one
of the two and minimise the error in satisfying the other one. In the first approach (based
on the original idea of Ritz
1
) solutions are proposed that satisfy the boundary conditions
exactly. The error in satisfying the differential equation is then minimised. This is the
well known Finite Element Method. In the alternative (proposed by Trefftz
2
), the
assumed functions satisfy the DE exactly and the error in the satisfaction of the
boundary conditions is minimised.
Most readers of this topic will be familiar with the finite element method. In the most
common version of this method we subdivide the domain into elements and approximate
