CN111967080A - Tunnel mechanics model construction method based on uniform strength theory - Google Patents
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Abstract
The invention relates to a tunnel mechanical model construction method based on a uniform strength theory, which comprises the following steps: establishing a circular tunnel mechanical model; selecting a uniform strength criterion as a rock strength criterion; solving a surrounding rock stress field and a displacement field of the tunnel after excavation and immediate supporting on the basis of an elastoplasticity theory by applying a strength criterion; correcting the result by considering the interaction of the surrounding rock and the supporting structure to obtain the radius r of the plastic zone of the surrounding rock, which considers the rigidity of the supporting structure and the initial radial elastic displacement of the surrounding rock and is supported in a short time after excavation in the actual construction process of the tunnelpsAnd the displacement delta R of the tunnel wall completes the construction of the tunnel mechanical model. Aiming at the actual stress and construction characteristics of the tunnel when the tunnel is immediately supported after excavation is finished, the method is based on the stress characteristics of surrounding rocks and the mutual interaction of the surrounding rocks and the supporting structureThe action mechanism is set out, the actual stress condition and the construction process of the circular tunnel can be better reflected, and reference is further provided for related engineering design.
Description
Technical Field
The invention belongs to the field of research on tunnel mechanical models, and particularly relates to a tunnel mechanical model construction method based on a uniform strength theory.
Background
The new Olympic method is widely applied to tunnel construction at present, the surrounding rock is considered to be not only a load source but also a bearing structure, and the aim of supporting is to fully transfer the bearing capacity of the surrounding rock. Therefore, the tunnel construction theory and practice are rapidly developed, and a plurality of scholars also carry out deep research on the interaction mechanism of the surrounding rock-supporting structure.
A foreign scholar Kastner provides a calculation model of interaction between a circular tunnel supporting structure and surrounding rocks based on an ideal rock elastic-plastic model and a Mohr-Coulomb (M-C) strength criterion, and a foundation is laid for theoretical research of tunnel supporting. However, with the development of production practice and theoretical research, the theory is thought to not reflect the mechanical characteristics of the actual tunnel well, and the main problem is that the influence of the middle main stress and the surrounding rock-supporting structure interaction on the mechanical characteristics of the tunnel cannot be considered well.
In order to solve the problems, the invention patent of Liuhong rock and the like, namely a circular tunnel mechanics calculation method considering the interaction of surrounding rock and a supporting structure (patent application number: 201811568876.0), well considers the problems in the two aspects, and plays a good role in promoting the tunnel mechanics calculation method. However, the method still needs to be improved in four aspects, firstly in order to take into account the intermediate principal stressesThe Drucker-Prager intensity criterion is adopted, but when the principle is used for solving corresponding equilibrium differential equation and stress boundary condition, the apparent stress lode parameter mu is determinedσIs constant, however this is not suitable in some cases because μσIs a parameter associated with the stress state, which is generally not constant; secondly, the secondary stress state after the tunnel surrounding rock is excavated and the tertiary stress state after the supporting are independently calculated, which is possible for the tunnel engineering with obviously delayed supporting, but is not applicable for most tunnels which are immediately supported after the excavation in the actual engineering. Thirdly, in the aspect of displacement calculation of the inner boundary and the outer boundary of the surrounding rock plastic zone, the method provided by the patent is obviously different from the applied patents such as Liuhong rock, and the method provided by the patent is based on the assumption that the volume of the rock in the plastic zone is unchanged, so that the calculation process is more reasonable, clear and understandable, and more conforms to the actual situation. Fourthly, due to the limitation of the Drucker-Prager criterion, the Drucker-Prager criterion cannot reflect the strength difference of different meridians of the rock, so that the strength characteristic of the rock cannot be well described.
Therefore, although a relatively rich research result is obtained in the aspect of a circular tunnel mechanical model at present, the defects still exist, so that the patent proposes that on the basis of the applied patent, a rock uniform strength theory is adopted to consider the influence of the middle main stress, and meanwhile, the interaction of the surrounding rock and the supporting structure is considered, and finally, a tunnel mechanical model construction method based on the uniform strength theory and the interaction of the surrounding rock and the supporting structure is provided.
Disclosure of Invention
Aiming at the defects of the prior art, the technical problems to be solved by the invention are as follows: a construction method of a tunnel mechanical model based on a uniform strength theory and surrounding rock-supporting structure interaction is provided. The method aims at the actual stress and construction characteristics of the tunnel when the tunnel is immediately supported after excavation is completed, and starts from the stress characteristics of surrounding rocks and the interaction mechanism of the surrounding rock-supporting structure, the mechanical characteristics of the surrounding rocks are deeply researched by adopting the ideal elastoplasticity theory of the axial symmetry problem, so that the actual stress condition and the construction process of the round tunnel can be better reflected, and reference is further provided for related engineering design.
The technical scheme adopted by the invention for solving the technical problems is as follows: a tunnel mechanical model construction method based on a uniform strength theory is provided, and comprises the following steps:
the first step is as follows: establishing a circular tunnel mechanical model;
the second step is that: selecting a rock strength criterion: selecting a uniform strength criterion to replace a Mohr-Coulomb (M-C) criterion or a Drucker-Prager (D-P) criterion so as to consider the influence of intermediate principal stress and overcome the defect that the D-P criterion cannot reflect the difference of different meridian strengths of the rock;
the third step: solving a surrounding rock stress field and a displacement field of the tunnel which is excavated and immediately supported by the tunnel on the basis of an elastoplasticity theory by applying the strength criterion of the second step;
the fourth step: correcting the result of the third step by considering the interaction of the surrounding rock and the supporting structure to obtain the radius r of the plastic zone of the surrounding rock, which considers the rigidity of the supporting structure and the initial radial elastic displacement of the surrounding rock and is supported in a short time after excavation in the actual construction process of the tunnelpsAnd the displacement delta R of the tunnel wall completes the construction of the tunnel mechanical model.
According to the method, the corresponding support resistance p is provided by considering the initial radial elastic displacement of the surrounding rock and the interaction of the surrounding rock and the support structure in the actual construction process of the tunnelsIs no longer considered as a constant force, but a force that varies with the displacement of the surrounding rock. And on the basis of the calculation result of the third step, providing a tunnel mechanical model considering the middle main stress and the interaction of the surrounding rock and the supporting structure, and obtaining the change rule of the surrounding rock plastic zone radius, the tunnel surrounding rock inner wall displacement, the supporting force and the like along with the middle main stress coefficient and the tunnel surrounding rock initial radial elastic displacement from the tunnel mechanical model, wherein the surrounding rock plastic zone radius, the tunnel surrounding rock inner wall displacement and the supporting force are functions of the middle main stress coefficient and the tunnel surrounding rock initial radial elastic displacement.
The expression of the theoretical criterion of uniform strength is (expressed by polar coordinates):
wherein:
c、
respectively the cohesive force and the internal friction of the rock; b is the middle principal stress coefficient; sigmaθ、σrRespectively the circumferential and radial normal stresses of the circular section.
The method for analyzing the stress field and the displacement field of the surrounding rock after the tunnel excavation and immediate supporting is solved based on the elastoplasticity theory comprises the following steps:
radial stress p at elastoplastic interfacefIs represented by formula (12):
excavated and immediately supported plastic zone radius r of surrounding rockpsIs represented by formula (13):
radial displacement u at the outer boundary of the plastic regionBIs represented by formula (21):
the tunnel wall displacement Δ R is:
in the above-mentioned description,
psfor supporting forces acting on the tunnel boundaries, r0Is the tunnel radius, E is the rock modulus of elasticity, p0And taking the self weight of the overlying soil body at the center of the tunnel.
Surrounding rock plastic zone radius r considering support structure rigidity and construction timepsAnd the tunnel wall displacement Δ R is respectively represented by formula (23) and formula (24):
wherein u is0Initial radial elastic displacement of tunnel wall, k, before supporting constructionsThe rigidity of the supporting structure is improved; v is the Poisson's ratio;
solving the numerical solution of the tunnel wall displacement delta R by adopting an iterative method, and then substituting the solved delta R for the formula (23) to calculate the plastic zone radius R of the surrounding rockps。
The analysis process for solving the stress field and the displacement field after the tunnel is excavated and immediately supported based on the elastoplasticity theory is as follows:
after tunnel excavation and strut, can form plastic region and elastic zone outward from the tunnel center in proper order, carry out the analysis to plastic region and elastic zone respectively:
and (3) plastic zone stress analysis: let the plastic region radius be rpThe plastic zone rock should simultaneously satisfy the uniform strong theoretical criterion, the equilibrium differential equation and the stress boundary condition, namely, the formula (7):
in the formula, the superscript "p" represents the physical quantity of the plastic region; r is0Is the tunnel radius; r ispsIs the plastic zone radius; r is any one of the surrounding rocksDistance of the point to the center of the tunnel; p is a radical ofsIs the supporting force acting on the tunnel boundary.
Therefore, the stress component of the plastic zone surrounding rock is obtained as follows:
assuming a radial stress at the elastoplastic interface of pfAccording to the elastic mechanics, the stress of the elastic zone of the surrounding rock is as follows:
stress sigma at elastoplastic interfacerAnd σθIs continuous, so that r is equal to rpsCan be obtained by substituting formulae (8) to (11):
the formula (13) is the plastic zone radius of the excavated and immediately supported surrounding rock; pressure p0And taking the self weight of the overlying soil body at the center of the tunnel.
The displacement of the plastic zone of the surrounding rock is found below. As shown in FIG. 1, let the displacements at the inner and outer boundaries of the plastic region be Δ R and u, respectivelyB。
First, the displacement u at the elasto-plastic interface is determinedB. Let the stress at the elasto-plastic interface be σr epAnd σθ epThen both should satisfy both the elastic condition and the plastic condition. When the elasticity condition is satisfied, the following expressions (10) to (11) can be obtained:
σr ep+σθ ep＝2p0 (14)
when the plasticity condition, i.e., the uniform strength criterion shown in equation (6), is satisfied, then:
the radial stress σ at the elastoplastic boundary can be obtained by solving equations (14) to (15)r epComprises the following steps:
the solution of the plastic zone displacement is related to the plastic zone volume deformation assumption, and if the plastic zone volume is not changed, the following are provided:
since the tunnel problem can be regarded as a plane strain problem, the geometric equation is:
can be substituted by the formula (17):
integrating and utilizing deformation coordination condition on elastic plastic zone interface to obtain radial displacement of plastic zone
For (because of the axial symmetry problem, the hoop displacement is 0):
wherein: e and ν are respectively the rock elastic modulus and Poisson's ratio.
Changing r to rpsAnd formula (16) can be substituted for formula (20) to obtain a radial displacement u at the outer boundary of the plastic regionBComprises the following steps:
the tunnel wall displacement Δ R is solved below. Since it has been assumed herein that the volume of the rock mass remains unchanged during deformation, i.e. the volume of the plastic zone rock before deformation is equal to the volume of the rock after deformation. This gives:
π(rps 2-r0 2)＝π[(rps-uB)2-(r0-ΔR)2] (22)
formula (21) is substituted by formula (22) and the formula is arranged to obtain:
substituting formula (13) for formula (23) yields:
(ΔR)2-2r0ΔR+r0 2B＝0 (24)
wherein:
solving the formula (24) to obtain the displacement Δ R of the tunnel wall as:
however, the above process does not take the surrounding rock-supporting structure interaction into good consideration, so the following is a process of correcting the above calculation result by taking the surrounding rock-supporting structure interaction into consideration:
assuming radial bracing resistance p provided by the bracing structuresLinear with its radial displacement at the inner diameter of the tunnel, Δ R (this assumption is reasonable since the bracing structure is generally stiff and Δ R is generally small):
ps＝ks·ΔR (26)
wherein: k is a radical ofsTo support the structural rigidity. Since only the radial equipartition bracing resistance is considered here, ksOnly the compressive (tensile) stiffness of the supporting structure.
The supporting structure is generally applied after the tunnel is excavated, so that the surrounding rock has certain initial displacement, but the patent considers that the surrounding rock is immediately supported after the tunnel is excavated, so that the displacement of the surrounding rock is considered to be small and in an elastic range in the situation, and the surrounding rock is considered to have small initial radial elastic displacement (set as u)0) Supporting resistance p of the supporting structuresThe relationship to its radial displacement can be expressed as:
ps＝ks·(ΔR-u0) (27)
then, the formula (27) is respectively substituted into the formula (13) and the formula (25) to obtain the plastic zone radius r of the surrounding rock considering the rigidity of the supporting structure and the initial radial elastic displacement of the surrounding rockpsAnd the tunnel wall displacement delta R is respectively as follows:
wherein u is0The initial radial elastic displacement of the tunnel wall before the support structure is constructed is based on the requirement of surrounding rock protection, so that the support structure is generally required to surround the tunnelMaximum elastic deformation of the surrounding rock of the tunnel before rock maximum elastic deformation is applied and before support (u)e)maxComprises the following steps:
therefore, 0 ≦ u may be obtained0≦(ue)max。
Since it is difficult to find the analytical solution because both ends of the equation contain Δ R, equation (29) can find the numerical solution by an iterative method, and then calculate the surrounding rock plastic region radius R by substituting the Δ R found in equation (28)ps。
The excavation and immediate supporting or the supporting within a short time after the excavation refers to the condition that the supporting is carried out before a secondary stress field is not formed after the excavation, and the supporting is generally carried out within minutes to hours according to the characteristics of surrounding rocks and different construction methods.
Compared with the prior art, the invention has the beneficial effects that:
(1) the invention introduces the uniform strength theory into the construction of the tunnel mechanical model, considers the influence of the intermediate main stress on the displacement and the plastic zone of the tunnel surrounding rock by introducing the uniform strength theory rule, overcomes the defect that the D-P rule can not reflect the different meridian strength differences of the rock, and is more suitable for the actual tunnel engineering. The intermediate principal stress is expressed as a function of the first principal stress, the third principal stress and the parameter b through the parameter b, so that the influence of the intermediate principal stress can be reflected through the parameter b, a tunnel surrounding rock plastic zone calculation formula considering the intermediate principal stress, namely formula (13), is obtained, and the result shows that when the intermediate principal stress is considered, the radius of the surrounding rock plastic zone is reduced to different degrees.
(2) As mentioned above, according to the different construction time and action mechanism of the supporting structure, the invention considers that the supporting structure can be immediately applied in a short time after tunnel excavation, the supporting force provided by the supporting structure is not constant, but is related to the interaction of surrounding rock and the supporting structure, the displacement of the inner wall and the outer wall of the plastic zone is determined under the condition of assuming that the volume is unchanged, and further the invention proposes to consider the initial surrounding rock of the tunnel before supportingTunnel mechanics model of radial elastic displacement and supporting structure rigidity, wherein supporting force psThe method is a function of the rigidity of the supporting structure and the displacement of the surrounding rock, and the fact that the supporting is timely is shown when the initial radial elastic displacement of the surrounding rock before supporting is small, otherwise, the supporting lags obviously. The method is particularly suitable for the situation that a plurality of tunnels are actually supported soon after excavation, namely the supporting is started when the stress of the surrounding rock is not balanced after the tunnel excavation, and the secondary stress state and the tertiary stress state of the surrounding rock do not need to be considered separately.
(3) When the initial elastic displacement of surrounding rocks before the supporting structure is constructed and the interaction of the surrounding rocks and the supporting structure are considered, the tunnel surrounding rock plastic area calculated by the method is gradually reduced along with the increase of the middle main stress coefficient b from 0 to 1, which shows that the middle main stress has influence on the surrounding rock plastic area.
(4) The invention researches the initial radial elastic displacement u of the surrounding rock0Rigidity k of supporting structuresInternal angle of friction of rock
And stress p of the original rock0And the like on the radial displacement of the inner wall of the tunnel surrounding rock, the radius of a plastic zone and the supporting force. The tunnel mechanical model construction method based on the unified strength theory and the surrounding rock-supporting structure interaction is found to be capable of reflecting the mechanical characteristics of the tunnel surrounding rock more objectively, and a new thought is provided for the mechanical analysis of the circular tunnel.
(5) The method considers the principal stress based on the uniform strength theory, overcomes the defect that the Drucker-Prager criterion cannot consider the rock strength difference on different meridians, and can objectively reflect the real mechanical characteristics of the rock; and meanwhile, a method capable of calculating the displacement of the inner wall and the outer wall of the plastic region of the tunnel more simply is adopted, so that the calculation result is more in line with the actual situation.
Description of the drawings:
FIG. 1 circular tunnel calculation model;
FIG. 2 is the influence of the central principal stress on the plastic zone of the tunnel surrounding rock;
FIG. 3(a) is the effect of initial radial elastic displacement of tunnel surrounding rock on the plastic zone;
FIG. 3(b) is the influence of the initial radial elastic displacement of the tunnel surrounding rock on the displacement of the tunnel wall;
FIG. 3(c) is the influence of the initial radial elastic displacement of the tunnel surrounding rock on the supporting force;
FIG. 4(a) shows the rigidity k of the supporting structuresThe effect on the plastic zone;
FIG. 4(b) shows the rigidity k of the supporting structuresInfluence on tunnel wall displacement;
FIG. 4(c) shows the rigidity k of the supporting structuresInfluence on the supporting force;
FIG. 5(a) shows the initial ground stress p0The effect on the plastic zone;
FIG. 5(b) shows the initial ground stress p0Influence on tunnel wall displacement;
FIG. 5(c) shows the initial ground stress p0Influence on the supporting force;
FIG. 6(a) shows the internal friction angle of rock
The effect on the plastic zone;
FIG. 6(b) shows the internal friction angle of the rock
Influence on tunnel wall displacement;
FIG. 6(c) shows the internal friction angle of the rock
Influence on the supporting force;
the specific implementation mode is as follows:
the present invention is further described with reference to the following drawings and examples, which should not be construed as limiting the scope of the present invention.
The invention relates to a tunnel mechanics model construction method based on a uniform strength theory, which comprises the following steps:
(1) and establishing a circular tunnel mechanical model.
The calculation model of the circular tunnel is shown in FIG. 1Assuming that the surrounding rock is a homogeneous isotropic body, and assuming that the periphery of the tunnel is uniformly pressurized, the pressure p0And taking the self weight gamma h of the overlying soil body at the center of the tunnel, wherein gamma is the rock weight and h is the burial depth of the center point of the section of the tunnel. Tunnels are generally linear structures, i.e., the dimension in the length direction is much larger than the cross-sectional dimension, and thus can be considered a planar strain problem. Therefore, due to the symmetry of the tunnel structure and the load, the problem is a typical axial symmetry problem, so that polar coordinates can be adopted for solving the problem conveniently. Let sigmaθ、σr、σz、θ、r、zThe stress and the strain are respectively the annular stress, the radial stress and the axial stress of a circular section, and the stress satisfies sigmaθ＞σz＞σr. Due to tunnel excavation, additional stress is generated on surrounding rocks, when the additional stress of the surrounding rocks exceeds the strength of rocks, a plastic area is generated in the surrounding rocks, and an elastic area is still arranged outside the plastic area. For this purpose, the physical quantities of the elastic and plastic regions are denoted by "e" and "p", respectively. U in FIG. 1BAnd delta R is radial displacement generated from the outer boundary and the inner boundary of the surrounding rock plastic region to the center of the tunnel after the tunnel is excavated; r is0Is the tunnel radius; r ispsIs the plastic zone radius; r is the distance from any point in the surrounding rock to the center of the tunnel; p is a radical ofsIs the supporting force acting on the tunnel boundary.
(2) And selecting an intensity criterion.
After the tunnel is excavated, the stress of the surrounding rock is adjusted, if the adjusted stress meets the plastic yield condition of the rock, the surrounding rock enters a plastic zone, and the range of the plastic zone is a key parameter for tunnel support design, so that the reasonable determination of the surrounding rock is very important. In order to judge whether the surrounding rock enters a plastic state, a corresponding strength criterion needs to be adopted, and different rock strength criteria can also result in different surrounding rock plastic zones. As mentioned above, the M-C criterion can well reflect the nature of compression-shear failure of geotechnical materials, but fails to reflect the sigma2And thus cannot explain the phenomenon of rock yield failure at high confining and hydrostatic pressures. While the D-P criterion takes sigma into account well2But does not reflect the rock differencesThe difference in meridian intensity. The application adopts a uniform strength theory applicable to rock-soil materials.
If set to σ1、σ2And σ3Respectively 3 principal stresses of the unit body, and σ1≥σ2≥σ3Here, the compressive stress is taken to be positive. The theory of uniform strength considers that when the unit body is in a failure state, the main stress should meet the following conditions:
firstly, when
② when
In the formula: c.
respectively the cohesive force and the internal friction angle of the rock; b is the intermediate principal stress coefficient and reflects the sigma in rock failure2When b is 0 and 1, it corresponds to the M-C intensity criterion and the double shear intensity criterion, respectively.
For the planar strain problem, the longitudinal axial stress is σ when the material enters the plastic state2Assuming that the volume of the rock remains unchanged during deformation, i.e. the poisson ratio v of the rock is 0.5, the assumed condition of plane strain, i.e. the longitudinal strain, becomes zero, i.e.:
when in use
If so, the formula (3) is substituted for the formula (1) to obtain:
if it is provided with
Equation (4) can be written as:
for a circular roadway with polar coordinates, there are:
(3) and (5) analyzing the mechanical properties of the surrounding rock after the tunnel is excavated and immediately supported.
Assuming that the tunnel is immediately supported after excavation, the original stress state in the rock mass will be adjusted due to the influence of excavation and support. At the moment, partial surrounding rocks around the tunnel enter a plastic state, and a radius r is formed in the surrounding rocks due to symmetrical structure and loadpsThe plastic region of (a) is still the elastic region outside the plastic region, as shown in fig. 1.
Stress analysis of surrounding rock
Firstly, analyzing the stress of the rock in the plastic region, wherein the stress component of the plastic region simultaneously satisfies the uniform strength theory, the equilibrium differential equation and the stress boundary condition, namely:
therefore, the stress component in the plastic zone of the surrounding rock is obtained as follows:
assuming a radial stress at the elastoplastic interface of pfAccording to the elastic mechanics, the stress of the elastic zone of the surrounding rock is as follows:
stress sigma at elastoplastic interfacerAnd σθIs continuous, so that r is equal to rpsCan be obtained by substituting formulae (8) to (11):
and the formula (13) is the plastic zone radius of the surrounding rock after excavation and immediate supporting.
Displacement of plastic region
As shown in FIG. 1, let the displacements at the inner and outer boundaries of the plastic region be Δ R and u, respectivelyB。
First, the displacement u at the elasto-plastic interface is determinedB. Let the stress at the elasto-plastic interface be σr epAnd σθ epThen both should satisfy both the elastic condition and the plastic condition. When the elasticity condition is satisfied, the following expressions (10) to (11) can be obtained:
σr ep+σθ ep＝2p0 (14)
when the plasticity condition, i.e., the uniform strength criterion shown in equation (6), is satisfied, then:
the radial stress σ at the elastoplastic boundary can be obtained by solving equations (14) to (15)r epComprises the following steps:
the solution of the plastic zone displacement is related to the plastic zone volume deformation assumption, and if the plastic zone volume is not changed, the following are provided:
since the tunnel problem can be regarded as a plane strain problem, the geometric equation is:
can be substituted by the formula (17):
integrating and utilizing deformation coordination condition on elastic plastic zone interface to obtain radial displacement of plastic zone
Is (circumferential displacement of 0):
changing r to rpsAnd formula (16) can be substituted for formula (20) to obtain a radial displacement u at the outer boundary of the plastic regionBComprises the following steps:
the tunnel wall displacement Δ R is solved below. Since it has been assumed herein that the volume of the rock mass remains unchanged during deformation, i.e. the volume of the plastic zone rock before deformation is equal to the volume of the rock after deformation. This gives:
π(rps 2-r0 2)＝π[(rps-uB)2-(r0-ΔR)2] (22)
formula (21) is substituted by formula (22) and the formula is arranged to obtain:
substituting formula (13) for formula (23) yields:
(ΔR)2-2r0ΔR+r0 2B＝0 (24)
wherein:
solving the formula (24) to obtain the displacement Δ R of the tunnel wall as:
(4) correction of the above-mentioned calculation results taking into account the surrounding rock-supporting structure interaction
The conventional method for tunnel support mainly comprises concrete support, anchor bolt support or a combined support structure thereof and the like, and the deformation of the concrete or steel support structure can be considered as linear elasticity relative to the mechanical property of surrounding rocks. Meanwhile, the supporting structure and the tunnel enclosure are assumed to be in the deformation process of the tunnel enclosure rockThe rock contact is tight and there is no relative displacement, so the total radial displacement of the support structure is equal to the radial displacement at the inner diameter of the tunnel. If the radial supporting resistance p provided by the supporting structure is assumedsIn linear relation to its radial displacement Δ R at the inner diameter of the tunnel, then:
ps＝ks·ΔR (26)
wherein: k is a radical ofsTo support the structural rigidity. Since only the radial equipartition bracing resistance is considered here, ksOnly the compressive (tensile) stiffness of the supporting structure.
Because the supporting structure is generally applied after the tunnel is excavated, a certain initial radial elastic displacement (set as u) of the surrounding rock occurs0) Supporting resistance p of the supporting structuresThe relationship to its radial displacement can be expressed as:
ps＝ks·(ΔR-u0) (27)
then, the formula (27) is respectively substituted into the formula (13) and the formula (25) to obtain the plastic zone radius r of the surrounding rock considering the rigidity of the supporting structure and the initial radial elastic displacement of the surrounding rockpsAnd the tunnel wall displacement delta R is respectively as follows:
wherein u is0The initial radial elastic displacement of the tunnel cave wall before the supporting structure is applied is based on the requirement of surrounding rock protection, so that the supporting structure is generally required to be applied before the tunnel surrounding rock generates the maximum elastic deformation, and the maximum elastic deformation of the tunnel surrounding rock before the supporting is applied is as follows:
therefore, 0 ≦ u may be obtained0≦(ue)max。
Since it is difficult to find the analytical solution because both ends of the equation contain Δ R, equation (29) can find the numerical solution by an iterative method, and then calculate the surrounding rock plastic region radius R by substituting the Δ R found in equation (28)ps。
Example 1
The calculation parameters of a certain deeply buried circular tunnel are as follows: radius r of tunnel03m, original rock stress p015 MPa; the elastic modulus E of the rock is 10GPa, the Poisson ratio v is 0.25, the cohesive force c is 1.0MPa, and the internal friction angle
Rigidity k of supporting structures150 MPa/m. Then from equation (30) one can derive: (u)e)max0.01125m, so that the initial radial elastic displacement of the tunnel wall before supporting is taken as u00.005 m. The mechanical properties of the tunnel surrounding rock under the supporting condition are researched.
Plastic zone of wall rock
By using rps/r0To indicate the size of the plastic zone, the radius of the plastic zone of the surrounding rock, which can be determined by the method herein, equation (28), is shown in fig. 2.
According to the calculation result, the following results are obtained: the radius of the surrounding rock plastic zone decreases with the increase of the value of the intermediate principal stress coefficient b, and when b is equal to 0 and 1, rps/r01.408 and 1.266, respectively, the surrounding rock plastic zone was reduced by 10.09%. At the same time, for comparative analysis, the calculation results are also given without support, i.e. k in equation (28)s·(ΔR-u0) Is 0. It is understood that when b is 0 or 1, r isps/r01.551 and 1.335, respectively, the surrounding rock plastic zone was reduced by 13.93%. Thus, it can be seen that: (1) the middle main stress has an influence on the plastic area of the surrounding rock, the influence amplitude is about 10-15%, and the influence amplitude is related to the size of the tunnel and the mechanical property of the surrounding rock; (2) the influence degree of the middle main stress on the plastic zone of the surrounding rock is closely related to the stress state of the surrounding rock. When the tunnel is supported, the radial stress of the surrounding rock at the radius of the tunnel is not zero, the surrounding rock is in a three-dimensional stress state, and the influence of the middle main stress on the plastic zone of the surrounding rock is small; when there is noneDuring supporting, the radial stress of the surrounding rock at the radius of the tunnel is zero, the surrounding rock at the radius is in a two-way stress state, and the influence of the middle main stress on the plastic area of the surrounding rock is large. This shows that the influence degree of the intermediate principal stress on the plastic zone of the surrounding rock is closely related to the stress state of the surrounding rock.
(II) analysis of parameter sensitivity
The initial radial elastic displacement u of the tunnel wall is discussed below using a parameter sensitivity analysis (i.e., only one of the parameters is changed at a time, and the other parameters are not changed)0Rigidity k of supporting structuresInitial ground stress p0And internal friction angle of rock
Equal pair of relative plastic zones r of surrounding rockps/r0Tunnel wall displacement delta R and supporting force psThe influence of (c).
(1) Initial radial elastic displacement u of tunnel wall0Influence on the calculation results
Taking initial radial elastic displacement u of tunnel wall 00, 0.002, 0.004, 0.006 and 0.008m, respectively, and the results of the calculations are shown in fig. 3(a) -3 (c), and it can be seen that: first from u0When u is viewed from the influence on the plastic zone of surrounding rock0And a certain time, the surrounding rock plastic area is gradually reduced along with the increase of b, which shows that the surrounding rock plastic area can be effectively reduced by the middle main stress, and the stability of the surrounding rock plastic area is further improved. With u0The plastic area of the tunnel surrounding rock is gradually increased, and the increasing amplitude of the plastic area of the tunnel surrounding rock is gradually increased. When u is 1, for example0When the plastic area r of the tunnel surrounding rock is gradually increased from 0m to 0.002m, 0.004m, 0.006m and 0.008mps/r0The value is gradually increased from 1.180 to 1.212, 1.246, 1.285 and 1.331, and the increase amplitudes are respectively 2.71%, 2.81%, 3.13% and 3.58%. This shows that although some initial elastic deformation of the surrounding rock can be allowed, it should be limited to a certain range, otherwise, too large initial elastic deformation of the surrounding rock would lead to rapid increase of the plastic zone of the surrounding rock, and further lead to damage of the surrounding rock. ② the second subordinate0When u is viewed from the influence on the displacement of the tunnel wall0At a certain time, with the increase of b, the tunnelThe tunnel wall displacement is gradually reduced, which shows that the tunnel wall displacement can be effectively reduced by the intermediate main stress. With u0The displacement of the tunnel wall is gradually increased, and the increase range is gradually increased. Therefore, in order to ensure that the tunnel has enough clearance, the tunnel should be supported as soon as possible, and the initial radial elastic displacement of the tunnel wall is reduced. ③ finally from u0When u is viewed as the influence on the supporting force0At a certain moment, the supporting force is gradually reduced along with the increase of the b, which shows that when the influence of the intermediate main stress is considered, the supporting engineering amount can be reduced, namely the intermediate main stress can improve the stability of the surrounding rock. And with u0The supporting force is gradually reduced, which shows that the supporting work amount can be effectively reduced if the surrounding rock is allowed to generate a certain initial radial elastic displacement. But the surrounding rock plastic area is correspondingly increased, and the tunnel section clearance is reduced, so that comprehensive consideration is needed.
(2) Rigidity k of supporting structuresInfluence on the calculation results
Rigidity k of support structuresRespectively 50, 100, 150, 200 and 250MPa/m, and the calculation results are shown in FIGS. 4(a) to 4(c), and it can be seen that: first from ksWhen k is considered to influence the plastic area of the surrounding rocksAnd a certain time, the surrounding rock plastic area is gradually reduced along with the increase of b, which shows that the surrounding rock plastic area can be effectively reduced by the middle main stress, and the stability of the surrounding rock plastic area is further improved. With ksThe plastic area of the tunnel surrounding rock is gradually reduced, and the reduction amplitude of the plastic area of the tunnel surrounding rock is gradually reduced. When k is 1, for examplesWhen the plastic area range r of the tunnel surrounding rock is gradually increased from 50MPa/m to 100, 150, 200 and 250MPa/mps/r0The reduction is gradually reduced from 1.304 to 1.283, 1.266, 1.250 and 1.241, and the reduction amplitude is 1.61%, 1.33%, 1.26% and 0.72%, respectively. This shows that although the surrounding rock plastic zone can be reduced by improving the rigidity of the supporting structure, when the rigidity of the supporting structure is increased to a certain degree, the influence on the surrounding rock plastic zone is not obvious. Second from ksWhen k is considered to influence the displacement of the tunnel wallsAt a certain time, the displacement of the tunnel wall is gradually reduced along with the increase of b, which shows that the intermediate main stress can be effectively reducedAnd (5) displacing the tunnel wall. And with ksThe displacement of the tunnel wall is gradually reduced, and the reduction amplitude is gradually reduced. ③ Final Slave ksWhen k is the influence on the retaining forcesAt a certain moment, the supporting force is gradually reduced along with the increase of the b, which shows that when the influence of the intermediate main stress is considered, the supporting engineering amount can be reduced, namely the intermediate main stress can improve the stability of the surrounding rock. And with ksThe supporting force is gradually increased, which shows that the improvement of the rigidity of the supporting structure can effectively bear the load of the surrounding rock and reduce the displacement and the plastic zone of the surrounding rock. Therefore, in the tunnel with higher safety requirement, a high-rigidity supporting structure can be considered for supporting.
(3) Initial ground stress p0(initial geostress is also called virgin rock stress) on the calculation results
Taking initial ground stress p010, 12.5, 15, 17.5 and 20MPa, respectively, and the calculation results are shown in fig. 5(a) to 5(c), and it can be seen that: first from p0When p is considered to influence the plastic area of the surrounding rock0And a certain time, the surrounding rock plastic area is gradually reduced along with the increase of b, which shows that the surrounding rock plastic area can be effectively reduced by the middle main stress, and the stability of the surrounding rock plastic area is further improved. With p0The plastic area of the tunnel surrounding rock is gradually increased, and the increase amplitude of the plastic area is gradually reduced. When b is 1, for example, p0When the plastic area r of the tunnel surrounding rock is gradually increased from 10MPa to 12.5, 15, 17.5 and 20MPa/mps/r0The voltage gradually increases from 1.236 to 1.252, 1.266, 1.275 and 1.281, and the increase is 1.29%, 1.12%, 0.71% and 0.47%, respectively. This shows that although an increase in stress initially will result in an increase in the surrounding rock plastic zone, when it is increased to a certain extent, its effect on the surrounding rock plastic zone is not significant. ② second from p0When p is considered to influence the displacement of the tunnel wall0At a certain time, the displacement of the tunnel wall is gradually reduced along with the increase of b, which shows that the displacement of the tunnel wall can be effectively reduced by the intermediate main stress. And with p0The displacement of the tunnel wall is gradually increased, which means that the larger the ground stress is, the larger the displacement of the surrounding rock is correspondingly. ③ finally from p0When p is considered to influence the supporting force0At a certain moment, the supporting force is gradually reduced along with the increase of the b, which shows that when the influence of the intermediate main stress is considered, the supporting engineering amount can be reduced, namely the intermediate main stress can improve the stability of the surrounding rock. And with p0The supporting force is gradually increased, which shows that when the initial stress is larger, correspondingly larger supporting force is needed to ensure the stability of the surrounding rock.
(4) Internal friction angle of rock
Influence on the calculation results
Since the rock internal friction angle and the cohesion belong to the shear strength parameters, the rock internal friction is only used as an example for illustration. Taking internal friction angle of rock
30 °, 32.5 °, 35 °, 37.5 ° and 40 °, respectively. As can be seen from the calculation results shown in fig. 6(a) to 6 (c): firstly, the following steps
The influence on the plastic area of the surrounding rock is as follows
And a certain time, the surrounding rock plastic area is gradually reduced along with the increase of b, which shows that the surrounding rock plastic area can be effectively reduced by the middle main stress, and the stability of the surrounding rock plastic area is further improved. With following
The plastic area of the tunnel surrounding rock is gradually reduced, and the reduction amplitude of the plastic area of the tunnel surrounding rock is gradually reduced. Take b as an example and 1 when
When the angle gradually increases from 30 degrees to 32.5 degrees, 35 degrees, 37.5 degrees and 40 degrees, the plastic region range r of the tunnel surrounding rockps/r0The current is gradually reduced from 1.446 to 1.380, 1.326, 1.277 and 1.236, and the reduction range is respectively 4.56 percent,3.91%, 3.70% and 3.21%. This shows that although the surrounding rock plastic region can be reduced by increasing the internal rock friction angle, the influence on the surrounding rock plastic region is not significant when the internal rock friction angle is increased to a certain degree. ② the second slave
When viewed from the influence on the displacement of the tunnel wall
At a certain time, the displacement of the tunnel wall is gradually reduced along with the increase of b, which shows that the displacement of the tunnel wall can be effectively reduced by the intermediate main stress. While following with
The displacement of the tunnel wall is gradually reduced, and the reduction amplitude is gradually reduced. ③ finally from
The influence on the supporting force is as follows
At a certain moment, the supporting force is gradually reduced along with the increase of the b, which shows that when the influence of the intermediate main stress is considered, the supporting engineering amount can be reduced, namely the intermediate main stress can improve the stability of the surrounding rock. While following with
The supporting force is gradually reduced, which shows that the strength of the rock is improved along with the increase of the internal friction angle of the rock, and the required supporting force is correspondingly reduced. Therefore, the rock strength can be improved by grouting and the like, and the supporting engineering quantity is reduced.
Nothing in this specification is said to apply to the prior art.
Claims (7)
1. A tunnel mechanics model construction method based on a uniform strength theory comprises the following steps:
the first step is as follows: establishing a circular tunnel mechanical model;
the second step is that: selecting a rock strength criterion: selecting a uniform strength criterion to replace a Mohr-Coulomb (M-C) or Drucker-Prager (D-P) criterion so as to consider the influence of the intermediate principal stress and reflect the difference of different meridian strengths of the rock;
the third step: solving a surrounding rock stress field and a displacement field of the tunnel which is excavated and immediately supported by the tunnel on the basis of an elastoplasticity theory by applying the strength criterion of the second step;
the fourth step: correcting the result of the third step by considering the interaction of the surrounding rock and the supporting structure to obtain the radius r of the plastic zone of the surrounding rock, which considers the rigidity of the supporting structure and the initial radial elastic displacement of the surrounding rock and is supported in a short time after excavation in the actual construction process of the tunnelpsAnd the displacement delta R of the tunnel wall completes the construction of the tunnel mechanical model.
2. The method of claim 1, wherein the unified intensity criteria is expressed by:
wherein σθ、σrRespectively the circumferential and radial positive stresses under a polar coordinate system;
c、
respectively the cohesive force and the internal friction angle of the rock; b is the intermediate main stress coefficient, and b is more than or equal to 0 and less than or equal to 1.
3. The calculation method according to claim 2, wherein the analysis of the stress field and the displacement field of the surrounding rock after the tunnel excavation and immediate supporting is solved based on the elasto-plastic theory comprises the following steps:
radial stress p at elastoplastic interfacefIs represented by formula (12):
excavated and immediately supported plastic zone radius r of surrounding rockpsIs represented by formula (13):
radial displacement u at the outer boundary of the plastic regionBIs represented by formula (21):
the tunnel wall displacement Δ R is:
in the above-mentioned description,
psfor supporting forces acting on the tunnel boundaries, r0Is the tunnel radius, E is the rock modulus of elasticity, p0And taking the self weight of the overlying soil body at the center of the tunnel.
4. Construction method according to claim 3, characterized in that the plastic zone radius r of the surrounding rock is taken into account the rigidity of the supporting structure and the timing of the constructionpsAnd the tunnel wall displacement Δ R is respectively represented by formula (23) and formula (24):
wherein u is0Initial radial elastic displacement of tunnel wall, k, before supporting constructionsThe rigidity of the supporting structure is improved; v is the Poisson's ratio;
solving the numerical solution of the tunnel wall displacement delta R by adopting an iterative method, and then substituting the solved delta R for the formula (23) to calculate the plastic zone radius R of the surrounding rockps。
5. The construction method according to claim 3, wherein the concrete process of analyzing the stress field and the displacement field of the surrounding rock after the tunnel is excavated and immediately supported based on the elasto-plastic theory is as follows:
assuming that the tunnel is immediately supported after excavation, the original stress state in the rock mass will be adjusted due to the influence of excavation and support. At the moment, partial surrounding rocks around the tunnel enter a plastic state, and a radius r is formed in the surrounding rocks due to symmetrical structure and loadpsThe plastic zone of (a) is still an elastic zone outside the plastic zone;
firstly, analyzing the stress of the rock in the plastic region, wherein the stress component of the plastic region simultaneously satisfies the uniform strength theory, the equilibrium differential equation and the stress boundary condition, namely:
in the formula, the superscript "p" represents the physical quantity of the plastic region; r is0Is the tunnel radius; r ispsIs the plastic zone radius; r is the distance from any point in the surrounding rock to the center of the tunnel; p is a radical ofsThe supporting force acting on the tunnel boundary;
thus, the stress component of the plastic zone surrounding rock is obtained as follows:
assuming a radial stress at the elastoplastic interface of pfAccording to the elastic mechanics, the stress of the elastic zone of the surrounding rock is as follows:
stress sigma at elastoplastic interfacerAnd σθIs continuous, so that r is equal to rpsObtained by substituting formulae (8) to (11):
the formula (13) is the plastic zone radius of the excavated and immediately supported surrounding rock; pressure p0And taking the self weight of the overlying soil body at the center of the tunnel.
The displacement of the plastic zone of the surrounding rock is obtained, and the displacements at the inner and outer boundaries of the plastic zone are respectively set as delta R and uB；
First, the displacement u at the elasto-plastic interface is determinedB: let the stress at the elasto-plastic interface be σr epAnd σθ epThen both should satisfy the elastic condition and the plastic condition at the same time, when the elasticity is satisfiedUnder sexual conditions, the following formulae (10) to (11):
σr ep+σθ ep＝2p0 (14)
when the plasticity condition, i.e., the uniform strength criterion shown in equation (6), is satisfied, then:
solving the radial stress σ at the elastoplastic boundary of equations (14) to (15)r epComprises the following steps:
the solution of the plastic zone displacement is related to the assumption of plastic zone volume deformation, and if the plastic zone volume is not changed, the following steps are carried out:
the tunnel problem is considered to be a planar strain problem, so its geometric equation is:
substituting formula (17) to obtain:
integrating and obtaining the radial displacement of the plastic zone by using the deformation coordination condition on the elastic-plastic zone interface
Is (circumferential displacement of 0):
changing r to rpsAnd formula (16) is substituted for formula (20) to obtain a radial displacement u at the plastic zone outer boundaryBComprises the following steps:
solving the displacement delta R of the tunnel wall: assuming that the volume of the rock body is kept unchanged in the deformation process, namely the volume of the plastic zone rock before deformation is equal to the volume of the rock after deformation, the following results:
π(rps 2-r0 2)＝π[(rps-uB)2-(r0-ΔR)2] (22)
formula (21) is substituted by formula (22) and the formula is arranged to obtain:
substituting formula (13) for formula (23) yields:
(ΔR)2-2r0ΔR+r0 2B＝0 (24)
wherein:
solving the formula (24) to obtain the displacement Δ R of the tunnel wall as:
6. construction method according to claim 3, characterized in that a surrounding rock-supporting structure interaction is introduced to the above-mentionedPlastic zone radius rpsAnd the displacement delta R of the tunnel wall is corrected,
relative to the mechanical properties of surrounding rocks, the deformation of the concrete or steel supporting structure is considered to be linear elasticity; meanwhile, in the deformation process of the tunnel surrounding rock, the supporting structure is tightly contacted with the tunnel surrounding rock without relative displacement, and the total radial displacement of the supporting structure is equal to the radial displacement of the inner diameter of the tunnel; if the radial supporting resistance p provided by the supporting structure is assumedsIn linear relation to its radial displacement Δ R at the inner diameter of the tunnel, then:
ps＝ks·ΔR (21)
wherein: k is a radical ofsThe rigidity of the supporting structure is improved; since only the radial equipartition bracing resistance is considered here, ksOnly the compressive (tensile) stiffness of the supporting structure;
the supporting structure is applied within a short time of tunnel excavation, and at the moment, certain initial radial elastic displacement u of the surrounding rock occurs0Supporting resistance p of the supporting structuresThe relationship to its radial displacement is expressed as:
ps＝ks·(ΔR-u0) (22)。
7. the construction method according to claim 1, characterized in that it is determined from the established tunnel mechanics model: the influence degree of the middle main stress on the plastic zone of the surrounding rock is closely related to the stress state of the surrounding rock;
initial radial elastic displacement u0Influences are exerted on surrounding rock plastic areas, tunnel wall displacement and supporting force;
the rigidity of the supporting structure influences the plastic area of surrounding rocks, the displacement of the tunnel wall and the supporting force;
stress p of original rock0Influences are exerted on surrounding rock plastic areas, tunnel wall displacement and supporting force;
internal friction angle of rock
Has shadow on surrounding rock plastic zone, tunnel wall displacement and supporting forceAnd (6) sounding.
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