## 7 Comparison of Methods for Instability Analysis of Liners  
### Local Instability of Penstocks and Instability of Aerial Pipes  
Local instability is the failure mode that may be applied both to the calculation of the steel shell span between stiffeners (in the case of penstocks) and the span of aerial pipes between supports.  
Earlier work included a comparison of some analytical methods, such as those of [PERSON], [PERSON] and [PERSON] [17]. The applied data set for that comparison is shown in Table 5.  
Table 6 shows the critical pressure values resulting both from analytical formulations and a three-dimensional linear FEM model. Two cases with different degrees of spacing of 2.5 m and 30 m (with the latter being practically unstiffened) between stiffeners are shown.  
It is notable that there is a significant spread in the results obtained, given that [PERSON] does not consider the failure mode (the number of lobes) but the minimum critical pressure of instability. Hence, the Roark formula provides the most conservative values for critical pressure.  
Figures 8 and 9 show how the Roark line is practically the tangent at the lower point of the von Mises curve.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline \(D\)/\(t\) & 285.71 \\ \(E\) & 210 MPa \\ \(\
u\) & 0.3 \\ \(L\)/\(t\) & 0.625 \\ \(t\) & 7.5 mm \\ \hline \hline \end{tabular}
\end{table}
Table 5: Data used in the comparison of the local instability analysis methods of penstocks.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{6}{c}{**Critical Buckling Pressure PCR [MPa]**} \\ \hline \hline \multicolumn{2}{c}{**Spacing of**} & \multicolumn{2}{c}{**Failure Mode**} & \multicolumn{3}{c}{**Analysis Method**} \\ \cline{3-6}
**Stiffeners [mm]** & **(Number of Lobes) \(n\)** & **FEM** & **[PERSON]** & **Donnel** & **[PERSON]** \\ \hline \multirow{6}{*}{30.000} & 2 & — & 0.050 & 0.092 & 0.078 \\  & 3 & 0.080 & 0.050 & 0.075 & 0.057 \\  & 4 & 0.111 & 0.050 & 0.111 & 0.100 \\  & 5 & 0.169 & 0.050 & 0.167 & 0.159 \\  & 6 & 0.244 & 0.050 & 0.238 & 0.231 \\  & 7 & 0.334 & 0.050 & 0.324 & 0.317 \\  & _Pcr_ & 0.080 & 0.050 & 0.075 & 0.057 \\ \hline \multirow{6}{*}{2.500} & 5 & — & 0.596 & 34.890 & 2.710 \\  & 6 & 2.350 & 0.596 & 14.220 & 1.236 \\  & 7 & 1.419 & 0.596 & 6.610 & 0.788 \\  & 8 & 1.051 & 0.596 & 3.520 & 0.649 \\  & 9 & 0.933 & 0.596 & 2.190 & 0.700 \\  & 10 & 0.940 & 0.596 & 1.617 & 0.783 \\  & 11 & 1.016 & 0.596 & 1.391 & 0.900 \\  & 12 & 1.136 & 0.596 & 1.340 & 1.049 \\  & 13 & 1.286 & 0.596 & 1.394 & 1.200 \\  & 14 & 1.420 & 0.596 & 1.504 & 1.374 \\  & 15 & 1.655 & 0.596 & 1.653 & 1.563 \\  & _Pcr_ & 0.940 & 0.596 & 1.340 & 0.783 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Critical pressure for linear elastic buckling of a thin-walled steel liner.  
Figure 8: Comparison of the calculated critical buckling pressure without consideration of radial constraint; \(D/t=285.7\) and \(L=30\) m.  
Given the results shown in Figures 8 and 9, although it seems suitable that the critical pressure be assessed through different formulations, the development of a finite element model, even though there is a linear eigenvalue analysis [30], is recommended. As it can be seen, different stiffener lengths (\(L=30\) m in Figure 8 and \(L=2.5\) m in Figure 9) show very different results of the critical buckling pressure. Some formulations are more appropriate and conservative than other for different geometries. Nevertheless, Roark is always the most conservative formulation, irrespective of the geometry used, and Donnel and the FEM offer the most optimistic values. This shows the importance of trying out different formulations.