The circular liner is generated about the circumference with a sufficient number of elements, and through the layer thickness with several element layers. The network refinement should be proved to be sufficient to be able to truly model also substantial wall thickness increases [Mielk97]. For instance, 8-node iso-parametric continuum elements (CPE8 [Hibbi95]) with quadratic geometry and displacement statements can be utilised so that also the curvature of the pipe wall can be approximated.
The comparative very stiff old pipe is presented as a rigid surface. In order to prevent the liner forcing its way through the old pipe, a modelled contact surface pair between liner external surface and old pipe internal surface can be defined with the aid of Lagrangian multipliers. Friction between liner and old pipe can mostly be ignored. The ideal modelling of the system without malfunctions and imperfections thus presented, leads, under external pressure, however, only to a reduction of the cross section whilst the geometry of the circular cross section remains the same. Thus, the actual available buoyancy of the liner should be simulated as near to reality as possible as a starting malfunction, so that the pipe deforms further under the second loading step, the external water pressure from all sides.
Because of the buoyancy, only an unsymmetrical gap figure in which the liner comes to rest against the crown is assumed in the above case. Thus, the invert region has a gap of double width ωs whereas the gap at the crown point disappears [Falte94b].
The results for the circular cross section under the influence of buoyancy and external pressure from all sides are shown in (Image 18.104.22.168.2.4.2-1) [Mielk97]. As is to be expected, the critical buckling pressure is reduced with increasing gap width whilst it increases as the wall thickness increases.
(Image 22.214.171.124.2.4.2-2) shows a typical load-displacement curve as well as some characteristic deformation figures of the circular cross section subject to buoyancy and external pressure, from the starting figure, through the failure case, up to the post-buckling behaviour [Mielk97]. The alteration in the rise of the curve at point * in the transition from the buoyancy pressure increase to the general increase of the external pressure from all sides can be clearly recognised. Under pure external pressure, the system behaviour is very resistant to bending which is caused by the neglected influence of the linear bending share. The non-linear share of the normal forces substantially emphasizes the deformation in this region.
It must be noted that, depending on the width of the gap with increasing t/Dm conditions, the curve no longer flattens up the point of a horizontal tangent but that the system, under growing deformation, can withstand further loading [Mielk97]. The breakdown problem then becomes a stressing problem. Investigations of stability should therefore be limited to wall thicknesses of approx. t/Dm ≤ 5 %.