CN106844942A - A kind of optimization method of tunnel lining design - Google Patents

Abstract

The present invention provides a kind of optimization method of tunnel lining design, it is characterised in that comprise the following steps：Step 1：Based on transparent clouds design theory, the lining cutting maximum crack width of different designs scheme is calculated, the raw data set that calculating parameter and result of calculation are analyzed as subsequent mathematical；Step 2：Raw data set to step 1 is pre-processed, first by initial data standardization；Data exception monitoring is done to the data set after standardization using mahalanobis distance theory to process；Step 3：Using the stepwise regression method backward in multi-variate statistical analysis, the regression equation of control variables and design parameter is set up successively, choose final optimization pass equation of the maximum regression equation of significance test value as lining design；Mathematical optimization models of the present invention consider influence of the design parameter to engineering safety, and influence of the quantitative analysis Different Optimization design measure to raising lining cutting security, model simple understands, design efficiency is improved, with wide applicability.

Description

A kind of optimization method of tunnel lining design

Technical field

The invention belongs to hydraulic structure field of structural design, more particularly to a kind of optimization method of tunnel lining design.

Background technology

Hydraulic tunnel typically regards concrete lining as impermeable material when designing, but in rich water engineering-environment high, The processing mode often causes that the lining thickness of design is larger, or even is difficult to be received by engineering.With a large amount of high pressures or buried tunnel The extensive use in hole, above mentioned problem becomes increasingly conspicuous.Lining cutting is increasingly taken seriously by the theory that water penetration is designed, and many researchs are opened Beginning is put in lining cutting seepage water pressure by body force, studies its influence to seepage field and forces in lining.

However, transparent clouds design theory is typically more complicated, related scholar is mostly in single parameter to lining cutting security Influence aspect theoretical to transparent clouds launch research；Meanwhile, current main direction of studying mostly concentrates on design parameter to lining The qualitative analysis aspect of security is built, research conclusion can not be engineered personnel and directly use.Permeable lining in Practical Project Building design will often consider the combined influence of multiple design parameters, be that a typical optimization design based on quantitative study is asked Topic.

Therefore, the achievement in research for carrying out tunnel lining design is primarily present problems with：(1) many design parameters are not accounted for To the combined influence of lining cutting security；(2) have ignored practicality of the achievement in research in Practical Project；(3) not from quantitative point The angle of analysis takes out intuitively optimization design equation, and design efficiency is relatively low.

The content of the invention

For the problem that prior art is present, the present invention intends providing a kind of tunnel lining Thoughts on Optimized Design, with permeable lining Build based on theory, with lining cutting maximum crack width as control variables, tunnel lining is carried out with reference to multi-variate statistical analysis mathematical method The optimization design of block, the present invention is as follows to solve the technical scheme that problems of the prior art are used：

A kind of optimization method of tunnel lining design, it is characterised in that comprise the following steps：

Step 1, based on transparent clouds design theory, calculates the lining cutting maximum crack width of different designs scheme, will calculate The raw data set that parameter is analyzed with result of calculation as subsequent mathematical；

Step 2, due to each design parameter dimension disunity of step 1 gained raw data set and data area difference it is larger, Raw data set to step 1 is pre-processed, including following sub-step：

Step 2.1, data normalization treatment, initial data is mapped to the value of interval [0,1], and computing formula is as follows：

Wherein, x^*Be it is normalized after mapping value, x is raw value, x_minFor initial data concentrates every kind of index Minimum value, x_maxThe maximum of every kind of index is concentrated for initial data；

Step 2.2, does data exception monitoring to the data set after standardization and processes using mahalanobis distance theory, calculates Formula is as follows

Wherein, D²It is the mahalanobis distance of each sample, X is initial data (standardized data) matrix, G^-1It is covariance square The inverse matrix of battle array,It is element average value；

D²Bigger, probability density is smaller, and when greatly to a certain extent, the probability density of its distribution will be small to certain journey Degree, so that the scope for being no longer belong to normal point outside this scope；In α confidence levels, critical value can be distributed with F and be determined：

Wherein, l is the dimension of sample, and n is sample size, and F is distributed as the conventional distribution that mathematical statistics does parameter Estimation, by Normal distribution is developed, and α is confidence level, and mathematical statistics belongs to term, and when sign estimates overall with sample, sample is close to totally Possibility, generally represented with 1- α.

Step 3, using the stepwise regression method backward in multi-variate statistical analysis, sets up control variables and design parameter successively Regression equation, choose final optimization pass equation of the maximum regression equation of significance test value as lining design；It is wherein main Including：

Step 3.1, sets up the full model of multivariate statistical regression.It is provided with m independent variable x₁,x₂...x_m, using m independent variable The model of fitting is referred to as full model, i.e.,

Y=β₀+β₁x₁+...+β_mx_m+ε (4)

Wherein, y is dependent variable, β₀It is constant, β₁,β₂...β_mIt is regression coefficient, ε is regression error；

Step 3.2, progressively rejects independent variable, sets up the multivariate statistical regression model of remaining independent variable；If from this m variable In leave out independent variable x_k(k=1...m), at this moment it is referred to as subtracting model with m-1 independent variable model of fit, i.e.,

y_k=β₀+β₁x₁+...+β_k-1x_k-1+β_k+1x_k+1+...+β_mx_m+ε (5)

Step 3.3, the significant difference value that model asks for fit equation is subtracted to full model with all,TakeFull model conspicuousness Difference value is F₀, reject independent variable x_kThe significant difference value for subtracting model afterwards is F_k(rejecting x_kWhen, this subtracts model and has rejected x₁, x₂...x_k-1K-1 independent variable), then the significant difference value for subtracting model is represented by

Wherein,

y_iIt is dependent variable reality Value, i=1...n；

F₀Ask method and F_kIt is identical；

Step 3.4, seeks the maximum F in all significant difference values_max, F_maxCorresponding multiple regression equation is finally Optimization design equation.

Lining cutting maximum crack width computational theory is as follows in step 1：

Lining cutting shows as armored concrete associated working before cracking, and as internal water pressure is continuously increased, lining cutting ring should Change is continuously increased, it is assumed that after lining cutting hoop strain is more than the critical axial tensile atrain of concrete, Lining cracks, i.e.,：

Wherein：ε_θLIt is lining cutting hoop strain；f_tIt is the design tensile strength of lining concrete；E_CIt is concrete elastic mould Amount；

As internal water pressure is continuously increased in tunnel running, lining cutting shows as first ftractureing, then departs from country rock, lining cutting With country rock without departing from when, hoop strain is：

Wherein：ΔP_wIt is tunnel hydrostatic pressure, Δ P_w=P_i-P₀, P_iIt is internal water pressure, P₀It is underground hydrostatic pressure；v_mFor Country rock Poisson's ratio；E_mIt is surrounding rock deformation monitor；E_sIt is reinforcing bar elastic modelling quantity；A is the mean radius after tunnel lining；t_sIt is reinforcing bar Equivalent thickness；

Now, the equivalent coefficient of permeability of lining cutting is：

Wherein：k_LIt is lining cutting equivalent coefficient of permeability；γ_wIt is the unit weight of water；U is the coefficient of dynamic viscosity of water；Between S is crack Away from；Wherein, the computational methods of fracture interval S are as follows：

Wherein：D is tension reinforcement diameter；Ρ is the ratio of reinforcement；α₁It is design factor (α during axial tension₁0.16 is taken, the data Can pass through《Hydraulic tunnel design specification》Inquiry)；V is that (spiral v takes 0.7 to the coefficient relevant with rebar surface shape, and this is Number represents rebar surface shape, because reinforcing bar can be divided into plain bar, spiral etc. by surface configuration, coefficient inquiry can be joined Examine《Hydraulic tunnel design specification》)；

When internal water pressure is larger, the water penetration of lining cutting is also larger, and lining cutting departs from country rock, according to waterpower continuity equation q_L=q_m：

Wherein：q_LIt is the seepage discharge that lining cutting is flowed into by lining cutting inner surface；q_mIt is that oozing for country rock is flowed into by lining cutting outer surface Flow；k_mIt is country rock infiltration coefficient；h₀It is underground flood peak；h_iIt is water head in tunnel；h_w1It is the outer water head of lining cutting；h₀It is underground Head；B is the external diameter of lining cutting；a₁It is the internal diameter of lining cutting；L=2h₀；

Can be obtained by formula (11)：

Again due to Δ P_L=γ_wΔh_L, Δ P_w=γ_wΔh_w, substituting into can obtain the net external water pressure of lining cutting in formula (12) and be：

Hydrostatic pressure passes to the thrust of country rock through lining cutting：

Wherein,

The condition that lining cutting departs from country rock is：

ΔP_w1>P_r (15)

Formula (13), (14) are substituted into formula (15) and obtained：

After lining cutting departs from country rock, the maximum hoop strain that lining cutting is born is：

Bring above formula result into formula (9) and obtain lining cutting maximum infiltration coefficient：

By incremental theory, lining cutting hoop strain is：

Bring formula (17)~formula (19) into formula (9), the lining cutting equivalent coefficient of permeability after lining cutting departs from country rock can be obtained：

Formula (20) is substituted into formula (13), the quantitative relationship of lining cutting parameters after lining cutting departs from country rock is obtained：

Wherein,

Δ P is tried to achieve by formula (21)_L/ΔP_w, substitute into formula (19), you can the ring for obtaining lining cutting when lining cutting departs from country rock should Become ε_θL, and then lining cutting maximum crack width w can be obtained_max, specifically refer to formula (22)：

w_max=2 ε_θLS (22)

The invention has the advantages that：

(1) based on tunnel lining design theory, the calculating data of different designs scheme are made full use of, with reference to rigorous mathematics Analysis, takes out the optimization method of control variables and design parameter, with tight logicality；

(2) mathematical optimization models consider influence of the design parameter to engineering safety, and quantitative analysis Different Optimization sets Influence of the meter measure to raising lining cutting security, achievement in research can directly for engineering is actually directly used；

(3) mathematical optimization models are concise, can be received by numerous engineers and technicians, improve design efficiency, have Wide applicability.

Brief description of the drawings

Fig. 1 is the technology of the present invention flow chart；

Fig. 2 is tunnel seepage calculation sketch of the present invention；

Fig. 3 is lining cutting maximum crack width of the present invention and equivalent coefficient of permeability Changing Pattern；

Fig. 4 is lining cutting maximum crack width of the present invention with ratio of reinforcement change curve；

Fig. 5 is lining cutting maximum crack width of the present invention with ratio of reinforcement change curve；

H in Fig. 2_i、h_w1、h₀It is followed successively by water head in tunnel, the outer water head of lining cutting and underground flood peak；P_i、P_w1、P₀It is followed successively by Tunnel internal water pressure, external water pressure on lining, underground hydrostatic pressure；ΔP_i、ΔP_w1、ΔP_LFor tunnel hydrostatic pressure, lining cutting are only outer The hydrostatic pressure that water pressure and lining cutting are undertaken, γ_wIt is the unit weight of water.

Specific embodiment

Below by embodiment, and with reference to accompanying drawing, technical scheme is described in further detail, such as Fig. 1- A kind of a kind of optimization method of tunnel lining design shown in 2, specific embodiment is as follows：

(1) design conditions

The present embodiment is based on certain pumping energy storage power station project reality, and its diversion tunnel is circular cross section, designs internal water pressure 5.0MPa, level of ground water 186m high.Lining concrete is C25, tunnel excavation hair hole radius 2.15m, infiltration coefficient 1 × 10^-9m/ S, deformation modulus 28GPa.Surrounding rock category is IV classes (Poisson's ratio 0.25 of country rock, infiltration coefficient 2~5 × 10^-6m/s).Reinforcing bar is leaned on Lining concrete disposed inboard, thickness of protection tier 50mm.Reinforcement scheme uses 6 Φ 22 (ratio of reinforcement 0.57%), initial lining Build thickness 0.4m.

In the runtime, heavy-duty pressure tunnel internal water pressure increases, and Lining cracks, infiltration coefficient increases.The larger stream that period causes Amount production run of the loss often to power station brings larger economic loss, therefore, it is necessary to study lining cutting maximum crack width and The Evolution of equivalent coefficient of permeability, as a result as shown in Figure 3.

From figure 3, it can be seen that tunnel is in water-filling running, with the increase of internal water pressure, lining cutting seepage field table Reveal more obvious three phase change：(1) lining cutting is not ftractureed；(2) Lining cracks but with country rock without departing from；(3) lining cutting with enclose Rock departs from.After cracking, maximum crack width constantly increases with lining cutting infiltration coefficient with the increase of internal water pressure for lining cutting, (2) phase change is most fast, and opposite (3) phase change speed substantially slows down.After this is primarily due to lining cutting and country rock disengaging, lining cutting Interior external water pressure difference increases smaller, and the development speed of fracture width is limited to a certain extent.

Lining cracks when internal water pressure reaches about 1.95MPa, lining cutting departs from country rock when reaching about 2.38MPa.Depart from lotus Load differs smaller with cracking load, only 0.43MPa.After illustrating Lining cracks, lining cutting will depart from rapidly with country rock, endochronic model The influence of effect is larger.When internal water pressure reaches 2.52MPa, Lining Crack width reaches 0.25mm, will more than crack-control design Ask.It is therefore desirable to adjusted design scheme.

(2) design procedure

Step 1, based on transparent clouds design theory, adjusted design parameter constructs the raw data set of multi-variate statistical analysis.

Keep the reinforcement scheme of 6 Φ 22 constant, research country rock Penetration Signature k_m=1~50 × 10^-7During m/s, lining thickness Influence to crack-control design.Result of calculation is as shown in Figure 4.

From fig. 4, it can be seen that keeping country rock osmotic coefficient k_mIt is constant, only increase lining thickness, maximum crack width only has Reduce by a small margin.Such as work as k_m=5 × 10^-6During m/s, lining thickness increases to 0.7m by 0.4m, and maximum crack width is only reduced 3%.And keep lining thickness constant, and improve the impervious characteristic of country rock, then the maximum crack that can largely reduce lining cutting is wide Degree.Work as k_mFrom by 5 × 10^-6M/s is reduced to 1 × 10^-7During m/s, lining cutting maximum crack width is greatly lowered and far away from 0.25mm Below.Illustrating to take the measure of the single raising impervious characteristic of country rock can make transparent clouds meet the requirement of crack-control design, but hold Easily there is design situation more than needed, and reduction country rock Penetration Signature is difficult to be guaranteed in construction grouting quality, also brings along Cost and duration problem, it is therefore necessary to further seek the other measures of optimization design of lining design.

Keep lining thickness 0.4m constant, the ratio of reinforcement is increased under the conditions of research different surrounding rock Penetration Signature to crack-control design Influence.Result of calculation is as shown in Figure 5.

From fig. 5, it can be seen that the impervious characteristic for improving country rock can effectively improve lining cutting with placing of reinforced bar in lining rate is increased Security, and improve the impervious characteristic of country rock become apparent.From the angle of optimization design, combine and take the increase ratio of reinforcement to be enclosed with raising The measure of the impervious characteristic of rock, can be reduced to below 0.25mm by lining cutting maximum crack width, to ensure that lining cutting fully meets limit Split the requirement of design.

To sum up, in the optimization design that lining cutting limit is split, reduce the relatively optimal measure of maximum crack width, carry first The impervious characteristic of country rock high, next to that improving placing of reinforced bar in lining rate, can finally increase lining thickness as supplementary means.

In the qualitative concrete measure for having inquired into optimization design of lining design, the concrete thought of optimization design of lining design process is given Afterwards, this part will be based on above-mentioned calculating data, with the mathematical method of multiple stepwise regression, the research ratio of reinforcement, country rock infiltration system Number and influence of the lining thickness to lining cutting maximum crack width.The ratio of reinforcement takes 0.57%, 0.71%, 0.85%, 0.99%, country rock Infiltration coefficient takes 5 × 10^-6m/s、1×10^-6m/s、5×10^-7m/s、1×10^-7M/s, lining thickness take 0.4m, 0.5m, 0.6m, 0.7m corresponds to 64 groups of maximum crack width result of calculations of scheme as the raw data set of multi-variate statistical analysis.

Step 2, the theoretical exceptional value monitoring of mahalanobis distance is carried out to the data set after data standardization.

After standardization is done to 64 groups of raw data sets, then 64 groups of data are done with mahalanobis distance calculating, wherein geneva It is 4.31 apart from maximum, minimum value is 0.64.D_0.01 ²(4,60)=3.87, in 64 groups of samples, have 11 groups to belong to exceptional value.Will Abnormality value removing, regression analysis is done with 53 groups of data.

Step 3, using the stepwise regression method backward in multi-variate statistical analysis, sets up control variables and design parameter successively Regression equation, choose final optimization pass equation of the maximum regression equation of significance test value as lining design.

By maximum crack width w_max, ratio of reinforcement ρ, country rock osmotic coefficient k_mLater data are standardized with lining thickness t to remember It is w_max ^*、ρ^*、k_m ^*And t^*.Using MATLAB, progressively Fitting Toolbox carries out multiple regression analysis.ρ is chosen first^*、k_m ^*And t^*It is right w_max ^*Carry out multiple regression.Shown in regression result such as formula (23).

w_max ^*=-0.2791 ρ^*+0.6733lgk_m ^*-0.01418t^*+0.1827 (23)

Wherein, regression coefficient R²=0.9575, conspicuousness F test values F₁=451, residual mean square (RMS) 0.05711.Multiple regression Analysis fitting degree is preferable, and F₁>F_0.01(4,49)=3.72, regression effect is notable.

Method gradually rejects variable backward, first rejects t^*, choose ρ^*、k_m ^*To w_max ^*Carry out multiple regression.Regression result such as formula (24) shown in：

w_max ^*=-0.2782 ρ^*+0.6720lgk_m ^*+0.1756 (24)

Wherein, R²=0.9571, conspicuousness F test values F₂=682, residual mean square (RMS) 0.05689.F₂>F₁, and R²It is equal with residual error Side's value is basically identical with formula (1), illustrates using the ratio of reinforcement and country rock infiltration coefficient to maximum crack width multiple regression effect Significantly.Then k is only chosen_m ^*To w_max ^*Carry out simple regression.Shown in regression result such as formula (25).

w_max ^*=0.6613lgk_m ^*+0.03648 (25)

Wherein, R²=0.8080, fitting degree is very poor, and illustrating can not be wide to lining cutting maximum crack from single interpretation The influence of degree.

To sum up, multiple regression of the ratio of reinforcement with country rock infiltration coefficient to lining cutting maximum crack width explains that effect preferably, is intended Conjunction degree is higher, conspicuousness degree highest.By standardized formula (23) reduction, obtain：

w_max=-0.1759 ρ+0.1049lgk_m+0.9706 (26)

Wherein, w_maxUnit is mm, and ρ units are %, k_mUnit is m/s.

As can be seen that the ratio of reinforcement often increases by 0.1%, lining cutting maximum crack width reduces 1.759 × 10-2mm, country rock infiltration Coefficient often reduces by 10 times, and lining cutting maximum crack width reduces 0.1049mm.

Protection scope of the present invention is not limited to the above embodiments, it is clear that those skilled in the art can be to this hair It is bright to carry out various changes and deform without deviating from scope and spirit of the present invention.If these are changed and deformation belongs to power of the present invention In the range of profit requirement and its equivalent technologies, then the intent of the present invention also comprising these changes and deforms.

Claims (2)

1. the optimization method that a kind of tunnel lining is designed, it is characterised in that comprise the following steps：

Step 1：Based on transparent clouds design theory, the lining cutting maximum crack width of different designs scheme is calculated, by calculating parameter The raw data set analyzed as subsequent mathematical with result of calculation；

Step 2：Raw data set to step 1 is pre-processed, including following sub-step：

Step 2.1, data normalization treatment, initial data is mapped to the value of interval [0,1], and computing formula is as follows：

$x^{*}=\frac{x-x_{min}}{x_{max}-x_{min}}---\left(1\right)$

Wherein, x^*Be it is normalized after mapping value, x is raw value, x_minThe minimum of every kind of index is concentrated for initial data Value, x_maxThe maximum of every kind of index is concentrated for initial data；

Step 2.2, does data exception monitoring to the data set after standardization and processes using mahalanobis distance theory, computing formula It is as follows

$D^2={(X-\stackrel{\‾}{X})}^{\′}{G}^{-1}(X-\stackrel{\‾}{X})---\left(2\right)$

Wherein, D²It is the mahalanobis distance of each sample, X is initial data (standardized data) matrix, G^-1It is covariance matrix Inverse matrix,It is element average value；

D²Bigger, probability density is smaller, when greatly to a certain extent, its distribution probability density will it is small to a certain extent so that In the scope for being no longer belong to normal point outside this scope；In α confidence levels, critical value can be distributed with F and be determined：

${D_{\mathrm{\α}}}^2(l,n-l)=\frac{(n^2-1)}{(n-l)n}{F}_{\mathrm{\α}}(l,n-l)---\left(3\right)$

Wherein, l is the dimension of sample, and n is sample size, and F is distributed as the conventional distribution that mathematical statistics does parameter Estimation, by normal state Distribution evolution, α is confidence level；

Step 3：Using the stepwise regression method backward in multi-variate statistical analysis, returning for control variables and design parameter is set up successively Return equation, choose final optimization pass equation of the maximum regression equation of significance test value as lining design；Wherein mainly include：

Step 3.1, sets up the full model of multivariate statistical regression.It is provided with m independent variable x₁,x₂...x_m, it is fitted using m independent variable Model be referred to as full model, i.e.,

Y=β₀+β₁x₁+...+β_mx_m+ε (4)

Wherein, y is dependent variable, β₀It is constant, β₁,β₂...β_mIt is regression coefficient, ε is regression error；

Step 3.2, progressively rejects independent variable, sets up the multivariate statistical regression model of remaining independent variable；If being deleted from this m variable Remove independent variable x_k(k=1...m), at this moment it is referred to as subtracting model with m-1 independent variable model of fit, i.e.,

y_k=β₀+β₁x₁+...+β_k-1x_k-1+β_k+1x_k+1+...+β_mx_m+ε (5)

Step 3.3, the significant difference value that model asks for fit equation is subtracted to full model with all,TakeFull model significant difference It is F to be worth₀, reject independent variable x_kThe significant difference value for subtracting model afterwards is F_k, then the significant difference value for subtracting model can represent For：

$F_k=\frac{S_{Rk}}{S_{ek}}\·\frac{n-m-1}{m}---\left(6\right)$

Wherein,

y_iIt is dependent variable actual value, i= 1...n；

F₀Ask method and F_kIt is identical；

Step 3.4, seeks the maximum F in all significant difference values_max, F_maxCorresponding multiple regression equation is final excellent Change design equation.

2. the optimization method that a kind of tunnel lining as claimed in claim 1 is designed, it is characterised in that：Lining cutting in the step 1 Maximum crack width computational theory is as follows：

Lining cutting shows as armored concrete associated working before cracking, and as internal water pressure is continuously increased, lining cutting hoop strain is not It is disconnected to increase, it is assumed that after lining cutting hoop strain is more than the critical axial tensile atrain of concrete, Lining cracks, i.e.,：

${\mathrm{\ϵ}}_{\mathrm{\θ}L}>\frac{f_t}{E_C}---\left(7\right)$

Wherein：ε_θLIt is lining cutting hoop strain；f_tIt is the design tensile strength of lining concrete；E_CIt is modulus of elasticity of concrete；

As internal water pressure is continuously increased in tunnel running, lining cutting shows as first ftractureing, then with country rock depart from, lining cutting with enclose Rock without departing from when, hoop strain is：

${\mathrm{\ϵ}}_{\mathrm{\θ}L}=\frac{{\mathrm{\ΔP}}_w(1+v_m)}{E_m+E_s\frac{t_s}{a}(1+v_m)E_m}---\left(8\right)$

Wherein：ΔP_wIt is tunnel hydrostatic pressure, Δ P_w=P_i-P₀, P_iIt is internal water pressure, P₀It is underground hydrostatic pressure；v_mFor country rock is moored Pine ratio；E_mIt is surrounding rock deformation monitor；E_sIt is reinforcing bar elastic modelling quantity；A is the mean radius after tunnel lining；t_sIt is the equivalent thickness of reinforcing bar Degree；

Now, the equivalent coefficient of permeability of lining cutting is：

$k_L=\frac{{\mathrm{\γ}}_w}{12u}{{\mathrm{\ϵ}}_{\mathrm{\θ}L}}^3{S}^2---\left(9\right)$

Wherein：k_LIt is lining cutting equivalent coefficient of permeability；γ_wIt is the unit weight of water；U is the coefficient of dynamic viscosity of water；S is fracture interval；Its In, the computational methods of fracture interval S are as follows：

$S=(60+{\mathrm{\α}}_1\frac{d}{\mathrm{\ρ}})v---\left(10\right)$

Wherein：D is tension reinforcement diameter；Ρ is the ratio of reinforcement；α 1 is design factor；V is the coefficient relevant with rebar surface shape；

When internal water pressure is larger, the water penetration of lining cutting is also larger, and lining cutting departs from country rock, according to waterpower continuity equation q_L= q_m：

$\left\{\begin{array}{c}{q}_L=\frac{2{\mathrm{\πk}}_L\left(h_i-h_{w1}\right)}{\ln \left(\frac{b}{a_1}\right)}\\ q_m=\frac{2{\mathrm{\πk}}_m\left(h_{w1}-h_0\right)}{\ln \left(\frac{L}{b}\right)}\end{array}\right.---\left(11\right)$

Wherein：q_LIt is the seepage discharge that lining cutting is flowed into by lining cutting inner surface；q_mIt is the seepage flow that country rock is flowed into by lining cutting outer surface Amount；k_mIt is country rock infiltration coefficient；h₀It is underground flood peak；h_iIt is water head in tunnel；h_w1It is the outer water head of lining cutting；h₀It is underground water Head；B is the external diameter of lining cutting；a₁It is the internal diameter of lining cutting；L=2h₀；

Can be obtained by formula (11)：

$\frac{{\mathrm{\Δh}}_L}{{\mathrm{\Δh}}_w}=\frac{k_{m}ln\frac{b}{a_1}}{k_{L}ln\frac{L}{b}+k_{m}ln\frac{b}{a_1}}---\left(12\right)$

Again due to Δ P_L=γ_wΔh_L, Δ P_w=γ_wΔh_w, substituting into can obtain the net external water pressure of lining cutting in formula (12) and be：

${\mathrm{\ΔP}}_{w1}={\mathrm{\ΔP}}_w-{\mathrm{\ΔP}}_L=\frac{k_{L}ln\frac{L}{b}}{k_{L}ln\frac{L}{b}+k_{m}ln\frac{b}{a_1}}{\mathrm{\ΔP}}_w---\left(13\right)$

Hydrostatic pressure passes to the thrust of country rock through lining cutting：

$P_r=\frac{{\mathrm{\ΔP}}_w}{M}---\left(14\right)$

Wherein,

The condition that lining cutting departs from country rock is：

ΔP_w1>P_r (15)

Formula (13), (14) are substituted into formula (15) and obtained：

$\frac{k_L}{k_m}>\frac{{\mathrm{aE}}_{m}ln\frac{b}{a_1}}{E_s{t}_s(1+v_m)ln\frac{L}{b}}---\left(16\right)$

After lining cutting departs from country rock, the maximum hoop strain that lining cutting is born is：

${\mathrm{\ϵ}}_{\mathrm{\θ}0}=\frac{{\mathrm{\ΔP}}_{w}a}{E_s{t}_s}---\left(17\right)$

Bring above formula result into formula (9) and obtain lining cutting maximum infiltration coefficient：

$k_{L0}=\frac{{\mathrm{\γ}}_w}{12u}{{\mathrm{\ϵ}}^3}_{\mathrm{\θ}0}{S}^2---\left(18\right)$

By incremental theory, lining cutting hoop strain is：

${\mathrm{\ϵ}}_{\mathrm{\θ}L}=\frac{{\mathrm{\ΔP}}_{L}a}{E_s{t}_s}---\left(19\right)$

Bring formula (17)~formula (19) into formula (9), the lining cutting equivalent coefficient of permeability after lining cutting departs from country rock can be obtained：

$k_L={\left(\frac{{\mathrm{\ΔP}}_L}{{\mathrm{\ΔP}}_w}\right)}^3{k}_{L0}---\left(20\right)$

Formula (20) is substituted into formula (13), the quantitative relationship of lining cutting parameters after lining cutting departs from country rock is obtained：

$C{\left(\frac{{\mathrm{\ΔP}}_L}{{\mathrm{\ΔP}}_w}\right)}^4+\frac{{\mathrm{\ΔP}}_L}{{\mathrm{\ΔP}}_w}-1=0---\left(21\right)$

Wherein,

Δ P is tried to achieve by formula (21)_L/ΔP_w, substitute into formula (19), you can obtain the hoop strain ε of lining cutting when lining cutting departs from country rock_θL, And then lining cutting maximum crack width w can be obtained_max, specifically refer to formula (22)：

w_max=2 ε_θLS (22)。