Characteristic curves, as well as finite element software, are useful tools for the design of underground works as well as for back analyses when defining rock parameters. Unfortunately, these tools do not consider one fundamental variable, namely time.
In the literature, however, both for the ‘convergenceconfinement’ method and for numerical analyses, there are procedures that take this variable into account, but only for supports.
The aim of this study is to define a new method that takes the time factor into account, not only for the supports, but also for the rock mass. Once the hypotheses have been verified, the study also aims to establish the effective capacity of the supports to absorb displacements as a function of excavation speed and to define tables for an initial evaluation of the support-rock mass system.
INTRODUCTION
The dimensioning of supports in tunnels is fundamental in the design of underground works.
Where the physical and mechanical characteristics of the rock mass are precisely known, the first objective is to predict the displacements acting on the shell formed of shotcrete (‘Spritzbeton’) in order to verify and/or select additional supports.
In this case, the choice of design type is also fundamental. In clearer terms, if the design approach uses characteristic curves, i.e. the ‘convergenceconfinement’ method, by taking the time factor into account it will be possible to determine an exact equilibrium point between the supports and the rock mass, while simultaneously permitting the variation of the physical and mechanical properties of the material to be considered for more thorough verification.
The consideration of time in the characteristic curve of the supports is, therefore, the right approach and the first step in correct design. Using a classic bi-linear curve to represent the physical model for concrete is in fact not really correct, as it assumes the instantaneous hardening of the material that, in turn, implies greater stiffness and strength than that which actually occurs during excavation.
The next step is to consider the excavation speed for representing the phenomenon. This speed, as will be seen later, may result in the breaking of the material used as a support despite the fact that an equilibrium point for the system has been reached.
HOW TO CONSIDER THE TIME FACTOR
The time factor is of fundamental importance in determining a valid equilibrium point for the system. Valid, here, means that the supporting shell is able to withstand the loads, i.e. displacements, of the mass and will not suffer cracks, fissures, detachments, movements between the parts of the section, etc.
The time factor is considered for the following construction phases: the hardening of the shotcrete after its application in the section; the relaxation of the rock mass (i.e. the determination of the variation in the internal fictitious pressure over time); and, the excavation speed, so as to determine the actual displacements acting on the support structure. These analyses are explained in detail below.
Shotcrete
If applied wet, shotcrete needs time to harden, and its mechanical properties are therefore a function of the age of the material. It follows that the shotcrete’s constitutive bond (σ-ε), and therefore the maximum nominal resistance of the material, will vary over time. In addition, the type of support considered is subject to phenomena that produce deterioration of the mechanical characteristics, such as shrinkage and creep, in turn inducing additional stress states that are added to the stress states caused by the rock mass.
This paper refers to the bond (σ-ε) as proposed and described in the CEB-FIP Model Code 1990 – Comité Euro-International du Beton1. Its equations were also calculated as a function of time, to determine their variation (see Figure 1).
The effects over time due to creep and shrinkage were studied according to P. Schubert’s method (Flow Rate Method2). This method makes it possible to calculate the deformation over time according to equation 1 , with reference to axial strain, axial stress, Young’s Modulus, plastic creep rate, reversible creep strain, and shrinkage strain:
Once the deformation state is known, it is possible to calculate the correlated stress as shown in equation 2 . For the meaning of the individual components of equations 1 and 2 , refer to reference2.
The results obtained, as further specified below, were compared with reality (through verification with measured displacements) and with analyses carried out using finite element software (Plaxis 2D). This software uses the ‘concrete’ model developed at the Technical University of Graz (B-Schädlich & H.F. Schweiger, 20143) for calculation of the material’s stress-strain state. The comparison revealed a high degree of coincidence in both cases.
Methods used to find the equilibrium point
This work used two methods – Method 1 and Method 2 – for determining the system’s effective equilibrium point. The first method is iterative, i.e. an incremental value is fixed for the displacement and the confinement pressure is calculated in order to check whether the point in question belongs to the characteristic curve of the mass. In this case, the time only affects the age of the concrete and the excavation speed, not however the relaxation of the rock mass.
The relaxation of the mass is a characteristic feature of Method 2, as will be seen below.
Three tests were carried out in which the rock mass parameters and excavation speeds were varied. The calculation parameters are shown in Table 1. The comparison is shown in Tables 2, 3 and 4.
Terms in the tables relate to displacements in the unsupported condition, at the face, and at the equilibrium point.
The tables show that the results of Method 1 approximate the results of the software used. It should be noted that Method 1 does not represent the reality as, depending on the excavation speed used, different equilibrium points should be obtained. To achieve this objective, Method 2 was implemented: this takes account of the relaxation of the rock mass over time to determine the actual displacements acting on the lining.
Relaxation of the rock mass
What is the relaxation of the rock mass and why is it necessary to know its trend over time? The ‘convergence-confinement’ method assumes that the internal pressure, equal to the lithostatic pressure (Overburden depth x Specific Weight), will decrease from its initial value to zero; the corresponding displacements are calculated for each internal pressure according to the theory used. In this study the theory proposed by Hoek4 was used to calculate the characteristic curve of the rock mass.
In order to calculate the relaxation over time it is necessary to know the displacement of the rock mass as a function of time. To achieve this, the equation of Panet, Sulem and Guenot5 was applied, as shown below, with reference to terms including radial displacement, distance from the face, and different function parameters:
For the meaning of the individual components of equation 3 , refer to reference5.
Equation 3 in its complete form describes the displacements over time, i.e. including the rheological phenomena of the rock mass. For simplicity, this work ignores these phenomena and reference is only made to the component derived from the speed of advancement.
A simple mathematical transposition permits each internal pressure-displacement pair on the characteristic curve to be linked to the displacementtime pair of equation 3 . It is therefore possible, following application of the shotcrete shell, to know which displacements take place and over what time they occur. The study was carried out using time intervals of one hour.
Using this mathematical transposition, it is possible to define a characteristic curve for the rock mass in three dimensions. Figures 2 and 3, respectively, show an example of rock mass relaxation and a three-dimensional characteristic curve (pressure, displacement, time).
Actual displacements acting on the supporting shell
Once the trio of pressure-time-displacement is known, each displacement can be associated with the time during which it occurs. It will then be possible to calculate the deformation on the support shell and the tension state of the constitutive law. Once the tension state is known, the stress state and therefore the confinement pressure exerted by the shell on the mass can be determined.
As mentioned, this procedure is carried out for each time interval considered, i.e. every hour.
Once the confinement pressure of the concrete and the relative displacement have been calculated, it is possible to verify whether an equilibrium point has been reached and whether the verification of the concrete has been satisfied. It is, of course, possible to draw the resulting, non-linear, characteristic curve.
This method has been extended to various types of support, including when in combination with others, such as shotcrete, anchors, deformation gaps and deformable elements. Applications have been developed for this purpose in Visual Basic [objectoriented programming language for the extensive calculations required, even for finite element modelling of the mobilisation forces) that also permit the calculation of other characteristics, such as rheological phenomena, partial excavation and the mobilisation forces between the concrete and the rock mass, concrete work rate, etc.
Figures 4 and 5 show the characteristic curves of the supports for an example carried out using Method 2 at different excavation speeds with the corresponding concrete work rate for its verification.
It can be seen from Figures 4 and 5 that in this example a valid equilibrium point is always reached, i.e. the concrete always has a work rate of less than 1.
Verification of procedure and of Method 2
Method 2, and the procedure set out in the previous paragraphs, was verified by comparison with realworld examples – through geotechnical measurements during excavation of the West-South main gallery of the Brenner Base Tunnel on the Austrian side in Lot H51, Pfons-Brenner. Comparison was made with the average of the measurement points of the section.
The parameters of the rock mass used for the verification calculation are: unit weight () = 27.00KN/ m3; friction angle () = 34.00°; cohesion (c) = 1.60MPa; Young’s Modulus (E) = 8,000.00MPa; and, Poisson’s coefficient (v) = 0.13. Also, radius (r0) = 5.520m; Chainages = 113m, 135m, 145m & 168m; overburden (H) = 490m; and, geology is phyllite limestone.
The comparison between the measurements made and the calculations appears in Figures 6, 7, 8 and 9. The average speed during excavation of the tunnel in question was approximately 6.8m/day.
Deformation energy following excavation
Once the method for calculating the equilibrium point has been established, it is possible to calculate the exact dissipative capacity of the supports with reference to the effective deformations of the rock mass, all as a function of the excavation speed. This, as has been seen, conditions the result of the verifications.
The dissipative capacity consists of the capacity of the shell to deform without suffering breakage or significant loss of strength.
Before determining the dissipative capacity of the shell as a function of the rock mass characteristics and excavation speeds, it is necessary to define and describe what happens to the rock mass from an energetic point of view.
The initial relaxation of the rock mass at the face, 0, is defined by equation 4 , with reference to the pressure of the rock at the face and the primary stress, respectively:
By noting the relaxation immediately after excavation, the corresponding deformation energy can be calculated. The deformation energy released in the absence of supports is known. If ‘PRE’ is the energy released at the face immediately after excavation and ‘GRE’ is the global energy released in the absence of supports, the PRE/GRE ratio can be defined. This ratio has been seen to be invariable with respect to the elastic modulus of the rock mass, but this depends on the physical parameters of the friction angle, cohesion and plastic radius.
The PRE/GRE energy ratio can be described by the following equations, with regard to the ratio of plastic radius to radius, and also the ratio of energy release types, and including friction angles, and 0 is a constant (=1):
The PRE/GRE energy ratio can also be calculated as a function of 0: the ratio is represented below with its graph when calculated according to equation 4 :
On the other hand, using equations 5 and 6 produces the following results:
Using Figures 10 and 11, it is possible to determine the PRE/GRE energy ratio as a function of the relaxation factor compared to 0 (initial relaxation) and vice versa.
Calculation simulations and definition of admissible limits
The objective of this work is to define the capacity of the supports to absorb displacements. This depends not only on the physical characteristics of the rock mass, but also on the speed at which advancement occurs and, therefore, also on the hardness (stiffness and compressive strength) of the confining material.
Various analyses were carried out to establish the maximum load-bearing capacity of the shotcrete as a function of these variables, varying the overburden, the characteristics of the rock mass and the speed of excavation; while for the support, the calculations always referred to a thickness of 20 cm and a compressive strength of 30MPa at 28 days. The overburdens were varied from 50m to 1000m, the cohesion from 0.5MPa to 1.25MPa and the friction angle from 30° to 40°.
The results for the different excavation speeds are shown in the following graphs. The calculations were made in the most unfavourable situation, i.e. always using an elastic modulus of 500MPa so as to obtain the maximum deformations and in this way to define a maximum limit.
The maximum deformation in unsupported conditions is given by the maximum displacement of the rock mass in the absence of support divided by the radius, while the energy ratio has been defined in the previous paragraphs. It can be seen from the graphs in Figure 12 that, as the speed of advancement increases, the verification points for the support decrease with a rock mass of the same physical characteristics.
The calculation iterations carried out permit the maximum limit deformation that the rock mass can withstand to be defined, including as a function of 0 (initial relaxation) for different excavation speeds.
All the iterations carried out were repeated for the case of partial section excavation. This work omits the development of the characteristic curve for the top heading alone, as well as the corresponding graphs for determining the maximum limit deformation as a function of the PRE/GRE energy ratio.
Correlations
Once the maximum limit deformation of the mass is known as a function of the deformation energy ratio at different excavation speeds, it is possible, with reference to the equations shown previously, to define graphs for an initial verification of the concrete shell.
These graphs appear in Figures 13 and 14. They are easy to use:
- All points below the functions can be considered to be valid equilibrium points, i.e. equilibrium points at which the work rate of the concrete is always in permissible conditions of verification with the strength satisfied.
- The points above the various functions are not equilibrium points or are not valid equilibrium points and require more in-depth analysis at the design stage.
CONCLUSION
Using Method 1, the assumptions for both the Flow Rate Method2 and the ‘Concrete’ model of the Plaxis software3 were found to be valid.
However, experience, observations and theories for tunnel excavations show that the equilibrium point of the rock mass-support system depends on the excavation speeds.
Method 2 was therefore employed: defining a relaxation curve for the rock mass as a function of time gives a better approximation of reality (see comparison with geotechnical measurements).
A more realistic equilibrium point does offer considerable advantages in both the design and construction phases.
In the former it is possible to optimise the statics of the sections (and the tender contract) by, for example, setting minimum and maximum speeds for the type of rock mass.
In addition, a true equilibrium point for the system also allows the final tunnel lining to be optimised via assessment of the actual loads on the concrete.
These considerations permit the costs of the work to be evaluated in the best possible way.
In the execution phase, on the other hand, defining the right excavation speed avoids the problem of having to restructure parts of the tunnel that have already been excavated, optimising the working rate of the concrete, and anticipating the formation of cracks, detachments and breakages in the external lining.
Once the equilibrium point has been defined and the deformation energy has been applied, the limit values can be established and, therefore, the tables for initial verification of the section.
The use of these tables permits the initial choice of excavation speed according to the type of rock mass (maximum limit deformation lim).
