錢營(yíng)孜煤2.4Mta新井設(shè)計(jì)【含CAD圖紙+文檔】
錢營(yíng)孜煤2.4Mta新井設(shè)計(jì)【含CAD圖紙+文檔】,含CAD圖紙+文檔,錢營(yíng)孜煤,mta,設(shè)計(jì),cad,圖紙,文檔
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Distinct analysis of fully grouted bolts around a circular tunnel considering the congruence of displacements between the bar and the rock
Abstract
This paper presents a new calculation procedure for the analysis and dimensioning of fully grouted radial bolts in tunnels. This procedure is based on the principle of congruency of the displacements, imposing the condition that the lengthening of the bolt, added to the displacements due to the shear strains in the bolt–rock interface, is equal to the expansion in the radial direction of the reinforced rock annulus. A comparison of the results obtained using this procedure with those obtained using the traditional numerical modelling method, which is more costly in computational terms, has led to satisfactory results.The application of the procedure to a real case has given results that are comparable with the in situ measurements. The proposed instrument is able to analyse the stress and strain state in rock in detail, in the presence of a bolting intervention, and to proceed with its correct dimensioning with the objective of improving the degree of stability of the rock around the tunnel.
1. Introduction
Fully grouted radial bolting is an intervention that is commonly used in the field of tunnelling in rock, as it is easy to carry out, reliable and efficacious in controlling the radial displacements of the tunnel walls and in improving the degree of stability of the rock around the tunnel. It is generally adopted in a systematic manner in rock masses with medium to good geomechanical quality (RMR varying from 35 to 70); in rock of optimal quality, spot bolting is instead used in order to stabilize any individual potentially unstable rock blocks.
This method involves the use of no pre-stressed rigid steel elements that are completely adherent to the medium into which they are inserted due to the mortar injected into the hole or through friction produced by the high contact pressures of the element on the hole walls ( swellex and split-set bolts ). They are arranged in a radial manner around the perimeter of the tunnel and are usually installed close to the excavation face. They are initially unloaded and it is only with the subsequent advancement of the face that they become loaded and they begin to function statically.
In civil infrastructure tunnels , where it is necessary to guarantee the functionality of the bolts for a sufficiently long time, fully grouted bolts are generally used . The cemented mortar usually has a water/ cement ratio of 0.3–0.35 and it has the duty of connecting the element to the surrounding rock and protecting it from the actions of corrosive agents .
Many different authors , starting from back in the 1980s, have studied the behaviour of passive radial bolts arranged in a systematic way around the perimeter of a circular tunnel [1–9] an d they have formulated both analytical and numeric al calculation method s that are able to evaluate the effect of the bolts on the stress and stra in behaviour of the rock around the tunnel. Unfortunately, however, the complexity of the problem and the necessity of always considering new aspects to adequately simulate the bolt–rock interaction have made such calculation methods difficult to use.Further more, most of the aforementioned methods refer to a continuous equivalent medium and do not allow the maximum force induced inside the bolts to be evaluated, a value that is necessary to obtain their correct dimensioning.
In this work, starting from a study of the results of an extensive parametrical analysis developed using a numer-ical calculation method and from experimental evidence in tunnels , the typical trend of the axial force along a bolt when the head of the bolt is constrained in a correct way on the tunnel walls, was first identified , from a qualitative point of view . Once the qualitative trend of the axial force was obtained, a calculation procedure was developed that is able to evaluate the stress and strain state in rock and the maximum force inside a bolt. For this purpose an iterative process was developed, comparing the lengthening of the bolt produced by the stresses induced on the inside of it with the radial expansion of the reinforced rock annulus around the tunnel. The lengthening of the bolt increases and the expansion of the reinforced annulus diminishes with an increase in the maximum force inside the bolt. The lengthening of the bolts and the expansion of the reinforced rock an nulus should be identical, less a difference due to the shear strains at the bolt–rock interface, due to congruency of the displacements. The set up calculation method, which has been implement ed in the Matlab programming language , was compared to a traditional numerical method that discretizes the continuous medium in great detail. The program can be accessed at http://staff. polito.it/pi erpaolo.orest e/. A comparison of the results of the eighty-one analyses that were performed has given satisfactory results. The application of the procedure to a real case has also given results that are very similar to the results of the measurements that were carried out.
2. Stress–strain analysis of the rock around a deep circular tunnel
Considering a homogeneous and isotropic material initially subjected to a lithostatic pressure of hydrostatic type it is possible to determine the stress and strains inside the medium and in particular on the tunnel perimeter supposed circular when the radial pressure acting on the perimeter is reduced to a value equal to p [10] . Another interesting result is given by the thickness of the plastic zone around the tunnel.
2.1. Stress and strain state in an elastic field
From the equilibrium of the forces in the medium outside a deep circular tunnel it is possible to obtain:
Where and are the radial and circumferential stresses (positive compression stresses ), r is the distance from the centre of the tunnel (radial coordinate) and is the circumferential angle.
It is possible to write the following two expressions of the radial and circumferential strains in function of the existing stress state according to the theory of elasticity in radial symmetry conditions:
where and are the variations of the stress state(radial and circumferential, respectively), with respects to the initial condition, before the excavation of the tunnel, when and are the radial and circumferential strains, respectively (positive compression strains); E and v are the elastic modulus and the Poisson ratio of the ground.
The following two equations, which connect the circumferential and radial strains to the radial displacements u (considered positive going towards the centre of the tunnel), are considered valid due to the congruency of the strains:
Resolving the differential system of equations constituted by Eqs. (1)–(4) , it is possible to obtain the circumferential and radial stresses and strains in an elastic field in closed form for the problem of the circular tunnel for r >R, where R is the radius for which the radial stressis known:
2.2. Stress and strain state in a plastic field
In the plastic field, the connection between the circumferential stresses (maximum principal stress) and radial stresses (minimum principal stress) is given by the strength criterion in residual conditions. Considering the Mohr–Coulomb criterion, we obtain:
and and are the residual cohesion and friction angle of the ground.
Inserting Eq. (8) into Eq. (1) and integrating and posing the boundary conditions (the tunnel radius ) we obtain:
where p is the radial pressure applied to the perimeter of the tunnel.
Eq. (9) describe the radial stresses in the ground inside the plastic zone. The circumferential stress calculated with the strength criterion on the plastic radius (the distance from the centre of the tunnel at which the plastic behaviour changes to elastic behaviour) should be equal to the circumferential stress calculated with Eq. (5) for the ground outside, where an elastic behaviour is present . From this condition it is possible to obtain the radial stress in correspondence to the plastic radius:
The ground shows a plastic behaviour for r and an elastic behaviour for r >.
The strains in the plastic field are made up of an elastic component and a plastic component. The relationship between the plastic components of the strains can be directly given, as a first approximation in a simplified approach , by the flow rule and it is therefore a function of the dilatancy angle :
The adopted simplified approach widespread in the technical literature [10] , allows us to obtain a closed- form solution for the elasto-plastic problem around a deep circular tunnel, without taking the unloading process of the rock mass into account [11] . For rock masses having an RMR value greater than 35 (low values of the dilatancy angle and large values of the rock mass strength ), the error in the displacement values using the simplified approach is estimated [11] to be lower than 5% compared to the results obtained by the rigorous app roach [12] . For
this reason, in the following the simplified approach to determine the plastic strains in the plastic zone is used.
The entity of the plastic components can therefore be expressed as a function of the plastic multiplier :
where an d are the elastic an d plastic components of the radial strains, and are the elastic and plastic components of the circumferential strains, is the dilatancy angle and is the plastic multiplier.
Summing Eq. (13) with Eq. (12) multiplied by the coefficient k, substituting the total strains and with Eqs. (3) and (4), and the radial and circumferential stresses with Eqs. (9) and (8), and placing the boundary condition for , we obtain
2.3. Evaluation of the tunnel stability conditions
It is possible to evaluate the stability conditions of a deep tunnel by analysing the radial displacement of the tunnel perimeter and the plastic radius . The latter is calculated using Eq. (11) : if , a plastic area forms around the tunnel, otherwise, if the medium around the tunnel has elastic behaviour throughout. In the first case, the radial displacement on the tunnel perimeter ( ) is simply calculated using Eq. (7) placing and :
In the second case, Eq. (14) is used imposing .
Conditions in which the radial displacement on the tunnel perimeter exceeds 1% of the tunnel radius and/or the thickness of the plastic zone () exceeds the tunnel radius itself, are usually considered detrimental for the stability. When there is a certain risk of instability of the rock around the tunnel, a rock reinforcement system can be used. The most commonly used rock reinforcement method at present is that of reinforcement with radial fully grouted bolts.
3. The numerical simulation of the rock reinforcement around the tunnel using radial fully grouted bolts
A 2D numerical model has been set up with a finite difference method (FLAC calculation code) to analyse the behaviour of a bolting system around a deep circular tunnel. The model is made up of 22,500 quadrilateral elements: only a quarter of the tunnel was considered thanks to the symmetry of the analysed problem . All the boundaries of the model are artificial: the lithostatic pressures that were present before constructing the tunnel were applied to these boundaries. The bolts were simulated with mono-dimensional elements connected to the bi-dimensional elements of the model through opportune constraints that were able to consider the shear stiffness of the bolt–rock interface. A total of three stages were foreseen in the calculation ( Table 1) after the initial reproduction of the stress conditions of the site. Each of these phases simulates the excavation and also the rock reinforcement operations. The tunnel excavation phases are considered in the model by reducing the internal pressure on the tunnel perimeter [13] . An analysis of the results was conducted in the last stage of the modelling when the internal pressure on the tunnel perimeter was nil (final calculation condition with the tunnel face far from the studied section ).
Three different types of rock quality (RMR index), three radius values and three tunnel depths were considered in order to evaluate the efficacy of the fully grouted radial bolts to stabilize the tunnel perimeter in some of the most frequent geometric, stress and geomechanical conditions that can be encountered in practice. The number of bolts at the tunnel perimeter was varied in order to obtain a bolting density that fell within the interval of values considered to be technically possible. The bolt length was defined in
function of the tunnel radius : 3 m for =2 , 4.5 m for =3.5, 6 m for =5 . A 32 mm bolt diameter was assumed in the numerical calculation. The bolt constraint on the tunnel wall has be en considered perfect by making the node of the bolt element coincide with the node of the element mesh: this situation occurs when the anchorage plate of the bolt head is inserted in a correct manner and has an adequate stiffness. Eighty- one numerical analyses were developed for each case that could be obtained permuting the values reported in Table 2.
The rock mass geomechanical parameters for the three RMR indexes examined are shown in Tables 3 and 4. The elastic modulus of the rock mass was estimated as function of RMR [14,15] . A Mo hr–Coulomb strength criterium was adopted; the cohesion, friction angle and tensile strength of the rock mass were obtained making a linear approximation of the Hoek–Brown criterium for a confining stress equal to the mean radial stress in the plastic zone (the m and s strength parameters were estimated directly by the RMR index [14] , while the material constant was put equal to 12 and the uniaxial compressive strength of the intact rock was considered equal to 45 MPa for RMR =35–60 MPa for RMR =50–75 MPa for RMR =65). The Mohr–Coulomb strength parameters turn out to be therefore a function of the RMR index and lithostatic pressure .
The most interesting results of the calculation concern the radial displacement s of the tunnel perimeter, the extension of the plastic zone around the tunnel and the maximum traction force induced inside the bolts . The radial displacements and the plastic zones around the tunnel are shown in Fig. 1 as an example for the case of a tunnel with a radius of 3.5 m, a lithostatic stress of 4 MPa , RMR=35 and the minimum foreseen bolting density. The radial wall displacements and the maximum
traction forces in the bolts obtained from the numerical calculation for a tunnel radius of 3.5 m are shown in Fig. 2.
As expected, a reduction occurs of the radial displacements of the tunnel walls with a diminishing of the spacing and an increase in the RMR geomechanical quality index, while an increase occurs with an increase in the lithostatic pressure (connected to the depth of the tunnel) and also in the tunnel radius . The maximum force in the bolts is influenced in the same way by the four fundamental parameters (bolt spacing , tunnel radius , lithostatic pressure and RMR ind ex of the rock mass ). It can be noticed how a systematic bolting is useful to contain tunnel convergence for an RMR index of the rock be low 55–60 and for lithostatic pressures above 1 M Pa (a value that corresponds to a depth of about 40 m). On the other hand , for lithostatic pressures of 7 MPa , the maximum admissible force for each bolt (equal to 0.25–0 .30 MN, considering the provisional purpose of the intervention) is exceeded for RMR values be low 45–50. On the basis of the performed parametric analysis, the systematic radial bolting would therefore result to be not very useful for low depth tunnels (=1 MPa ) excavated in good or very good rock masses (RM R >450); it cannot operate alone due to the very high values of the traction forces in the bolts for deep tunnels ( =7 MPa ) excavated in medium to poor rock mass es (RMR<50). The spot bolting, used in the stabilization of rock blocks around a tunnel, has to be considered instead where fractured rock masses can isolate potentially unstable rock blocks. The present paper deals with only systematic bolting without considering spot bolting .
The numerical calculation has also made it possible to determine the behaviour of the axial force along the bolts. It generally assumes the same shape, which is reported in Fig. 3 for the case of a tunnel with a radius of 3.5 m, a lithostatic stress of 4 MPa, an RMR =35 and the minimum foreseen bolting density: the maximum value is reached at a certain distance from the tunnel wall (equal to about 1/6 of the length of the bolt) and the force in correspondence to the head of the bolt is equal to about 2/3 of the maximum force.
4. The presence of the fully g routed radial bolts in the stress–strain analysis
In order to study the contribution of the bolting on the stress and strain state of the rock around the tunnel perimeter, it is necessary to resort the finite difference method so as to be able to integrate the differential equations that govern the problem. The portion of rock into which the bolts are inserted is divided into thin concentric annulus of constant thickness ( Fig. 4 ). The calculation proceeds from the terminal point of the bolts (at a distance +L from the centre of the tunnel, with L the length of the bolts) up to the tunnel perimeter, evaluating the stresses , strains and radial displacements in the rock for each annulus.
4.1. Discretization of the stress and strain state in an elastic field
Eq. (1), pertaining to the equilibrium of the forces, can be expressed in incremental terms refer ring to a generic thin annulus and then Eq. (15) can be obtained:
Eq. (2), rewritten for the generic annulus, becomes on the internal surface,
4.2. Discretization of the stress and strain state in a plastic field
In the plastic zone Eq. (15) can be associated to Eq. (8) expressed for the internal surface (i ) of a generic thin annulus. Resolving the system, it is also possible to obtain the radial and circumferential stresses on the internal surface ( i ) in function of only the stress state that exists on the external surface (i +l):
Eq. (27) allows the radial displacement on the internal surface of the generic annulus to be obtained, if the radial displacement on the external surface and the previously defined existing stress state are known.
4.3. Influence of the bolts on the stress and strain state of the rock mass
The mathematical treatment that is given hereafter refers to fully grouted bolts , the widespread passive bolts in civil underground works .From the analysis of the in situ measurements and the results of the bi-dimensional numerical calculation in the performed parametric study it is possible to note how the traction force along the bolts can be represented, with a certain approximation, by a simple two-line graphic( Fig. 5), with the maximum value at the distance ( L /2 from the tunnel wall, where L is the length of the bolt and is the distance of the maximum from the mean point of the bolt. The axial force in the bolt head () has the value of a certain aliquot of the maximum force: , when a stiff face plate tightened to the bolt head (perfect constraint) is foreseen on the bolt head. When a perfect constraint is not guaranteed the efficiency of the bolt is drastically reduced and can be nil.
The shear stresses on the lateral surface of the bolts considering the simplified behaviour of the traction force of Fig. 5 are simply given by Eqs. (28) and (29) in function of the value of the maximum traction force .In zone A, that nearest the tunnel perimeter, the bolt blocks the surrounding rock which would tend to show higher radial displacements and for t
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