Convergence-confinement theory in practical use
The first blog of this series explained what the ‘Convergence-Confinement’ theory is, and mentioned that a key aspect of convergence-confinement is the share of resistance between the structure and the ground itself. A tunnel will keep converging on a steady trend until intercepted by the supporting structure (the confinement) at some point.
The second blog of this series explained the differences in the effect on convergence due to stiffness and installation timing of the confinement (i.e. structural support).
This blog will complete this series by discussing some practical uses of the Convergence-Confinement theory.
Choice of tunnelling and support strategy
As the tunnel deforms progressively, it usually allows time for observation and reaction, which means the convergence can be monitored, and based on the observation and re-assessment, the strategy for support installation can be adjusted. This observation-reaction loop falls under the framework of ‘Observational Method’.
For example, when the convergence rate is low, installation of support can be deferred (to speed up tunnelling), or even omitted. A thinner or less stiff support can be chosen as well. This tends to be the case for ‘hard’ ground tunnelling, e.g. in competent rock. When this observation-reaction loop is achieved with sprayed concrete and associated ductile supports (such as rock bolt), it is often called NATM (New Austrian Tunnelling Method). In contrast, in ‘soft’ ground, particularly in an urban environment, the time window allowed for the observation-reaction loop is very limited, and the tolerance for error is much more stringent. Observational Method has relatively less useful scope in this situation. Instead, prescriptive and pre-emptive supporting strategy is normally adopted. This usually means much thicker and stiffer support, installed as early as possible. This strategy minimises ground movement due to convergence and therefore provides best protection to existing assets nearby.
A design approach guidance
A key use of the convergence-confinement method is to estimate the relative weighting between the contribution from the ground and that from the support, in the process of compensating for the lost balance of stresses due to excavation. The share by the ground in total is called ‘ground relaxation factor’. This factor is particularly useful for 2D cross-section based numerical models.
A tunnel engineer usually finds the wished-in-place 2D analysis to be overly conservative in that the tunnel lining takes on too much stress. As Convergence-confinement theory tells us, a large part of the ground stress would have been taken by the ground itself before the tunnel lining comes into existence. By discounting this part, we will be able to reduce the conservatism in the analysis and arrive at a much more economic design.
Why do 2D analysis when we could do 3D? 2D analysis are multiple times quicker to build and to change than 3D, hence within the limited timeframe of a consulting engineer is usually given, you could run many sensitivity tests. In most cases these tests are more important to the safety and accuracy of the design than getting the perfect geometry.
Estimation of the relaxation ratio
One of the most useful things about the convergence-confinement theory is that it can be used to estimate the relaxation ratio (stress taken by the ground itself vs. total in-situ stress), to guide the design. The ratio can be calibrated to historical data, ideally from similar construction in similar ground conditions in the past. A best available estimation can also be done with equations provided in relevant published papers.
It should be noted that the ratio may be different for different design purposes, such as estimating ground movement and structural design of tunnel lining. In practice, most of the time the factor is not black and white, but falls within a range or spectrum. The engineer must be careful about the direction of conservatism – i.e. whether upper or lower bound is more conservative. For example, as explained in the previous blog, a higher relaxation ratio corresponds to high ground movement (settlement), which means the ground is taking on relatively higher proportion of the load compared to the supporting structure. Therefore, a higher relaxation ratio leads to more conservative estimation of ground settlement, but less conservative estimation of the lining stress.
When does monitoring start
It should be noted that in real life, only part of the convergence can be monitored. Usually, a tunnel is excavated first, followed by installation of structural support at certain distance behind the face. It is only after the installation of support, can monitoring equipment be installed and the convergence reading start. When doing the calibration, we must bear in mind which part of convergence is actually recorded, so as not to misalign data and reality.
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