## 7 Comparison of Methods for Instability Analysis of Liners
### Instability of Unstiffened Steel Liners  
Across the analytical formulations, the basic parameters involved in the design of liners are the relations \(D/t\) (slenderness) and \(g/R\) (relative gap), provided that the steel modulus of elasticity is 210 MPa and Poisson's ratio \(\
u=0.3\). Figure 10 shows the values for the critical buckling pressure assessed by various methods depending on the ratio \(D/t\) and for a gap ratio of \(g/R=1.5\times 10^{-3}\).  
The similarity of the [PERSON] equation to that of [PERSON] for unrestrained cylinders [31; 32] and to other formulations is notable. Remember that [PERSON]'s equation is expressed as follows:  
\[P_{C}r=b\cdot E^{\prime}\bigg{(}\frac{D}{t}\bigg{)}^{m}, \tag{6}\]  
whereas for the case of a free cylinder (Euler), \(b=2\), \(m=-3\), for that of [PERSON], \(b=1\), \(m=2.2\). Such values are the formal limits of the parameters, with the real solution therefore varying with respect to a free cylinder and a completely constrained one (without a gap).  
It should be noted that whereas the formulations offered by [PERSON], [PERSON], and [PERSON] are applicable when failure is produced by yielding [17], the remainder consider only failure due to geometric instability, which is uncommon for \(D/t\) ratios of between 100 and 300.  
Figure 9: Comparison of the calculated critical buckling pressure without consideration of radial constraint; \(D/t=285.7\); \(L=2.5\) m and \(\epsilon=14\) mm.  
One simple manner for comparing such formulations involves determining the factor \(K\), which depends on the material of the liner, geometry, and size of the gap [33; 34]. Thus, with \(Per^{ideal}\) being the critical pressure of Euler, \(K\) may be defined so that:  
\[Per=K\cdot Per^{ideal} \tag{7}\]  
Figure 11 shows the parameter \(K\), for the various formulations assuming a gap of \(g=3\) mm. Nonlinearity is apparent for those obtained through the FEM, as well as the strong dependence of all on the parameter \(D/t\).  
Figure 11: Parameter \(K\) with regard to [PERSON]’s \(Per^{ideal}\) (\(g=3\) mm).  
Figure 10: Comparison of stability calculation methods for cylindrical shells with radial constraint; \(g=3\) mm and \(R=2\) m.  
As may be observed, both the value of \(K\) and its scatter are greater as the liner slenderness increases (\(D/t\)). Moreover, given that in some approaches the relation is nonlinear, an approximate formulation of the following type could be suitable:  
\[Pcr=c_{1}\left(\frac{D}{t}\right)Pcr^{ideal} \tag{8}\]  
which may resemble the variability of \(K\) with \(D/t\). Indeed, this relation is equivalent to that expressed above in Equation (6). Table 7 shows the estimate of parameters \(b\) and \(m\) in the aforementioned formulations from results obtained upon considering two values of the gap \(g\): 8 mm and 3 mm.  
From a comparison of Figures 10 and 11 and Table 7 with the values of the parameters obtained, it may be deduced that the influence of the initial gap, \(g\), in the determination of the critical pressure should be considered, with the influence of the parameter \(D/t\) being insufficient. Additionally, it may be observed that the relation is not linear but exponential:  
\[Pcr=b_{0}\Big{(}\frac{g}{D}\Big{)}^{b_{1}}E^{\prime}\Big{(}\frac{D}{t}\Big{)}^ {n_{0}(\frac{g}{D})^{n_{1}}} \tag{9}\]  
in which \(b_{0}\), \(b_{1}\), \(n_{0}\), and \(n_{1}\) are constants.  
In another case study [17], for a gap of 0.6 mm, the results obtained by different methods were compared with those achieved through a finite element model. Figure 12 shows that the latter follow the values obtained by [PERSON] and [PERSON]. The reason is that the failure is due to steel yielding.  
There is a significant scatter across the results of the models traditionally used ([PERSON] and [PERSON]) and those obtained through elastic finite element models or the subsequent formulations derived. The reason is that the former assume that failure occurs by yielding, whereas the latter consider the geometric instability of the cylinder.