CN1731177A - Method for determination of engineering rock mass damage - Google Patents

Abstract

The invention discloses a method for quoting the project rock damage, which dose quantification by the subject orientation characteristic factor of the rock engineer and the quoting of the real rock index, then it introduces the quantification value to the subject factor weakness model to adjust it so as to obtain the rock characteristic index adjusting value, at last it quotes the rock damage by the rock characteristic index.

Description

The method of determination of engineering rock mass damage

Technical field

The invention belongs to the engineering geology technical field, particularly in engineering rock mass, destroy the method for judging.

Background technology

Destruction for engineering rock mass judges that the Hoek-Brown criterion is the decision criteria of present best rock mass damage of generally acknowledging, and this criterion can also further provide every mechanics parameter of rock mass both at home and abroad.Yet, because the part input parameter of this criterion need obtain by professional's artificial subjective determination, therefore, based on the stability analysis of the actual engineering rock mass of this criterion and the influence of the individual subjective factor difference that parameter estimation certainly will be subjected to the slip-stick artist, DIFFERENT SOIL slip-stick artist's subjective factor difference may cause result of determination distortion or inconsistent present situation.

Summary of the invention

In order to solve the influence of artificial subjective factor difference in the destruction of engineering rock mass is judged, the object of the present invention is to provide a kind of method of determination of engineering rock mass damage, this method has been subdued the interference effect of the individual subjective psychological factor of Geotechnical Engineering teacher greatly, and it is more accurate that final rock mass damage is judged.

The technical solution adopted in the present invention is, by individual subjective tendency characteristic factor and slip-stick artist quantification is carried out in the judgement of actual rock mass rating to Geotechnical Engineering teacher, bringing the quantification value into subjective factor weakens in the model and revises, obtain rock mass characteristic index modified value, carry out the judgement of rock mass damage according to this index modified value, this method is carried out according to the following steps:

At first, Geotechnical Engineering's teacher psychological tendency feature is carried out before measurement

Makings, mood, sensitivity, 4 subjective tendency characteristic factors of the structure of knowledge are carried out systemic Pretesting, obtain the quantitative target of the different people's difference on these 4 kinds of subjective tendency characteristic factors of reflection;

Carry out the result of determination quantification of Geotechnical Engineering teacher again to actual rock mass rating

Allow the slip-stick artist that surface condition index, structural index and perturbation index are judged successively, requiring the slip-stick artist that same rock mass rating is carried out five times continuously judges, and five times the result can not be all identical, respectively according to the surface condition index reference system of engineering rock mass, structural index reference system and perturbation index reference system, with its result of determination quantification;

Set up subjective factor and weaken model

At first the before measurement data are carried out pre-service, foundation reduces the sensitivity differentia influence, reduces the makings differentia influence, reduces the partial model that excited level difference influences, reduces structure of knowledge differentia influence respectively again, last comprehensive each partial model obtains the whole synthesis subjective factor and weakens model;

Weaken model by subjective factor and obtain rock mass characteristic index modified value

To the result of determination of 3 characteristic index of engineering rock mass, bring subjective factor into and weaken model in every test result that psychological before measurement is obtained and the last step, obtain the modified value of rock mass characteristic index;

Carry out the judgement of rock mass damage at last

To bring broad sense Hoek-Brown criterion in 2002 by the modified value that subjective factor weakens the rock mass characteristic index that model obtains into, whether judge rock mass damage.

The present invention can eliminate or reduce the uncertainty of the result of determination that individual subjective factor difference causes in the artificial quantification decision process effectively under existence conditions, thereby provides data target more reliably for further analyzing.

Description of drawings

Fig. 1 is that surface condition is abstract in gray scale striped comparison diagram;

Fig. 2 is that structural index is abstract in the full comparison diagram of cake;

Fig. 3 is a gray scale striped contrast reference system, and wherein a is with reference to being 0, and b is with reference to being 50, and the c reference is 100;

Fig. 4 is the full contrast reference system of cake, and a is with reference to being 50, and the b reference is 100;

Fig. 5 is a surface condition indices quantification reference system;

Fig. 6 is structural indices quantification reference system;

Fig. 7 is the continuous value reference system of rock mass perturbation index D;

Embodiment

The present invention is described in detail below in conjunction with the drawings and specific embodiments.

Method of the present invention has independently considered to produce the subjective factor of negative interference, has made up method of testing, mathematical model and the whole weakening model of each subjective factor, and has provided the method that complete rock mass damage is judged on this basis.

Wherein: Geotechnical Engineering's teacher psychological before measurement---obtain the relevant tendentiousness characteristic information of Geotechnical Engineering's teacher subjective psychology state;

Actual rock mass rating judgement---the characteristic index to actual engineering rock mass quantizes to judge;

Subjective factor weakens model---and utilize this mathematical model Geotechnical Engineering's teacher of rock mass characteristic index subjective determination result to be weakened the correcting process of interference caused by subjective factors influence;

Rock mass damage is judged---according to the above-mentioned rock mass characteristic index modified value that obtains, judge the stability of engineering rock mass.

The present invention carries out according to the following steps:

At first, the relevant tendentiousness feature of Geotechnical Engineering's teacher subjective psychology state is carried out before measurement

Be chosen in 4 subjective tendency features that produce interference effect in people's the subjective quantity decision process: makings, mood, sensitivity, the structure of knowledge are carried out systemic Pretesting, obtain the quantitative target that can reflect different people difference on these 4 kinds of subjective tendency features, so that bring mathematical model subsequently into, weaken the interference of its factor and individual subjective factor, Geotechnical Engineering's teacher original amount decision content is converted into the modified value that approaches true value more.In 4 kinds of subjective tendency characteristic factors, gas quality index and sentiment indicator directly obtain by corresponding before measurement, and sensitivity index and structure of knowledge index obtain indirectly by data analysis.

1. the test of gas quality index

According to Denmark scholar's classical temperament type figure people's makings is divided into 4 types of the quality of bile, sanguine temperament, mucilaginous substance, melancholy, the slip-stick artist selects of the most approaching oneself reaction tendency in these 4 types.Tolerance height according to different temperament type correspondences, and the real process of judging in conjunction with rock mass rating, the gas quality index that each typical temperament type is defined its quantification is as follows: the quality of bile---0.6, sanguine temperament---0.6, mucilaginous substance---0.8, melancholy---0.4.

2. the test of sentiment indicator

Adopt in the experimental psychology the most ripe scaling method Test Engineer's sentiment indicator.

Require to select in slip-stick artist's scale below the predicate of approaching current psychological condition, the score that slip-stick artist's predicate is selected is accumulated then, obtains this project's teacher sentiment indicator.

Option number	Life is	Life is	Be future	The people is
					The predicate score	The predicate score	The predicate score	The predicate score
	Option one	Very well 0.1	0.1 of inanition	Likely 0.1					Easily old 0.1
Option 2					0.3	0.3	0.3	0.3
	Option 3	More smoothly 0.5	Than 0.5 of inanition	More likely 0.5					Easily old 0.5
Option 4					0.7	0.7	0.7	0.7
	Option 5	General 0.9	General 0.9	General 0.9					General 0.9
Option 6					0.7	0.7	0.7	0.7
	Option 7	More not smooth 0.5	Enrich 0.5	More hopeless 0.5					Be difficult for old 0.5
Option 8					0.3	0.3	0.3	0.3
	Option 9	Very not smooth 0.1	Enrich 0.1	Hopeless 0.1					Be difficult for old 0.1

3. the test of sensitivity index and structure of knowledge index

Obtain the quantification decision data of slip-stick artist by test, the data basis of slip-stick artist's sensitivity difference and structure of knowledge difference is provided for follow-up mathematical model these abstraction graphs to the every performance index dependence test of rock mass figure.

Shown in Fig. 1,2,3,4, abstract respectively to surface condition and structural impression be impression to gray scale striped comparison diagram and the full comparison diagram of cake.

Stipulate: stimulus intensity is in 0～100 closed interval herein

Strong stimulation is meant a stimulus intensity value that generates at random in 60～80 closed interval scope;

The weak stimulation is meant a stimulus intensity value that generates at random in 20～40 closed interval scope.

Determining that through related experiment the subject is 2.3 for the sensation difference limen of fringe gray level contrast, is 3.4 to the sensation difference limen of the full degree of cake.So to continuous several times bar graph and pie chart test that the subject implements, the difference of its adjacent twice stimulus intensity separately can not be less than above-mentioned threshold.

By tested slip-stick artist the test of pie chart and bar graph is obtained the intensity subjective determination value of a series of strong stimulation and weak stimulation, interpersonal individual sensitivity difference and structure of knowledge difference just contain in this.

Carry out the result of determination quantification of Geotechnical Engineering teacher again to actual rock mass rating

Finish after the aforesaid test, respectively " surface condition " index, " structural " index and " perturbation " index of actual engineering rock mass quantized to judge successively again.

According to the reference system that Fig. 5, Fig. 6, Fig. 7 provided, allow Geotechnical Engineering teacher that same rock mass rating (surface condition, structural or perturbation) is carried out independently subjective determination continuously 5 times, require 5 results can not be all identical.

Set up subjective factor and weaken model

1. at first the before measurement data are carried out pre-service

Order: P ₁∈ (P ₁, P ₂, P ₃... P _n), Pi is the slip-stick artist that obtains in certain before measurement continuous several times evaluation of estimate to master drawing, requiring among the present invention is 5 evaluations of estimate;

P is P _iArithmetic mean;

P is the true value of this time before measurement;

G _i∈ (G ₁, G ₂, G ₃... G _n), n=5 herein, Gi is the continuous several times evaluation of estimate of slip-stick artist to a certain index of actual rock mass, requiring among the present invention is 5 evaluations of estimate;

G is G _iArithmetic mean;

G is the true value of this rock mass rating;

Suppose that the before measurement data satisfy: 0≤P _i≤ 2P, (1)

Reflect the dispersion index of every input value of this time before measurement with respect to arithmetic mean value dispersion degree,

${I_P}^{\′}=\frac{1}{n}\sqrt{\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{\left(\frac{P_i-\stackrel{\‾}{P}}{\stackrel{\‾}{P}}\right)}^2},0\≤{I_P}^{\′}\≤1---\left(2\right)$

Reflect the dispersion index of every input value of this time before measurement with respect to the true value dispersion degree,

$I_P=\frac{1}{n}\sqrt{\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{\left(\frac{P_i-P}{P}\right)}^2},0\≤I_P\≤1---\left(3\right)$

Every input that this project teacher estimates this rock mass rating is with respect to the dispersion index of arithmetic mean value dispersion degree,

${I_G}^{\′}=\frac{1}{n}\sqrt{\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{\left(\frac{G_i-\stackrel{\‾}{G}}{\stackrel{\‾}{G}}\right)}^2},0\≤{I_G}^{\′}\≤1---\left(4\right)$

Every input that this project teacher estimates this rock mass rating is with respect to the dispersion index of true value dispersion degree,

$I_G=\frac{1}{n}\sqrt{\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{\left(\frac{G_i-G}{G}\right)}^2},0\≤I_G\≤1---\left(5\right)$

Relatively stable degree and the relative degree that misfits that shows rule in before measurement to the rock mass rating evaluation are referred to as the coefficient of alienation, with respect to the coefficient of alienation of arithmetic mean,

I′＝I _G′/I _p′ (6)

With respect to the coefficient of alienation of true value, formula 7 is arranged,

I＝I _G/I _P≈I _G′/I _P。(7)

2. set up the model that reduces the sensitivity differentia influence

Model is as follows:

$G=\underset{i=1}{\overset{n}{\mathrm{\Σ}}}\left({\mathrm{\ω}}_{p1}{G}_1\right)---\left(8\right)$

Weight wherein

${\mathrm{\ω}}_{\mathrm{pi}}=\frac{\underset{j=1}{\overset{n}{\mathrm{\Σ}}}|P_j-P|-|P_i-P|}{(n-1)\underset{j=1}{\overset{n}{\mathrm{\Σ}}}|P_j-P|}---\left(9\right)$

$0\≤{\mathrm{\ω}}_i\≤\frac{1}{n-1}---\left(10\right)$

Weight depends on the performance of slip-stick artist in relevant perceptibility before measurement, reflected each time input of being contained in the continuous before measurement evaluation of estimate of slip-stick artist the relative size regularity of distribution with respect to true value, and this average weighted form, can offset a part of stochastic error.

Reflected [P ₁, P ₂, P ₃... P _n] in each member's relative importance, promptly to the influence degree of P.

Consider G _i/ G and P _iThe influence of/P relative size difference, the computing formula of true value is transformed to

$G=\underset{i=1}{\overset{n}{\mathrm{\Σ}}}\left({\mathrm{\ω}}_{\mathrm{pi}}{G_i}^{\′}\right)---\left(14\right)$

Wherein

${G_i}^{\′}=\left(\frac{G_i}{P_i}\right)\·P$

3. set up the model that reduces the makings differentia influence

Considered the rock mass rating true value after slip-stick artist's temperament type exerts an influence to the exercise fatigue effect, its model is:

$G^{\left(1\right)}=\underset{k=1}{\overset{m-1}{\mathrm{\Σ}}}[\left(\frac{1-V1}{m}\right)\·{\left(\frac{k}{m-1}\right)}^{V1}\·G\left(t_k\right)]+(\frac{1-V1}{m}+V1)\·G\left(t_m\right)---\left(15\right)$

If order

$Q\left(t_k\right)=\left(\frac{1-V1}{m}\right)\·{\left(\frac{k}{m-1}\right)}^{V1},(k=\mathrm{1,2,3},...m-1)---\left(16\right)$

$Q\left(t_m\right)=\frac{1-V1}{m}+V1---\left(17\right)$

Then, [Q] is the weight factor of reflection historical law influence.

So, the rock mass rating evaluation of estimate after the effect of consideration makings

$G^{\left(1\right)}=\underset{k=1}{\overset{m-1}{\mathrm{\Σ}}}[Q\left(t_k\right)\·G\left(t_k\right)]+Q\left(t_m\right)\·G\left(t_m\right)---\left(18\right)$

$=\underset{k=1}{\overset{m}{\mathrm{\Σ}}}[Q\left(t_k\right)\·G\left(t_k\right)]$

Wherein, G ⁽¹⁾For having considered the achievement after practice effect (makings) influences;

V1 is the makings influence index value that current slip-stick artist obtains in this before measurement;

t _k(k=1,2,3 ... m) expression this project teacher uses t historical the k time _mRepresent current use;

G (t _k) expression last time the slip-stick artist to the evaluation of rock mass rating, after the rule of having applied mechanically gained before measurement data when extracting on this project's teacher history the k time login system is revised, the historical true value that obtains.

4. set up the model that reduces excited level difference influence

Order: G ⁽²⁾For having considered the achievement behind the emotion influence;

V2 is current slip-stick artist's an emotion influence index;

Then have:

G ⁽²⁾＝(1-V2′)·G ⁽¹⁾+V2′· G (19)

Wherein: V2 '=α V2 is the modified value of emotion influence index V2, considers the finiteness of good mood to the cognitive activities facilitation, herein α=0.5;

G ⁽¹⁾For having considered the achievement after practice effect influences;

G is the arithmetic mean value of rock mass rating input.

5. set up the model that reduces structure of knowledge differentia influence

In the rock mass rating decision process, the effect of the structure of knowledge may be in particular in two following aspects:

1. when the structure of knowledge is unfavorable for the decision process of rock mass rating, the decision content of slip-stick artist's input will depart from true value generally more;

2. under the ideal state, the coefficient of alienation I of the rock mass rating decision content of slip-stick artist's input _GWith before measurement data coefficient of alienation I _pBasically identical, the mathematical model how appropriate of at this moment aforementioned each subjective factor of consideration is higher, that is: various rules of refining from the before measurement data can be applicable to the decision process of rock mass rating on the whole.But actual conditions are that before measurement is a summary, abstract from practical problems, is different from actual rock mass rating and judges.Consider slip-stick artist's practical intelligence structure, the coefficient of alienation I of the rock mass rating decision content of its input _GWith before measurement data coefficient of alienation I _pMay be inconsistent, and the slip-stick artist with different structures of knowledge, its I _GWith I _pBetween departure degree with inconsistent.

Depart from the problem of true value for slip-stick artist's input value integral body of first rock mass rating, can pass through following model, promptly the weight coefficient to the before measurement data is weighted calculating, directly offsets slip-stick artist's input value departing from true value to a certain extent:

$G=\underset{i=1}{\overset{n}{\mathrm{\Σ}}}\left({\mathrm{\ω}}_{\mathrm{pi}}{G}_i\right)---\left(20\right)$

${\mathrm{\ω}}_{\mathrm{pi}}=\frac{\underset{j=1}{\overset{n}{\mathrm{\Σ}}}|P_j-P|-|P_i-P|}{(n-1)\underset{j=1}{\overset{n}{\mathrm{\Σ}}}|P_j-P|}---\left(21\right)$

For second I that causes by structure of knowledge difference _GDepart from I _pThe problem of degree varies, can set up following model and pay attention to:

${\mathrm{\ω}}_{P1}=\frac{\underset{j=1}{\overset{n}{\mathrm{\Σ}}}{|P_j-P|}^{\mathrm{\η}}-{|P_i-P|}^{\mathrm{\η}}}{(n-1)\underset{j=1}{\overset{n}{\mathrm{\Σ}}}{|P_j-P|}^{\mathrm{\η}}}---\left(22\right)$

Exponential term

$\mathrm{\η}=1/I=I_P/I_G$

$\≈I_P/{I_G}^{\′}=\sqrt{\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{\left(\frac{P_i-P}{P}\right)}^2}/\sqrt{\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{\left(\frac{G_i-\stackrel{\‾}{G}}{\stackrel{\‾}{G}}\right)}^2}---\left(23\right)$

6. comprehensive above each partial model obtains following whole synthesis model:

$G=(1-V{2}^{\′})\·\left[\underset{k=1}{\overset{m-1}{\mathrm{\Σ}}}\right[(\frac{1-V1}{m})\·{\left(\frac{k}{m-1}\right)}^{V1}\·G\left(t_k\right)]+(\frac{1-V1}{m}+V1)\·G\left(t_m\right)]+V{2}^{\′}\·\stackrel{\‾}{G}---\left(24\right)$

$G\left(t\right)=\underset{i=1}{\overset{n}{\mathrm{\Σ}}}\left\{\right[(\underset{j=1}{\overset{n}{\mathrm{\Σ}}}{|P_j\left(t\right)-P\left(t\right)|}^{\mathrm{\η}}-{|P_i\left(t\right)-P\left(t\right)|}^{\mathrm{\η}})/\left((n-1)\underset{j=1}{\overset{n}{\mathrm{\Σ}}}{|P_j\left(t\right)-P\left(t\right)|}^{\mathrm{\η}}\right)\left]\left(\frac{G_i}{P_i\left(t\right)}\right)P\left(t\right)\right\}---\left(25\right)$

$\mathrm{\η}=\sqrt{\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{\left(\frac{P_1\left(t\right)-P\left(t\right)}{P\left(t\right)}\right)}^2/\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{\left(\frac{G_1-\stackrel{\‾}{G}}{\stackrel{\‾}{G}}\right)}^2}---\left(26\right)$

Weaken model by subjective factor and obtain the rock mass characteristic desired value

To the result of determination of 3 characteristic index of engineering rock mass, bring subjective factor into and weaken model in every test result that psychological before measurement is obtained and the abovementioned steps, obtain rock mass characteristic index modified value, this modified value will be more near true value.

Undertaken by following two steps:

1. the test result P that obtains for Pretesting _i∈ (P ₁, P ₂, P ₃... P _n), (P that records when adopting strong stimulation and weak stimulation respectively ^H ₁, P ^H ₂, P ^H ₃... P ^H _n) and (P ^L ₁, P ^L ₂, P ^L ₃... P ^L _n) the corresponding G of calculating ^HAnd G ^L:

Wherein:

60 and 40 is respectively the threshold value that is used to divide strong stimulation and weak stimulation in the before measurement.

2. for three rock mass ratings (surface condition, structural, be subjected to level of disruption), except that the before measurement of the before measurement of the corresponding fringe gray level contrast of surface condition and the full degree of structural corresponding cake, rock mass is subjected to the before measurement of level of disruption correspondence to be meant the joint test achievement of " fringe gray level contrast " and " the full degree of cake ", brings following formula into and calculates:

Wherein:

G _StripedThe optimal value that the before measurement data computation that the expression employing contrasts about fringe gray level obtains;

G _CakeThe optimal value that expression adopts the before measurement data computation about the full degree of cake to obtain.

By aforementioned whole synthesis model, can eliminate the slip-stick artist when rock mass experience index is judged, the influence of its sensitivity, mood, makings and the structure of knowledge obtains one more near the optimal value of objective reality situation.

Carry out the judgement of rock mass damage at last

To bring broad sense Hoek-Brown criterion in 2002 more near the rock mass characteristic index modified value of true value by what subjective factor weakened that model obtains, wherein:

σ _CiBe the sillar uniaxial compressive strength, directly obtain by experiment;

m _bBe the rock material constant, the method that provides according to Hoek directly calculates;

D is the engineering perturbation index of rock mass, takes from through subjective factor to weaken the modified value that model obtains;

GSI is the rock mass geology intensity index, weakens " surface condition " index that model obtains and the modified value of " structural " index by subjective factor.According to the following table value:

According to following formula, whether the decidable rock mass damage to be:

The critical value of the big principle stress of rock mass ${\mathrm{\σ}}_1^{\′}={\mathrm{\σ}}_3^{\′}+{\mathrm{\σ}}_{\mathrm{ci}}{(m_b\frac{{\mathrm{\σ}}_3^{\′}}{{\mathrm{\σ}}_{\mathrm{ci}}}+s)}^a---\left(56\right)$

M wherein _bBe rock mass materials constant m _iReduced value,

$m_b=m_i\exp \left(\frac{\mathrm{GSI}-100}{28-14D}\right)---\left(57\right)$

S and a are the rock mass constants,

$s=\exp \left(\frac{\mathrm{GSI}-100}{9-3D}\right)---\left(58\right)$

$a=\frac{1}{2}+\frac{1}{6}(e^{-\mathrm{GSI}/15}-e^{-20/3})---\left(59\right)$

D is the engineering excitation parameter of reflection construction disturbance, and its value changes to 1 (extremely strong disturbance) from 0 (without the protolith of disturbance).

Uniaxial compressive strength: σ _c=σ _Cis ^a(60)

Tensile strength:

${\mathrm{\σ}}_t=-\frac{s{\mathrm{\σ}}_{\mathrm{ci}}}{m_b}---\left(61\right)$

Normal stress:

${\mathrm{\σ}}_n^{\′}=\frac{{\mathrm{\σ}}_1^{\′}+{\mathrm{\σ}}_3^{\′}}{2}-\frac{{\mathrm{\σ}}_1^{\′}-{\mathrm{\σ}}_3^{\′}}{2}\·\frac{d{\mathrm{\σ}}_1^{\′}/d{\mathrm{\σ}}_3^{\′}-1}{d{\mathrm{\σ}}_1^{\′}/d{\mathrm{\σ}}_3^{\′}+1}---\left(62\right)$

Shear stress:

$\mathrm{\τ}=({\mathrm{\σ}}_1^{\′}-{\mathrm{\σ}}_3^{\′})\frac{\sqrt{d{\mathrm{\σ}}_1^{\′}/d{\mathrm{\σ}}_3^{\′}}}{d{\mathrm{\σ}}_1^{\′}/d{\mathrm{\σ}}_3^{\′}+1}---\left(63\right)$

Wherein

$d{\mathrm{\σ}}_1^{\′}/d{\mathrm{\σ}}_3^{\′}=1+a{m}_b{(m_b{\mathrm{\σ}}_3^{\′}/{\mathrm{\σ}}_{\mathrm{ci}}+s)}^{a-1}---\left(64\right)$

The deformation modulus of rock mass,

$E_m\left(\mathrm{GPa}\right)=(1-\frac{D}{2})\sqrt{\frac{{\mathrm{\&sigma;}}_{\mathrm{<figure-callout\; id="100"\; label="ci"\; filenames="A20051004305400282.png"\; state="\{\{state\}\}">ci</figure-callout>}}}{100}}\&CenterDot;10^{((\mathrm{GSI}-10)/40)}---\left(65\right)$

Angle of equivalent internal friction and cohesion,

${\mathrm{\&phi;}}^{\&prime;}={\sin }^{-1}\left[\frac{{6\mathrm{am}}_b{(s+m_b{\mathrm{\&sigma;}}_{3n}^{\&prime;})}^{a-1}}{2(1+a)(2+a)+6a{m}_b{(s+m_b{\mathrm{\&sigma;}}_{3n}^{\&prime;})}^{a-1}}\right]---\left(66\right)$

$c^{\&prime;}=\frac{{\mathrm{\&sigma;}}_{\mathrm{ci}}[(1+2a)s+(1-a)m_b{\mathrm{\&sigma;}}_{3n}^{\&prime;}]{(s+m_b{\mathrm{\&sigma;}}_{3n}^{\&prime;})}^{a-1}}{(1+a)(2+a)\sqrt{1+\left(6a{m}_b{(s+m_b{\mathrm{\&sigma;}}_{3n}^{\&prime;})}^{a-1}\right)/\left((1+a)(2+a)\right)}}---\left(67\right)$

In the formula:

σ _3n＝σ3 _max′/σ _ci′ (68)

The stress that produces on the border surpasses σ _cThe time, excavation will produce destruction on the border.Destruction to a biaxial stress regional spread, and surpasses caused σ when local intensity from initial point ₁' and σ ₃In ' time, finally reach stable.

When to the overall permanence of rock mass and non local destruction when judging, the bulk strength of rock mass is obtained by following formula:

${\mathrm{\&sigma;}}_{\mathrm{cm}}^{\&prime;}=\frac{2c^{\&prime;}\cos {\mathrm{\&phi;}}^{\&prime;}}{1-\sin {\mathrm{\&phi;}}^{\&prime;}}---\left(69\right)$

If equivalent cohesion c ', angle of equivalent internal friction φ ' are at σ _t＜σ ₃'＜σ _CiScope in obtain, formula 70 then arranged:

${\mathrm{\&sigma;}}_{\mathrm{cm}}^{\&prime;}={\mathrm{\&sigma;}}_{\mathrm{ci}}\&CenterDot;\frac{(m_b+4s-a(m_b-8s)){(m_b/4+s)}^{a-1}}{2(1+a)(2+a)}---\left(70\right)$

σ _3MAX' minute following two kinds of situation values:

Tunnel---σ here _3MAX' provided two kinds of failure criteria for the equivalent characteristic of buried chamber or the equivalent sedimentation outline line of shallow embedding chamber;

Side slope---safety coefficient, destruction face shape and the position of calculating here all must be by equivalences.

Under the situation that does not have slump to take place, for the shallow tunnel of buried depth less than 3 times of cave footpaths, the comparison numerical value research that destroys development has all provided the relation identical with deep tunnel with the size of ground settlement:

$\frac{{\mathrm{\&sigma;}}_{3\max }^{\&prime;}}{{\mathrm{\&sigma;}}_{\mathrm{cm}}^{\&prime;}}=0.47{\left(\frac{{\mathrm{\&sigma;}}_{\mathrm{cm}}^{\&prime;}}{\mathrm{\&gamma;H}}\right)}^{-0.94}---\left(71\right)$

σ wherein _Cm' be rock mass strength, γ is the unit weight of rock mass, H is the buried depth of chamber.

At the horizontal stress position higher, should replace γ h with the horizontal stress value than vertical stress.

For side slope, utilize the general Bishop principle of being the very image of, analyze according to following formula:

$\frac{{\mathrm{\&sigma;}}_{3\max }^{\&prime;}}{{\mathrm{\&sigma;}}_{\mathrm{cm}}^{\&prime;}}=0.72{\left(\frac{{\mathrm{\&sigma;}}_{\mathrm{cm}}^{\&prime;}}{\mathrm{\&gamma;H}}\right)}^{-0.91}---\left(72\right)$

Wherein H is the height of side slope.

By input pointer in Hoek-Brown rock mass experience criterion, geology intensity index GSI and perturbation index D, experiment in the artificial decision process has proved that method of the present invention can be eliminated effectively or has significantly reduced the negative effect that quantizes subjective factor individual in the decision process.

Judgement to the rock surface condition

The subjective determination of surface condition achievement of the present invention

Tested

The mean value modified value

1 56.2 55.2

2 68.6 52.86

3 43.2 48.66

Judgement to the rock mass structure index

Structural subjective determination achievement of the present invention

Tested

The mean value modified value

1 77.2 67.72

2 83.2 62.48

3 62.6 64.45

Rock mass is subjected to the judgement of disturbance index

To the subjective determination of perturbation achievement of the present invention

Tested

The mean value modified value

1 0.38 0.39

2 0.54 0.42

3 0.24 0.33

The system prediction achievement of rock mass physical mechanics parameter

Tested	Input	The initial index of Hoek-Brown criterion						The rock mass parameter that the Hoek-Brown criterion is calculated
														GSI	D	σ ci	Mi	Unit weight	Buried depth	C	Phi	σ t	σ c	σ cm	E m
		-	-	MPa	-	Kg/m 3	m	-	°	MPa	MPa	MPa	MPa
														1	Mean value	53.17	0.38	30	10	0.026	50	0.33	47.9	0.06	1.48	4.55	5324
Optimal value	48.75	0.39	0.28	46.4	-0.04	1.09	4.03	4103
									2	Mean value	61.49	0.54	30		10	0.026	50					0.41	48.9	-0.11	2.18	5.13	7747
Optimal value	46.00	0.42	0.25	45	-0.03	0.87	3.66	3437
										3	Mean value	41.9		0.24				30	10	0.026	50	0.25	45.7	-0.03	0.84	3.77	3023
Optimal value	44.98	0.33	0.26	45.7	-0.03	0.92	3.82	3425

The calculating achievement comparative analysis

		C	Phi	σ t	σ c	σ cm	E m
								-	°	MPa	MPa	MPa	MPa
		The slip-stick artist directly imports	max	0.41	48.9	-0.03	2.18							5.13	7747
min	0.25							45.7	-0.11	0.84	3.77	3023
			|max-min|	0.16	3.2	0.08	1.34						1.36	4724
After reducing interference caused by subjective factors	max							0.28	46.4	-0.03	1.09	4.03			4103
		min	0.25	45	-0.04	0.87	3.66						3425
	|max-min|							0.03	1.4	0.01	0.22	0.37		678
		| max-min| reduces relatively	81.25％	56.25％	87.50％	83.58％	72.79％						85.65％

As can be seen from the above table, under the conventional situation Hoek-Brown criterion according to 3 slip-stick artists separately to the subjective determination of GSI and D, each the rock mass parameter that calculates, its uncertain region is bigger.After adopting method of the present invention tested interference caused by subjective factors separately weakens to 3, the Hoek-Brown criterion method that the rock mass parameter that calculates is more traditional is more reliable, and uncertain region has reduced 50%～80% respectively.

Claims (9)

1. the method for a determination of engineering rock mass damage, by individual subjective tendency characteristic factor and slip-stick artist quantification is carried out in the judgement of actual rock mass rating to Geotechnical Engineering teacher, bringing the quantification value into subjective factor weakens in the model and revises, obtain rock mass characteristic index modified value, carry out the judgement of rock mass damage according to this index modified value, it is characterized in that this method is carried out according to the following steps:

At first, Geotechnical Engineering's teacher psychological tendency feature is carried out before measurement

Makings, mood, sensitivity, 4 subjective tendency characteristic factors of the structure of knowledge are carried out systemic Pretesting, obtain the quantitative target of the different people's difference on these 4 kinds of subjective tendency characteristic factors of reflection;

Carry out the result of determination quantification of Geotechnical Engineering teacher again to actual rock mass rating

Allow the slip-stick artist that surface condition index, structural index and perturbation index are judged successively, requiring the slip-stick artist that same rock mass rating is carried out 5 times continuously judges, and 5 times the result can not be all identical, respectively according to the surface condition index reference system of engineering rock mass, structural index reference system and perturbation index reference system, with its result of determination quantification;

Set up subjective factor and weaken model

At first the before measurement data are carried out pre-service, foundation reduces the sensitivity differentia influence, reduces the makings differentia influence, reduces the partial model that excited level difference influences, reduces structure of knowledge differentia influence respectively again, last comprehensive each partial model obtains the whole synthesis subjective factor and weakens model;

Weaken model by subjective factor and obtain rock mass characteristic index modified value

To the result of determination of 3 characteristic index of engineering rock mass, bring subjective factor into and weaken model in every test result that psychological before measurement is obtained and the last step, obtain rock mass characteristic index modified value;

Carry out the judgement of rock mass damage at last

To weaken the broad sense Hoek-Brown criterion that rock mass characteristic index modified value that model obtains was brought into 2002 by subjective factor, whether judge rock mass damage.

2. method according to claim 1 is characterized in that, described makings, mood, sensitivity, the structure of knowledge are carried out systemic Pretesting is to carry out by the following method,

The test of gas quality index

Makings is divided into 4 types of the quality of bile, sanguine temperament, mucilaginous substance, melancholy, the slip-stick artist selects of the most approaching oneself reaction tendency in above 4 types, the index that each temperament type is defined its quantification is: the quality of bile---0.6, sanguine temperament---0.6, mucilaginous substance---0.8, melancholy---0.4;

The test of sentiment indicator

Adopt the sentiment indicator of the scaling method Test Engineer in the experimental psychology, allow the slip-stick artist in scale, select the predicate of approaching current psychological condition, then the score of slip-stick artist's predicate selection is accumulated, obtain this project's teacher sentiment indicator;

The test of sensitivity index and structure of knowledge index

Obtain the quantification decision data of slip-stick artist by test to these abstraction graphs to the every performance index dependence test of rock mass figure, abstract respectively to surface condition and structural impression be that the gray scale striped is contrasted and the impression of the full degree of cake, regulation stimulus intensity scope is between 0～100, strong stimulation is a stimulus intensity value that generates at random in 60～80 closed interval scope, the weak stimulation is a stimulus intensity value that generates at random in 20～40 closed interval scope, the slip-stick artist is 2.3 to the sensation difference limen of fringe gray level contrast, sensation difference limen to the full degree of cake is 3.4, by tested slip-stick artist the impression of resolution chart contrast is obtained strong stimulation and weak stimulus intensity value, promptly obtain sensitivity index and structure of knowledge index.

3. method according to claim 1 is characterized in that, the described subjective factor of setting up weakens and in the model before measurement data to be carried out pre-service and be meant,

Order: P _i∈ (P ₁, P ₂, P ₃... P _n) be the slip-stick artist that obtains in the before measurement evaluation of estimate, n=5 to master drawing;

P is P _iArithmetic mean;

P is the true value of this time before measurement;

G _i∈ (G ₁, G ₂, G ₃... G _n) be the evaluation of estimate of slip-stick artist to certain rock mass rating, n=5;

G is G _iArithmetic mean;

G is the true value of this rock mass rating;

The before measurement data are satisfied: 0≤P _i≤ 2P, (1)

Reflect the dispersion index of every input value of this time before measurement with respect to arithmetic mean value dispersion degree,

${I_P}^{\&prime;}=\frac{1}{n}\sqrt{\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\left(\frac{P_i-\stackrel{\&OverBar;}{P}}{\stackrel{\&OverBar;}{P}}\right)}^2},0\&le;{I_P}^{\&prime;}\&le;1----\left(2\right)$

Reflect the dispersion index of every input value of this time before measurement with respect to the true value dispersion degree,

$I_P=\frac{1}{n}\sqrt{\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\left(\frac{P_i-P}{P}\right)}^2},0\&le;I_P\&le;1----\left(3\right)$

Every input that this project teacher estimates this rock mass rating is with respect to the dispersion index of arithmetic mean value dispersion degree,

${I_G}^{\&prime;}=\frac{1}{n}\sqrt{\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\left(\frac{G_i-\stackrel{\&OverBar;}{G}}{\stackrel{\&OverBar;}{G}}\right)}^2},0\&le;{I_G}^{\&prime;}\&le;1----\left(4\right)$

Every input that this project teacher estimates this rock mass rating is with respect to the dispersion index of true value dispersion degree,

$I_G=\frac{1}{n}\sqrt{\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\left(\frac{G_i-G}{G}\right)}^2},0\&le;I_G\&le;1----\left(5\right)$

Relatively stable degree and the relative degree that misfits that shows rule in before measurement to the rock mass rating evaluation are referred to as the coefficient of alienation, with respect to the coefficient of alienation of arithmetic mean,

I′＝I _G′/I _p′ (6)

The coefficient of alienation with respect to true value has

I＝I _G/I _P≈I _G′/I _P。(7)

4. method of estimation according to claim 1 is characterized in that, the described sensitivity differentia influence model of setting up in the subjective factor weakening model that reduces of setting up is:

$G=\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}\left({\mathrm{\&omega;}}_{\mathrm{pi}}{G}_i\right)----\left(8\right)$

Weight wherein

${\mathrm{\&omega;}}_{\mathrm{pi}}=\frac{\underset{j=1}{\overset{n}{\mathrm{\&Sigma;}}}|P_j-P|-|P_i-P|}{(n-1)\underset{j=1}{\overset{n}{\mathrm{\&Sigma;}}}|P_j-P|}----\left(9\right)$

$0\&le;{\mathrm{\&omega;}}_i\&le;\frac{1}{n-1}.----\left(10\right)$

5. method of estimation according to claim 1 is characterized in that, the described makings differentia influence model of setting up in the subjective factor weakening model that reduces of setting up is:

$G^{\left(1\right)}=\underset{k=1}{\overset{m-1}{\mathrm{\&Sigma;}}}[\left(\frac{1-V1}{m}\right)\&CenterDot;{\left(\frac{k}{m-1}\right)}^{V1}\&CenterDot;G\left(t_k\right)]+(\frac{1-V1}{m}+V1)\&CenterDot;G\left(t_m\right)----\left(15\right)$

If order $Q\left(t_k\right)=\left(\frac{1-V1}{m}\right)\&CenterDot;{\left(\frac{k}{m-1}\right)}^{V1},(k=\mathrm{1,2,3},...m-1)----\left(16\right)$

$Q\left(t_m\right)=\frac{1-V1}{m}+V1----\left(17\right)$

Then, [Q] is the weight factor of reflection historical law influence,

So, the rock mass rating evaluation of estimate after the effect of consideration makings

$G^{\left(1\right)}=\underset{k=1}{\overset{m-1}{\mathrm{\&Sigma;}}}[Q\left(t_k\right)\&CenterDot;G\left(t_k\right)]+Q\left(t_m\right)\&CenterDot;G\left(t_m\right),----\left(18\right)$

$=\underset{k=1}{\overset{m}{\mathrm{\&Sigma;}}}[Q\left(t_k\right)\&CenterDot;G\left(t_k\right)]$

Wherein, G ⁽¹⁾For having considered the achievement after practice effect (makings) influences;

V1 is the makings influence index value that current slip-stick artist obtains in this before measurement;

t _k(k=1,2,3 ... m) expression this project teacher uses t historical the k time _mRepresent current access times;

G (t _k) expression last time the slip-stick artist to the evaluation of rock mass rating, after the rule of having applied mechanically gained before measurement data when extracting on this project's teacher history the k time login system is revised, the historical true value that obtains.

6. method according to claim 1 is characterized in that, the described subjective factor of setting up weakens the excited level difference of setting up in the model that reduces and influences model and be:

Order: G ⁽²⁾For having considered the achievement behind the emotion influence;

V2 is current slip-stick artist's an emotion influence index,

Then have:

G ⁽²⁾＝(1-V2′)·G ⁽¹⁾+V2′· G (19)

Wherein: V2 '=α V2 is the modified value of emotion influence index V2, herein α=0.5;

G ⁽¹⁾For having considered the achievement after practice effect influences;

G is the arithmetic mean value of rock mass rating input.

7. method according to claim 1 is characterized in that, the described structure of knowledge differentia influence model of setting up in the subjective factor weakening model that reduces of setting up is:

The slip-stick artist is departed from the situation of true value to the input value integral body of rock mass rating, has:

$G=\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}\left({\mathrm{\&omega;}}_{\mathrm{pi}}{G}_i\right)----\left(20\right)$

${\mathrm{\&omega;}}_{\mathrm{pi}}=\frac{\underset{j=1}{\overset{n}{\mathrm{\&Sigma;}}}|P_j-P|-|P_i-P|}{(n-1)\underset{j=1}{\overset{n}{\mathrm{\&Sigma;}}}|P_j-P|};----\left(21\right)$

To I _GDepart from I _pThe situation of degree varies has:

${\mathrm{\&omega;}}_{\mathrm{pi}}=\frac{\underset{j=1}{\overset{n}{\mathrm{\&Sigma;}}}{|P_j-P|}^{\mathrm{\&eta;}}-{|P_i-P|}^{\mathrm{\&eta;}}}{(n-1)\underset{j=1}{\overset{n}{\mathrm{\&Sigma;}}}{|P_j-P|}^{\mathrm{\&eta;}}}----\left(22\right)$

Exponential term

η＝1/I＝I _p/I _G

$\&ap;\frac{I_p}{{I_G}^{\&prime;}}=\sqrt{\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\left(\frac{P_i-P}{P}\right)}^2}/\sqrt{\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\left(\frac{G_i-\stackrel{\&OverBar;}{G}}{\stackrel{\&OverBar;}{G}}\right)}^2}.----\left(23\right)$

8. method according to claim 1 is characterized in that, the described whole synthesis subjective factor weakening model of setting up in the subjective factor weakening model of setting up is:

$G=(1-{V2}^{\&prime;})\&CenterDot;\left[\underset{k=1}{\overset{m-1}{\mathrm{\&Sigma;}}}\right[\left(\frac{1-V1}{m}\right)\&CenterDot;{\left(\frac{k}{m-1}\right)}^{V1}\&CenterDot;G\left(t_k\right)]+(\frac{1-V1}{m}+V1)\&CenterDot;G\left(t_m\right)]+{V2}^{\&prime;}\&CenterDot;\stackrel{\&OverBar;}{G}----\left(24\right)$

$G\left(t\right)=\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}\left\{\right[(\underset{j=1}{\overset{n}{\mathrm{\&Sigma;}}}{|P_j\left(t\right)-P\left(t\right)|}^{\mathrm{\&eta;}}-{|P_i\left(t\right)-P\left(t\right)|}^{\mathrm{\&eta;}})/\left((n-1)\underset{j=1}{\overset{n}{\mathrm{\&Sigma;}}}{|P_j\left(t\right)-P\left(t\right)|}^{\mathrm{\&eta;}}\right)\left]\left(\frac{G_i}{P_i\left(t\right)}\right)P\left(t\right)\right\}----\left(25\right)$

$\mathrm{\&eta;}=\sqrt{\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\left(\frac{P_i\left(t\right)-P\left(t\right)}{P\left(t\right)}\right)}^2/\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\left(\frac{G_i-\stackrel{\&OverBar;}{G}}{\stackrel{\&OverBar;}{G}}\right)}^2}.----\left(26\right)$

9. method according to claim 1 is characterized in that, whether described broad sense Hoek-Brown criterion according to 2002 judges rock mass damage, is to adopt following method:

When judging the local characteristics destruction of rock mass,

The critical value of the big principle stress of rock mass

${\mathrm{\&sigma;}}_1^{\&prime;}={\mathrm{\&sigma;}}_3^{\&prime;}+{\mathrm{\&sigma;}}_{\mathrm{ci}}{(m_b\frac{{\mathrm{\&sigma;}}_3^{\&prime;}}{{\mathrm{\&sigma;}}_{\mathrm{ci}}}+s)}^a----\left(56\right)$

M wherein _bBe rock mass materials constant m _iReduced value,

$m_b=m_i\exp \left(\frac{\mathrm{GSI}-100}{28-14D}\right)----\left(57\right)$

S and a are the rock mass constants,

$s=\exp \left(\frac{\mathrm{GSI}-100}{9-3D}\right)----\left(58\right)$

$a=\frac{1}{2}+\frac{1}{6}(e^{-\mathrm{GSI}/15}-e^{-20/3})----\left(59\right)$

D is the engineering excitation parameter of reflection construction disturbance, and its value changes to 1 (extremely strong disturbance) from 0 (without the protolith of disturbance),

Uniaxial compressive strength: σ _c=σ _Cis ^a(60)

Tensile strength:

${\mathrm{\&sigma;}}_t=-\frac{s{\mathrm{\&sigma;}}_{\mathrm{ci}}}{m_b}----\left(61\right)$

Normal stress:

${\mathrm{\&sigma;}}_n^{\&prime;}=\frac{{\mathrm{\&sigma;}}_1^{\&prime;}+{\mathrm{\&sigma;}}_3^{\&prime;}}{2}-\frac{{\mathrm{\&sigma;}}_1^{\&prime;}-{\mathrm{\&sigma;}}_3^{\&prime;}}{2}\&CenterDot;\frac{\frac{d{\mathrm{\&sigma;}}_1^{\&prime;}}{d{\mathrm{\&sigma;}}_3^{\&prime;}}-1}{\frac{d{\mathrm{\&sigma;}}_1^{\&prime;}}{d{\mathrm{\&sigma;}}_3^{\&prime;}}+1}----\left(62\right)$

Shear stress:

$\mathrm{\&tau;}=({\mathrm{\&sigma;}}_1^{\&prime;}-{\mathrm{\&sigma;}}_3^{\&prime;})\frac{\sqrt{\frac{d{\mathrm{\&sigma;}}_1^{\&prime;}}{d{\mathrm{\&sigma;}}_3^{\&prime;}}}}{\frac{d{\mathrm{\&sigma;}}_1^{\&prime;}}{d{\mathrm{\&sigma;}}_3^{\&prime;}}+1}----\left(63\right)$

Wherein

$\frac{d{\mathrm{\&sigma;}}_1^{\&prime;}}{d{\mathrm{\&sigma;}}_3^{\&prime;}}=1+a{m}_b{(m_b\frac{{\mathrm{\&sigma;}}_3^{\&prime;}}{{\mathrm{\&sigma;}}_{\mathrm{ci}}}+s)}^{a-1}----\left(64\right)$

The deformation modulus of rock mass,

$E_m\left(\mathrm{GPa}\right)=(1-\frac{D}{2})\sqrt{\frac{{\mathrm{\&sigma;}}_{\mathrm{ci}}}{100}}\&CenterDot;{10}^{((\mathrm{GSI}-10)/40)}----\left(65\right)$

Angle of equivalent internal friction and cohesion,

${\mathrm{\&phi;}}^{\&prime;}={\sin }^{-1}\left[\frac{6a{m}_b{(s+m_b{\mathrm{\&sigma;}}_{3n}^{\&prime;})}^{a-1}}{2(1+a)(2+a)+6a{m}_b{(s+m_b{\mathrm{\&sigma;}}_{3n}^{\&prime;})}^{a-1}}\right]----\left(66\right)$

$c^{\&prime;}=\frac{{\mathrm{\&sigma;}}_{\mathrm{ci}}[(1+2a)s+(1-a)m_b{\mathrm{\&sigma;}}_{3n}^{\&prime;}]{(s+m_b{\mathrm{\&sigma;}}_{3n}^{\&prime;})}^{a-1}}{(1+a)(2+a)\sqrt{1+\left(6a{m}_b{(s+m_b{\mathrm{\&sigma;}}_{3n}^{\&prime;})}^{a-1}\right)/((1+a)(2+a))}}----\left(67\right)$

In the formula:

σ _3n＝σ _3max′/σ _ci′， (68)

When judging the overall permanence destruction of rock mass,

The rock mass bulk strength is:

${\mathrm{\&sigma;}}_{\mathrm{cm}}^{\&prime;}=\frac{2c^{\&prime;}\cos {\mathrm{\&phi;}}^{\&prime;}}{1-\sin {\mathrm{\&phi;}}^{\&prime;}}----\left(69\right)$

If equivalent cohesion c ', angle of equivalent internal friction φ ' are at σ _t'＜σ ₃'＜σ _CiScope in obtain, then have:

${\mathrm{\&sigma;}}_{\mathrm{cm}}^{\&prime;}={\mathrm{\&sigma;}}_{\mathrm{ci}}\&CenterDot;\frac{(m_b+4s-a(m_b-8s)){(m_b/4+s)}^{a-1}}{2(1+a)(2+a)}.----\left(70\right)$

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