CN104408022B - The method for solving of shallow tunnel surrouding rock stress and the Explicit Analytic Solutions of displacement - Google Patents
The method for solving of shallow tunnel surrouding rock stress and the Explicit Analytic Solutions of displacement Download PDFInfo
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Abstract
The invention discloses shallow tunnel surrouding rock stress and the method for solving of the Explicit Analytic Solutions of displacement, this method is based on Complex Function Method, the progression form of z-plane analytical function is solved using inverse mapping functions, and cauchy riemann equation (C R conditions) is combined to the analytical function derivation in shallow tunnel surrouding rock stress and the implicit analytic solutions of displacement, draw the series explicit expression of shallow tunnel stress and displacement function；Compared with implicit analytic solutions, the Explicit Analytic Solutions are directly perceived, are easy to be used by engineering staff, and its program calculation amount is also smaller, and engineering measurement formation displacement and the result of calculation goodness of fit of display analytic solutions are high, demonstrate the correctness and practicality of Explicit Analytic Solutions.
Description
Technical field
The present invention relates to Urban underground Tunnel engineering, more particularly to the explicit of a kind of shallow tunnel surrouding rock stress and displacement
The method for solving of analytic solutions.
Background technology
, will for building for the city tunnel of representative with city underground and road tunnel etc. with the continuous development of urban transportation
It can greatly increase, often buried depth is shallower in this kind of tunnel, and comprehensive by Complicated Loads such as the constructions of structures in ground and stratum makees
With the formation displacement influence factor in work progress is more, and Accurate Prediction formation displacement is more difficult.With people's environmental consciousness
Constantly enhancing, formation displacement caused by Construction of City Tunnel and its influence control problem to surrounding environment oneself filled by related scholar
Divide and pay attention to, and accurately prediction formation displacement is effective premise for controlling Construction on Environment influence degree before constructing tunnel.It is existing
Using Complex Function Method analysis shallow tunnel peripheral rock stress and the thinking of displacement in technology, the calculating of bipolar coordinate method is avoided
On difficulty, it also avoid in image method on singular point assume deficiency, also directly give displacement while stress field is provided
The Exact of field.But the solution is related to the mappings repeatedly of two complex planes when solving, stress and displacement solution can not directly by
The function representation of R domains internal coordinate (x, y) before conformal projection, some following inconvenience be present：1. the solution needs certain specialty reason
By knowledge, if not the professional of research complex function, no image of Buddha Peck empirical equations are directly used by engineering staff like that,
And stress and displacement can be tried to achieve by directly inputting formation parameter for Explicit Analytic Solutions, engineering staff, this just makes engineering
Personnel are unnecessary to be taken a lot of time to study its complicated mapping principle；2. Verruijt solution procedurees are related to two complex planes
Mapping repeatedly, computational efficiency is low, and solution procedure is complicated, and program calculation amount is big, and Explicit Analytic Solutions, can directly by stress and
Displacement is programmed for the function of R domains internal coordinate (x, y) before conformal projection, and amount of calculation is smaller；3. Verruijt solutions are not directly perceived, solving
Surrouding rock stress caused by constructing tunnel and during displacement under the conditions of bearing formation, it is poor that it is superimposed applicability.
Accordingly, it is desirable to provide the method for solving of a kind of shallow tunnel surrouding rock stress and the Explicit Analytic Solutions of displacement, to overcome
Above-mentioned problems of the prior art.
The content of the invention
The technical problem to be solved in the present invention is to provide a kind of shallow tunnel surrouding rock stress and the Explicit Analytic Solutions of displacement
Method for solving, to overcome above-mentioned problems of the prior art.
In order to solve the above technical problems, the present invention uses following technical proposals
The method for solving of shallow tunnel surrouding rock stress and the Explicit Analytic Solutions of displacement, it is characterised in that the step of this method
Including
S1, based on Complex Function Method, establish the nothing of shallow tunnel surrouding rock stress and displacement on z-plane and embody form
Analytical functionAnd ψ1, and shallow tunnel reservoir stress and the calculation formula of displacement (z)：
Wherein, σxx, σyy, σxyFor reservoir stress, ux, uyFor stratum position
Move,And ψ1(z) two nothings for z-plane embody the analytical function of form, and μ is modulus of shearing, and κ is Poisson's ratio phase
Relation number (κ=3-4 υ), υ are Poisson's ratio, and i is imaginary number constant；
S2, utilize conformal mapping functionWherein, α is by cavern radius r and the cavern center of circle
The parameter that buried depth h ratio defines,Nothing on z-plane is embodied to the analytical function of formAnd ψ1
(z) analytical function for embodying form being converted into ζ planesWith ψ (ζ), the relation of two groups of analytical functions isWithAnalytical function φ (ζ) and ψ (ζ) concrete form is respectivelyWithWherein, a0, ak, bk, c0, ck, dkPhysics meaning
Justice is tunnel cavern Boundary condition coefficient；
S3, using inverse mapping functions by ζ plane analytical functionsIt is converted into Complex Function Method on z-plane and has with ψ (ζ)
There is the analytical function of concrete formAnd ψ1(z), and to the progression form of z-plane analytical function solve；
S4, based on Cauchy-Riemann equations, to the analytical function with the form that embodiesAnd ψ1(z) asked
Lead；
S5, by the analytical function with concrete formAnd ψ1(z) derived function substitutes into the shallow tunnel stratum should
In power and the calculation formula of displacement, the display analytic solutions of shallow tunnel reservoir stress and displacement are obtained.
Preferably, the step S3 includes
To formulaInverse mapping is carried out, obtains inverse mapping functions
If z=x+iy, then the formula can be deformed into：Wherein,
By formulaIt is converted into plural ζ triangular form
By the correlation method of the power computing of plural number
Then understand：
By analytical functionWith
The form of real and imaginary parts is converted into, if a0=a '0i,
ak=a 'ki,bk=b 'ki,c0=c '0i,ck=c 'ki,dk=d 'ki., then have,
After simplified,
Wherein,
Preferably, the step S4 includes
It is right when f (z)=u (x, y)+iv (x, y) can be micro- in point z=x+iy
With
Derivation is carried out, is obtained
Preferably, the step S5 includes
Will
With
Substitute into formula
Obtain
Beneficial effects of the present invention are as follows：
Technical scheme of the present invention utilizes inverse mapping functions by ζ plane analytical functionsZ-plane is converted into ψ (ζ)
Analytical function with the form that embodiesAnd ψ1(z), with reference to Cauchy-Riemann equations (C-R conditions) to based on multiple change letter
Analytical function derivation in number method shallow tunnel surrouding rock stress and the implicit analytic solutions of displacement, has drawn stress function and displacement function
Series explicit expression form；The Explicit Analytic Solutions are directly perceived, are easy to be used by engineering staff, and the programming meter of more implicit analytic solutions
Calculation amount is small.
Technical scheme of the present invention is based on case history, and shallow tunnel surrouding rock stress and displacement Explicit Analytic Solutions are carried out
Application, and the calculated results of Explicit Analytic Solutions and engineering measurement formation displacement have been subjected to comparative analysis, demonstrate aobvious
The correctness and practicality of formula analytic solutions.
Brief description of the drawings
The embodiment of the present invention is described in further detail below in conjunction with the accompanying drawings；
Fig. 1 shows a kind of schematic diagram of the method for solving of the Explicit Analytic Solutions of shallow tunnel surrouding rock stress and displacement；
Fig. 2 shows the schematic diagram of " black box theory "；
Fig. 3-a are shown in embodiment to the analytic solutions and actual measurement vertical displacement comparison diagram of No. 4 lines in San Francisco；
Fig. 3-b are shown in embodiment to the analytic solutions and measured level displacement comparison figure of No. 4 lines in San Francisco；
Fig. 4-a are shown in embodiment to the analytic solutions and actual measurement vertical displacement comparison diagram of No. 2 lines in San Francisco；
Fig. 4-b are shown in embodiment to the analytic solutions and measured level displacement comparison figure of No. 2 lines in San Francisco.
Embodiment
The invention discloses a kind of shallow tunnel surrouding rock stress and the method for solving of the Explicit Analytic Solutions of displacement, based on multiple change
Function method, the progression form of z-plane analytical function is solved using inverse mapping functions, and combine Cauchy-Riemann equations (C-R conditions)
To the analytical function derivation in shallow tunnel surrouding rock stress and the implicit analytic solutions of displacement, shallow tunnel stress field and displacement letter are drawn
Several series Explicit Analytic Solutions.By with engineering measurement date comprision, verify the correctness and practicality of Explicit Analytic Solutions.
First, based on Complex Function Method, be converted into the solution of plane problem the problem of shallow tunnel stress and displacement, can be with
The nothing for being expressed as parsing everywhere in region R (the half-plane y ＜ 0 for removing hole) embodies the analytical function of formWith
ψ1(z), shallow tunnel stress solution and displacement solution can determine according to formula (1a), formula (1b) and formula (1c).
Wherein, σxx, σyy, σxyFor reservoir stress, ux, uyFor formation displacement,ψ1(z) for two of z-plane without tool
The analytical function of body expression-form, μ are modulus of shearing, and κ is Poisson's ratio coefficient correlation (κ=3-4 υ), and υ is Poisson's ratio, and i is imaginary number
Constant；
Due to z-plane analytical functionAnd ψ1(z) there is no concrete form, therefore shallow tunnel stress solution and the meter of displacement
Formula (1a) to (1c) solution do not have concrete outcome, it is necessary to map that in ζ planes, obtain with embodying form
Analytical functionWith ψ (ζ) concrete form, then the analytical function of z-plane is converted back into by inverse mapping functions, obtains parsing letter
NumberAnd ψ1(z) embody form.
The conformal mapping function that z-plane is converted into ζ planes is：
In formula, α is the parameter defined by cavern radius r and cavern center of circle buried depth h ratio (r/h), can be represented
For：
Nothing on z-plane is embodied to the analytical function of formψ1(z) it is converted into having specifically in ζ planes
The analytical function of expression-formWith ψ (ζ), the relation of two groups of analytical functions isWithAnalytical functionThere is concrete form with ψ (ζ), determined by following two formula：
Wherein, a0, ak, bk, c0, ck, dkPhysical significance be tunnel cavern Boundary condition coefficient, these coefficients are passed by following
Apply-official formula determines：
Wherein：
u0Physical significance be：The value of the uniform radial contraction on tunnel cavern border.
udPhysical significance be：The value of the ovalization deformation on tunnel cavern border.
However, as shown in Fig. 2 the coordinate of every can use (x, y) to represent in mapping forefoot area R, in mapping rear region γ often
The coordinate of point can use (ζ, η) to represent.Solution procedure is related to the mapping of two complex planes, i.e., if desired for the stress for asking a bit (x, y) place
Or shift value, phase must be re-mapped back after obtaining value first with corresponding point (ζ, η) is obtained in mapping function to region γ
(x, the y) point answered.That is, in solution procedure stress and displacement solution can not directly by R domains internal coordinate before conformal projection (x,
Y) function representation, for engineering staff application be inconvenient.
Shallow tunnel surrouding rock stress and the Explicit Analytic Solutions of displacement derive
The present invention using in the γ domains after conformal projection as a "black box", as shown in Figure 2, it is intended to draw and use conformal projection
The stress and the explicit expression function of displacement field that preceding R domains internal coordinate (x, y) represents, thereby retaining prior art solving precision
Meanwhile the form of a brief and practical is provided for engineer applied personnel.Therefore, using inverse mapping functions by ζ plane analytical functionsψ (ζ) is converted into the analytical function for embodying form of z-planeψ1(z), with reference to Cauchy-Riemann equations
(C-R conditions) carries out derivation to the analytical function in shallow tunnel surrouding rock stress in the prior art and the implicit analytic solutions of displacement, will
Analytical function with concrete formAnd ψ1(z) derived function substitutes into the shallow tunnel reservoir stress and the meter of displacement
Calculate in formula, obtain the display analytic solutions of shallow tunnel reservoir stress and displacement.What Explicit Analytic Solutions derived comprises the following steps that
(1) solution of z-plane analytical function progression form
Inverse mapping functions are formula (2) accordingly：
If setting z=x+i y, formula (5) can be deformed into：
Wherein:
Then conversion type (6) is that plural ζ triangular form is as follows：
From the related rule of the power computing of plural number：
According to the symmetry of problem, all coefficients are pure imaginary number, for convenience of the derivation after (simplification), by analytical function
The form of real and imaginary parts is converted into, can be set：
a0=a '0i,ak=a 'ki,bk=b 'ki,c0=c '0i,ck=c 'ki,dk=d 'ki。 (9)
Then analytical function formula (4a) and formula (4b) can be deformed into：
Simplified style (10a) and formula (10b), and obtained in view of formula (4a) and formula (4b)：
Wherein：
(2) according to Cauchy-Riemann equations to analytical function derivation
Becauseψ (ζ) is analytical function, is understood according to Cauchy-Riemann equations (C-R conditions)：When f (z)=u (x,
Y) when point z=x+iy can be micro-, its derivative can be calculated+iv (x, y) by following formula：
Then can be rightAnd ψ 1 (z) [i.e. formula (11a) and formula (11b)] derivation, derivation obtain：
(3) shallow tunnel surrouding rock stress and displacement solve
Formula (13a), formula (13b) and formula (13c) are substituted into formula (1a), formula (1b) and formula (1c) and obtain stress function and displacement letter
Several explicit expressions:
In view of the foregoing it is apparent that in formula (14a)-formula (14e), will be " black as one in the γ domains after conformal projection
Case ", stress and displacement function Explicit Analytic Solutions, its right and left are all the forms in R domains before mapping, for engineering staff, no
Need to study its theoretical principle, can be as Peck empirical equations, by directly inputting parameter, you can directly pass through coordinate
(x, y) tries to achieve the value of respective point.In formula (14a)-formula (14e), contained input parameter can be divided into two groups：1. tunnel geometry and stratum
Parameter：E, μ, ν, κ, h, r, a.Formally, because it is containing the main body function that all coordinates (x, y) are Explicit Analytic Solutions expression formula
Part, it is determined that the geometry of function；2. deformation boundaries conditional correlation coefficient：ɑ0, ɑk, bk, c0, ck, dk, can be by tunnel border
Deformation condition determines.Formally, its coefficient divided for main body correspondence department, it is determined that the physical dimension size of function.Therefore,
Explicit Analytic Solutions can clearly study the affecting laws of Different Strata parameter and deformation boundaries condition to analytic solutions.Formula (14a)-formula
In (14e), programming process pertains only to map preceding complex plane, and the more implicit analytic solutions of computational efficiency are high, and solution procedure is simple, Ke Yitong
Cross and directly input formation parameter to try to achieve the stress of gained and displacement.Stress and displacement directly can be programmed for by Explicit Analytic Solutions
With the function of coordinate (x, y) in R domains before conformal projection, amount of calculation is smaller, and computational efficiency is high.
Below by one group of embodiment, the present invention will be further described：
To illustrate the practicality of Explicit Analytic Solutions, now quote following two groups of tunnel examples and analyzed.Table 1 gives two groups
The physical dimension in tunnel, Physical And Mechanical Indexes of The Typical etc..In engineering, the displacement of eyeball measures frequently with extension meter and inclinator, can obtain
The level of monitoring point and vertical displacement on horizontal and vertical survey line in stratum.With derivation Explicit Analytic Solutions formula formula (14a)-
Formula (14e), a large amount of stratum each point displacement datas succinctly can be easily proposed, now carry out engineering measurement data and analytic solutions pair
Than interpretation of result is as follows：
The tunnel physical dimension of table 1 and formation parameter
(1) the threaded list road N-2 bid sections of San Francisco 4
The threaded list road N-2 bid sections of San Francisco 4 build a diameter of 3.56m tunnel using a diameter of 3.7m EPB shield machines.
During the engineering construction, using the vertical and horizontal displacement of 23 measuring point measurement stratums of instrument layout, as shown in Figure 3.Red spots
For formula measured data, solid black lines are the curve that Explicit Analytic Solutions are made every 1m extraction data bus connections.
(2) the threaded list road N-2 bid sections of San Francisco 2
Not lose the generality of analytic solutions, same method goes to study the threaded list road N-2 bid section Explicit Analytic Solutions of San Francisco 2
With surveying stratum settlement, as shown in Figure 4.
From Fig. 3 and Fig. 4, most eyeballs all overlap with Explicit Analytic Solutions, show the present invention derive based on
The Explicit Analytic Solutions of Complex Function Method can predict formation displacement caused by constructing tunnel well.
Analyzed more than visible, show that the advantage of analytic solutions is engineering staff can be facilitated to make a plurality of earth's surface and ground
The internal survey line of layer, to system research stratum intrinsic displacement changing rule, practicality is good, and the settlement prediction to Construction of City Tunnel has
Significance.Moreover, if the reservoir stress under the load actions such as the construction of structures in ground and stratum and displacement are studied, more implicitly
For analytic solutions, various loads can be more easily superimposed, so as to greatly improve complex condition constructing tunnel reservoir stress
And the computational efficiency of displacement.
In summary, technical scheme of the present invention utilizes inverse mapping functions by ζ plane analytical functionsTurn with ψ (ζ)
Turn in Complex Function Method z-plane analytical function in R domainsAnd ψ1(z), with reference to Cauchy-Riemann equations (C-R conditions) to shallow
The analytical function derivation in tunnel surrounding stress and the implicit analytic solutions of displacement is buried, has drawn the series of stress function and displacement function
Explicit expression form；The Explicit Analytic Solutions are directly perceived, are easy to be used by engineering staff, and the program calculation amount of more implicit analytic solutions
It is small.
Technical scheme of the present invention is based on case history, and shallow tunnel surrouding rock stress and displacement Explicit Analytic Solutions are carried out
Application, and the calculated results of Explicit Analytic Solutions and engineering measurement formation displacement have been subjected to comparative analysis, demonstrate aobvious
The correctness and practicality of formula analytic solutions.
Obviously, the above embodiment of the present invention is only intended to clearly illustrate example of the present invention, and is not pair
The restriction of embodiments of the present invention, for those of ordinary skill in the field, may be used also on the basis of the above description
To make other changes in different forms, all embodiments can not be exhaustive here, it is every to belong to this hair
Row of the obvious changes or variations that bright technical scheme is extended out still in protection scope of the present invention.
Claims (3)
1. the method for solving of shallow tunnel surrouding rock stress and the Explicit Analytic Solutions of displacement, it is characterised in that wrap the step of this method
Include
S1, based on Complex Function Method, establish the solution that the nothing of shallow tunnel surrouding rock stress and displacement on z-plane embodies form
Analyse functionAnd ψ1, and shallow tunnel reservoir stress and the calculation formula of displacement (z)：
Wherein, σxx, σyy, σxyFor reservoir stress, ux, uyFor formation displacement,And ψ1(z) two nothings for z-plane embody the analytical function of form, and μ is modulus of shearing, and κ is Poisson's ratio phase relation
Number, κ=3-4 υ, υ are Poisson's ratio, and i is imaginary number constant；
S2, utilize conformal mapping functionWherein, α is by Tunnel chamber radius r and tunnel cavern
The parameter that center of circle buried depth h ratio defines,Nothing on z-plane is embodied to the analytical function of formWith
ψ1(z) analytical function for embodying form being converted into ζ planesWith ψ (ζ), the relation of two groups of analytical functions isWith ψ (ζ)=ψ1(ω (ζ))=ψ1(z), analytical functionConcrete form with ψ (ζ) is respectivelyWithWherein, a0, ak, bk, c0, ck, dkPhysical significance
For tunnel cavern Boundary condition coefficient；
S3, using inverse mapping functions by ζ plane analytical functionsBeing converted into ψ (ζ) in Complex Function Method on z-plane has tool
The analytical function of body formAnd ψ1(z), and to the progression form of z-plane analytical function solve；
S4, based on Cauchy-Riemann equations, to the analytical function with the form that embodiesAnd ψ1(z) derivation is carried out；
S5, by the analytical function with concrete formAnd ψ1(z) derived function substitute into the shallow tunnel reservoir stress and
In the calculation formula of displacement, the display analytic solutions of shallow tunnel reservoir stress and displacement are obtained；
The step S3 includes
To formulaInverse mapping is carried out, obtains inverse mapping functionsIf z=x
+ i y, then the formula can be deformed into：Wherein,
By formulaIt is converted into plural ζ triangular form
By the phase of the power computing of plural number
Rule is closed to understand：
By analytical functionWithIt is converted into real and imaginary parts
Form, if a0=a '0i,ak=a 'ki,bk=b 'ki,c0=c '0i,ck=c 'ki,dk=d 'kI, then have,
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<mi>a</mi>
<mi>r</mi>
<mi>c</mi>
<mi>c</mi>
<mi>o</mi>
<mi>s</mi>
<mrow>
<mo>(</mo>
<mfrac>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
<mo>+</mo>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
<mo>-</mo>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
<mo>+</mo>
<msup>
<mi>y</mi>
<mn>4</mn>
</msup>
<mo>+</mo>
<msup>
<mi>a</mi>
<mn>4</mn>
</msup>
<mo>+</mo>
<mn>2</mn>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
<mo>+</mo>
<mn>2</mn>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
<mo>-</mo>
<mn>2</mn>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mfrac>
<mo>)</mo>
</mrow>
<mo>&rsqb;</mo>
<mo>,</mo>
</mrow>
<mrow>
<msub>
<mi>W</mi>
<mn>2</mn>
</msub>
<mo>=</mo>
<msup>
<mrow>
<mo>&lsqb;</mo>
<mfrac>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
<mo>+</mo>
<msup>
<mrow>
<mo>(</mo>
<mi>y</mi>
<mo>+</mo>
<mi>a</mi>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
<mo>+</mo>
<msup>
<mrow>
<mo>(</mo>
<mi>y</mi>
<mo>-</mo>
<mi>a</mi>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mfrac>
<mo>&rsqb;</mo>
</mrow>
<mrow>
<mo>-</mo>
<mfrac>
<mi>k</mi>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
<mi>c</mi>
<mi>o</mi>
<mi>s</mi>
<mo>&lsqb;</mo>
<mrow>
<mo>(</mo>
<mo>-</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mi>a</mi>
<mi>r</mi>
<mi>c</mi>
<mi>c</mi>
<mi>o</mi>
<mi>s</mi>
<mrow>
<mo>(</mo>
<mfrac>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
<mo>+</mo>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
<mo>-</mo>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
<mo>+</mo>
<msup>
<mi>y</mi>
<mn>4</mn>
</msup>
<mo>+</mo>
<msup>
<mi>a</mi>
<mn>4</mn>
</msup>
<mo>+</mo>
<mn>2</mn>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
<mo>+</mo>
<mn>2</mn>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
<mo>-</mo>
<mn>2</mn>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mfrac>
<mo>)</mo>
</mrow>
<mo>&rsqb;</mo>
<mo>,</mo>
</mrow>
<mrow>
<msub>
<mi>W</mi>
<mn>3</mn>
</msub>
<mo>=</mo>
<msup>
<mrow>
<mo>&lsqb;</mo>
<mfrac>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
<mo>+</mo>
<msup>
<mrow>
<mo>(</mo>
<mi>y</mi>
<mo>+</mo>
<mi>a</mi>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
<mo>+</mo>
<msup>
<mrow>
<mo>(</mo>
<mi>y</mi>
<mo>-</mo>
<mi>a</mi>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mfrac>
<mo>&rsqb;</mo>
</mrow>
<mfrac>
<mi>k</mi>
<mn>2</mn>
</mfrac>
</msup>
<mi>sin</mi>
<mo>&lsqb;</mo>
<mi>k</mi>
<mi>arccos</mi>
<mrow>
<mo>(</mo>
<mfrac>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
<mo>+</mo>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
<mo>-</mo>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
<mo>+</mo>
<msup>
<mi>y</mi>
<mn>4</mn>
</msup>
<mo>+</mo>
<msup>
<mi>a</mi>
<mn>4</mn>
</msup>
<mo>+</mo>
<mn>2</mn>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
<mo>+</mo>
<mn>2</mn>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
<mo>-</mo>
<mn>2</mn>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mfrac>
<mo>)</mo>
</mrow>
<mo>&rsqb;</mo>
<mo>,</mo>
</mrow>
<mrow>
<msub>
<mi>W</mi>
<mn>4</mn>
</msub>
<mo>=</mo>
<msup>
<mrow>
<mo>&lsqb;</mo>
<mfrac>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
<mo>+</mo>
<msup>
<mrow>
<mo>(</mo>
<mi>y</mi>
<mo>+</mo>
<mi>a</mi>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
<mo>+</mo>
<msup>
<mrow>
<mo>(</mo>
<mi>y</mi>
<mo>-</mo>
<mi>a</mi>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mfrac>
<mo>&rsqb;</mo>
</mrow>
<mrow>
<mo>-</mo>
<mfrac>
<mi>k</mi>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
<mi>sin</mi>
<mo>&lsqb;</mo>
<mrow>
<mo>(</mo>
<mo>-</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mi>arccos</mi>
<mrow>
<mo>(</mo>
<mfrac>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
<mo>+</mo>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
<mo>-</mo>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
<mo>+</mo>
<msup>
<mi>y</mi>
<mn>4</mn>
</msup>
<mo>+</mo>
<msup>
<mi>a</mi>
<mn>4</mn>
</msup>
<mo>+</mo>
<mn>2</mn>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
<mo>+</mo>
<mn>2</mn>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
<mo>-</mo>
<mn>2</mn>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mfrac>
<mo>)</mo>
</mrow>
<mo>&rsqb;</mo>
<mo>.</mo>
</mrow>
2. method for solving according to claim 1, it is characterised in that the step S4 includes
It is right when f (z)=u (x, y)+iv (x, y) can be micro- in point z=x+iy
With
Derivation is carried out, is obtained
<mrow>
<msubsup>
<mi>&psi;</mi>
<mn>1</mn>
<mo>&prime;</mo>
</msubsup>
<mrow>
<mo>(</mo>
<mi>z</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mo>{</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>&infin;</mi>
</munderover>
<msubsup>
<mi>c</mi>
<mi>k</mi>
<mo>&prime;</mo>
</msubsup>
<mfrac>
<mo>&part;</mo>
<mrow>
<mo>&part;</mo>
<mi>x</mi>
</mrow>
</mfrac>
<msub>
<mi>W</mi>
<mn>1</mn>
</msub>
<mo>+</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>&infin;</mi>
</munderover>
<msubsup>
<mi>d</mi>
<mi>k</mi>
<mo>&prime;</mo>
</msubsup>
<mfrac>
<mo>&part;</mo>
<mrow>
<mo>&part;</mo>
<mi>x</mi>
</mrow>
</mfrac>
<msub>
<mi>W</mi>
<mn>2</mn>
</msub>
<mo>}</mo>
<mi>i</mi>
<mo>-</mo>
<mo>{</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>&infin;</mi>
</munderover>
<msubsup>
<mi>c</mi>
<mi>k</mi>
<mo>&prime;</mo>
</msubsup>
<mfrac>
<mo>&part;</mo>
<mrow>
<mo>&part;</mo>
<mi>x</mi>
</mrow>
</mfrac>
<msub>
<mi>W</mi>
<mn>3</mn>
</msub>
<mo>+</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>&infin;</mi>
</munderover>
<msubsup>
<mi>d</mi>
<mi>k</mi>
<mo>&prime;</mo>
</msubsup>
<mfrac>
<mo>&part;</mo>
<mrow>
<mo>&part;</mo>
<mi>x</mi>
</mrow>
</mfrac>
<msub>
<mi>W</mi>
<mn>4</mn>
</msub>
<mo>}</mo>
<mo>,</mo>
</mrow>
3. method for solving according to claim 2, it is characterised in that the step S5 includes
Will
With
Substitute into formula
In, obtain
<mrow>
<msub>
<mi>&sigma;</mi>
<mrow>
<mi>x</mi>
<mi>x</mi>
</mrow>
</msub>
<mo>=</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>&infin;</mi>
</munderover>
<mo>&lsqb;</mo>
<mrow>
<mo>(</mo>
<mo>-</mo>
<mn>2</mn>
<msubsup>
<mi>a</mi>
<mi>k</mi>
<mo>&prime;</mo>
</msubsup>
<mo>+</mo>
<msubsup>
<mi>c</mi>
<mi>k</mi>
<mo>&prime;</mo>
</msubsup>
<mo>)</mo>
</mrow>
<mfrac>
<mo>&part;</mo>
<mrow>
<mo>&part;</mo>
<mi>x</mi>
</mrow>
</mfrac>
<msub>
<mi>W</mi>
<mn>3</mn>
</msub>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mo>-</mo>
<mn>2</mn>
<msubsup>
<mi>b</mi>
<mi>k</mi>
<mo>&prime;</mo>
</msubsup>
<mo>+</mo>
<msubsup>
<mi>d</mi>
<mi>k</mi>
<mo>&prime;</mo>
</msubsup>
<mo>)</mo>
</mrow>
<mfrac>
<mo>&part;</mo>
<mrow>
<mo>&part;</mo>
<mi>x</mi>
</mrow>
</mfrac>
<msub>
<mi>W</mi>
<mn>4</mn>
</msub>
<mo>-</mo>
<mi>y</mi>
<mrow>
<mo>(</mo>
<msubsup>
<mi>a</mi>
<mi>k</mi>
<mo>&prime;</mo>
</msubsup>
<mfrac>
<msup>
<mo>&part;</mo>
<mn>2</mn>
</msup>
<mrow>
<mo>&part;</mo>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mfrac>
<msub>
<mi>W</mi>
<mn>1</mn>
</msub>
<mo>+</mo>
<msubsup>
<mi>b</mi>
<mi>k</mi>
<mo>&prime;</mo>
</msubsup>
<mfrac>
<msup>
<mo>&part;</mo>
<mn>2</mn>
</msup>
<mrow>
<mo>&part;</mo>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mfrac>
<msub>
<mi>W</mi>
<mn>2</mn>
</msub>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>x</mi>
<mrow>
<mo>(</mo>
<msubsup>
<mi>a</mi>
<mi>k</mi>
<mo>&prime;</mo>
</msubsup>
<mfrac>
<msup>
<mo>&part;</mo>
<mn>2</mn>
</msup>
<mrow>
<mo>&part;</mo>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mfrac>
<msub>
<mi>W</mi>
<mn>3</mn>
</msub>
<mo>+</mo>
<msubsup>
<mi>b</mi>
<mi>k</mi>
<mo>&prime;</mo>
</msubsup>
<mfrac>
<msup>
<mo>&part;</mo>
<mn>2</mn>
</msup>
<mrow>
<mo>&part;</mo>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mfrac>
<msub>
<mi>W</mi>
<mn>4</mn>
</msub>
<mo>)</mo>
</mrow>
<mo>&rsqb;</mo>
<mo>,</mo>
</mrow>
<mrow>
<msub>
<mi>&sigma;</mi>
<mrow>
<mi>y</mi>
<mi>y</mi>
</mrow>
</msub>
<mo>=</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>&infin;</mi>
</munderover>
<mo>&lsqb;</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>2</mn>
<msubsup>
<mi>a</mi>
<mi>k</mi>
<mo>&prime;</mo>
</msubsup>
<mo>+</mo>
<msubsup>
<mi>c</mi>
<mi>k</mi>
<mo>&prime;</mo>
</msubsup>
<mo>)</mo>
</mrow>
<mfrac>
<mo>&part;</mo>
<mrow>
<mo>&part;</mo>
<mi>x</mi>
</mrow>
</mfrac>
<msub>
<mi>W</mi>
<mn>3</mn>
</msub>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>2</mn>
<msubsup>
<mi>b</mi>
<mi>k</mi>
<mo>&prime;</mo>
</msubsup>
<mo>+</mo>
<msubsup>
<mi>d</mi>
<mi>k</mi>
<mo>&prime;</mo>
</msubsup>
<mo>)</mo>
</mrow>
<mfrac>
<mo>&part;</mo>
<mrow>
<mo>&part;</mo>
<mi>x</mi>
</mrow>
</mfrac>
<msub>
<mi>W</mi>
<mn>4</mn>
</msub>
<mo>+</mo>
<mi>y</mi>
<mrow>
<mo>(</mo>
<msubsup>
<mi>a</mi>
<mi>k</mi>
<mo>&prime;</mo>
</msubsup>
<mfrac>
<msup>
<mo>&part;</mo>
<mn>2</mn>
</msup>
<mrow>
<mo>&part;</mo>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mfrac>
<msub>
<mi>W</mi>
<mn>1</mn>
</msub>
<mo>+</mo>
<msubsup>
<mi>b</mi>
<mi>k</mi>
<mo>&prime;</mo>
</msubsup>
<mfrac>
<msup>
<mo>&part;</mo>
<mn>2</mn>
</msup>
<mrow>
<mo>&part;</mo>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mfrac>
<msub>
<mi>W</mi>
<mn>2</mn>
</msub>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mi>x</mi>
<mrow>
<mo>(</mo>
<msubsup>
<mi>a</mi>
<mi>k</mi>
<mo>&prime;</mo>
</msubsup>
<mfrac>
<msup>
<mo>&part;</mo>
<mn>2</mn>
</msup>
<mrow>
<mo>&part;</mo>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mfrac>
<msub>
<mi>W</mi>
<mn>3</mn>
</msub>
<mo>+</mo>
<msubsup>
<mi>b</mi>
<mi>k</mi>
<mo>&prime;</mo>
</msubsup>
<mfrac>
<msup>
<mo>&part;</mo>
<mn>2</mn>
</msup>
<mrow>
<mo>&part;</mo>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mfrac>
<msub>
<mi>W</mi>
<mn>4</mn>
</msub>
<mo>)</mo>
</mrow>
<mo>&rsqb;</mo>
<mo>,</mo>
</mrow>
<mrow>
<msub>
<mi>&sigma;</mi>
<mrow>
<mi>x</mi>
<mi>y</mi>
</mrow>
</msub>
<mo>=</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>&infin;</mi>
</munderover>
<mo>&lsqb;</mo>
<msubsup>
<mi>c</mi>
<mi>k</mi>
<mo>&prime;</mo>
</msubsup>
<mfrac>
<mo>&part;</mo>
<mrow>
<mo>&part;</mo>
<mi>x</mi>
</mrow>
</mfrac>
<msub>
<mi>W</mi>
<mn>1</mn>
</msub>
<mo>+</mo>
<msubsup>
<mi>d</mi>
<mi>k</mi>
<mo>&prime;</mo>
</msubsup>
<mfrac>
<mo>&part;</mo>
<mrow>
<mo>&part;</mo>
<mi>x</mi>
</mrow>
</mfrac>
<msub>
<mi>W</mi>
<mn>2</mn>
</msub>
<mo>+</mo>
<mi>x</mi>
<mrow>
<mo>(</mo>
<msubsup>
<mi>a</mi>
<mi>k</mi>
<mo>&prime;</mo>
</msubsup>
<mfrac>
<msup>
<mo>&part;</mo>
<mn>2</mn>
</msup>
<mrow>
<mo>&part;</mo>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mfrac>
<msub>
<mi>W</mi>
<mn>1</mn>
</msub>
<mo>+</mo>
<msubsup>
<mi>b</mi>
<mi>k</mi>
<mo>&prime;</mo>
</msubsup>
<mfrac>
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| 地面荷载下浅埋隧道围岩应力的复变函数解法;陆文超;《江南大学学报（自然科学版）》;20021225(第4期);第409-413页 * |
| 浅埋隧道围岩应力场的解析解;陆文超等;《力学季刊》;20030330;第24卷(第1期);第50-54页 * |
| 浅埋隧道围岩应力场的计算复变函数求解法;王志良等;《岩土力学》;20100815;第31卷(第S1期);第86-90页 * |
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