On the effect of segment curvature on joint rotations in segmental linings

Previously I wrote about a few approaches to modelling segment joints for the design of segmental tunnel linings. The model that I discussed is typically applied in a particular way where it is assumed that all of the deformation of the ring occurs due to rotation at the segment joints. This model is relatively simple to derive and importantly the outputs can be clearly understood. The simplicity does however hide the complexity of the problem that it is trying to address.

The approaches I previously discussed for calculating the bearing stress at segment joints typically result in eccentric loads being applied at the joints. If we consider a simple case of a single segment with large joint rotations at each end, if the rotations are in the same direction and of the same magnitude then there will be constant bending moment along the length of the segment due to the eccentricity of the hoop force in the segment. This constant bending moment will result in a constant curvature along the entire length of the segment in the same direction as the rotation of the joints. Under the right conditions bending in the segments caused by the eccentricity of the force at the joints can result in a reduced joint rotation… all the deformation of the ring is no longer assumed to be due to joint rotation but can include curvature of the segment as well.

Sketch of a segment in constant bending due to eccentric hoop force applied to segment at both joints

Sketch of a segment in constant bending due to eccentric hoop force applied to segment at both joints

Obviously having a high rotation and hence high bending moment across the entire length of a segment is unlikely in most linings where there is a change in radial deformation and and bending moment around the tunnel profile. The simple case considered above with a uniform bending moment is however instructive in considering this problem and how a typical assessment of joint rotation may overestimate the rotation and hence bearing and bursting stresses.

So how much of an impact might the segment rotation have on the results? Unfortunately building an analytical model of the problem is not trivial because of the interaction between the joint rotations and segment rotations. There are also a lot of feasible combinations of results so you need to define additional criterion to be used when finding the solution you are looking for.

Because of this I’ve just looked at a relatively simple model to estimate the impact of including segment curvature in the rotation calculation. I’ve assumed that the curvature in the segment is defined by a simple bending moment diagram with full eccentricity at one end and no eccentricity at the other which gives half the rotation of the constant curvature case considered above. The following results are taken from the lining I detailed in the last post about joint stress models at joints.

Mobilised bearing stress capacity for varying number of joints in the ring for cases where the curvature of the segment is and is not taken into account.

Mobilised bearing stress capacity for varying number of joints in the ring for cases where the curvature of the segment is and is not taken into account.

Mobilised bursting stress capacity for varying number of joints in the ring for cases where the curvature of the segment is and is not taken into account.

Mobilised bursting stress capacity for varying number of joints in the ring for cases where the curvature of the segment is and is not taken into account.

From these results you can clearly see that considering curvature of the segment does result in a reduced bearing and bursting stress at the joints, the reduction being very significant where relatively few joints are considered in a large diameter tunnel. The behaviour of the jointed lining is still consistent with the expected result that increasing the number of joints decreases the stresses at the joint but the effect is much less pronounced. The results, whilst still worse than the case when you consider half the segment length, are now more consistent with this model. Note that I have not applied the segment curvature model to the half a segment length model as I believe that this would be double counting the curvature of the segment.

So where does this leave us? Over the years I have encountered many situations where some adjustments to the joint bursting calculation are applied because it is felt that the design is too conservative such as doing the check at SLS rather than ULS. Joint bursting of course is typically the critical design case and therefore whilst these modifications may be beneficial at times they could be considered ill advised. If the model considered above is considered reasonable then there is a clear source of a potential reduction in the stresses at the joint that that should remain code compliment and not require any adjustments.

Of course this is not the only source of complexity that you might want to consider. In the coming weeks I’ll consider the design of a bolted joint and how this can impact the design of a ring.

On the method of designing joints in segmental linings

Amongst the different methods for designing the joints for segmental linings there are a few different approaches based on the assumption of a triangular stress distribution at the joint. In these models the rotational stiffness of the joint is derived by assuming that the concrete to concrete contact can be approximate to linear elastic springs. The stiffness of the springs is defined by the stiffness of the concrete and by a depth of concrete beneath the joint over which the strains in the concrete distribute.

It should be fairly obvious that these triangular stress distributions are not strictly correct because of the assumption of linear elasticity, however it is a model used by many designers and appears to be a reasonable approach for the design of joints. The application of the method is generally relatively simple, however there is one unknown, the depth of concrete to use when defining the stiffness of the springs. I have seen a number of different values used for this length from the thickness of the segmental lining through to half the circumferential length of the segment. All of these approaches have some justification and are probably more or less appropriate in different circumstances. In this post I am not going to go into why these different lengths are used, but what I would like to consider instead is how these different approaches perform when applied to the design as a whole. In particular I’m going to consider a few different design cases and look at how the bearing and bursting stress vary at the joints when different numbers of segments are adopted for each approach.

In this post I’m going to consider three basic development lengths

  • half the circumferential length of a segment
  • the thickness of the segment
  • twice the thickness of the segment capped at half the circumferential length of a segment
    I have seen all three of these values applied to reasonable designs for segmental linings before and so I would like to see how the results of the different approaches compare.

To do the comparison I’ve run a typical design for an 11.65m ID tunnel in soft ground. This is indicative of a typical twin lane road tunnel crossing a river. All the other parameters used for the assessment are at the end of this post1.

The following chart illustrates how the mobilised bearing capacity at the joints varies as the number of segments increases. There are a few few obvious points in this chart. The use of half the circumferential length of a segment is clearly the least cautious approach. This is actually a very consistent result with most design checks exhibiting similar behaviour which often leads to it being a preferred approach where the design loads on a lining are significant. The problem with this approach is that the curve of mobilised capacity with the number of joints in the ring shows a slight upward trend i.e. as the number of joints increases in a ring, the bearing stresses at the joint also increase. This is not consistent with observed behaviour of tunnels and essentially invalidates a large amount of the benefit of having joints in a segmental lining.

Considering the other two models there is a clear reduction in the total stress at the joints as the number of joints increases which is the behaviour that would be expected. The use of a longer development length of twice the segment thickness rather than the segment thickness gives a clear reduction in the bearing stress at the joint.
A smilar chart can be produced for bursting stress rather than bearining stress but which gives a similar behaviour to the bearing stress calculation.

At this point it is clear that the case for using the development length based on a proportion of the thickness of the segment is clear to me. The exact dimension remains to be addressed, however the behaviour of the model using the circumferential length is certainly flawed in its behaviour and appears to be far from ideal.

In a future post I will consider the development length in more detail to try and understand what proportion of the segment thickness may be correct.

On the importance of risk assessments in tunnel crossings

Urban tunnelling poses some specific challenges, often the biggest being asset protection. Tunnelling inevitably causes ground movements and assessing the impact of those movements, controlling them and mitigating them can be a major part of any urban tunnelling project.

For major asset owners, especially those with tunnelling or similar experience, they often have extensive documentation in place describing the critical elements of an assessment for their assets. But may asset owners do not. They may not have the experience or the scale for such documentation to be produced. In that case we often have to advise the asset owner what documentation they should expect to ensure that a piece of tunnelling does not damage their asset.

In that case, what documentation should we be requesting, what is the most critical document that those planning and undertaking the tunnelling can produce?

My view is the by far the most important document is the risk assessment. It is from this document that almost all other documentation is produced. Those undertaking the tunnelling must undertake a site specific, construction method specific, asset specific and comprehensive risk assessment highlighting where the risks associated with their chosen approach are.

The risk assessment should highlight by what mechanism the asset is at risk. From this should flow the analytical assessments of the asset covering the mechanisms highlighted by the risk assessment. These mechanisms and the enumerated values will direct the control methods and monitoring that needs to be used. The risk assessment will highlight unknown information that is critical, directing any further ground investigation or surveys that must be undertaken.

Of course, all the documentation needs to be produced correctly and be representative of the situation being assessed. But for me, the document that should always guide the whole assessment is the risk assessment.

On the use of moment thrust envelopes at segment joints

Most modern approaches to segmental lining design explicitly consider the behaviour and capacity of the joints. There are however a lot of different approaches to incorporating the joints into the analysis.

One common approach in the UK is to calculate the rotation of the segments at a longitudinal joint based on the as built tolerance and the ground induced deformations of the ring. Given the rotation of the segment and the hoop force it is then possible to derive a size and pressure distribution of the compressive contact patch between the two segments at the joint. The problem is, given this pressure distribution, what checks need to be undertaken.

Typically checks might include compressive bearing stress checks at the joint and tensile bursting stress checks. These are reasonable, code compliant and consistent with past experience of the types of failures that occur at segment joints. There is however also a trend for undertaking bending capacity checks using a moment thrust envelope based on the magnitude of the hoop thrust and the eccentricity of the centre of force in the contact patch. My view is this check is both not necessary and is not representative of the behaviour of the forces either around the joint or in the ring.

The key issue is that there is a critical assumption in the derivation of the moment thrust envelope, ’plane sections remain plane’. This can be practically considered in a few different ways but for the purposes of this discussion I’m going to consider it as meaning that the flow lines of stress in the structure being assessed are parallel to each other and the neutral axis. This is typically a reasonable assumption in the middle section of a segment away from the joints. At the joints however it clearly is not the case; the whole reason for the bursting stress check is the divergence of the compressive stress flow lines around the joint. This means that any check on the moment capacity in the region of the joint using a moment thrust envelope is invalid. Individual checks as a minimum on bearing and bursting should be undertaken at joints and in appropriate cases on other potential failures such as indirect tensile stresses around the joint.

There is an argument that bending checks based on the eccentricity of the forces at the joint are still reasonably valid for the mid-point of a segment, however this approach can also be problematic. Joint rotations will not be the same at either end of the segment and so the moments will also not be the same. The change in force eccentricity from the region near the joint to the region where the compressive flow lines are parallel can be very significant.

Whilst in some cases the approach of checking moment capacity at the joints is a common approach, I believe that it is a flawed and checking effort at the joints should focus on the checks that are reasonable and valid.

On case histories of tunnel settlement

It is very much a personal opinion but I’m a big fan of case history in engineering. Our engineering instincts are important but all too often they need to be trained and case history is critically important in training our judgement and instinct.

Over the years I’ve collated a lot of case history of tunnels. Originally I kept the information in an excel spreadsheet but unfortunately that has proved to be too prone to 'improvements' being made by colleagues. Whilst this is always done with the best intention, if the improvement is not in keeping with my approach to managing the data it can cause problems and usually a loss of data. Now I keep it in an online database.

After a discussion with a colleague on Friday about tunnel settlement and volume losses I’ve exposed some of the data, a list of tunnel settlements and trough width values which you can view [here]. It’s not comprehensive, it’s not even my complete data set. It is also imperfect with data coming from a mix of public sources including scaling from graphs, but I do find it is better than nothing.

As it is my personal data source please don’t expect it to be permanently available. Feel free to use it as you wish but it is a live document so if you reference it please make sure that you include the date.

Histogram of volume losses in the database

Histogram of volume losses in the database

On ground arching in closed form solutions

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:

Parameter Value
Internal dimeter 11.65m
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.

Fenner Packer curve for Overconsolidated Clay for the the classic plastic solution and the solutions derived using Einstein and Schwatrz and Duddeck and Erdmann.

Fenner Packer curve for Overconsolidated Clay for the the classic plastic solution and the solutions derived using Einstein and Schwatrz and Duddeck and Erdmann.

Fenner Packer curve for Weak Rock for the the classic plastic solution and the solutions derived using Einstein and Schwatrz and Duddeck and Erdmann.

Fenner Packer curve for Weak Rock for the the classic plastic solution and the solutions derived using Einstein and Schwatrz and Duddeck and Erdmann.

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.