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英文原文Input to the application of the convergence connement method with time-dependent material behaviour of the supportAbstract The convergence connement 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 connement method is usually employed in the preliminary design of underground structures. This article investigates the various methods of the convergence connement 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 connement method.1.IntroductionUnderground 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 connement 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 connement method by implementing a linear elasticideal 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 MohrCoulomb failure criterion. In 1992Carranza-Torres and Fairhurst (Carranza-Torres, 2004; Carranza- Torres and Fairhurst, 2000) developed an application to use an elasticperfectly plastic rock masses with HoekBrown failure criterion.In the last view years the convergence connement method experienced a revival. Newer publications such as (Alejano et al.,2010) from Alejano, describe the implementation of HoekBrown strain-softening behaviour into the convergence connement method.An important part of the CCM is the support characteristic curve. It describes the strainstress 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 connement method has to offer. 2. Basics2.1. Assumptions and preconditions Determining the ground characteristic curve requires an analytical solution, which usually makes use of the theory of an innite plate with a circular hole. For the analytical solution the following assumptions are made: the theory of an innite plate is a 2D model with plane strain conditions and innite 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 simplication.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 elasticideal plastic MohrCoulomb (MC) media. Extensions have also been derived for HoekBrown (HB) media and with more com- plex behaviour after a failure, specically linear elasticideal 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 connement 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 elasticideal plastic and linear elasticbrittle ideal plastic material behaviour. To compare the two material models MohrCoulomb(MC) and HoekBrown (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 MohrCoulomb (MC) and the HoekBrown (HB) material model as well as the geo- metrical parameter of the underground structure and the specic 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 distributionAccording 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 tunnels 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 denite by the angle of dilatation. Some difcul- 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, rpthe plastic radius, ur the radial displacements, m the poissons ratio,and rUCS is the unconned 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), Federelasto-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 modication of the Youngs Modulus or the V-modulus is disregarded in most methods, although its inuence can be grave.3. General remarks on the convergence connement methodIn this paper the convergence connement 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 proles (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); includingthe important points (tunnel face, support installation, and point of equilibrium between rock mass and support) for a support calculation.4. Ground characteristic curveThe 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 tunnelThe longitudinal deformation prole can be used to determine the radial displacements along the tunnel in association with the distance () from tunnel face. A temporal relationship can be taken into account by considering a constant advance rate () (Eq. (2).Temporal relationship between the advance rate and distance from the tunnel faceThe in numerical simulations the pre-deformations are considered by a pre-relaxation factor.The installed support and/or the construction process has inuence 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 MohrCoulomb 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 proles (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 Radoncic , 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 inuence 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 curveThe support characteristic curve acts as a tool to depict the sup- port measures in the convergence connement method (bearing capacity curve of the support measures). The support characteristic curve or the actual support pressure pi at a predened 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 supportpressure 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 failurewhere 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. ShotcreteThe 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 SpC (t ) is the shotcrete compressive strength at the time t,SpC (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 (0.010.05), 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 2,3 is the strain at point (time) 2 and 3, 2,3 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, d2 the delayed elastic strain, C the time approach for the progress of the viscous strain, Q the constant; from the velocity of the reversible creep deformation, sh the change of the shrink-age-strain, t the change of the temperature-strain, 2 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 compression strain forced on the shotcrete is too large at a particular time, yielding elements can be installed(Schubert et al., 1996 ). In the convergence confinement method the yielding elements are implemented by their load bearing capacity ( FSE), at which the displacements occur at a defined load level. This load bearing capacity must be suited to the time-dependent load-bearing behaviour of the shotcrete as well as the time-dependent loading of the rock mass due to the deformations of the excavation work (Eq. (13) ). Load bearing capacity for the calculation of yielding elements where pi ,SE is the activating support pressure of the yielding element, FSE the load bearing capacity, rSpC the radius in the middle of the shotcrete shell,ASpC the cross-sectional area of the shotcrete shell, and SpC is the shotcrete compressive strength. The friction-bond between the shotcrete shell and the rock mass has to be taken into account too. Based on that, the maximum utilization of the shotcrete does not occur directly at the yielding element (Pottler, 1990 ), and as a result neither does the failure in the shotcrete shell. Furthermore, the system stiffness is different between a support system with yielding elements, and one with-out them ( Radoncic et al., 2009). Fig. 6 shows a simplified CCM example of a shotcrete shell with and without yielding elements, time-dependent material behaviour for the shotcrete afterOreste (2003)and two different advance rates of 2 and 5 m per day. For this calculation a constant LDP after Hoek et al. (1983), with consideration of the maximum radial displacements, was used. For the maximum displacements and the GCC the theory after Salencon ( Itasca Consulting Group, 2006), as in Eq. (1), was used. The parameter for the shotcrete and the yielding elements can bee seen in Table 2.Fig. 6. Shotcrete with yielding elements; dashed (related to the right-hand side): duration from/until passing the face in dependence on the advance rate (Aldrian, 1991;Kainrath-Reumayer et al., 2009; Kienberger, 1999; Oreste, 2003 ). In Fig. 6 it is evident that the SCC of the shotcrete shell without any yielding element does not reach the point of equilibrium. The consequence is a failure of the support and a collapse of the tunnel. The additional application of yielding elements in the shotcret shell allows the support construction to absorb more deformation. In this case the SCC reaches the point of equilibrium. Fig. 6 shows the effect of different advance rates (2 and 5 m per day) on the SCC. In the convergence confinement method rock bolts are categorised into two groups: end-anchored and fully grouted. End-anchored rock bolts are fixed to the rock mass at the full depth of the hole and at the edge of the cavity. Conventional end-anchored rock bolts are seldom used in deep tunnelling, and there are anchor systems able to absorb large displacements ( Kainrath-Reumayer and Galler, 2008a). In this case the support pressure is calculated with a homogenized external force as in Eq. (14) . The system stiffness ( KSN,A) therefore depends on many different parameters like total length diameter, Youngs modulus of the rock bolt and the rock bolt installation grid (Eq. (15) ).Maximum support pressure of end-anchored rock boltswhere pi,A is the support pressure, FA,A the supporting force for one rock bolt, and AA, is the rock bolt assigned area.Stiffness of end-anchored rock boltswhere dA is the rock bolt diameter, lA the rock bolt length, eA, fA the radial and circumferential distance between the rock bolts, EA the youngs modulus of the rock bolt rod, and Q is the factor for the displacements of the rock bolt head (rock bolt plate).On the other hand, the effect of the rock bolts can be implemented by a smeared effect on the anchored rock mass area, which is mostly common for fully grouted rock bolts. Normally an improvement of rock mass properties (strength and deformation characteristics) can be assumed. This is usually done by increasing the cohesion in the anchored area, which is influenced by a number of factors: strength and deformation ability of rock bolts, rock bolt mortar and the rock mass on the one hand, and geometric factors such as rock bolt length and diameter on the other hand. The size of the increase of cohesion c can be determined by laboratory and in-situ tests. Several different semi-empirical and/or analytical calculation methods addressing this topic have been developed. The best known examples are the works of Bjurstrom(1974), Schubert (1984) and Spang (1988).In case of the material behaviour after Mohr-Coulomb, including the strength parameters ( c) and () and the uni-axial compressive strength (dg), the increase of the cohesion can be taken into account in two different ways as shown in Figs. 7 and 8 . Both ways use the effective support pressure 3(resulting from the rock bolts) to increase the rock mass strength from (d)g to (dg,A).Fig. 7. Passive approach for the increase of the cohesion in the u s diagram (Kainrath-Reumayer and Dolsak, 2008; Kainrath-Reumayer and Galler, 2008a).Fig. 8. Active approach for the increase of the cohesion in the u s diagram (Kainrath-Reumayer and Dolsak, 2008; Kainrath-Reumayer and Galler, 2008a). The common approach for increasing the cohesion is the passive approach ( Fig. 7), in which the original conditions of the rock mass will be increased by (cp), subsequently resulting in a vertical shift of the line of failure in the MohrCoulomb diagram. The second and more conservative approach (normally for u 45 )is the active approach. In this case cohesion increase is implemented by a horizontal shift of the s -axis in the MohrCoulomb diagram. Fig 9 represents a summary of results from analytical (index ) and numerical (index ) calculations from a discoid model. The case studies were: analytical solution with no rock bolts, analytical solution with a passive cohesion increase (cp), analytical solution with a active cohesion increase (ca), analytical solution with support pressure from the rock bolts (pi,a), numerical simulation with support pressure from rock bolts (pi,n).Fig. 9. Summary of results for analytical and numerical stress distributions(circumferential and radial) for a circular opening with and without rock bolts(Kainrath-Reumayer and Galler, 2008b). For the numerical calculation FLAC 2D 6.0 was used. The results between the analytical and numerical calculation without rock bolts shows comparable results for the stress distribution and the size of the plastic zone. The analytical calculation, with the simplified approaches of the system-bolting, shows a minor difference between the active and passive approach for the increase of the cohesion. For fully grouted rock bolts calculating the stiffness is much more complicated and depends on many factors, like the displacements distribution (including the displacements rate) the excavation edge and the hardening speed of the rock bolt mortar. Schubert (1984) and Blumel (1996) have conducted a pull-out test of rock bolts in hardened mortar. The results of these tests (Fig. 10 ) are used to develop a highly simplified model of the behaviour of fully grouted rock bolts. This essentially considers: the load rate of the fully grouted rock bolt, the development of strength behaviour of the rock bolt mortar as well as the failure load, the breaking elongation of the rock bolt rod. The displacement rate at the excavation edge is calculated via the LDP as shown in Eq. (16) . Displacement rate at the excavation edge where u12 is the displacement rate time 1 to time 2, ur ,1 the radial displacement at time 1, ur ,2 the radial displacement at time 1, tmin,1 the time 1, and tmin,2 the time 2.Fig. 10. 3D diagram of a pull-out test afterBlumel (1996) . The maximum stress and thus the effective force in the rock bolt rod can be assumed as a simplification proportional to the displacement rate and is determined by Hookes Law. The strength and deformation development of the rock bolt mortar is taken intoaccount with a similar approach after Oreste (2003)for shotcrete Comparison with manufacturers data shows that the strength of the rock bolt mortar is overestimated at a young age, but reproduces the results of the experiments from Blumel (1996) suffi-ciently accurate in the first approach (Eq. (17) ).Rock bolt mortar strength development model after Oreste(2003)where FA ,zul is the acceptable rock bolt force, FA ,max the maximum rock bolt force, t(h) the mortar age in h, and fT is the factor for the strength development (0.01 and 0.05). In consideration of the criteria for the pull-out resistance (shear strength) as a function of the displacement rate and the capacity utilization of the rock bolt tensile force the effective support pressure, the increasing of the cohesion can be calculated with the equations in Fig. 7 or Fig. 8 . The parameters for the fully grouted rock bolt are shown in the following Table 3.Fig. 11shows a fully grouted rock bolt system as a rock mass improving measure. At the point A in the diagram is the rock bolt installation and the start where the increase of the cohesion is taken into account. The consequence is a improvement of the rock mass strength and a change in the GCC.Fig. 11. Example for increasing cohesion in the convergence confinement method. The deformation of the rock bolt plate, from point B to C, is taken into account too by keeping the rock mass strength at a constant level. Some effects of steel tubes installed between the rock bolt plate and the rock mass can also be considered, like yielding elements for the shotcrete shell. During the calculation two inquiries, regarding the maximum tensile strength and the maximum tolerable displacement of the rock bolt, are made. If any of the two specified criteria is exceeded, some additional support measures are needed, otherwise the rock bolt system will fail. Mathematically, this can occur at the same time at all points of the excavation edge, but since this is only a theoretical result, it can be disregarded for the purpose at hand.7. ConclusionThe convergence confinement method is a demonstrative and quick tool for roughly estimating ground and system behaviour,which can be used as a first approximation of the fundamental requirements for support measures. This approach has a limited scope of application due to the assumptions that have to be made for analytical methods. These methods vary substantially from each other. This is why a fundamental understanding of system behaviour in general and failure processes in particular is essential for correctly applying the convergence confinement method. Also the combination of the three curves in the CCM plays a major role.This paper dine refers on four different theories for the GCC, six different theories for the LDP and many more for the SCC depending on the support measures. Based on the high amount of different basic approaches and theories the calculation and graphical results deviate from each other.One factor that should not be neglected in the application is the time dependence, which has to be taken into account during the determination of the ground behaviour and the support characteristic curves.The time-dependent material behaviour, including the process of the strength and Young s modulus determination of shotcrete,is very important, especially the distinctive modification in the early stage after the installation of the shotcrete shell ( Fig. 6). A largely new development in the CCM is the implementation of the time-dependent increase of the cohesion of fully grouted rockbolts ( Fig. 11). This article also gives a short overview of the diversity of application possibilities of the different theories from the convergence confinement method and shows a huge amount of prospects for further research work.中文译文应用收敛约束法对与时间相关的支护材料特性的深入论述摘要收敛约束法是一种应用在地下结构设计及描述地层系统关系的二维分析方法。其目的是从以下数值的组合，得出所需要的支护方法：地面特性曲线；在轴向方向的巷道断面径向变形的发展模型；支护特性曲线；支护方式的安装地点及安装时间。收敛约束法通常用在地下结构的初步设计中。本文探讨收敛约束的各种方法，包括对可能的一些应用方案的探讨。本文有两个观点：一，考虑锚杆与岩块支护系统的整体及岩石加固的方法。二，喷射混凝土材料的随时间变化的性质及其变化过程是否符合收敛约束法。1、介绍地下结构的设计可以采用许多不同的计算方法。初步设计以解析方法和经验方法为主，当今采用的比较详细的设计是在数值计算过程中充分模拟整个施工过程。此外，在各个设计和施工过程的阶段中，数值分析方法可以作为一个快速确定数值分析方法和系统特性评价的工具。基于这些分析，设计可作相应调整。这种分析方法的一个例子是收敛约束法（CCM）。收敛位移约束法的一个重要发展是由帕赫完成的。他在一个实验巷道中用威严的变形来描述整个地层的变化特性。菲德和阿瓦尼塔基斯 将一个线性的理想的弹塑性材料的特性曲线施加到地面特性曲线中，进一步发展了收敛约束法。此外，他们的一个重要成果是：在这个实验中，圆形断面和沿中心均匀分布的各向同性的应力状态对于此项实验不作要求。试验中，大部分方法都是采用的莫尔库伦准则。1992年，卡兰萨、托雷斯和费尔赫斯特根据胡克布朗破坏准则用理想的弹塑性岩块发展了一个新的应用方法。在最后的几年内里收敛约束法重新兴起。新的出版物中，比如作者亚历哈诺，在位移约束收敛方法中描述了胡克布朗应变软化的围岩特性。收敛约束方法的一个重要部分是支护特性曲线。他描述了对岩石支护后支护方法材料的应力应变关系。对于收敛约束法在真实案例中的应用，2001的AFTES可以看做一个根本的工作。对于喷射混凝土的数值计算以及其随时间变化的特性的重要发展是由奥雷斯特作出的。本文对于位移收敛约束方法应该给出的一些可能也作了大致阐述。2、基础内容2.1.假设和前提条件确定地面特性曲线通常需要充分利用圆孔无限长板理论的分析方法。基于此分析方法，做出如下假设：无限长板理论是一个平面受力状态的二维模型并无维数限制圆形断面中心对称的均匀受力状态受恒压作用岩块为均质材料无材料的流变特征满足各向同性材料规律只有一些少数模型与以上的假设有部分不同，比如像菲德和阿瓦尼塔基斯 发表的一个模型，但对于提供任何状态的主应力和椭圆形腔几何特性有一定的局限性。上述提到的大部分假设只与现实中的情况在某一定程度上相符合。精确一点来讲，不同的地面特性曲线和支护特性曲线必须依开切面上的每一点而确定，此外，建设顺序在计算中不能够被分解，并且只能作一些简化。2.2 应力分布Lame和基尔希已经确定出在弹性介质中孔穴周围的应力分布。然而，如果开挖面的切向应力超出岩块的强度，岩石会展现出塑性和软化特性。卡斯特纳解决了在线性的理想的弹塑性莫尔库伦媒介中确定孔穴周围的应力分布的微分方程的问题。同样在胡克布朗媒介中得到进一步延伸发展并得到了一个在破坏之后呈现的更加复杂的规律，准确的说是在突然或逐渐软化的理想的弹塑性体中得到的。在收敛约束法中可以在预报应力应变状态时考虑这个软化机制。从塑性特性向弹性特性的转化过程发生在塑性区半径（rpl）时。图1揭示了具有线性理想弹性塑性以及线性弹脆性理想塑性的岩石材料在其塑性区发展时其圆形开挖面的应力分布状态情况。为了比较这两种材料模型（即莫尔库伦和胡克布朗型两种模型），采用区域替换方法对材料的参数进行了转换。尽管应力分布有一点不一样，产生这种现象的原因是由于莫尔库伦和胡克布朗型两种模型材料的破坏线的不同形状。图1 径向（R）和切向（T）（HB和MC）的圆形孔洞模型周围的应力分布比较表一 材料参数和几何参数的计算几何参数计算容重（kN/m3）25巷道深度(m)200直径（m）11材料参数E（GPa）8460.35G(MPa)313.33莫尔库伦参数c(MPa)27.35（）0.382ucs（MPa）1.26胡可布朗参数mb0.6625s0.000220.5Ci(MPa)15GSI24表1分别揭示了莫尔库伦和胡克布朗型两种模型材料的参数和地下结构的几何参数以及准确的重力。表1的参数作为本篇文章中描述岩块和底层特性曲线的一个例子。2.3 位移分布依据菲德和阿瓦尼塔基斯开挖面的加剧的变形由三个部分组成：弹性部分塑性部分塑性区域体积的增大通过应力场与材料准则的结合确定出位移大小。当给出微分方程时，需要注意个别的应变分量。在大多数情况下，威猛假设出平面位移状态并且将巷道轴向方向分量设置为0.这意味着，只需考虑塑性区中的切向和径向分量。体积的增加是由松散的因素造成的，同时也可以由岩石的膨胀角确定。位移分布的困难在于：原先的假设和采用的结合常量难以确定。作为计算的例子， Salencon之后的径向位移分布在方程1给出了说明。Salencon之后的径向位移分布G时剪切模量，K是被动区压力系数，是松散系数，P0是主应力，Pi支撑压力，r范围控制变量，r0开切半径，rp塑性半径，ur径向位移，泊松比，为不受限制的压缩强度。在塑性区中开挖面的位移和位移分布依据采用方法的变化而变化。研究过这些理论的人有Sulemet (1987)等人, Salencon (1969) , 菲德 、阿瓦尼塔基斯 (1976)和胡克(1983)等人, 胡克(2007)，如图2中所示。从上述的理论中得出的计算结果与塑性区域中几乎一样。图2 开挖面的不同变形分布(Gschwandtner, 2010)，依据Sulem（1987年）等人，Salencon（1969），HB（Gschwandtner，2010），：菲德尔弹塑性（Feder及Arwanitakis，1976年，Feder，1978年），HB弹塑性模型（扩张）卡兰萨，托雷斯（2004）。位移分布的塑性部分的波动是由于像边界条件和不同的假设和扩散因子引起的。此外，弹性模量和体积模量与应力有关。 尽管影响很大，但是大部分方法中都忽略了对杨氏模量或者v-模型的修改。3、收敛约束法概论在本文中收敛约束法是作为一个分析二维的工具，并且能够根据不同的曲线推断出地层和整个系统的特性：地层特性曲线（GCC）支护特性曲线（SCC）纵向变形曲线（LDP）在下面的章节中将对这些曲线进行详细的解释。最重要的是地层特性曲线和支护特性曲线的交点，在这个交点处岩石卸载应力和锚杆安装的恒定应力达到平衡点。此外，为了以一种简化的方式模拟施工过程，并需将二维系统转换成三维系统。这可以通过运用描述巷道纵线方向的径向位移分析模型获得。LDP可以用来确定巷道面的位置以及支护装备的安装地点。这三种曲线的交互示意图见图3.图3。对于在CCM中的三条曲线的相互作用的示例;包括的要点（掌子面，支护安装，岩体和支持之间的平衡点）的支持计算。4、地层特性曲线地层特性曲线反映出了有效的内部支撑压力和开切面的径向位移的关系。地层特性曲线可以通过将主应力的支撑力减小到0而得到。当支撑压力减小后，岩石的弹性特性达到支撑压力的临界值Pi，crit。如果有效的支撑压力低于临界支撑压力，就会出现塑性材料或软岩特性。可以根据岩石的时间特性考虑改变围岩强度和变形参数。5、巷道轴线方向的径向位移纵向变形曲线可以用来确定沿着巷道方向与到巷道开切面距离有关的径向位移。可以依据时间关系考虑到持续的进展速率。巷道断面处的进展速率和距离之间的时间关系：数值模拟中依据预松弛因子考虑到围岩的预变形。安装后的支护以及施工过程对位移沿巷道分布产生一定影响。变形速率的准则是储存在岩石中的能量。在莫尔库伦图中这中能量可以在地层特性曲线和支护特性曲线之间看出来。图4展现了一些采用不同分析方法得到的纵向变形曲线。沿着巷道的径向变形的不同曲线本质上彼此都不相同，尤其是支护设备安装的端头处。基于这个原因，支护利用率的计算取决于在此过程中将采用何种方法。图4. 比较各种理论的径向变形的发展： Panet 和Guenot (1982) , Corbetta和Nguyen-Minh (1992), Unlu 和Gercek (2003), Hoek (2007) , Vlachopoulos 和Diederichs (2009) , Pilgerstorfer 和Radoncic(2009).支护设备的安装及其施工过程同样会对巷道断面后的预变形和沿巷道的位移分布产生影响。一般而言这个过程是一个隐式的计算过程。在当前的文章中将考虑采用渐进的方法。6、支护特性曲线利用支护特性曲线座椅一个描述在收敛约束法（支护方式的承受能力）中描述支护方法的的工具。在预先给定的开切面边缘的位移中，支护特性曲线或真实的支撑力Pi 可以通过材料特性，如刚度Ksn ，可持续最大应力以及应变以以数学公式描述出来。在每一个计算步骤中，计算出的支撑压力Pi与必须小于Pi，ult的通过支护承载能力得到的可持续最大应力相比较。支护失效时，加剧的位移变化由三部分组成：支护设安装时产生的位移支护的弹性变形支护的塑性变形图5揭示了支护特性曲线和不同变形部分的示意图。图5 支持特性曲线示意图; Y轴是支撑压力和x轴显示的位移的支持。方程3和方程4显示了当位移为Ur时有效安装压力的计算过程和支护失效时总的位移的计算。有效支撑压力：Pi为支撑压力，Ksn为刚度，Ur为径向位移，r0为开掘半径。 Ur，ult，pl 是径向位移，Ur，s是安装支护设备时产生的位移，Ur，pl（塑性区的）径向位移，Ksn是刚度，Ur为径向位移，r0为开掘半径。6.1 锚喷支护在收敛约束法中锚喷支护的周面形状定义为一个圆环。混合时间的影响可以通过锚喷支护形成的周面的厚度计算出来。此外，锚喷方法和岩体的给定距离也可以考虑到。在现实中，锚喷支护呈现出与时间有关的强度和变形不断变化的材料特性。这包括，蠕变，松弛，收缩作用。在本文中，在收敛约束法中与时间有关的材料特性既舒伯特 (1988)、奥尔德林和奥雷斯特之后再次采用。锚喷支护后，锚喷支护壳有一恒定的厚度，此时系统刚度Ksn，spc可以用方程5计算。当喷浆厚度（e）远小于巷道半径（r）时，支护系统刚度可由方程6求得。锚喷支护壳的刚度（Gesta等人）。Ksn,spc是壳锚喷支护系统的刚度，Espc为锚喷支护的杨氏模量，v为泊松比，ra的锚喷支护壳的外表面半径，ri是锚喷支护壳的内面半径。锚喷支护壳的刚度当er（Gesta等人）。Ksn,spc是壳锚喷支护系统的刚度，Espc为锚喷支护的杨氏模量，v为泊松比，rSPC的锚喷支护壳的半径，eSPC是锚喷支护壳的厚度。（7）式说明如何计算最大有效支持压力。锚喷支护壳的最大有效支持压力（Gesta等人）。其中pi，ult，spc是锚喷支护壳的最大有效支持压力，Spc是喷射混凝土抗压强度，ra锚喷支护壳的外表面半径，ri是锚喷支护壳的内表面半径。任何时候计算与时间相关的支护材料的性能时，锚喷支护壳的支撑压力都要小于最大支撑压力，是这世间的推移，锚喷支护壳的最大支撑压力取决于材料强度的变化。鉴于此，Aldrian (1991)对以前的结论进行了修订。材料强度随时间的变化Aldrian (1991)。其中SpC（t）是喷射混凝土在时间t时的抗压强度，SpC（28D）喷浆后28天的抗压强度，t的单位是小时。由于抗压强度随时间变化，这就导致了杨氏模量也随时间而改变。这就影响到了支撑刚度，因为支撑压力在随时间变化。以下