錢(qián)營(yíng)孜煤礦3.0 Mta新井設(shè)計(jì)含6張CAD圖-采礦工程.zip
錢(qián)營(yíng)孜煤礦3.0 Mta新井設(shè)計(jì)含6張CAD圖-采礦工程.zip,錢(qián)營(yíng)孜煤礦3.0,Mta新井設(shè)計(jì)含6張CAD圖-采礦工程,錢(qián)營(yíng)孜,煤礦,3.0,Mta,設(shè)計(jì),CAD,采礦工程
英文原文
Input to the application of the convergence con?nement method with time-dependent material behaviour of the support
Abstract
The convergence con?nement method is a two-dimensional, analytical method used in the design of sub- surface structures and for the description of ground and system behaviour. Its purpose is to derive the required support measures from the combination of the following values: the ground characteristic curve; a model of the development of the radial deformations of the excavation surface in the axial direction of the tunnel; the support characteristic curve; and the installation time and location of the support measures. The convergence con?nement method is usually employed in the preliminary design of underground structures. This article investigates the various methods of the convergence con?nement method and includes comments on possible application scenarios. One point of focus considers the system-bolting of rock mass as a supporting as well as a reinforcement measure. Another view is taken on the time-dependent material behaviour of shotcrete and its adaptation to the convergence con?nement method.
1.Introduction
Underground structures can be designed by using many different calculation methods. While the preliminary design is dominated by analytical and empirical methods, fully modelling the entire construction process in the course of numerical calculations represents the standard procedure for detail design today. Further- more analytical methods can act as a tool for quickly verifying the numerical calculations and assessing the system behaviour during all stages of the design and construction process. Based on that knowledge, the design can be adjusted accordingly. An example for such an analytical procedure is the convergence con?nement method (CCM).
A major development of the CCM was done by Pacher (1964). He investigated the deformation behaviour in an experimental tunnel to describe the ground behaviour.
Feder and Arwanitakis (1976) improved the convergence con?nement method by implementing a linear elastic–ideal plastic material behaviour into the ground characteristic curve. Further- more, a very important achievement is, that in this case the circular opening and the central symmetrical homogenous stress state is not a requirement for the analytical solution. Most of those solutions use the Mohr–Coulomb failure criterion. In 1992Carranza-Torres and Fairhurst (Carranza-Torres, 2004; Carranza- Torres and Fairhurst, 2000) developed an application to use an elastic–perfectly plastic rock masses with Hoek–Brown failure criterion.
In the last view years the convergence con?nement method experienced a revival. Newer publications such as (Alejano et al.,
2010) from Alejano, describe the implementation of Hoek–Brown strain-softening behaviour into the convergence con?nement method.
An important part of the CCM is the support characteristic curve. It describes the strain–stress relationship of the support measures against the rock mass. For the application of the CCM in real projects, AFTES (Panet et al., 2001) from 2001 can be seen as a fundamental work. A major improvement for the calculation of shotcrete lining and the time dependent behaviour was developed by Oreste (2003). This paper gives also an overview of the possibilities which the convergence con?nement method has to offer.
2. Basics
2.1. Assumptions and preconditions
Determining the ground characteristic curve requires an analytical solution, which usually makes use of the theory of an in?nite plate with a circular hole. For the analytical solution the following assumptions are made:
the theory of an in?nite plate is a 2D model with plane strain conditions and in?nite dimension,
circular opening,
central symmetrical homogenous stress state (hydrostatic stress),
constant primary stress,
homogenous material properties of the rock mass,
non rheological material behaviour,
isotropic material law.
Only few models partially differ from these assumptions, like for example the one delivered by Feder and Arwanitakis (1976), who – with limitations – provides geometry for any state of pri- mary stress and oval cavity in his calculations. Most of the assump- tions stated above are only met to a certain extent in reality. To be precise, different ground characteristic curves and different sup- port characteristic curve would have to be determined for each point on the excavation surface; in addition, construction se- quences cannot be factored out in the calculation, and can only be considered as a simpli?cation.
2.2. Stress distribution
The stress distribution around a cavity in an elastic medium has been determined by Lame and Kirsch. If, however, the circumferen- tial stresses at the excavation surface exceed the rock mass strength, then a zone with plastic material behaviour or softening develops. Kastner (1962) solved the differential equation for the determination of the stress distribution around cavities in linear elastic–ideal plastic Mohr–Coulomb (MC) media. Extensions have also been derived for Hoek–Brown (HB) media and with more com- plex behaviour after a failure, speci?cally linear elastic–ideal plas- tic with sudden or gradual softening (Carranza-Torres, 2004; Feder and Arwanitakis, 1976; Hoek et al., 1983; Sharan, 2008). The soft- ening can be taken into account in the convergence con?nement method by altering the strength or strain properties. The transition from plastic to elastic behaviour takes place at the plastic radius (rpl). Fig. 1 shows the stress distribution around a circular excava- tion with the development of a plastic zone for rock with linear elastic–ideal plastic and linear elastic–brittle ideal plastic material behaviour. To compare the two material models – Mohr–Coulomb(MC) and Hoek–Brown (HB) – the material parameters were con- verted using the area replacement method. Although there is still a small difference in the stress distribution, the reason for this is the different shape of the line of failure between the MC and HB material model.
Table 1 shows the material parameters for the Mohr–Coulomb (MC) and the Hoek–Brown (HB) material model as well as the geo- metrical parameter of the underground structure and the speci?c weight.
The parameters of Table 1 are used in the examples for the description of the rock mass and the ground characteristic in this paper.
2.3. Displacement distribution
According to Feder and Arwanitakis (1976) the aggregated deformations around the excavation are made up of three components:
– elastic component,
– plastic component,
– volume increase in the plastic zone,
The displacements are determined through the integration of the stress ?eld in combination with a material law. When writing the differential equation attention needs to be paid to the individ- ual strain components. In most cases, a planar displacement state is assumed and the component in the tunnel’s axial direction is set to zero (Seeber, 1999). This means that only the circumferential and the radial strain components in the plastic zone are consid- ered. The volume increase is determined by a loosening factor, which also can be de?nite by the angle of dilatation. Some dif?cul- ties in the displacement distribution are the initial assumptions and the integration constants used in the calculation. As a calcula- tion example the radial displacement distribution after Salencon (Itasca Consulting Group, 2006) is illustrated in Eq. (1).
Radial displacement distribution after Salencon (Itasca Consult- ing Group, 2006)
Fig. 1. Comparison of radial (r) and circumferential ( t ) stress distribution around a circular cavity for various material models (HB and MC);Itasca Consulting Group, 2006;
Salencon, 1969 .
where G is the shear modulus, k the passive side pressure coef?- cient, kw the loosening factor, p0 the primary stress, pi the support pressure, r the range control variable, r0 the excavation radius, rp
the plastic radius, ur the radial displacements, m the poisson’s ratio,
and rUCS is the uncon?ned compressive strength.
The displacements at the surface of the excavation and the dis- placement distributions varying in the plastic zone according to the method used. However, the theories investigated are Sulem et al. (1987), Salencon (1969), Feder and Arwanitakis (1976) and Hoek et al. (1983), Hoek (2007) as shown in Fig. 2. The calculation results from the mentioned theories are approximately the same in the plastic area. The variation in the plastic part of the displacement distributions results from the different assumptions like the boundary conditions and the dispersal factor.
Fig. 2. Different deformation distribution at the excavation surface (Gschwandtner, 2010) according to Sulem et al. (1987) , Salencon (1969),HB(Gschwandtner, 2010), Feder
elasto-plastic ( Feder and Arwanitakis, 1976; Feder, 1978), HB elasto-plastic (with dilatation) ( Carranza-Torres, 2004) .
In addition, the E-modulus and the V-modulus are stress-dependent. The modi?cation of the Young′s Modulus or the V-modulus is disregarded in most methods, although its in?uence can be grave.
3. General remarks on the convergence con?nement method
In this paper the convergence con?nement method (CCM) Fenner, 1938; Gesta et al., N/A; Pacher, 1964 is treated as an analytical, two-dimensional method that is able to deduced the ground and system behaviour from three different curves:
ground characteristic curve (GCC),
support characteristic curve (SCC),
longitudinal deformation pro?les (LDP).
These curves will be explained in detail in the next chapters.The most important part is the point of intersection, between the ground characteristic curve (GCC) and the support characteristic curve (SCC),where the loading forces of the rock mass and the stabilizing forces of the installed support reaches the point of equilibrium.
Moreover, to simulate the construction process in a simplified way the two dimensional system has to be transformed into a three dimensional system. This can be achieved by utilizing an analytical model describing the radial displacements in the longitudinal direction of the tunnel. In particular LDP will be used to declare the location of the tunnel face and the installation of the support. The combination and interaction of all three curves is shown schematically in Fig. 3.
Fig. 3. Example for the interaction of the three curves in the CCM (ground characteristic curve, support characteristic curve and longitudinal displacemen ts profile); including
the important points (tunnel face, support installation, and point of equilibrium between rock mass and support) for a support calculation.
4. Ground characteristic curve
The ground characteristic curve represents the relationship between the effective internal support pressure and the radial deformation at the excavation surface. The ground characteristic curve is created by reducing the support pressure of the primary stress level to zero. When the support pressure is reduced, the rock behaves elastically up to the critical support pressure pi,crit. If the effective support pressure falls below the critical support pressure, plastic material behaviour or the softening occurs. Also, the time dependent behaviour of the rock can be taken into account by altering the strength and deformation parameters.
5. Radial deformations in the axial direction of the tunnel
The longitudinal deformation pro?le can be used to determine the radial displacements along the tunnel in association with the distance (χ) from tunnel face. A temporal relationship can be taken into account by considering a constant advance rate (ν) (Eq. (2)).
Temporal relationship between the advance rate and distance from the tunnel face
The in numerical simulations the pre-deformations are considered by a pre-relaxation factor.
The installed support and/or the construction process has in?uence on the displacement distribution along the tunnel. The criterion for the deformation velocity is the energy which is stored in the rock mass. In the Mohr–Coulomb diagram this energy can be seen between the ground characteristic curve and the support characteristic curve. Fig. 4 shows the comparison of several longitudinal deformation pro?les (LDP) with different analytic solutions (Panet and Guenot, 1982; Corbetta and Nguyen-Minh, 1992; Unlu and Gercek, 2003; Hoek, 2007; Vlachopoulos and Diederichs, 2009; Pilgerstorfer and Radoncˇic′ , 2009). The various curves of the radial deformation along the tunnel vary from each other substantially, particularly in the heading area where the support is installed. For this reason the analytical calculation of the utilization ratio of the support depends on which theory will be used in the calculation process.
Fig. 4. Comparison of various theories for the development of radial deformations; Panet and Guenot (1982) , Corbetta and Nguyen-Minh (1992), Unlu and Gercek (2003),
Hoek (2007) , Vlachopoulos and Diederichs (2009) , Pilgerstorfer and Radoncic (2009).
Also, the support installation and the construction process have some in?uence on the pre-deformations behind the tunnel face and on the displacement distribution along the tunnel. In general this process is subjected to an implicit calculation process. How- ever, in the present article the process will be taken into account with an incremental approach.
6. Support characteristic curve
The support characteristic curve acts as a tool to depict the sup- port measures in the convergence con?nement method (bearing capacity curve of the support measures). The support characteristic curve or the actual support pressure pi at a prede?ned displace- ment of the excavation edge can be expressed mathematically through material parameters like stiffness (KSN), maximum sustainable stress (pi,ult) and strain (ur ,max)(Gesta et al., N/A; Panetet al., 2001).
In every calculation step the calculated support pressure pi is compared with the maximum sustainable stress, which can be obtained through the load-bearing capacity of the support, and has to be less than (pi,ult). The aggregated displacements at the sup- port failure (ur,ult,pl) consist of the three:
– displacements occurring at the support installation (ur,s),
– the elastic deformations (ur,el) of the support,
– plastic deformations of the support (ur,pl).
Fig. 5 illustrates a schematic diagram of the support character- istic curve and the different parts of the deformation.
Fig. 5. Schematic image of the support characteristic curve; y-axis is the support
pressure and the x-axis shows the displacements of the support.
Eqs. (3) and (4) show the calculation process for the effective support pressure at a displacement (ur) and the calculation of the total displacements at support failure.
Effective support pressure (Gesta et al., N/A)
where pi is the support pressure, KSN the stiffness, ur the radial dis- placements, and r0 is the excavation radius.
Total displacements at the support failure
where ur ,ult,pl is the radial displacements, ur,S the displacements occurring at the support installation, ur ,pl the radial displacements(plastic part), KSN the stiffness, ur the radial displacements, and r0 is the excavation radius.
6.1. Shotcrete
The shotcrete shell is defined in the convergence confinement method as a circular ring. The influence of the bending moment can be calculated through the thickness of the shotcrete shell. Furthermore, a defined bound between the shotcrete measures and the rock mass can be taken into account too. In reality shotcrete displays a time-dependent material behaviour with strength and deformational behaviour changing over time. This also includes creeping, relaxation and shrinkage effects. In this paper the time- dependent material behaviour in the convergence confinement method can be utilized after Schubert (1988) or Aldrian (1991) and afterOreste (2003).
The system stiffness KSN,SpC of the shotcrete shell, with the shell having a constant thickness (outer face of the shotcrete shell minus-outer face of the shotcrete shell minus), is calculated as shown in Eq. (5). In the case of a thin shell ( e) is much smaller than the tunnel radius (r) the stiffness can be obtained via Eq. (6).
Stiffness of shotcrete shell ( Gesta et al., N/A)
KSN,SpC is the stiffness of the shotcrete shell, ESpC the Youngs Modulus of the shotcrete, vSpC the Poisson′ s ratio of the shotcrete, ra the outer face of the shotcrete shell, and ri is the inner face of the shotcrete shell.
Stiffness of shotcrete shell with (e 《r)(Gesta et al., N/A)
KSN,SpC is the stiffness of the shotcrete shell, ESpC the Youngs Modulus of the shotcrete, vSpC the Poisson′ s ratio of the shotcrete, rSpC the radius in the middle of the shotcrete shell, and eSpC is the thickness of the shotcrete shell.
Eq. (7) shows how to calculate the maximum effective support pressure.
Maximum support pressure for a shotcrete shell (Gesta et al., N/A )
where pi ,ult,SpC is the maximum sustainable stress of the shortcrete shell, bSpC the shotcrete compressive strength, ra the outer face of the shotcrete shell, ri is the inner face of the shotcrete shell.At every point during the calculation process with a time-dependent material behaviour, the normal stress in the shotcrete shell has to be smaller than the maximum support pressure(pi ,ult,SpC). The size of the maximum support pressure depends on the strength-development over time. In this case the formulation after Aldrian (1991) can be used.
Time dependent strength-development ( Aldrian, 1991)
where βSpC (t ) is the shotcrete compressive strength at the time t,βSpC (28D) the shotcrete compressive strength after 28 days, and t is the time in hours.
In addition to the time dependent compressive strength a time dependet Youngs Modulus can also be taken into account. This influences the support stiffness subsequently during the development of the support pressure over time. The following Eqs. (9)and (10) are two examples how the time dependency of the Youngs Modulus can be implemented in the calculation process.
Time dependent Young′ s Modulus ( Aldrian, 1991)
where ESpC (t ) is the Youngs Modulus of the shotcrete at the time t,ESpC (28D) the Youngs Modulus of the shotcrete after 28 days, and t is the time in hours.
Time dependent Youngs Modulus ( Oreste, 2003 )
where ESpC (t ) is the Young ′ s Modulus of the shotcrete at the time t,ESpC (28D) the Youngs Modulus of the shotcrete after 28 days, a the factor (0.01–0.05), and t is the time in hours.
After Aldrian (1991) the time-dependent material behaviour of the shotcrete, with the load history taken into account, is calculated as follows (Eq. (11) ).
Calculation of the strain in shotcrete after Aldrian (1991)
where ε2,3 is the strain at point (time) 2 and 3, δ2,3 the stress at point (time) 2 and 3,E28 the youngs modulus after 28 days, V*（t;a）the ordered deformation modulus, F the constant; factor for load relieving, εd2∞ the delayed elastic strain, ΔC the time approach for the progress of the viscous strain, Q the constant; from the velocity of the reversible creep deformation, Δεsh the change of the shrink-age-strain, Δεt the change of the temperature-strain, α2 the load factor, and Cd∞ is the limit of the reversible creep deformation.
Calculation of the stress by predetermined strain in shotcrete after Aldrian (1991)
By utilizing a converted form of Eq.(8) the existing stress is calculated through the enforced displacements of the surrounding rock mass (Eq. (12) ).
6.2. Yielding elements
In deep tunnelling large displacements of rock mass can occur. Despite the rapid hardening, the shotcrete can initially accept large strains (up to 1%). If the compre
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