There is a view within the tunnelling industry that our typical closed form solutions are good for basic design calculations, but they are conservative and you can get reduced lining forces by looking at a more advanced solution such as numerical modelling that can explicitly include soil arching. This approach is even written into some major technical documents such as the following statement:

'... it discounts the soil structure interaction pocess by which the ground continues to arch as the lining deforms thus over estimating the load on the lining and under-estiamting the load transferred to the ground'

At first glance these statements appear reasonably valid, the only problem is they are incorrect. Not only do our typical closed form solutions model arching in the ground, because these methods do not allow for palsticity in the ground they can over estamate the amount of arching in ground which is weak or where large deformations have occured.

By way of example I'm going to compare two different methods, two closed form solutions, Einstein and Schwartz, and Duddeck and Erdmann with the Fenner Packer curve, a method that is well recognized to model ground arching. It's not a perfect compariosn because the Fenner Packer curve can only model ground where Ko is 1, but it is sufficient to demonstrate how the closed form solutions model arching.

To compare these two methods I've run the closed form solutions whith a variety of different E values for the lining. The high values model a case where the support is stiff or in Fenner Packer terms the supporting pressure is high. The low E values model the case where the support is flexible or the support pressure is low. We can get the radial displacement from the closed form solutions using their standard equations. To get an equivalent support pressure we just find the axial hoop force and divide it by the tunnel radisu which gives an estiamte of the equivalent vericla load applied to the excavation.

For comparison purposes I've modelled two different cases with the following parameters:

Case | Case 1 | Case 2 |
---|---|---|

Typical strata | Overconsolidated clay | Weak rock |

Tunnel diameter | 6m | 12m |

Vertical pressure | 700kPa | 2000kPa |

Elastic modulus | 50MPa | 500MPa |

Poisson's ratio | 0.2 | 0.3 |

Friction angle | 22° | 39° |

Effective cohesion | 15kPa | 100kPa |

Lining thickness | 200mm | 200mm |

Plotting the Fenner Packer curve for these two different cases we can see how much arching the models assume for a given convergence. When the supporting pressure is the same as the in-situ vertical pressure there is no arching in the ground. As the supporting pressure decreases the ground has to take more of the load with the support taking less of the load.

It is clear from these two plots that in both cases, where the strains are relatively large, the assumptions of the closed form solutions break down and the solutions over predict the ability of the ground to arch and hence under predict the amount of load on the lining. Any approach which considers that the closed form solutions under predict arching and especially those that introduce additional arching by reducing the effective pressure put into the model may under predict the load on the lining.

Of course the good news is that in both of these cases large deformations had to occur prior to the plasticity occurring. In most uses of the closed form solutions we tend not to assume large amounts of relaxation as the tunnel is installed and we tend to consider relatively stiff lining systems so I don't think it needs to change the way most of us apply the closed form solutions.