## 6 Actions to Consider in Calculations Associated with Pressure Penstocks
### External Pressure  
External pressure due to hydrostatic load may be present in pressure tunnels which, for inspection and maintenance purposes, are quickly emptied. There may also be conditions of external pressure when a tunnel subject to low pressure crosses an area with a high phreatic level or when excessive grouting pressure has been applied in the backfill during construction [5]. In certain cases, an external drainage system has been constructed, which consists of pipes that run parallel to the tunnel axis. In addition, sometimes pressure relief valves have been installed too [1; 15]. Regardless, a liner should be designed to withstand a proportion of the total external hydrostatic load given that drainage cannot fully dissipate the external pressure and that it may become blocked or even not function correctly. Environmental factors should also be taken into account when implementing a permanent drainage system in the tunnel since they may result in permanent drainage at the ground surface or even in the depletion of aquifers. The most frequent failure modes of steel liners due to external pressure are overall instability or local instability of the span between stiffeners; both often entail the formation of a single lobe parallel to the tunnel axis [5; 16]. The term instability encompasses both elastic and inelastic buckling. Other types of failure can be localised yielding of the steel and weld breakage. Figure 7 shows the overall buckling of a steel liner under external pressure. Figure 6: Percentage of internal pressure carried by the steel liner (for \(D/t=285.7\) and \(Rfs=1.5 Rs\)). Yielding of the most stressed cross-section is the most common failure mode for thin walled unstiffened liners with a ratio \(D/t\) between 100 and 300 [16]. Only for very slender liners with high-strength steels may geometric instability occur. In this case, the material is not used to the full. Regardless, geometric nonlinearity should be considered in the model [17]. Calculation Methods for a Liner Submitted to External Pressure  
In the case of steel-lined pressure shafts and tunnels, the liner is confined by the concrete backfill, which prevents outward displacements once the gap is closed. This involves a resistance to instability significantly higher (up to 20 times greater) than that of an unconfined or an aerial pipeline with similar characteristics. Since the 1950s, several researchers have examined the theory of instability of steel liners in hydraulic shafts and tunnels. As the various solutions have produced different results, it is necessary to carry out a comparative analysis of the respective methodologies to determine the critical pressure at which buckling occurs. There are three main approaches associated with the phenomenon: the theory of multiple-lobe instability, that of radially symmetric buckling, and that of single-lobe buckling [5]. * Multiple-lobe instability approach  
This is used to define the critical pressure at which instability occurs in the span between two consecutive stiffeners or adjacent supports for aerial pipes. It considers a cylindrical steel pipeline with unrestrained radial displacements and subjected to uniform external pressure. The pipeline tends to buckle symmetrically with two or more lobes. The formulations of [PERSON] [5] and [PERSON] [12] serve as references for the calculation of local buckling in piping circumferentially fixed at both ends. Figure 7: Liner failure as a consequence of external pressure. (Photo taken by the authors). * Radially symmetric buckling approach This takes into account the radial confinement provided by the concrete backfill encasing the cylinder. The existing gap is assumed uniform around the entire perimeter. Its use is less frequent given that it provides less conservative results than single-lobe buckling theories. The formulations of [PERSON] (1956) (12) and [PERSON] (12) enable calculation of the instability under radial restraint.