基于富集有限元方法的全锚固灌浆锚杆模型外文文献翻译

英文原文Modelling of fully grouted rock bolt based on enriched finite element methodDebasis Deb a,n, Kamal C. Das b（Department of Mining Engineering, India；b Department of Mathematics, India）Abstract：Analysis of mechanical behaviour of rock mass reinforced by fully grouted rock bolt is introduced based on the enriched finite element method (EFEM). A solid element intersected by a rock bolt along any arbitrary direction is defined asenrichedel ement and it has additional degrees of freedom at each node for estimating displacements and stresses in bolt rod. Numerical procedures of EFEM are developed to form enriched stiffness matrix using constitutive relations of rock mass, properties of bolt rod and grout material and orientation of the bolt. Decoupling of rock bolt and elasto-plastic behaviour of rock mass has been incorporated into the EFEM procedures. Released displacement of rock mass prior to bolt installation has also been considered in the procedures to provide realistic solution. The results of this method are verified with an analytical pull-out test model. In addition, a numerical example of a bolted tunnel is provided to demonstrate the efficacy of the proposed method for practical applications.Keywords: EFEM Grouted rock bolt Decoupling Released displacement MohrCoulomb yield criterion Supported tunnels1 .IntroductionRock bolts have been widely used as the primary support system to stabilize the rock masses around tunnel, underground mine galleries, slopes and others structures made in rock masses. In general, rock bolts reinforce rock masses through restraining the deformation within rock masses 1 and reduces the yield region around the excavation boundary. Axial load distribution along a fully grouted passive rock bolts shows that a neutral point exists on the bolt rod where shear stress at the interface between the bolt and grout material vanishes. The pickup length is defined as the length between free end from the tunnel boundary to the neutral point, and the shear stress along this bolt length drags the bolt towards the tunnel 1. The bolt length between the neutral point and the other free end of the bolt (inside the rock mass) is designated as anchor length and the developed shear stress drags the bolt towards the rock mass or in other words, anchors the bolt into the rock mass. Based on these concepts, shear stresses and axial loads developed along a bolt rod are analytically formulated by many researchers 1,2. Considering a bolt density factor, Indraratna and Kaiser 3 established an analytical model for the design of boltgrout interactions around a circular tunnel according to the elasto-plastic law. Cai et al. 4 derived an analytical solution of rock bolts for describing the interaction behaviours of rock bolt, grout material and rock mass using shear lag model (SLM). Brady and Lorig 5 numerically analyzed the interactions of boltgrout in MohrCoulomb media using the finite difference method (FDM) technique and showed that radial displacement and yielded region reduced due to the installation of grouted bolts around a circular tunnel.Limited works have been published for analysis of the interaction between fully grouted rock bolt installed in elasto-plastic rock mass using the finite element method. A few literatures deal with finite element procedure involving combination of decoupled rock bolts and elasto-plastic rock mass 6,7. In this study, the concept of enriched element 8 has been introduced in which a bolt can intersect a regular element at any arbitrary direction. Each node of an “enriched element” has additional degrees of freedom to determine displacements and stresses of the bolt rod. The stiffness of an enriched element comprises constitutive properties of rock mass, physical andmaterial properties of bolt and grout, orientation of bolt and bore hole diameter. The enriched element can be applied for analysis of elasto-plastic rock mass as well as decoupling of the bolt and/or both. Released displacement of rockmass, often signifying the delay in bolt installation, is an important factor of bolt performance. In this study, a released displacement factor has been incorporated into the EFEM procedures to provide realistic solution of displacements and stresses in rock mass as well as in bolt rod. MohrCoulomb yield criterion has been applied to determine the onset of yielding of rock mass. Decoupling at boltgrout interface has been modelled based on the peak shear strength of the grout material. For verification of the proposed method, results of pull-out test presented by Li and Stillborg 1 are compared with those obtained from the numerical models. This method has also been applied to evaluate rock mass behaviour and bolt performance for a circular bolted tunnel. Results presented in this paper show the efficacy of the proposed method, which can easily be applied for practical applications.2. Finite element formulation of boltgrout interactionConsider a body of rock mass, , with the boundary T = Tu U Tt is subjected to a traction t at boundary Tt and is constrained at boundary Tu (Fig. 1a). A fully grouted rock bolt of length Tb is installed at a free boundary as shown in Fig. 1(a) and (b). The displacement fields in the rock mass and in the bolt are represented by ut and ub, respectively. The axis, x0 denotes the longitudinal axis of the bolt. The potential energy developed in the body with a rock bolt can be expressed 9 as the sum of strain energy developed due to (i) straining of rock mass, (ii) extension of the bolt rod, (iii) shearing at the boltgrout interface and (iv) shear resistance provided by bolt rod in the transverse direction, and work done by traction at the boundary and by body force. Therefore, the total potential energy without body force leads to an expression: where er and rr are Cauchys strain and stress tensors in rock mass, , ,Ab is the cross sectionalarea of the bolt rod, Eb is the modulus of elasticity of bolt steel, Gb is the shear modulus of bolt steel, and k is the stiffness of grout material and generally depends on grout properties, geometry of the bolt, bore hole and spacing of bolts. The superscripts a and t denote the axial and transverse direction, respectively. The virtual work in terms of admissible displacement variations for rock mass, r, and admissible displacement variations for rock bolt, b, can be written aswhere are relative admissible displacement variations of rock and bolt at the boltgrout interface.2.1. Discretized enriched finite element equationsthat a rock bolt intersects a three-node triangular element at an angle a with the x axis, as shown in Fig. 2. The displacement vectors of rock mass and bolt at ith node are denoted by respectively. Two additional degrees of freedom are incorporated at every node to determine displacements of bolt rod in x and y directions, as shown in Fig. 2a Expressing variations r and b in terms of variations of nodal displacement ar and ab, respectively, as r= Nr ar , b =Nbab and gradient of variations as , the elemental equations of an enriched element are obtained from Eq. (2) aftertaking variations of gr and gb separately asWhere Nr and Nb represent matrices of the nodal shape functions for rock mass and bolt, respectively. The first term of Eq. (3a) equates with the stiffness matrix of rock mass. The second and third terms in Eq. (3a) and first and second terms in Eq. (3b) are formulated along x- y coordinate system and give the corresponding stiffness matrices of the bolt element. The stiffness matrix in x-y coordinate system can be written aswhere K is symmetric 8*8 square matrix and each components matrices are 4*4 square matrix. The component matrices are derived from Eq. (8) of 6, and are given below:where the straindisplacement matrices are Br= 1/Lb 0 -1 0 1 , Bb =1/Lb-1 0 1 0 and NTr = NTb =P1 0 P2 0 , where P1 and P2 denote the shape functions of inter section points (1) and (2) of the bolt element, as shown in Fig. 2b. In one-dimensional parametric space, shape function can be expressed as P1=1/2(1-r) and P2=1/2(1+r), where -1=r=1.Finally, in full discretized form, the combined stiffness matrix of rockbolt element as given in Eq. (4) can be expressed by adding appropriate components of Eq. (5ad) as given below:where kb= EbAb/lb, kv = GbAb=lb and lb is the length of bolt in the enriched element. The parameter k in Eq. (5ad) is determined from loaddeformation curve of pull-out test results. If pull-out test data are not available then k can be estimated as k =Gg/ ln(rh/rb) 1,4, where Gg is the shear modulus of grout material, rh is the radius of bore hole and rb is the radius of bolt rod. The relation between k and kg is then kg =klb/3 = (lbGg/3ln(rh/rb).2.2. Transformation of stiffness matrixThe stiffness matrix given in Eq. (6) is transformed into global x-y coordinates system before it is added with the stiffness matrix of rock mass. First, the matrix K0 is transformed into x-y coordinate system asWhere Then, the matrix K is transformed into nodal space of the element as follows:Wheren is the number of nodes per element and Ni denotes the shape function matrix of intersection points as given below:Hence, the combined stiffness matrix of an enriched element is Robtained by adding the stiffness matrix of rock mass Kr = , where D is the constitutive matrix of rock mass, as3. MohrCoulomb yield criterion for rock massOnset of yielding of rock mass is evaluated using the Mohr Coulomb criterion:where 1 and 3 are the major and minor principal stresses,respectively, N= (1+sin/1-sin)=tan2(/4+/2) is called triaxial factor, c is the cohesion of rock mass and j is the angle of internal friction. Numerical models are analyzed based on elastic-perfect plastic conditions assuming plane strain conditions. Cohesion and angle of internal friction of rock mass are reduced after each iteration based on accumulated plastic strain increments and the updated values are used in next iteration to determine the onset of yielding. NewtonRaphson iterative solution method is used to evaluate nodal displacements. The procedure for elastoplastic correction is well described elsewhere 1012, and hence it is omitted here.4. Decoupling between bolt rod and grout materialThe onset of yielding of grout material is expressed using the following yield function as (Fig. 3a)where t = kus/2rb and p denotes the peak shear strength of grout material. If F is positive, decoupling occurs at the boltgrout interface. In case of shear yielding (decoupling), the plastic component of displacement increment () is divided into elastic and plastic parts asIn this study, elastic-perfect plastic condition is assumed and hence The plastic displacement increment is added to the effective cumulative plastic displacement (useff) asFig. 3a shows how the shear stress falls and attains a residual, and Fig. 3b shows how the shear modulus falls in the grout material. Shear strength and shear modulus of grout material for next iteration are reduced based on useff of the current load step as shown schematically in Fig. 3b. Hence, the system of equations for solution of incremental displacement vector of an enriched element having material nonlinearity and decoupling at boltgrout interface is expressed aswhere is the updated elasto-plastic stressstrain matrix, fext is the applied external force vector, f rint is the updated internal reaction force vector of rock mass and f bint is the updated internal reaction force vector of bolt.The above numerical procedures are implemented using a three node triangular element and can easily be extended for higher order elements. As mentioned before, each enriched node has four degrees of freedom: two for the rock mass and the other two for bolt rod. Fig. 4 shows a finite element mesh intersected by a rock bolt. The enriched nodes are marked by hollow circles and others are regular nodes.An enriched element is numerically integrated with one Gauss point for CST and with two points Gauss integration rule for bolt element. The integration procedure of an enriched element is adopted based on the work proposed in 13 for cohesive rock joint analysis. It is found that the procedure is also well suited for grouted rock bolt.5. Displacement release of rock mass before bolt installationBolts are generally installed after (i) clearance of blast fumes,(ii) dressing of the face and its surrounding and (iii) mucking of blasted material. In a TBM face, however, bolt installation is much faster than conventional drilling and blasting face. Fig. 5 shows a schematic diagram of a typical ground reaction curve and responsecurve of bolt support 14.In EFEM procedure, a parameter, released displacement (ud) is introduced as 100(u0=uwr )%, where urw denotes radial displacement of rock mass without bolt support, to analyze the effect of bolt installation time on the performance of rock bolts. The parameter ud is a unique user defined parameter applied to each bolt. A ud=0% would mean the bolt is installed immediately after the excavation is made and ud=100% signifies that bolt is installed after the entire rock displacement has been released. The bolt stiffness is added to the enriched element if rock displacement at the bolt head exceeds the corresponding ud.6. Verification of EFEM using pull-out test resultThe pull-out test is intended to measure the short-term strength of rock bolt anchor 15. The strength is measured by pulling the head of a bolt using an instrument that also records head displacement as a function of applied load. Li and Stillborg 1 have derived an analytical solution of axial force and shear stress as a function of applied pull load, 1.24 m or every 15 interval from the horizontal. Rock bolts are of 4.0 m length and 20.0 mm diameter. The geomechanical and MohrCoulomb parameters of the rock masses are as follows: tunnel radius ra=4.75 m, Youngs modulus Er=5 GPa, Poissons ratio Vr=0.25, cohesion before yielding c=1.5 MPa, residual value of cohesion cr=0.5 MPa, angle of internal friction =30, residual value of angle of internal friction r=20, dilation angle =20, residual value dilation angle r=10 and tensile strength t=1 MPa. The rock bolt parameters are taken to be radius rb=10 mm, bore hole radius rh=20 mm, Youngs modulus of bolt steel Eb=210 GPa, shear modulus Gb=84 GPa and circumferential spacing=1.24 m. The grout material properties are taken as shear modulus Gg=1 GPa, residual values of the shear modulus Gg r =0.1 GPa, peak shear strength p=1.5 MPa and residual shear strength p r = 1 MPa.Stress and displacement distributions of elasto-plastic rock mass around the tunnel boundary, axial force and shear stress response of rock bolts and the effect of ud on bolt performance are discussed in details. For comparison purpose, a model without bolts is also analyzed and the corresponding displacements and stresses in the rock mass are plotted wherever appropriate. The tunnel model without bolt support is referred later as no-bolt tunnel. The bolted tunnel model with and without decoupling options are termed as decoupled tunnel and coupled tunnel, respectively. For all models, released displacement of 40% and peak obtained from the analytical solution (Fig. 6a). The numerical model also accurately predicts the location of the peak shear stress for different pull-out loads (Fig. 6b). However, the magnitude of peak shear stresses predicted by EFEM is lower than those obtained from the analytical model. After decoupling of the bolt, the analytical model overestimates the peak shear stress since shear stress may not reach to the peak shear strength of the grout material once it yields, as predicted reasonably by EFEM.7. Numerical examples of a bolted circular tunnelA bolted circular tunnel of 4.75 m radius and subjected to a hydrostatic far-field stress (p0) of 8.0 MPa has been modelled using EFEM (Fig. 7a). Since the problem is axisymmetric, only one quarter of the tunnel has been studied. Models are made up with 2358 CST elements having 1254 nodes and are analyzed in plane strain condition considering MohrCoulomb yield criterion for rock mass. Fig. 7b shows the enlarged view near the tunnel boundary depicting six installed bolts at the circumferential spacing of grout shear strength of 1.5 MPa are considered. If any other values are used in the models, they are mentioned specifically in the paper.7.1. Reduction of plastic zone around the tunnelThe tangential and radial (confining) stresses are plotted with respect to ratio of radial distance from tunnel wall (r) and radius of tunnel (a) for no-bolt, coupled and decoupled tunnels, as shown in Fig. 8. The location of peak tangential stress is higher and closer to tunnel wall for coupled and decoupled tunnels as compared to nobolt tunnel. The confining stress is also higher for the former tunnels making them safer.The distributions of confining stress (s3) 0.5 minside the tunnel boundary (aa0 curve) and near neutral point for decoupled bolt, 1.3 minside (bb0 curve) are shown in Fig. 9a and b, respectively. The coupled bolts have increased the confinement in the rock mass noticeably as seen in the figures and also mentioned in 16. Fig. 10 shows the change in confining stress in bolted tunnels with respect to no-bolt tunnel. It is seen that about 52% increase in confining stress can be achieved if grout-bolt interfaces do not yield. However, if the grout yields, bolts cannot provide necessary confinement to the rock mass as evident from Fig. 10. For the given tunnel example having residual grout shear strength of 0.5 MPa, confining stress at the tunnel boundary may decrease sl