## 6 Actions to Consider in Calculations Associated with Pressure Penstocks
### External Pressure
* Single-lobe instability approach Here, it is assumed that the cylinder is unlikely to remain centered with an initial uniform and all-round gap. The cylinder will finally lean against the surrounding material at some point and will have twice the radial gap on the opposite side. When the external pressure reaches a critical value, the cylinder will buckle with a single lobe. However, it should be noted that [PERSON] (18) proposes two lobes for this case. The principal theories for the buckling calculation through formation of a single lobe are the following:
* [PERSON] (7). This analytical method is applicable to unstiffened liners under the assumptions of elastic behavior and complete containment (gap = 0). \[Pcr=E^{\prime}\left(\frac{t}{R}\right)^{2.2},\] (1) where \(Pcr\) is the critical pressure and \(E^{\prime}\) is the modulus of elasticity under plane strain assumption. * [PERSON] (18). This analytical method modifies the Glock equation to account for an initial gap and applies this to unstiffened liners. * [PERSON] (6). This study entails an analytical method assuming inelastic behavior; it is applicable to both stiffened and unstiffened liners. * [PERSON] (19). This contribution assumes plane strain and involves an analytical approach that considers inelastic response and is, despite its practical complexity, the most extensively used today. It is applicable both to stiffened liners and, with some simplifications, to unstiffened liners. The approach requires three non-linear simultaneous equations to be addressed. [PERSON] establishes that stress in the extreme fibre of the liner reaches the yielding value. This does not explicitly entail either the elastic or inelastic stability of the penstock, but assumes that the critical buckling pressure corresponds to the condition of plastification of a first fiber at any point on the liner. In order to validate the calculated critical pressure that causes instability, tangential stresses in the liner must not exceed 80% of the yield stress of the steel. When the stress level exceeds this value, the stress-strain behavior of steel tends to become nonlinear. \[\sigma\cdot FS=\frac{Pcr\cdot R}{t}FS\leq 0.8\ f_{sy},\] (2) In the above expression, \(f_{sy}\) is the yield stress of the steel and \(FS\) is the required safety factor, with \(\sigma\) being the design stress. * [PERSON] (20). This work, which develops an empirical formula assuming inelastic behavior, is restricted to unstiffened liners. * [PERSON] and [PERSON] (21). This research entails a finite element method (FEM) approach and provides a parametric study of unstiffened liners that involves a formulation for failure through elastic buckling. \[Pcr=\frac{E^{\prime}}{4}\cdot\left(\frac{t}{R}\right)^{3}\left[\frac{25+350 \frac{t}{R}+315\frac{g}{R}}{0.15+65\frac{t}{R}+350\left(\frac{t}{R}\right)^{2} +145\frac{g}{R}}\right],\] (3)
FEM has become the most used alternative in the past decade. It is widely used in the literature to carry out simulations [22; 23; 24; 25]. * [PERSON] [26]. This work develops an approach for unstiffened liners by application of the FEM and assuming elastic buckling. \[\frac{Pcr}{E^{\prime}}=\frac{0.218}{1.0+0.41\delta-0.006\delta^{2}}\left(\frac{ t}{R}\right)^{2.2},\] (4) \[\delta=3^{0.6}\left(\frac{g}{R}\right)\left(\frac{R}{t}\right)^{1.2}\] (5) with \(\delta\) being a dimensionless parameter, a function of the initial gap \(g\). The most referred-to theories for instability analysis in both stiffened and unstiffened liners (overall instability) are those provided by [PERSON] (1970) [5; 12] and [PERSON] [19]. In addition, in a parametric study [PERSON] [27], determined values for inelastic instability (yielding) of unstiffened liners. Finally, other valuable studies that involve the application of FEM are as those of [PERSON] [28] and [PERSON] [29].