Cross Sectional Deformation Due to Longitudinal Bending
A theoretical introduction and bibliography is given in [Axelr80].
In the condition of buckling deformation, there occurs a deformation of the pipe cross section (ovalisation) that must not be ignored in connection with the elastic kinking. This non-linear bending of pipes - the so-called Brazier problem [Brazi27] - was usually viewed in the past without reference to the buckling stability [Reiss59] [Wood58].
The geometric statics system of the longitudinally bent pipe is determined by the wall thickness d, the pipe radius R, the bending radius r, the internal pressure p, as well as modulus of elasticity E and the bending resistance B.
The technical operating requirements of sewer sections provide a constant fall as well as a straight run of pipe, already in the planning, in connection with the utilisation as a gravity flow pipe [Stein95j].
Stressing as a result of longitudinal bending can thus be seen as negligible for flexible piping [Larjo80] so that in measuring practice this behaviour is only taken into account in extraordinary cases of necessity during the laying [AWWAM23] or in proof for plastics piping on a transport roller [Janso89].
Deviations from planning, conditioned by unforeseen settling differences, subsidence or incorrect laying, however, can never be wholly excluded. In the case of unplanned longitudinal bending, one can assume that the pipe is in a pre-buckling condition, which can be described sufficiently accurately by means of the relationships of the non-linear bending of the pipe [Bosse97] [Reiss59].
Reissner in [Reiss59] also provides an equation for calculating the vertical deformation σv
The vertical deformation σv in a pressureless case is thus independent of the modulus of elasticity E, and is influenced only slightly by the lateral expansion number ν.
The solution provided by the Reissner equation mentioned above, as well as other more accurate shell theory solutions, were also compared by Bosseler with FEM calculations in [Bosse97].
The cross sectional figures arising from various buckles a of the pressureless pipe are depicted in (Image 2.8.8-2).
The results according to [Bosse97] underline the fact that the quality of the Reissner statement should not be placed on the same level with the numerous approximations of a higher order. However, the deviations in the pre-buckling region are negligibly small. Further information on determining the carrying loads of pipes under different limiting conditions, pre-buckling conditions and imperfections can be found in [Axelr83a].
Practice-oriented investigations of sewers laid in the ground [Bosse97], although they can lead to an expectation of a negligibly small influence of the longitudinal bending relationship on the cross sectional deformation, a generalisation is not recommended due to the small amount of data available. For the case of planned longitudinal bending, the influence of the actual earth pressure distribution as well as the bedding reaction forces on the cross sectional deformation must be taken into account. An extension of the reflexions on the types of pipe with profiled walls now increasingly appearing on the market presents special cause for consideration.