Numerical methods of calculation are very important in many sectors of engineering today. Because of the manifold possibilities of its application, the Finite Element Method is the method used most often even though the border element method (BEM), the Discrete Element Method (DEM) as well as the Finite Difference Method (FDM) in some cases present good alternatives [Schwe94].
In the Finite Element Method, the unknowns within an element are approximated by statement functions or form functions. The primary unknowns - in this case the displacements - are determined at the element nodes. An important advantage of the FEM that can be mentioned is that irregular geometries and edge conditions can easily be accounted for. The most important limitation of the finite element method that must also be mentioned is that it is based on continuum mechanical fundamentals and thus break conditions that are characterised by a continuum-discontinuum transition are not, or are insufficiently, described [Schwe94]. The method is described in detail in the literature [Hibbi95] [Schwe94] [Galla76] [Zienk75].
The results of a supporting framework analysis according to the Finite Element Method are always only an approximation of the actual supporting framework behaviour. The errors that occur become smaller the closer the elements are spaced. However, the amount of calculation rises substantially with an increase in the number of elements so that a compromise must always be struck between accuracy and the amount of calculation it involves.
The ABAQUS Version 5.6 FEM program system was utilised by the Workgroup of Pipeline Construction and Maintenance of the Ruhr-Universität Bochum (Germany) for numerical investigation of the bearing behaviour of liners. The important input data of the program refer to the modelling of the structural system and the description of the loading steps. The modelling data include:
The loading steps are determined by:
The results of the calculations can be prepared and displayed for viewing with the aid of a post processor after completion of the individual calculation steps or load increments. The edge conditions can easily be varied and thus the possibility of a targeted variation of parameters is offered.
In order to limit the amount of calculation, the investigations are limited to the geometric non-linear elastic bearing behaviour of the liner. The equality is determined by the deformed configuration also under large displacements and distortion. The contact problem requires the inclusion of variable edge conditions.
In the determination of the stability relationship of a static system there must be differentiated between two processes: the linear and the non-linear buckling analysis.
The linear buckling analysis (also known as eigenwert analysis), assumes infinitesimally small supporting framework deformations, equilibrium of the deformed system due to displacement attempts, directionally true external loads as well as linear elastic material behaviour. These limiting conditions do not change during the course of the load cycle. Linear buckling analysis is generally unsuitable for the calculation of the critical liner loading because edge conditions of the pipe-in-pipe model are only taken from the un-deformed configuration. In a calculation with gap without initial contact with the old pipe, only the buckling pressure of the free pipe is determined. Without an initial gap, however, the results are overstated high buckling pressures, as the resulting gap formation from the normal force deformation is not considered. The following discussion will therefore deal with the non-linear buckling theory for the pipe-in-pipe model.
Numerical solution algorithms that can be applied for the non-linear buckling analysis are the Newton-Raphsen method and the so-called RIKS method. In the Newton-Raphsen method, the deformation relationship in the vicinity of a load increment is linearised and the solution of each load increment is brought to convergence by the attempt of the disequilibrium forces with reference to the equilibrium. The RIKS method is based on a modification of the force control of the Newton-Raphsen method with which the breakdown relationship of a system after it has reached the maximum load point can be determined [Krätz94].