A high level strategy for automated tunnel design

Previously I had argued that tool sets are now available that mean that now is the right time to look at automated tunnel design. In this post I will discuss a high lever strategy for automated tunnel design. For the sake of simplicity I will consider a simple shaft for the connection to a sewer but I believe that the same or similar principals could be applied to much more complex problems.

One of the core elements to the following process is the use of the programming language Python. Python is a fairly modern language that has been widely embraced by the technical community (scientists, mathematicians and engineers). As such it is built into many of the advanced programmes that we might be working with as well as having frameworks to do things like producing high quality graphs and other technical images.

  • the first element is a BIM tool to hold all of the data and to integrate it with other structures. In this case I have initially chosen REVIT for this. It may not be the ideal solution but it is an easy tool to use to integrate with other designs.
  • the next element is a set of design protocols for all elements of the works that need to be designed. This would include not only the structural design but also the design of elements such as space proofing. For space proofing of the simple shaft this might include rules about the length and size of ladders and requirements for the number of ladders. It might also include elements such as the minimum size of shafts. For the structural design this would include design methods for the shaft lining as well as base slabs and cover slabs.
  • to undertake the space proofing I am considering a bespoke tool. Any space proofing tool for automated design need to consider aspects such as constructability and standardisation of components. I do not believe that most current space proofing tools are capable of this. An application to undertake the layout and spaceproofing of structures would have to be built within Python. Macros in REVIT can be called to undertake run a bespoke tool such as this and to generate the structure within REVIT.

  • the structural design will also be undertaken using bespoke code in Python. Critically the intention is not to undertake detailed numerical analysis and optimisation of a structure but to undertake the design using and efficient but simple and robust method. This approach is key to ensuring that the design can easily be followed and changes made where necessary.

  • the final component is reporting. With any complex system the human to machine interface is well know to be a weak point. With any automated design approach you must expect that there will be times when the automated design will fail and it must be manually overridden. It is only with high quality human readable output that the designer will know when to override the system. This therefore is an essential part of the system and may also be the most expensive and difficult part of the system to build. Again Python can be leveraged in a few different ways to produce the report with technologies such as matplotlib and Mathjax.

In future posts I want to look at how to build some of these components both in a theoretical sense and in writing some example code. I also want to look at some of the higher level isssues with this approach such as some of the constraints our industry has to this sort of technology actually being adopted.

On automated tunnel design

I had a big change of heart over the past year.I used to believe that it would be a long time before tunnel designs were automated. Designs would always be bespoke, every tunnel had some subtle difference from the rest that meant that a singular protocol could never be followed.

But over the last year I’ve come to a very much opposing view. Not only do I think that designs can be automated but in fact they should be, and the sooner they are the better and more consistent the designs produced by our industry will be.

Two things have changed my mind over the last year. The first is that I have been writing protocols for the design of various tunnel elements. These protocols are prescriptive and detailed descriptions of how to design a specific structural element. It is not essential that the designer follows the process in the protocol but if they do then there would be good consistency between the designs of one structure and the next.

The second thing I have realised is that in modern tunnel engineering speed is everything. Clients are requesting faster and faster turn around and yet due process, and the increasing complexity of design standards and requirements leads to friction in delivering fast turn arounds.

So imagine a world where you could enter details of what you would like a shaft to do, what are the constraints and then the shaft is automatically designed for you, the layout, the structural design and all the detailing. This sounds a long way off from where we are today but I’m going to try and explain over the course of a few posts about my strategy for how to achieve this sing current technology as well as serious implications for our industry if we were able to achieve this for all the major structures that we design.

On the effect of ground stiffness on the forces in a tunnel lining with internal pressures

I have previously looked at the equation that relates the internal pressure in a segmental lining to the forces in the linng and the opening at the joints. In this post I want to look at the implications of this equation in terms of the ground stiffness. Is there a typical minimum ground stiffness that is required for a benefical effect in terms of the internal pressure?

To asses this I've run two sumple check cases through the equation for variations in ground stiffness. The equation is a strong function of the tunnel radius and so I'vve looked at two concrete linings, one 9m in diameter and one 3m in diameter. I've selected typical lining properties and joint configuratins for each case and had a look at how the ground stiffness affects the results.

The followng two plots show the varaition in axial forces for the two lining types:

Hoop force changes in a typical 9m diameter tunnel for an internal pressure 250kPa. 

Hoop force changes in a typical 9m diameter tunnel for an internal pressure 250kPa. 

Hoop force changes in a typical 3m diameter tunnel for an internal pressure 250kPa. 

Hoop force changes in a typical 3m diameter tunnel for an internal pressure 250kPa. 

The results of the two cases are very similar when considered as normalised results. This indicates that a strong rule of thumb could be applied to this problem for tpical segmental linings.

Up to around 100MPa ground sitffness (≈stiff clay ) there is very little beneficial effect in considering the ground reaction to an internal pressure. Above 100MPa there is a gradual beneficial effect, increasing with increasing ground stiffness. At around 1000MPa (≈chalk) the load is shared approximately equally. At around 10,000MPa stiffness the vast majority of the load is taken by the ground reaction.

A similar chart can be produced for hte raidal displacement and jint opening of the lining under the action of internal pressure:

Displacement and joint opening changes in a typical 9m diameter tunnel for an internal pressure of 250kPa. 

Displacement and joint opening changes in a typical 9m diameter tunnel for an internal pressure of 250kPa. 

Displacement and joint opening changes in a typical 3m diameter tunnel for an internal pressure of 250kPa. 

Displacement and joint opening changes in a typical 3m diameter tunnel for an internal pressure of 250kPa. 

The displacement charts give very similar results to the foop force results in terms of the impact of the ground stiffness. A crude rule of thumb therefore appears to be appropriate for the design of segmental linings under the action of internal pressures.

Under 100MPa stiffness there is no signficiant benefit from ground support.
Above 100MPa there is some benefit increasing to about 50% of the unsupported condition at 1000MPa.
At around 10,000MPa ground support takes the majority of the internal pressure.

On the derivation of forces in a segmental lining with internal pressure and ground support

This post has the derivation of a function that I have derived in the last couple of weeks. if you have an existing segmental lining and then fill the lining with a liquid under pressure the simple design approach is to reduce the hoop force in the lining the internal pressure of the fluid multiple by the tunnel radius. This is a simple and cautious approach to addressing the problem but at times it may not be sufficient. If the tunnel is well grouted into a stiff ground the ground reaction could result in a significant reduction in the axial force that need to be applied due to the internal pressure. This derivation considers a relatively simple model for this lining where a bolted segmental lining is considered in elastic ground.

Considering the effective pressures on the lining at the point of balance the internal pressure is equal to the external ground reaction plus the tensile force in the lining converted to an equivalent pressure:

Equation ①

Equation ①

where:

pi = Internal pressure in the tunnel
pl = Equivalent pressure due to generated axial force in the lining
pg = Ground reaction pressure

Initially considering the ground reaction, this can be defined using the sub-grade reaction approach. Muir-Wood 1 gave a spring stiffness for tunnel lining design of:

Equation ②

Equation ②

where:

ν = Poisson's ratio of the ground
E = Elastic modulus of the ground
ro = Radius of the excavation

Given a radial deformation of the tunnel lining and the ground of δradial the reaction pressure of the ground is given by:

Equation ③

Equation ③

Considering the flexibility of the tunnel lining both the axial stiffness of the tunnel lining and the stiffness of the bolted joints will need to be considered. The hoop force in the lining is given by:

Equation ④

Equation ④

Given the hoop force in the lining this can be converted into axial extension of the lining due to the hoop force. Initially consider the tunnel lining along. The circumfrential increase in the length of the lining by treating it simple as a column:

Equation ⑤

Equation ⑤

where:
El = Elastic modulus of the tunnel lining.
Al = Cross section area fo the tunnel lining.

A similar function can be derived for the circumfrential expansion of the lining due to the expansion of the bolted joints:

Equation ⑥

Equation ⑥

where:
n = the number of segmental lining joints in a ring
lbolt = the effective length of the bolts at the segmental lining joints
Ebolt = the elastic modulus of the bolts at the segmental lining joints
Abolt = the cross sectional area of the bolts at the segmental lining joints

The circumfrential displacement can be converted to a radial displacement by dividing by 2π. The total radial displacement is then given by:

Equation ⑦

Equation ⑦

Substitute ③ and ⑦ into ①:

Equation ⑧ 

Equation ⑧ 

⑧ can be rearranged to get a value for δ radial _ from known values:

Equation ⑨ 

Equation ⑨ 

Having derived a value for δradial the equivalent ground reaction can be determinned from Equation ③. Equation ① can then be used to determine the hoop force in the lining. The expansion of the lining and the opening of the bolts can then be determinned from Equations ⑤ and ⑥.

1. A. M. Wood, “Tunnelling: Management by Design,” pp. 1–320, Nov. 2002.