Falter investigated the bearing relationship of liners on the basis of non-linear geometric calculations on rigid frame models [Falte94a]. The calculation was carried out on the assumption of constant loading over the length of the pipe, ignorable friction between liner and old pipe and true-direction loads for "small displacements", whereby also deformation and the formation of gaps were considered. The starting point is the Glock buckling formula (5-14) (Bild 220.127.116.11.1.2.3), which is adapted by means of reduction factors so that the following equation for determining the critical buckling pressure of full-walled pipes is [Falte94a] :
κv - Reduction factor for pre-deformation (Image 18.104.22.168.2.3.8-2)
κs - Reduction factor for gap formation (Image 22.214.171.124.2.3.8-3).
The gap is defined as a constant spacing ws between liner outer and the old pipe inner wall (Image 126.96.36.199.2.3.8-1). This reduces the load that can be borne by the liner. The reduction of the maximum load that can be borne by the liner is dependent on the radius/wall thickness relationship rL /sL of the liner and is taken into consideration in the proof of stability by the reduction factors κv and κs (Image 188.8.131.52.2.3.8-2) (Image 184.108.40.206.2.3.8-3).
Pre-deformation is assumed to be a cos2 function and is fully described by the following three factors:
The opening angle 2φ1, decisive for buckling, was determined for the lowest buckling pressure so that for the proof of stability only the position φv and the depth of the deformation wv are still freely selectable sizes. The reduction factor κv determined by variations of parameter can be taken from (Image 220.127.116.11.2.3.8-2). The cases occurring in practice should be covered by 1% v L/sL
According to [Falte94a], besides the failure of stability, a failure of the important edge fibres can also occur due to external water pressure. For a first estimate of wall thicknesses, for trend investigations and for checking electronic calculation results for the geometric non-linear case, numerous m- and n- diagrams are offered in Appendix 1/T3 of the reworked draft of ATV-A 127 [ATVA127b].