Stress analysis in mitre bends embedded in rock - new derivation and extension of existing formulae for use in penstocks

In the construction of pressurized conduits for hydropower plants, segmented bends (mitre bends) are widely used where ducts change direction. In practical applications, different approaches are used to consider the effects for exposed mitre bends in structural steel design. However, formulae for mitre bends embedded in rock or concrete for penstocks in underground installations have not been available so far.
In view of these restrictions on the calculation of underground mitre bends, the authors have developed analytical formulae for mitre bends embedded in rock by introducing terms for the elastic foundation of the rock mass to extend the classical differential equation for cylindrical shells. These formulae allow the whole local stress state in the bend to be calculated, in particular for the stress concentrations at the intersection of the mitre. The formulae were then validated by comparing them with the results of non-linear FE analyses.