System reliability and sensitivity analysis method for double piles in spatial variation expansive soil

Abstract

The invention provides a system reliability and sensitivity analysis method for double piles in spatial variation expansive soil, and belongs to the technical field of pile foundation engineering reliability analysis. The method comprises the following steps: establishing a double-pile foundation model and a soil property parameter random field model; element reliability indexes and element sensitivity indexes of the three failure modes of the double piles are calculated; and calculating a double-pile system reliability index and a system sensitivity index. According to the method, three failure modes of double-pile pulling resistance or compression resistance failure and double-pile settlement differential failure under complex conditions are given, a system reliability and sensitivity analysis method of the double-pile foundation is established, and the influence of expansive soil space variability, double-pile interaction, pile-soil interface nonlinear relation and matric suction change is considered. The method has high calculation precision and low calculation cost, and is beneficial for rock and soil practitioners to carry out safer reliability evaluation and design on the double-pile foundation under complex geological conditions.

Description

Technical Field

[0001] The invention belongs to the technical field of reliability analysis of pile foundation engineering, and in particular relates to a system reliability and sensitivity analysis method of double piles in spatially variable expansive soil. Background Art

[0002] Performance evaluation of pile groups in unsaturated expansive soils has always been a major challenge in geotechnical engineering due to the influence of internal and external factors. Internal factors mainly include the complex interactions between piles at different locations and at the pile-soil interface, spatial variability of soil properties, etc. External factors mainly include different levels of loads, rainfall and seepage, etc. Therefore, it is necessary to study the various failure modes of pile groups under complex conditions and analyze their system reliability and system sensitivity of soil parameters. It helps engineers understand the failure mechanism, failure mode and potential risks of pile groups, so as to better guide pile foundation engineering design and disaster prevention and mitigation.

[0003] The complex internal pile-soil interaction makes it difficult to accurately evaluate the bearing capacity of pile groups. At the same time, the spatial variability of soil parameters increases the uncertainty of bearing capacity, leading to risk assessment of pile foundation performance. In addition, the difference in soil properties at different locations will cause settlement differences between different piles, which will cause settlement difference damage in severe cases. Rainfall and seepage will reduce the matrix suction of the soil, causing the expansive soil to produce lateral expansion pressure on the pile side, thereby reducing the shear strength of the pile-soil interface, weakening the bearing capacity and pull-out resistance of the pile group, and reducing the performance of the pile group. Under different levels of load and rainfall infiltration time, the pile group can move upward or downward, and in severe cases, pull-out failure, compression failure or other failure modes will occur.

[0004] Reference 1, “Simplified non-linear approaches for response of a single pile and pile groups considering progressive deformation of pile–soil system” (Zhang Q., Liu S., Zhang S., Zhang J., Wang K., Soils and Foundations, 2016, 56(3), 473–484) proposes a simplified method for analyzing the load-displacement nonlinear response of single piles and pile groups in multi-layered soils. A hyperbolic model of each single pile in the pile group is established to describe the nonlinear relationship between the pile-soil interface lateral friction and relative displacement and between the pile end resistance and pile end displacement. The interaction between piles is also considered. However, this method is based on homogeneous soil conditions and does not take into account the spatial variability of soil properties, which will lead to incorrect risk assessment of pile foundations and cause safety accidents.

[0005] Reference 2, “Probabilistic considerations for the design of deep foundations against excessive differential settlement” (Naghibi F., Fenton G.A., Griffiths DV, 2016, Canadian Geotechnical Journal, 53(7), 1167–1175), proposed a settlement calculation method that takes into account the interaction between piles by establishing a double-pile finite element model. The variance formula of the settlement difference was also derived, and the relationship between the target maximum total settlement and the settlement difference under different loads and pile spacings was analyzed. However, this method only uses the finite element numerical simulation method under the condition of homogeneous soil to analyze the difference in bearing capacity between the two piles, but cannot analyze the settlement difference caused by the spatial variability of soil parameters, and the calculation process of this method is relatively complicated.

[0006] Reference 3, "Load displacement analysis of a single pile in an unsaturated expansive soil" (Liu Y., Vanapalli SK, 2019, Computers and Geotechnics, 106, 83–98) proposed a load transfer method calculation procedure for a single pile in an unsaturated expansive soil. The procedure takes into account the shear strength of the pile-soil interface affected by the change of matrix suction and the expansion of the expansive soil, and analyzes the two movement modes of the single pile, the upward and downward movement, and the distribution of the lateral friction and axial force. However, the calculation procedure is only applicable to single piles and has not been further extended to the analysis of pile groups.

[0007] Reference 4, “Reliability analysis of pile group in spatially variable unsaturated expansive soil based on load transfer method” (Dong X., Tan X., Guo W., Zhong K., Hou X., Ma H., Georisk: Assessment and Management of Risk for Engineered Systems and Geohazards, 2023, 18(3), 1–19) proposed a reliability analysis method for vertically loaded pile groups in spatially variable unsaturated expansive soil, which takes into account the interaction between different piles, the nonlinear relationship between the pile-soil interface, the influence of matrix suction changes, the expansion of expansive soil, and the spatial variability of soil parameters. However, this method only considers the vertical spatial variability and only establishes a simple system without considering multiple complex failure modes at the same time.

[0008] In summary, it is very important to analyze the system reliability and sensitivity of pile groups in spatially variable expansive soils. It is necessary to study the various failure modes of pile groups under complex conditions and their interrelationships, and further calculate the system reliability index of pile groups and the system sensitivity index of soil parameters. However, in the existing technology in the field of pile foundation engineering reliability analysis technology, the influence of multiple complex conditions and the analysis of the system reliability and system sensitivity of pile groups under multiple failure modes have not been considered at the same time. Summary of the invention

[0009] The technical problem to be solved by the present invention is the deficiency of the prior art. Specifically, the purpose of the present invention is to provide a system reliability and sensitivity analysis method of double piles in spatially variable expansive soil, which can simultaneously consider the interaction between different piles, the interaction between piles and soil, the spatial variability of expansive soil, the effect of rainfall infiltration, the load level, and the three failure modes of compressive failure, tensile failure and settlement difference failure of the piles. The higher calculation accuracy and lower calculation cost make it more suitable for the system reliability and sensitivity analysis of double pile foundations under complex conditions in actual engineering.

[0010] In order to solve the above technical problems, the present invention provides a system reliability and sensitivity analysis method of double piles in spatially variable expansive soil, comprising the following steps:

[0011] Step 1: Establish a double pile foundation model and a soil parameter random field model

[0012] Step 1.1, the double pile foundation model consists of two piles vertically inserted into the expansive soil foundation and the expansive soil foundation. Assume that the upper surface of the expansive soil foundation is flush with the ground surface, and take a cross section perpendicular to the earth's surface as the expansive soil foundation. The expansive soil foundation includes a soil rectangle and two pile rectangles symmetrically embedded with their axis from the outside to the inside. The left pile is pile No. 1 and the right pile is pile No. 2. The sizes and materials of the two piles are exactly the same; the length of the double piles embedded in the soil is L, the pile diameter is D, and the distance between the double piles is r p , the elastic modulus is E p , the horizontal side length of the soil rectangle is K times the pile diameter D, and the vertical height is H times the pile diameter D;

[0013] With the lower left corner vertex of the soil rectangle as the origin, a plane coordinate system is established and each pile rectangle is divided into N p A square grid with a side length of pile diameter D is constructed, and N p The grids are sorted, and the serial numbers are k=1, 2, ..., N. p ; Divide the soil rectangle into N e A square grid with a side length of pile diameter D is constructed from bottom to top and from left to right. e The grids are sorted, and the coordinates of the center point of each expansive soil grid are used as the coordinates of the expansive soil grid;

[0014] Step 1.2, the soil parameters are 3, namely effective cohesion c a ', effective friction angle δ' and saturated elastic modulus E sat , any one of them is recorded as soil parameter Γ h, h is the serial number of soil parameter, h = 1, 2, 3; the soil parameter random field model is the probability distribution model of soil parameter random field, the probability distribution model includes probability distribution type and autocorrelation function type, record the three soil parameters Γ h All are log-normal distribution and square exponential autocorrelation function;

[0015] The Karhunen–Loève expansion method is used to calculate the three soil parameters Γ in each grid. h The lognormal random field discrete value And recorded as discrete values Its expression is:

[0016]

[0017] Where ξ is a random vector in an independent standard normal space and is denoted as random vector ξ, ξ is a 1×3M matrix, M is the number of truncation terms; λ j is the jth eigenvalue in the eigenvalue matrix λ, where j is the number of cutoff terms, j = 1, 2, …, M; f j (x) is the characteristic function of the jth column in the characteristic function matrix f(x); μ h is the soil parameter Γ h The mean of h is the soil parameter Γ h The mean square error of ; L is the lower triangular matrix obtained by Cholesky decomposition of the random field cross-correlation coefficient matrix in the relevant standard normal space, and the superscript T represents the transpose of the matrix;

[0018] Step 2: Calculate the component reliability index and component sensitivity index of the three failure modes of double piles

[0019] Step 2.1, based on the three soil parameters Γ in step 1 h The lognormal random field discrete value The displacement s of the bottom No. 1 grid of a given set of double piles bi , i is the pile number, i = 1, 2, a set of grid displacement s of the double pile bi Based on this, the displacement s of the grid No. 1 in this group is calculated using the double-pile load transfer method that considers the change of matrix suction and the expansion of expansive soil. bi The corresponding loads on the tops of the two piles are Q i and displacement s i , and draw the relationship curve between the load Q and displacement s of each pile in a plane coordinate system;

[0020] Step 2.2, based on the load Q and displacement s relationship curve drawn in step 2.1, calculate the functional function values ​​of the three failure modes of the double piles under the vertical fixed load Q1 at the top of pile No. 1 and the vertical fixed load Q2 at the top of pile No. 2;

[0021] The first failure mode is the compression failure or pull-out failure of pile 1, which is denoted as element FM1. The expression of its functional value g1(ξ) is:

[0022]

[0023] In the formula, G p is the pile gravity; R down1 (ξ) is the ultimate bearing capacity of pile No. 1, according to the preset maximum allowable settlement s mdown Obtained from the load Q and displacement s relationship curve of pile No. 1; R up1 (ξ) is the ultimate pull-out capacity of pile No. 1, according to the preset maximum allowable upward displacement s mup It is obtained from the relationship curve between the load Q and displacement s of pile No. 1; s1(ξ) is the displacement of the pile top of pile No. 1, s1(ξ)≥0 means that the pile top moves downward, and s1(ξ)<0 means that the pile top moves upward;

[0024] The second failure mode is the compression failure or pull-out failure of pile 2, which is denoted as element FM2. The expression of its functional value g2(ξ) is:

[0025]

[0026] In the formula, R down2 (ξ) is the ultimate bearing capacity of pile 2, according to the preset maximum allowable settlement s mdown Obtained from the load Q and displacement s relationship curve of pile 2; R up2 (ξ) is the ultimate pull-out capacity of pile No. 2, according to the preset maximum allowable upward displacement s mup It is obtained from the relationship curve between the load Q and displacement s of pile No. 2; s2(ξ) is the displacement of the pile top of pile No. 2, s2(ξ)≥0 means that the pile top moves downward, and s2(ξ)<0 means that the pile top moves upward;

[0027] The third failure mode is the settlement difference Δ caused by the double piles s Exceeding the maximum allowable settlement difference Δ m , denoted as element FM3, the expression of its functional value g3(ξ) is:

[0028] g3(ξ)=Δ m -Δ s =Δ m -|s1(ξ)-s2(ξ)|

[0029] Step 2.3, based on the three failure mode function values ​​g calculated in step 2.2 q(ξ), q is the serial number of the three failure modes, q = 1, 2, 3, and the first-order reliability method is used to calculate the component reliability index β corresponding to each failure mode. q , component reliability index β q The higher it is, the safer the component is;

[0030] Step 2.4, calculate the three soil parameters Γ respectively h For each component reliability index β q The sensitivity index S q h The higher the sensitivity index, the stronger the soil property parameter Γ h The variability of the component reliability index β q The greater the influence, the calculation formula is as follows:

[0031]

[0032] In the formula, α q (j) is the direction cosine vector α of the qth failure mode q The j-th direction cosine value in ;

[0033] Step 3: Calculate the reliability index and system sensitivity index of the dual pile system

[0034] The three failure modes of the double pile foundation are in series relationship. The component reliability index β of the three failure modes calculated in step 2 is q , using the sequential component method to calculate the system reliability index β sys ;

[0035] Step 3.1, combine components FM1 and FM2 into a combined component E, and calculate the reliability index β of the combined component E 1or2 , the calculation formula is:

[0036] β 1or2 =-Φ -1 [Φ(-β1)+Φ(-β2)-Φ(-β 1|2 )Φ(-β2)]

[0037] Where Φ is the standard normal cumulative distribution function, Φ -1 (·) is the inverse function of the standard normal cumulative distribution; β1 is the reliability index of component FM1; β2 is the reliability index of component FM2; β 1|2 is the reliability index of component FM1 in the case of component FM2 failure, and the calculation formula is: F2 is parameter 1, F2 = φ(-β2) / Φ(-β2), where φ is the standard normal probability density function, W2 is parameter 2, W2 = F2(-β2+F2), ρ 12is the correlation coefficient between element FM1 and element FM2, ρ 12 =α1 T α2, α1 is the direction cosine vector of element FM1, α2 is the direction cosine vector of element FM2;

[0038] Step 3.2, record the combination of component E and component FM3 as a system, and the system reliability index is marked as β sys , and its calculation formula is:

[0039] β sys =-Φ -1 [Φ(-β 1or2 )+Φ(-β3)-Φ(-β 1or2|3 )Φ(-β3)]

[0040] Where, β3 is the reliability index of component FM3; β 1or2|3 is the reliability index of the combined component E in the case of component FM3 failure, and the calculation formula is: F3 is parameter No. 3, F3 = φ(-β3) / Φ(-β3), W3 is parameter No. 4, W3 = F3(-β3+F3), ρ (1or2)3 is the correlation coefficient between the combined element E and the element FM3;

[0041] Calculate the system reliability index β sys Afterwards, the three soil parameters Γ are calculated respectively. h System reliability index β sys The sensitivity index S sys h , first we need to calculate the mean matrix μ of the random vector ξ ξ System reliability index β sys The system sensitivity index matrix S s , the calculation formula is:

[0042]

[0043] In the formula, record is the first partial derivative parameter, is the partial derivative parameter No. 2; α3 is the direction cosine vector of element FM3;

[0044] Calculate three soil parameters Γ separately h Sensitivity index S to system reliability index sys h , the calculation formula is as follows:

[0045]

[0046] In the formula, S s (j) is the system sensitivity index matrix S sThe j-th system sensitivity index in .

[0047] Preferably, the calculation formula of the square exponential autocorrelation function in step 1.2 is as follows:

[0048]

[0049] Where x1 and x2 are the two-dimensional coordinate vectors of the centers of any two expansive soil grids, x1 is the coordinate vector x1, and x2 is the coordinate vector x2; ρ(x1, x2) is the soil property parameter Γ h The correlation coefficient between the coordinate vector x1 and the coordinate vector x2; 11 is the coordinate component of the coordinate vector x1 in the horizontal direction; 12 is the coordinate component of the coordinate vector x1 in the vertical direction; 21 is the coordinate component of the coordinate vector x2 in the horizontal direction; 22 is the coordinate component of the coordinate vector x2 in the vertical direction; L H is the autocorrelation distance in the horizontal direction; L V is the autocorrelation distance in the vertical direction.

[0050] Preferably, the calculation steps of the eigenvalue matrix λ and the eigenfunction matrix f(x) in step 1.2 are as follows:

[0051] Step 1.2.1, according to the Galerkin method, the characteristic function f j (x)(j=1,2,…,M) is decomposed into M orthogonal basis functions h m Linear combination of (x)(m=1,2,…,M):

[0052]

[0053] Where D (j) is an M×1 column vector, D (j) =[d1 (j) ...d m (j) ...d M (j) ] T ; h(x) is a column vector of M×1, h(x)=[h1(x)…h m (x)…h M (x)] T ;

[0054] Step 1.2.2, substitute the linear expansion of the above characteristic function into the Fredholm integral equation ∫ Ω ρ(x1,x2)f j (x1)dx1=λ j f j(x2), the following matrix equation is obtained:

[0055] AD=DΛ

[0056] Where A is the generalized stiffness matrix, which is an M×M matrix; D is the characteristic matrix of the generalized stiffness matrix A, D = [D (1) …D (j) …D (M) ], which is an M×M matrix; Λ is a diagonal matrix, and the jth value on its diagonal is the eigenvalue λ j ; The element A of the generalized stiffness matrix A jm (j,m=1,…,M) is determined by the following formula:

[0057] A jm =∫ Ω ∫ Ω ρ(x1,x2)h j (x2)h m (x1)dx1dx2

[0058] Step 1.2.3, let the one-dimensional study area be Ω = [x min ,x max ], using the standard Legendre polynomial function in the interval [-1,1] to represent [x min ,x max ] interval orthogonal basis function h m (x), which is:

[0059]

[0060] Where, L m-1 (x) is the Legendre polynomial; T is the translation coefficient, T = (x max +x min ) / 2; T' is the scaling factor, T'=(x max -x min ) / 2;

[0061] Step 1.2.4, transform the element A of the generalized stiffness matrix A jm Further rewritten as:

[0062]

[0063] Where x1 and x2 are the one-dimensional coordinate values ​​of any two expansive soil grid centers in the horizontal or vertical direction;

[0064] After solving the generalized stiffness matrix A from the above formula, the eigenvalue matrix λ and the eigenfunction matrix f(x) are solved according to the above matrix equation AD=DΛ. The final eigenfunction matrix f(x) is as follows:

[0065]

[0066] Preferably, the double pile load transfer method considering the change of matrix suction and the expansion of expansive soil in step 2.1 calculates the displacement s of the grid No. 1 in group bi The corresponding loads on the tops of the two piles are Q i and displacement s i The steps are as follows:

[0067] Step 2.1.1, given the pile number i, i = 1, 2; given the displacement s of the bottom grid No. 1 of the double pile bi ;

[0068] Step 2.1.2, when s bi When it is negative, it means that the bottom of the pile moves upward and the end resistance P bi is zero, when s bi When it is positive, it means that the bottom of the pile moves downward and there is end resistance P bi , end resistance P bi The calculation formula is as follows:

[0069]

[0070] In the formula, A i is the first empirical coefficient; B i is the second empirical coefficient, B i =R bf / P bui , where R bf is the empirical failure ratio of pile tip resistance, P bui is the limit end resistance; the first empirical coefficient A i is the pile end spring stiffness k bi The reciprocal of is calculated as:

[0071]

[0072] In the formula, k b1i is the first spring stiffness, calculated as k b1i =4G bi / πr0(1-v), where v is Poisson's ratio, r0 is the pile radius, G bi is the shear modulus of the expansive soil grid directly below the pile tip, calculated as G bi =E u-bi / [2(1+v)], where E u-bi is the unsaturated elastic modulus E of the expansive soil grid just below the pile tip unsat , unsaturated elastic modulus E unsat The calculation formula is as follows:

[0073]

[0074] In the formula, E sat is the saturated elastic modulus, which conforms to the calculation formula of the random field discrete value; U1 is the first fitting parameter; U2 is the second fitting parameter; ψ is the matrix suction; S r is saturation; P a The atmospheric pressure is 101.3 kPa;

[0075] k b2i is the second spring stiffness, and the calculation formula is

[0076] Step 2.1.3, calculate the displacement s of the expansive soil grid on the right side of the double pile No. 1 grid bi ',s bi '=s bi +Δh i , where Δh i The expansion amount of the expansive soil grid is calculated as follows:

[0077]

[0078] Where Δψ is the reduction of matrix suction ψ; E a is the average unsaturated elastic modulus of expansive soil from unsaturated state to saturated state;

[0079] Step 2.1.4: record the initial value s of the pile-soil relative displacement between the midpoint of the double pile No. 1 grid and the expansive soil grid on its right side. si The displacement of the expansive soil grid is s bi ' is equal, the lateral friction resistance τ of the midpoint of the double pile No. 1 grid si The calculation formula is as follows:

[0080]

[0081] In the formula, a i is the third empirical coefficient, which is the pile side spring stiffness k si The inverse of is calculated as follows:

[0082]

[0083] In the formula, k 1i is the third spring stiffness, calculated as k 1i =G i / r0ln(r mi / r0), where Gi is the expansive soil mesh shear modulus, r mi is the maximum influence radius of the i-th pile, and the calculation formula is Among them G ik is the shear modulus of the kth expansive soil grid on the right side of the i-th pile, and the calculation formula is G ik =E u-ik / [2(1+v)], where Eu-ik is the unsaturated elastic modulus E of the kth expansive soil grid unsat , G mi is the maximum shear modulus of the first column of expansive soil grid on the right side of the i-th pile; k 2i is the fourth spring stiffness, calculated as k 2i =Gi / r0 ln(r mi-1 / r p ), where r mi-1 is the maximum influence radius of the neighboring piles of the i-th pile, and the calculation formula is the same as r mi The same as k, but the shear modulus mentioned in the formula is the shear modulus of the expansive soil grid on the right side of the adjacent pile of the i-th pile; 3i is the fifth spring stiffness, calculated as k 3i =r p G i / r0 2 ln(r mi-1 / r p );

[0084] b i is the fourth empirical coefficient, and its calculation formula is b i =R sf / τ sui , where R sf is the empirical failure ratio of lateral friction, τ sui is the limit lateral friction. When the matrix suction changes from state ψ1 to state ψ2, the limit lateral friction τ sui The calculation formula is:

[0085]

[0086] In the formula, c ai ′ is the effective cohesion of the pile-soil interface, and this parameter conforms to the calculation formula of the random field discrete value; δ i ′ is the effective friction angle of the pile-soil interface, and this parameter conforms to the calculation formula of the random field discrete value; P S( ψ 1–0) represents the vertical expansion pressure generated from the matrix suction state ψ1 to the saturation state; E a( ψ 1–0) is the average unsaturated elastic modulus of expansive soil from matrix suction state ψ1 to saturated state; P S( ψ 2–0) represents the vertical expansion pressure generated from the matrix suction state ψ2 to the saturation state; E a( ψ 2–0) is the average unsaturated elastic modulus of the expansive soil from the matrix suction state ψ2 to the saturated state; θ is the volumetric water content; θ s is the saturated moisture content; θ r is the residual moisture content; σ vs is the effective earth pressure;

[0087] Step 2.1.5, record the lateral friction τ at different positions in each double pile grid si The load P on the top surface of the grid No. 1 of the double pile is calculated. ti , P ti =P bi +πD 2 τ si , assuming that the axial force in each double pile grid is linearly distributed, calculate the elastic compression s at the midpoint of the double pile grid No. 1 cmi , the calculation formula is:

[0088]

[0089] Then the pile-soil relative displacement correction value s at the midpoint of the double pile No. 1 grid is si ′ is s s ' i =s bi +s cmi , if the pile-soil relative displacement correction value s si ′ and the initial value of relative displacement between pile and soil s si If the difference is less than the first set threshold, proceed to the next step, otherwise return to step 2.1.4;

[0090] The pile-soil relative displacement correction value s of the midpoint of the double pile No. 1 grid is si ′ is the initial value of relative displacement of pile and soil s si Repeat steps 2.1.4 to 2.1.5 until the difference is less than the first set threshold;

[0091] Step 2.1.6, based on the final iterative stable pile-soil relative displacement correction value s si Calculate the side friction correction value τ si ′, and calculate the load P on the top surface of the double pile No. 1 grid ti ′,P′ ti =P bi +πD 2 τ′ si Then calculate the displacement of the top surface of the double pile No. 1 grid, the calculation formula is s ti =s bi +s ci , where s ci is the elastic compression of the double pile No. 1 grid, calculated as s ci =2(P′ ti +P bi ) / πDE p ;

[0092] The end resistance of the bottom surface of the double pile No. 2 grid is the load P on the top surface of the double pile No. 1 grid. ti′, the displacement of the bottom surface of the double pile No. 2 grid is the displacement of the top surface of the double pile No. 1 grid s ti , return to step 2.1.2 and calculate the load and displacement of the top surface of the grid No. 2 of the double pile according to steps 2.1.2 to 2.1.6. Repeat the above steps to calculate the load and displacement of the top surface of the grid No. 3, 4, ..., N of the double piles. p The load and displacement of the top surface of the grid, and finally the Nth double pile p The load Q on the top surface of the grid i and displacement s i , which is the load Q on the top of the double pile i and displacement s i .

[0093] Preferably, the first-order reliability method is used to calculate the component reliability index β in step 2.3. q The steps are as follows:

[0094] Step 2.3.1, let the initial vector ξ0 of the random vector ξ be a 1×3M zero matrix;

[0095] Step 2.3.2, calculate the initial reliability index β0 based on the initial vector ξ0, the calculation formula is:

[0096]

[0097] In the formula, this formula is recorded as Formula No. 1, g(ξ0) is the performance function value corresponding to the initial vector ξ0, is the partial derivative of the performance function g(ξ0) with respect to the initial vector ξ0, μ ξ is the mean matrix of the random vector ξ, C ξ is the covariance matrix of the random vector ξ;

[0098] Step 2.3.3, calculate the iteration vector ξ of the random vector ξ t ,ξ t =μ ξ +σ ξ α0β0, where this formula is recorded as Formula No. 2, α0 is the initial direction cosine vector, This formula is recorded as Formula No. 3, ρ ξ is the correlation coefficient matrix of the random vector ξ;

[0099] Step 2.3.4, use formula 1 to calculate the iteration vector ξ t The corresponding iterative reliability index β t , where all initial vectors ξ0 in formula 1 are replaced by the iteration vector ξ t , if the initial reliability index β0 and the iterative reliability index β t The difference is less than the second set threshold, and this iteration vector ξ t That is the verification point vector ξ* , proceed to the next step, otherwise use formula 2 to calculate the new iteration vector ξ t , where the initial direction cosine vector α0 is replaced by the iterative direction cosine vector α t , and replace all initial vectors ξ0 in formula 3 with the iteration vector ξ t , and then calculate the new iterative reliability index β t , and the calculation is repeated in this way until the initial reliability index β0 and the iterative reliability index β t The difference is less than a second set threshold;

[0100] Step 2.3.5, based on the verification point vector ξ * Use formula 1 to calculate the final component reliability index β q , and use formula 3 to calculate the final direction cosine vector α q , where all initial vectors ξ0 in formulas 1 and 3 are replaced by the verification point vector ξ * .

[0101] Preferably, the correlation coefficient ρ between the combination element E and the element FM3 in step 3.2 is (1or2)3 By inverse solving the following formula:

[0102]

[0103] Where Φ2 is the two-dimensional standard normal cumulative distribution function; β 1|3 is the reliability index of component FM1 in the case of component FM3 failure, and the calculation formula is: where ρ 13 is the correlation coefficient between element FM1 and element FM3, ρ 13 =α1 T α3, α3 is the direction cosine vector of element FM3; β 2|3 is the reliability index of component FM2 in the case of component FM3 failure, and the calculation formula is: where ρ 23 is the correlation coefficient between element FM2 and element FM3, ρ 23 =α2 T α3;

[0104] Step 3.2 The mean matrix μ of the random vector ξ ξ System reliability index β sys The system sensitivity index matrix S s In the calculation formula, the first partial derivative parameter The calculation formula is:

[0105]

[0106] Step 3.2 The mean matrix μ of the random vector ξ ξ System reliability index β sys The system sensitivity index matrix S s In the calculation formula, the second partial derivative parameter The calculation formula is:

[0107]

[0108] Compared with the prior art, the beneficial effects of the present invention include:

[0109] 1. A calculation method for the system reliability and sensitivity analysis of double piles in spatially variable expansive soils was established, which mainly includes five parts: using the Karhunen–Loève expansion method to establish a random field model of soil parameters, using the double-pile load transfer method considering the change of matrix suction and the expansion of expansive soil to calculate the double-pile load-displacement relationship curve, using the first-order reliability method to calculate the component reliability index of different failure modes, using the sequential component method to calculate the system reliability index, and calculating the sensitivity index of soil parameters to components and systems.

[0110] 2. In the present invention, three failure modes of the double pile foundation are analyzed, and the functional formulas of the three failure modes are established respectively. The first failure mode is that the No. 1 pile undergoes compression failure or pull-out failure, the second failure mode is that the No. 2 pile undergoes compression failure or pull-out failure, and the third failure mode is that the settlement difference produced by the double piles exceeds the maximum allowable settlement difference.

[0111] 3. The calculation method of the present invention has high calculation accuracy and calculation efficiency. After obtaining the double pile foundation model parameters, soil property parameters and random field model parameters, the present invention can be used to quickly calculate the component reliability index and component sensitivity index of different failure modes, as well as the system reliability index and system sensitivity index. At the same time, the high calculation efficiency can quickly analyze the influence of different parameters and different conditions on the reliability index and sensitivity index. BRIEF DESCRIPTION OF THE DRAWINGS

[0112] Figure 1 is a flow chart of the method of the present invention;

[0113] Figure 2 Schematic diagram of a double pile foundation model in spatially variable expansive soil according to an embodiment of the present invention;

[0114] Figure 3 Three failure modes and double pile load and displacement relationship curves provided by the present invention. DETAILED DESCRIPTION

[0115] The technical solution of the present invention is further described in detail below in conjunction with the accompanying drawings.

[0116] Figure 1 is a flow chart of the system reliability and sensitivity analysis method of double piles in spatially variable expansive soil of the present invention, Figure 1 It can be seen that the analysis method of the present invention comprises the following steps:

[0117] Step 1: Establish a double pile foundation model and a soil parameter random field model

[0118] Step 1.1, the double pile foundation model consists of two piles vertically inserted into the expansive soil foundation and the expansive soil foundation. Assume that the upper surface of the expansive soil foundation is flush with the ground surface, and take a cross section perpendicular to the earth's surface as the expansive soil foundation. The expansive soil foundation includes a soil rectangle and two pile rectangles symmetrically embedded with their axis from the outside to the inside. The left pile is pile No. 1 and the right pile is pile No. 2. The sizes and materials of the two piles are exactly the same; the length of the double piles embedded in the soil is L, the pile diameter is D, and the distance between the double piles is r p , the elastic modulus is E p , the horizontal side length of the soil rectangle is K times the pile diameter D, and the vertical height is H times the pile diameter D;

[0119] With the lower left corner vertex of the soil rectangle as the origin, a plane coordinate system is established and each pile rectangle is divided into N p A square grid with a side length of pile diameter D is constructed, and N p The grids are sorted, and the serial numbers are k=1, 2, ..., N. p ; Divide the soil rectangle into N e A square grid with a side length of pile diameter D is constructed from bottom to top and from left to right. e The grids are sorted, and the coordinates of the center point of each expansive soil grid are used as the coordinates of the expansive soil grid;

[0120] Step 1.2, the soil parameters are 3, namely effective cohesion c a ', effective friction angle δ' and saturated elastic modulus E sat , any one of them is recorded as soil parameter Γ h , h is the serial number of soil parameter, h = 1, 2, 3; the soil parameter random field model is the probability distribution model of soil parameter random field, the probability distribution model includes probability distribution type and autocorrelation function type, record the three soil parameters Γ h All are log-normal distribution and square exponential autocorrelation function;

[0121] In this embodiment, the calculation formula of the square exponential autocorrelation function in step 1.2 is as follows:

[0122]

[0123] Where x1 and x2 are the two-dimensional coordinate vectors of the centers of any two expansive soil grids, x1 is the coordinate vector x1, and x2 is the coordinate vector x2; ρ(x1, x2) is the soil property parameter Γ h The correlation coefficient between the coordinate vector x1 and the coordinate vector x2; 11 is the coordinate component of the coordinate vector x1 in the horizontal direction; 12 is the coordinate component of the coordinate vector x1 in the vertical direction; 21 is the coordinate component of the coordinate vector x2 in the horizontal direction; 22 is the coordinate component of the coordinate vector x2 in the vertical direction; L H is the autocorrelation distance in the horizontal direction; L V is the autocorrelation distance in the vertical direction.

[0124] The Karhunen–Loève expansion method is used to calculate the three soil parameters Γ in each grid. h The lognormal random field discrete value And recorded as discrete values Its expression is:

[0125]

[0126] Where ξ is a random vector in an independent standard normal space and is denoted as random vector ξ, ξ is a 1×3M matrix, M is the number of truncation terms; λ j is the jth eigenvalue in the eigenvalue matrix λ, where j is the number of cutoff terms, j = 1, 2, …, M; f j (x) is the characteristic function of the jth column in the characteristic function matrix f(x); μ h is the soil parameter Γ h The mean of h is the soil parameter Γ h The mean square error of ; L is the lower triangular matrix obtained by Cholesky decomposition of the random field cross-correlation coefficient matrix in the relevant standard normal space, and the superscript T represents the transpose of the matrix;

[0127] In this embodiment, the calculation steps of the eigenvalue matrix λ and the eigenfunction matrix f(x) described in step 1.2 are as follows:

[0128] Step 1.2.1, according to the Galerkin method, the characteristic function f j (x)(j=1,2,…,M) is decomposed into M orthogonal basis functions h m Linear combination of (x)(m=1,2,…,M):

[0129]

[0130] Where D (j)is an M×1 column vector, D (j) =[d1 (j) ...d m (j) ...d M (j) ] T ; h(x) is a column vector of M×1, h(x)=[h1(x)…h m (x)…h M (x)] T ;

[0131] Step 1.2.2, substitute the linear expansion of the above characteristic function into the Fredholm integral equation ∫ Ω ρ(x1,x2)f j (x1)dx1=λ j f j (x2), the following matrix equation is obtained:

[0132] AD=DΛ

[0133] Where A is the generalized stiffness matrix, which is an M×M matrix; D is the characteristic matrix of the generalized stiffness matrix A, D = [D (1) …D (j) …D (M) ], which is an M×M matrix; Λ is a diagonal matrix, and the jth value on its diagonal is the eigenvalue λ j ; The element A of the generalized stiffness matrix A jm (j,m=1,…,M) is determined by the following formula:

[0134] A jm =∫ Ω ∫ Ω ρ(x1,x2)h j (x2)h m (x1)dx1dx2

[0135] Step 1.2.3, let the one-dimensional study area be Ω = [x min ,x max ], using the standard Legendre polynomial function in the interval [-1,1] to represent [x min ,x max ] interval orthogonal basis function h m (x), which is:

[0136]

[0137] Where, L m-1 (x) is the Legendre polynomial; T is the translation coefficient, T = (x max +x min) / 2; T' is the scaling factor, T'=(x max -x min ) / 2;

[0138] Step 1.2.4, transform the element A of the generalized stiffness matrix A jm Further rewritten as:

[0139]

[0140] Where x1 and x2 are the one-dimensional coordinate values ​​of any two expansive soil grid centers in the horizontal or vertical direction;

[0141] After solving the generalized stiffness matrix A from the above formula, the eigenvalue matrix λ and the eigenfunction matrix f(x) are solved according to the above matrix equation AD=DΛ. The final eigenfunction matrix f(x) is as follows:

[0142]

[0143] Figure 2 Schematic diagram of a double pile foundation model in spatially variable expansive soil according to an embodiment of the present invention.

[0144] In this embodiment, the length L of the double pile model embedded in the soil is 4m, the pile diameter D is 0.5m, and the distance between the double piles is r p is 1.5m, elastic modulus E p The horizontal length of the clay rectangle is K = 9 times the pile diameter D, which is 4.5 m, and the vertical height is H = 9 times the pile diameter D, which is 4.5 m. The number of grids N for piles 1 and 2 p The number of grids for expansive soil is N. e is 65. In addition Figure 2 The load Q1 applied to the top of pile 1 is 180kN, and the load Q2 applied to the top of pile 2 is 180kN.

[0145] Three soil parameters Γ h The random field statistics are shown in Table 1. The mean and standard deviation of the three random fields in different expansive soil grids are equal, and the autocorrelation function selects the square exponential autocorrelation function type. The unit weight γ of the soil is 21kN / m 3, Poisson's ratio v is 0.4, and the double pile foundation experiences rainfall seepage for 5 days. The soil matrix suction at the burial depths of 0.23m, 0.74m, 1.24m, 1.74m, 2.24m, 2.75m, 3.25m, 3.75m and 4.00m are 577.52kPa, 497.07kPa, 354.06kPa, 190.60kPa, 10.30kPa, 6.84kPa, 1.13kPa, 0.98kPa and 0.20kPa respectively. The soil-water characteristic curve information is obtained when the matrix suction is 1.37kPa, 5.17 kPa, 19.51 kPa, 68.50 kPa, 203.09 kPa, 508.50 kPa, 1304.32 kPa, 3187.84 kPa, 7246.69 kPa, 17711.40 kPa, 47678.40 kPa, 131000.00 kPa and 473000.00 kPa correspond to water contents of 0.49, 0.48, 0.47, 0.45, 0.42, 0.38, 0.33, 0.27, 0.22, 0.17, 0.11, 0.07 and 0.02, respectively. This embodiment considers both the horizontal and vertical spatial variability of soil properties, where the horizontal autocorrelation distance L H Distance between piles p Same, 1.5m, vertical autocorrelation distance L V is 1m.

[0146] Table 1 Random field statistics of soil parameters

[0147] Soil parameters <![CDATA[Mean (μ h )]]> <![CDATA[Mean square error (σ h )]]> <![CDATA[Effective cohesive force (c a ')]]> 9.8kPa 32.7kPa Effective friction angle (δ') 16.7° 55.7° <![CDATA[Saturated elastic modulus (E sat )]]> 550kPa 1833kPa

[0148] Step 2: Calculate the component reliability index and component sensitivity index of the three failure modes of double piles

[0149] Step 2.1, based on the three soil parameters Γ in step 1 h The lognormal random field discrete value The displacement s of the bottom No. 1 grid of a given set of double piles bi , i is the pile number, i = 1, 2, a set of grid displacement s of the double pile bi Based on this, the displacement s of the grid No. 1 in this group is calculated using the double-pile load transfer method that considers the change of matrix suction and the expansion of expansive soil. bi The corresponding loads on the tops of the two piles are Q i and displacement s i , and draw the relationship curve between the load Q and displacement s of each pile in a plane coordinate system;

[0150] In this embodiment, the double pile load transfer method considering the change of matrix suction and the expansion of expansive soil in step 2.1 calculates the displacement s of the grid No. 1 in group 1. biThe corresponding loads on the tops of the two piles are Q i and displacement s i The steps are as follows:

[0151] Step 2.1.1, given the pile number i, i = 1, 2; given the displacement s of the bottom grid No. 1 of the double pile bi ;

[0152] Step 2.1.2, when s bi When it is negative, it means that the bottom of the pile moves upward and the end resistance P bi is zero, when s bi When it is positive, it means that the bottom of the pile moves downward and there is end resistance P bi , end resistance P bi The calculation formula is as follows:

[0153]

[0154] In the formula, A i is the first empirical coefficient; B i is the second empirical coefficient, B i =R bf / P bui , where R bf is the empirical failure ratio of pile tip resistance, P bui is the limit end resistance; the first empirical coefficient A i is the pile end spring stiffness k bi The reciprocal of is calculated as:

[0155]

[0156] In the formula, k b1i is the first spring stiffness, calculated as k b1i =4G bi / πr0(1-v), where v is Poisson's ratio, r0 is the pile radius, G bi is the shear modulus of the expansive soil grid directly below the pile tip, calculated as G bi =E u-bi / [2(1+v)], where E u-bi is the unsaturated elastic modulus E of the expansive soil grid just below the pile tip unsat , unsaturated elastic modulus E unsat The calculation formula is as follows:

[0157]

[0158] In the formula, E sat is the saturated elastic modulus, which conforms to the calculation formula of the random field discrete value; U1 is the first fitting parameter; U2 is the second fitting parameter; ψ is the matrix suction; S r is saturation; P aThe atmospheric pressure is 101.3 kPa;

[0159] The first fitting parameter U1 takes a value of 0.1, and the second fitting parameter U2 takes a value of 2.

[0160] k b2i is the second spring stiffness, and the calculation formula is

[0161] Step 2.1.3, calculate the displacement s of the expansive soil grid on the right side of the double pile No. 1 grid bi ',s bi '=s bi +Δh i , where Δh i The expansion amount of the expansive soil grid is calculated as follows:

[0162]

[0163] Where Δψ is the reduction of matrix suction ψ; E a is the average unsaturated elastic modulus of expansive soil from unsaturated state to saturated state;

[0164] Step 2.1.4: record the initial value s of the pile-soil relative displacement between the midpoint of the double pile No. 1 grid and the expansive soil grid on its right side. si The displacement of the expansive soil grid is s bi ' is equal, the lateral friction resistance τ of the midpoint of the double pile No. 1 grid si The calculation formula is as follows:

[0165]

[0166] In the formula, a i is the third empirical coefficient, which is the pile side spring stiffness k si The inverse of is calculated as follows:

[0167]

[0168] In the formula, k 1i is the third spring stiffness, calculated as k 1i =Gi / r0 ln(r mi / r0), where Gi is the expansive soil mesh shear modulus, r mi is the maximum influence radius of the i-th pile, and the calculation formula is Among them G ik is the shear modulus of the kth expansive soil grid on the right side of the i-th pile, and the calculation formula is G ik =E u-ik / [2(1+v)], where E u-ik is the unsaturated elastic modulus E of the kth expansive soil grid unsat , G miis the maximum shear modulus of the first column of expansive soil grid on the right side of the i-th pile; k 2i is the fourth spring stiffness, calculated as k 2i =G i / r0 ln(r mi-1 / r p ), where r mi-1 is the maximum influence radius of the neighboring piles of the i-th pile, and the calculation formula is the same as r mi The same as k, but the shear modulus mentioned in the formula is the shear modulus of the expansive soil grid on the right side of the adjacent pile of the i-th pile; 3i is the fifth spring stiffness, calculated as k 3i =r p G i / r0 2 ln(r mi-1 / r p );

[0169] b i is the fourth empirical coefficient, and its calculation formula is b i =R sf / τ sui , where R sf is the empirical failure ratio of lateral friction, τ sui is the limit lateral friction. When the matrix suction changes from state ψ1 to state ψ2, the limit lateral friction τ sui The calculation formula is:

[0170]

[0171] In the formula, c ai ′ is the effective cohesion of the pile-soil interface, and this parameter conforms to the calculation formula of the random field discrete value; δ i ′ is the effective friction angle of the pile-soil interface, and this parameter conforms to the calculation formula of the random field discrete value; P S( ψ 1–0) represents the vertical expansion pressure generated from the matrix suction state ψ1 to the saturation state; E a( ψ 1–0) is the average unsaturated elastic modulus of expansive soil from matrix suction state ψ1 to saturated state; P S( ψ 2–0) represents the vertical expansion pressure generated from the matrix suction state ψ2 to the saturation state; E a( ψ 2–0) is the average unsaturated elastic modulus of the expansive soil from the matrix suction state ψ2 to the saturated state; θ is the volumetric water content; θ s is the saturated moisture content; θ r is the residual moisture content; σ vs is the effective earth pressure;

[0172] Step 2.1.5, record the lateral friction τ at different positions in each double pile grid si The load P on the top surface of the grid No. 1 of the double pile is calculated. ti , P ti =P bi +πD 2 τ si , assuming that the axial force in each double pile grid is linearly distributed, calculate the elastic compression s at the midpoint of the double pile grid No. 1 cmi , the calculation formula is:

[0173]

[0174] Then the pile-soil relative displacement correction value s at the midpoint of the double pile No. 1 grid is si ′ is s s ' i =s bi +s cmi , if the pile-soil relative displacement correction value s si ′ and the initial value of relative displacement between pile and soil s si If the difference is less than the first set threshold, proceed to the next step, otherwise return to step 2.1.4;

[0175] The first threshold value is set to 10 -6 ;

[0176] The pile-soil relative displacement correction value s of the midpoint of the double pile No. 1 grid is si ′ is the initial value of relative displacement of pile and soil s si Repeat steps 2.1.4 to 2.1.5 until the difference is less than the first set threshold;

[0177] Step 2.1.6, based on the final iterative stable pile-soil relative displacement correction value s si Calculate the side friction correction value τ si ′, and calculate the load P on the top surface of the double pile No. 1 grid ti ′,P′ ti =P bi +πD 2 τ′ si Then calculate the displacement of the top surface of the double pile No. 1 grid, the calculation formula is s ti =s bi +s ci , where s ci is the elastic compression of the double pile No. 1 grid, calculated as s ci =2(P′ ti +P bi ) / πDE p ;

[0178] The end resistance of the bottom surface of the double pile No. 2 grid is the load P on the top surface of the double pile No. 1 grid.ti ′, the displacement of the bottom surface of the double pile No. 2 grid is the displacement of the top surface of the double pile No. 1 grid s ti , return to step 2.1.2 and calculate the load and displacement of the top surface of the grid No. 2 of the double pile according to steps 2.1.2 to 2.1.6. Repeat the above steps to calculate the load and displacement of the top surface of the grid No. 3, 4, ..., N of the double piles. p The load and displacement of the top surface of the grid, and finally the Nth double pile p The load Q on the top surface of the grid i and displacement s i , which is the load Q on the top of the double pile i and displacement s i .

[0179] Step 2.2, based on the load Q and displacement s relationship curve drawn in step 2.1, calculate the functional function values ​​of the three failure modes of the double piles under the vertical fixed load Q1 at the top of pile No. 1 and the vertical fixed load Q2 at the top of pile No. 2;

[0180] The first failure mode is the compression failure or pull-out failure of pile 1, which is denoted as element FM1. The expression of its functional value g1(ξ) is:

[0181]

[0182] In the formula, G p is the pile gravity; R down1 (ξ) is the ultimate bearing capacity of pile No. 1, according to the preset maximum allowable settlement s mdown Obtained from the load Q and displacement s relationship curve of pile No. 1; R up1 (ξ) is the ultimate pull-out capacity of pile No. 1, according to the preset maximum allowable upward displacement s mup It is obtained from the relationship curve between the load Q and displacement s of pile No. 1; s1(ξ) is the displacement of the pile top of pile No. 1, s1(ξ)≥0 means that the pile top moves downward, and s1(ξ)<0 means that the pile top moves upward;

[0183] The second failure mode is the compression failure or pull-out failure of pile 2, which is denoted as element FM2. The expression of its functional value g2(ξ) is:

[0184]

[0185] In the formula, R down2 (ξ) is the ultimate bearing capacity of pile 2, according to the preset maximum allowable settlement s mdown Obtained from the load Q and displacement s relationship curve of pile 2; R up2 (ξ) is the ultimate pull-out capacity of pile No. 2, according to the preset maximum allowable upward displacement s mupIt is obtained from the relationship curve between the load Q and displacement s of pile No. 2; s2(ξ) is the displacement of the pile top of pile No. 2, s2(ξ)≥0 means that the pile top moves downward, and s2(ξ)<0 means that the pile top moves upward;

[0186] The third failure mode is the settlement difference Δ caused by the double piles s Exceeding the maximum allowable settlement difference Δ m , denoted as element FM3, the expression of its functional value g3(ξ) is:

[0187] g3(ξ)=Δ m -Δ s =Δ m -|s1(ξ)-s2(ξ)|

[0188] Figure 3 The three failure modes and the load-displacement relationship curves of the double piles in this embodiment are the results of a set of random sample calculations based on the random vector ξ. Figure 3 The maximum allowable settlement of piles 1 and 2 is s mdown The maximum allowable upward displacement of piles 1 and 2 is 30 mm. mup Both are 2mm.

[0189] Step 2.3, based on the three failure mode function values ​​g calculated in step 2.2 q (ξ), q is the serial number of the three failure modes, q = 1, 2, 3, and the first-order reliability method is used to calculate the component reliability index β corresponding to each failure mode. q , component reliability index β q The higher it is, the safer the component is;

[0190] In this embodiment, the first-order reliability method is used to calculate the component reliability index β in step 2.3. q The steps are as follows:

[0191] Step 2.3.1, let the initial vector ξ0 of the random vector ξ be a 1×3M zero matrix;

[0192] Step 2.3.2, calculate the initial reliability index β0 based on the initial vector ξ0, the calculation formula is:

[0193]

[0194] In the formula, this formula is recorded as Formula No. 1, g(ξ0) is the performance function value corresponding to the initial vector ξ0, ▽g(ξ0) is the partial derivative of the performance function value g(ξ0) with respect to the initial vector ξ0, μ ξ is the mean matrix of the random vector ξ, C ξ is the covariance matrix of the random vector ξ;

[0195] Step 2.3.3, calculate the iteration vector ξ of the random vector ξ t ,ξ t =μ ξ +σ ξ α0β0, where this formula is recorded as Formula No. 2, α0 is the initial direction cosine vector, This formula is recorded as Formula No. 3, ρ ξ is the correlation coefficient matrix of the random vector ξ;

[0196] Step 2.3.4, use formula 1 to calculate the iteration vector ξ t The corresponding iterative reliability index β t , where all initial vectors ξ0 in formula 1 are replaced by the iteration vector ξ t , if the initial reliability index β0 and the iterative reliability index β t The difference is less than the second set threshold, and this iteration vector ξ t That is the verification point vector ξ * , proceed to the next step, otherwise use formula 2 to calculate the new iteration vector ξ t , where the initial direction cosine vector α0 is replaced by the iterative direction cosine vector α t , and replace all initial vectors ξ0 in formula 3 with the iteration vector ξ t , and then calculate the new iterative reliability index β t , and the calculation is repeated in this way until the initial reliability index β0 and the iterative reliability index β t The difference is less than a second set threshold;

[0197] The second threshold value is set to 0.01;

[0198] Step 2.3.5, based on the verification point vector ξ * Use formula 1 to calculate the final component reliability index β q , and use formula 3 to calculate the final direction cosine vector α q , where all initial vectors ξ0 in formulas 1 and 3 are replaced by the verification point vector ξ * .

[0199] Step 2.4, calculate the three soil parameters Γ respectively h For each component reliability index β q The sensitivity index S q h The higher the sensitivity index, the stronger the soil property parameter Γ h The variability of the component reliability index β q The greater the influence, the calculation formula is as follows:

[0200]

[0201] In the formula, α q (j) is the direction cosine vector α of the qth failure mode q The j-th direction cosine value in ;

[0202] In this embodiment, the reliability index of component FM1 is calculated to be 5.68, and the functional function value g1(ξ) is calculated a total of 366 times; the reliability index of component FM2 is calculated to be 5.65, and the functional function value g2(ξ) is calculated a total of 366 times; the reliability index of component FM3 is calculated to be 3.36, and the functional function value g3(ξ) is calculated a total of 305 times.

[0203] Three soil parameters Γ under three failure modes h The calculation results of the component sensitivity index are shown in Table 2. From the data, it can be seen that the effective cohesion c of the three failure modes a The sensitivity indexes of ' are almost 0, that is, their variability has almost no effect on the reliability indexes of the components of the three failure modes. For components FM1 and FM2, the sensitivity of the effective friction angle δ' is much greater than the saturated elastic modulus E sat The sensitivity of the effective friction angle δ' has a greater impact on the reliability index of the component. For component FM3, the saturated elastic modulus E sat has the greatest impact.

[0204] Table 2 Component sensitivity indexes of soil parameters under three failure modes

[0205]

[0206] Step 3: Calculate the reliability index and system sensitivity index of the dual pile system

[0207] The three failure modes of the double pile foundation are in series relationship. The component reliability index β of the three failure modes calculated in step 2 is q , using the sequential component method to calculate the system reliability index β sys ;

[0208] Step 3.1, combine components FM1 and FM2 into a combined component E, and calculate the reliability index β of the combined component E 1or2 , the calculation formula is:

[0209] β 1or2 =-Φ -1 [Φ(-β1)+Φ(-β2)-Φ(-β 1|2 )Φ(-β2)]

[0210] Where Φ is the standard normal cumulative distribution function, Φ -1(·) is the inverse function of the standard normal cumulative distribution; β1 is the reliability index of component FM1; β2 is the reliability index of component FM2; β 1|2 is the reliability index of component FM1 in the case of component FM2 failure, and the calculation formula is: F2 is parameter 1, F2 = φ(-β2) / Φ(-β2), where φ is the standard normal probability density function, W2 is parameter 2, W2 = F2(-β2+F2), ρ 12 is the correlation coefficient between element FM1 and element FM2, ρ 12 =α1 T α2, α1 is the direction cosine vector of element FM1, α2 is the direction cosine vector of element FM2;

[0211] Step 3.2, record the combination of component E and component FM3 as a system, and the system reliability index is marked as β sys , and its calculation formula is:

[0212] β sys =-Φ -1 [Φ(-β 1or2 )+Φ(-β3)-Φ(-β 1or2|3 )Φ(-β3)]

[0213] Where, β3 is the reliability index of component FM3; β 1or2|3 is the reliability index of the combined component E in the case of component FM3 failure, and the calculation formula is: F3 is parameter No. 3, F3 = φ(-β3) / Φ(-β3), W3 is parameter No. 4, W3 = F3(-β3+F3), ρ (1or2)3 is the correlation coefficient between the combined element E and the element FM3;

[0214] In this embodiment, the correlation coefficient ρ between the combination element E and the element FM3 in step 3.2 is (1or2)3 By inverse solving the following formula:

[0215]

[0216] Where Φ2 is the two-dimensional standard normal cumulative distribution function; β 1|3 is the reliability index of component FM1 in the case of component FM3 failure, and the calculation formula is: where ρ 13 is the correlation coefficient between element FM1 and element FM3, ρ 13 =α1 T α3, α3 is the direction cosine vector of element FM3; β 2|3 is the reliability index of component FM2 in the case of component FM3 failure, and the calculation formula is: where ρ 23is the correlation coefficient between element FM2 and element FM3, ρ 23 =α2 T α3;

[0217] Calculate the system reliability index β sys Afterwards, the three soil parameters Γ are calculated respectively. h System reliability index β sys The sensitivity index S sys h , first we need to calculate the mean matrix μ of the random vector ξ ξ System reliability index β sys The system sensitivity index matrix S s , the calculation formula is:

[0218]

[0219] In the formula, record is the first partial derivative parameter, is the partial derivative parameter No. 2; α3 is the direction cosine vector of element FM3;

[0220] Step 3.2 The mean matrix μ of the random vector ξ ξ System reliability index β sys The system sensitivity index matrix S s In the calculation formula, the first partial derivative parameter The calculation formula is:

[0221]

[0222] Step 3.2 The mean matrix μ of the random vector ξ ξ System reliability index β sys The system sensitivity index matrix S s In the calculation formula, the second partial derivative parameter The calculation formula is:

[0223]

[0224] Calculate three soil parameters Γ separately h Sensitivity index S to system reliability index sys h , the calculation formula is as follows:

[0225]

[0226] In the formula, S s (j) is the system sensitivity index matrix S s The j-th system sensitivity index in .

[0227] In this embodiment, the three soil parameters Γh The calculation results of the system sensitivity index are shown in Table 3. From the data, we can see that the effective cohesion c a The system sensitivity index of ' is almost 0, that is, its variability has almost no effect on the system reliability index. Saturated elastic modulus E sat The system sensitivity index of is greater than the system sensitivity index of the effective friction angle δ', that is, the saturated elastic modulus E sat The impact on the system reliability index is greater than that on the effective friction angle δ'.

[0228] Table 3 System sensitivity index of soil parameters

[0229]

Claims

1. A system reliability and sensitivity analysis method for double piles in spatially variable expansive soil, characterized in that: The steps include: Step 1: Establish a double pile foundation model and a soil parameter random field model Step 1.1, the double pile foundation model consists of two piles vertically inserted into the expansive soil foundation and the expansive soil foundation. Assume that the upper surface of the expansive soil foundation is flush with the ground surface, and take a cross section perpendicular to the earth's surface as the expansive soil foundation. The expansive soil foundation includes a soil rectangle and two pile rectangles symmetrically embedded with their axis from the outside to the inside. The left pile is pile No. 1 and the right pile is pile No.

2. The sizes and materials of the two piles are exactly the same; the length of the double piles embedded in the soil is L, the pile diameter is D, and the distance between the double piles is r p , the elastic modulus is E p , the horizontal side length of the soil rectangle is K times the pile diameter D, and the vertical height is H times the pile diameter D; With the lower left corner vertex of the soil rectangle as the origin, a plane coordinate system is established and each pile rectangle is divided into N p A square grid with a side length of pile diameter D is constructed, and N p The grids are sorted, and the serial numbers are k=1, 2, ..., N. p ; Divide the soil rectangle into N e A square grid with a side length of pile diameter D is constructed from bottom to top and from left to right. e The grids are sorted, and the coordinates of the center point of each expansive soil grid are used as the coordinates of the expansive soil grid; Step 1.2, the soil parameters are 3, namely effective cohesion c a ', effective friction angle δ' and saturated elastic modulus E sat , any one of them is recorded as soil property parameter Γ h , h is the serial number of soil parameter, h = 1, 2, 3; the soil parameter random field model is the probability distribution model of soil parameter random field, the probability distribution model includes probability distribution type and autocorrelation function type, record the three soil parameters Γ h All are log-normal distribution and square exponential autocorrelation function; The Karhunen–Loève expansion method is used to calculate the three soil parameters Γ in each grid. h The lognormal random field discrete value And recorded as discrete values Its expression is: Where ξ is a random vector in an independent standard normal space and is denoted as random vector ξ, ξ is a 1×3M matrix, M is the number of truncation terms; λ j is the jth eigenvalue in the eigenvalue matrix λ, where j is the number of cutoff terms, j = 1, 2, …, M; f j (x) is the characteristic function of the jth column in the characteristic function matrix f(x); μ h is the soil parameter Γ h The mean of h is the soil parameter Γ h The mean square error of ; L is the lower triangular matrix obtained by Cholesky decomposition of the random field cross-correlation coefficient matrix in the relevant standard normal space, and the superscript T represents the transpose of the matrix; Step 2: Calculate the component reliability index and component sensitivity index of the three failure modes of double piles Step 2.1, based on the three soil parameters Γ in step 1 h The lognormal random field discrete value The displacement s of the bottom No. 1 grid of a given set of double piles bi , i is the pile number, i = 1, 2, a set of grid displacement s of the double pile bi Based on this, the double-pile load transfer method considering the change of matrix suction and the expansion of expansive soil is used to calculate the displacement s of the grid No. 1 in this group. bi The corresponding loads on the tops of the two piles are Q i and displacement s i , and draw the relationship curve between the load Q and displacement s of each pile in a plane coordinate system; Step 2.2, based on the load Q and displacement s relationship curve drawn in step 2.1, calculate the functional function values ​​of the three failure modes of the double piles under the vertical fixed load Q1 at the top of pile No. 1 and the vertical fixed load Q2 at the top of pile No. 2; The first failure mode is the compression failure or pull-out failure of pile 1, which is denoted as element FM1. The expression of its functional value g1(ξ) is: In the formula, G p is the pile gravity; R down1 (ξ) is the ultimate bearing capacity of pile No. 1, according to the preset maximum allowable settlement s mdown Obtained from the load Q and displacement s relationship curve of pile No. 1; R up1 (ξ) is the ultimate pull-out capacity of pile No. 1, according to the preset maximum allowable upward displacement s mup It is obtained from the relationship curve between the load Q and displacement s of pile No. 1; s1(ξ) is the displacement of the pile top of pile No. 1, s1(ξ)≥0 means that the pile top moves downward, and s1(ξ)<0 means that the pile top moves upward; The second failure mode is the compression failure or pull-out failure of pile 2, which is denoted as element FM2. The expression of its functional value g2(ξ) is: In the formula, R down2 (ξ) is the ultimate bearing capacity of pile 2, according to the preset maximum allowable settlement s mdown Obtained from the load Q and displacement s relationship curve of pile 2; R up2 (ξ) is the ultimate pull-out capacity of pile No. 2, according to the preset maximum allowable upward displacement s mup It is obtained from the relationship curve between the load Q and displacement s of pile No. 2; s2(ξ) is the displacement of the pile top of pile No. 2, s2(ξ)≥0 means that the pile top moves downward, and s2(ξ)<0 means that the pile top moves upward; The third failure mode is the settlement difference Δ caused by the double piles s Exceeding the maximum allowable settlement difference Δ m , denoted as element FM3, the expression of its functional value g3(ξ) is: g3(ξ)=Δ m -D s =D m -|s1(ξ)-s2(ξ)| Step 2.3, based on the three failure mode function values ​​g calculated in step 2.2 q (ξ), q is the serial number of the three failure modes, q = 1, 2, 3, and the first-order reliability method is used to calculate the component reliability index β corresponding to each failure mode. q , component reliability index β q The higher it is, the safer the component is; Step 2.4, calculate the three soil parameters Γ respectively h For each component reliability index β q The sensitivity index S q h The higher the sensitivity index, the stronger the soil property parameter Γ h The variability of the component reliability index β q The greater the influence, the calculation formula is as follows: In the formula, α q (j) is the direction cosine vector α of the qth failure mode q The j-th direction cosine value in ; Step 3: Calculate the reliability index and system sensitivity index of the dual pile system The three failure modes of the double pile foundation are in series relationship. The component reliability index β of the three failure modes calculated in step 2 is q , using the sequential component method to calculate the system reliability index β sys ; Step 3.1, combine components FM1 and FM2 into a combined component E, and calculate the reliability index β of the combined component E 1or2 , the calculation formula is: b 1or2 =-Φ -1 [Φ(-β1)+Φ(-β2)-Φ(-β 1|2 )Φ(-β2)] Where Φ is the standard normal cumulative distribution function, Φ -1 (·) is the inverse function of the standard normal cumulative distribution; β1 is the reliability index of component FM1; β2 is the reliability index of component FM2; β 1|2 is the reliability index of component FM1 in the case of component FM2 failure, and the calculation formula is: F2 is parameter No. 1, F2 = φ(-β2) / Φ(-β2), where φ is the standard normal probability density function, W2 is parameter No. 2, W2 = F2(-β2+F2), ρ12 is the correlation coefficient between element FM1 and element FM2, ρ 12 =α1 T α2, α1 is the direction cosine vector of element FM1, α2 is the direction cosine vector of element FM2; Step 3.2, record the combination of component E and component FM3 as a system, and the system reliability index is marked as β sys , and its calculation formula is: b sys =-Φ -1 [Φ(-β 1or2 )+Φ(-β3)-Φ(-β 1or2|3 )Φ(-β3)] Where, β3 is the reliability index of component FM3; β 1or2|3 is the reliability index of the combined component E in the case of component FM3 failure, and the calculation formula is: F3 is parameter No. 3, F3 = φ(-β3) / Φ(-β3), W3 is parameter No. 4, W3 = F3(-β3+F3), ρ (1or2)3 is the correlation coefficient between the combined element E and the element FM3; Calculate the system reliability index β sys Afterwards, the three soil parameters Γ are calculated respectively. h System reliability index β sys The sensitivity index S sys h , first we need to calculate the mean matrix μ of the random vector ξ ξ System reliability index β sys The system sensitivity index matrix S s , the calculation formula is: In the formula, record is the first partial derivative parameter, is the partial derivative parameter No. 2; α3 is the direction cosine vector of element FM3; Calculate three soil parameters Γ separately h Sensitivity index S to system reliability index sys h , the calculation formula is as follows: In the formula, S s (j) is the system sensitivity index matrix S s The j-th system sensitivity index in .

2. The system reliability and sensitivity analysis method of double piles in spatially variable expansive soil according to claim 1 is characterized in that: The calculation formula of the square exponential autocorrelation function described in step 1.2 is as follows: Where x1 and x2 are the two-dimensional coordinate vectors of the centers of any two expansive soil grids, x1 is the coordinate vector x1, and x2 is the coordinate vector x2; ρ(x1, x2) is the soil property parameter Γ h The correlation coefficient between the coordinate vector x1 and the coordinate vector x2; 11 is the coordinate component of the coordinate vector x1 in the horizontal direction; 12 is the coordinate component of the coordinate vector x1 in the vertical direction; 21 is the coordinate component of the coordinate vector x2 in the horizontal direction; 22 is the coordinate component of the coordinate vector x2 in the vertical direction; L H is the autocorrelation distance in the horizontal direction; L V is the autocorrelation distance in the vertical direction.

3. A system reliability and sensitivity analysis method for double piles in spatially variable expansive soil according to claim 1 or 2, characterized in that: The calculation steps of the eigenvalue matrix λ and the eigenfunction matrix f(x) described in step 1.2 are as follows: Step 1.2.1, according to the Galerkin method, the characteristic function f j (x)(j=1,2,…,M) is decomposed into M orthogonal basis functions h m Linear combination of (x)(m=1,2,…,M): Where D (j) is an M×1 column vector, D (j) =[d1 (j) ...d m (j) ...d M (j) ] T ; h(x) is a column vector of M×1, h(x)=[h1(x)…h m (x)…h M (x)] T ; Step 1.2.2, substitute the linear expansion of the above characteristic function into the Fredholm integral equation ∫ Ω ρ(x1,x2)f j (x1)dx1=λ j f j (x2), the following matrix equation is obtained: AD=DΛ Where A is the generalized stiffness matrix, which is an M×M matrix; D is the characteristic matrix of the generalized stiffness matrix A, D = [D (1) …D (j) …D (M) ], which is an M×M matrix; Λ is a diagonal matrix, and the jth value on its diagonal is the eigenvalue λ j ; The element A of the generalized stiffness matrix A jm (j,m=1,…,M) is determined by the following formula: A jm =∫ Ω ∫ Ω ρ(x1,x2)h j (x2)h m (x1)dx1dx2 Step 1.2.3, let the one-dimensional study area be Ω = [x min ,x max ], using the standard Legendre polynomial function in the interval [-1,1] to represent [x min ,x max ] interval orthogonal basis function h m (x), which is: Where, L m-1 (x) is the Legendre polynomial; T is the translation coefficient, T = (x max +x min ) / 2; T' is the scaling factor, T'=(x max -x min ) / 2; Step 1.2.4, transform the element A of the generalized stiffness matrix A jm Further rewritten as: Where x1 and x2 are the one-dimensional coordinate values ​​of any two expansive soil grid centers in the horizontal or vertical direction; After solving the generalized stiffness matrix A from the above formula, the eigenvalue matrix λ and the eigenfunction matrix f(x) are solved according to the above matrix equation AD=DΛ. The final eigenfunction matrix f(x) is as follows:

4. The system reliability and sensitivity analysis method of double piles in spatially variable expansive soil according to claim 1, characterized in that: The displacement s of the grid No. 1 in this group is calculated by the double pile load transfer method considering the change of matrix suction and the expansion of expansive soil as described in step 2.

1. bi The corresponding loads on the tops of the two piles are Q i and displacement s i The steps are as follows: Step 2.1.1, given the pile number i, i = 1, 2; given the displacement s of the bottom grid No. 1 of the double pile bi ; Step 2.1.2, when s bi When it is negative, it means that the bottom of the pile moves upward and the end resistance P bi is zero, when s bi When it is positive, it means that the bottom of the pile moves downward and there is end resistance P bi , end resistance P bi The calculation formula is as follows: In the formula, A i is the first empirical coefficient; B i is the second empirical coefficient, B i =R bf / P bui , where R bf is the empirical failure ratio of pile tip resistance, P bui is the limit end resistance; the first empirical coefficient A i is the pile end spring stiffness k bi The reciprocal of is calculated as: In the formula, k b1i is the first spring stiffness, calculated as k b1i =4G bi / πr0(1-v), where v is Poisson's ratio, r0 is the pile radius, G bi is the shear modulus of the expansive soil grid directly below the pile tip, calculated as G bi =E u-bi / [2(1+v)], where E u-bi is the unsaturated elastic modulus E of the expansive soil grid just below the pile tip unsat , unsaturated elastic modulus E unsat The calculation formula is as follows: In the formula, E sat is the saturated elastic modulus, which conforms to the calculation formula of the random field discrete value; U1 is the first fitting parameter; U2 is the second fitting parameter; ψ is the matrix suction; S r is saturation; P a The atmospheric pressure is 101.3 kPa; k b2i is the second spring stiffness, and the calculation formula is Step 2.1.3, calculate the displacement s of the expansive soil grid on the right side of the double pile No. 1 grid bi ',s bi '=s bi +Δh i , where Δh i The expansion amount of the expansive soil grid is calculated as follows: Where Δψ is the reduction of matrix suction ψ; E a is the average unsaturated elastic modulus of expansive soil from unsaturated state to saturated state; Step 2.1.4: record the initial value s of the pile-soil relative displacement between the midpoint of the double pile No. 1 grid and the expansive soil grid on its right side. si The displacement of the expansive soil grid is s bi 'Equal, the lateral friction resistance τ of the midpoint of the double pile No. 1 grid si The calculation formula is as follows: In the formula, a i is the third empirical coefficient, which is the pile side spring stiffness k si The inverse of is calculated as follows: In the formula, k 1i is the third spring stiffness, calculated as k 1i =G i / r0ln(r mi / r0), where G i is the mesh shear modulus of expansive soil, r mi is the maximum influence radius of the i-th pile, and the calculation formula is Among them G ik is the shear modulus of the kth expansive soil grid on the right side of the i-th pile, and the calculation formula is G ik =E u-ik / [2(1+v)], where E u-ik is the unsaturated elastic modulus E of the kth expansive soil grid unsat , G mi is the maximum shear modulus of the first column of expansive soil grid on the right side of the i-th pile; k 2i is the fourth spring stiffness, calculated as k 2i =G i / r0ln(r mi-1 / r p ), where r mi-1 is the maximum influence radius of the neighboring piles of the i-th pile, and the calculation formula is the same as r mi The same as k, but the shear modulus mentioned in the formula is the shear modulus of the expansive soil grid on the right side of the adjacent pile of the i-th pile; 3i is the fifth spring stiffness, calculated as k 3i =r p G i / r0 2 ln(r mi-1 / r p ); b i is the fourth empirical coefficient, and its calculation formula is b i =R sf / τ sui , where R sf is the empirical failure ratio of lateral friction, τ sui is the limit lateral friction. When the matrix suction changes from state ψ1 to state ψ2, the limit lateral friction τ sui The calculation formula is: In the formula, c ai ′ is the effective cohesion of the pile-soil interface, and this parameter conforms to the calculation formula of the random field discrete value; δ i ′ is the effective friction angle of the pile-soil interface, and this parameter conforms to the calculation formula of the random field discrete value; P S (ψ 1–0 ) represents the vertical expansion pressure generated from the matrix suction state ψ1 to the saturation state; E a (ψ 1–0 ) is the average unsaturated elastic modulus of expansive soil from matrix suction state ψ1 to saturated state; P S (ψ 2–0 ) represents the vertical expansion pressure generated from the matrix suction state ψ2 to the saturation state; E a (ψ 2–0 ) is the average unsaturated elastic modulus of the expansive soil from the matrix suction state ψ2 to the saturated state; θ is the volumetric water content; θ s is the saturated moisture content; θ r is the residual moisture content; σ vs is the effective earth pressure; Step 2.1.5, record the lateral friction τ at different positions in each double pile grid si The load P on the top surface of the grid No. 1 of the double pile is calculated. ti , P ti =P bi +πD 2 τ si , assuming that the axial force in each double pile grid is linearly distributed, calculate the elastic compression s at the midpoint of the double pile grid No. 1 cmi , the calculation formula is: Then the pile-soil relative displacement correction value s at the midpoint of the double pile No. 1 grid is si ′ is s s ' i =s bi +s cmi , if the pile-soil relative displacement correction value s si ′ and the initial value of relative displacement between pile and soil s si If the difference is less than the first set threshold, proceed to the next step, otherwise return to step 2.1.4; The pile-soil relative displacement correction value s of the midpoint of the double pile No. 1 grid is si ′ is the initial value of relative displacement of pile and soil s si Repeat steps 2.1.4 to 2.1.5 until the difference is less than the first set threshold; Step 2.1.6, based on the final iterative stable pile-soil relative displacement correction value s si Calculate the side friction correction value τ si ′, and calculate the load P on the top surface of the double pile No. 1 grid ti ′,P′ ti =P bi +πD 2 τ′ si Then calculate the displacement of the top surface of the double pile No. 1 grid, the calculation formula is s ti =s bi +s ci , where s ci is the elastic compression of the double pile No. 1 grid, calculated as s ci =2(P′ ti +P bi ) / πDE p ; The end resistance of the bottom surface of the double pile No. 2 grid is the load P on the top surface of the double pile No. 1 grid. ti ′, the displacement of the bottom surface of the double pile No. 2 grid is the displacement of the top surface of the double pile No. 1 grid s ti , return to step 2.1.2 and calculate the load and displacement of the top surface of the grid No. 2 of the double pile according to steps 2.1.2 to 2.1.

6. Repeat the above steps to calculate the load and displacement of the top surface of the grid No. 3, 4, ..., N of the double piles. p The load and displacement on the top surface of the grid, and finally the Nth double pile p The load Q on the top surface of the grid i and displacement s i , which is the load Q on the top of the double pile i and displacement s i .

5. The system reliability and sensitivity analysis method of double piles in spatially variable expansive soil according to claim 1, characterized in that: Calculate the component reliability index β using the first-order reliability method as described in step 2.3 q The steps are as follows: Step 2.3.1, let the initial vector ξ0 of the random vector ξ be a 1×3M zero matrix; Step 2.3.2, calculate the initial reliability index β0 based on the initial vector ξ0, the calculation formula is: In the formula, this formula is recorded as Formula No. 1, g(ξ0) is the performance function value corresponding to the initial vector ξ0, is the partial derivative of the performance function g(ξ0) with respect to the initial vector ξ0, μ ξ is the mean matrix of the random vector ξ, C ξ is the covariance matrix of the random vector ξ; Step 2.3.3, calculate the iteration vector ξ of the random vector ξ t ,ξ t =μ ξ +σ ξ α0β0, where this formula is recorded as Formula No. 2, α0 is the initial direction cosine vector, This formula is recorded as Formula No. 3, ρ ξ is the correlation coefficient matrix of the random vector ξ; Step 2.3.4, use formula 1 to calculate the iteration vector ξ t The corresponding iterative reliability index β t , where all initial vectors ξ0 in formula 1 are replaced by the iteration vector ξ t , if the initial reliability index β0 and the iterative reliability index β t The difference is less than the second set threshold, and this iteration vector ξ t That is the verification point vector ξ * , proceed to the next step, otherwise use formula 2 to calculate the new iteration vector ξ t , where the initial direction cosine vector α0 is replaced by the iterative direction cosine vector α t , and replace all initial vectors ξ0 in formula 3 with the iteration vector ξ t , and then calculate the new iterative reliability index β t , and the calculation is repeated in this way until the initial reliability index β0 and the iterative reliability index β t The difference is less than a second set threshold; Step 2.3.5, based on the verification point vector ξ * Use formula 1 to calculate the final component reliability index β q , and use formula 3 to calculate the final direction cosine vector α q , where all initial vectors ξ0 in formulas 1 and 3 are replaced by the verification point vector ξ * .

6. The system reliability and sensitivity analysis method of double piles in spatially variable expansive soil according to claim 1, characterized in that: Step 3.2 Correlation coefficient ρ between the combination element E and the element FM3 (1or2)3 By inverse solving the following formula: Where Φ2 is the two-dimensional standard normal cumulative distribution function; β 1|3 is the reliability index of component FM1 in the case of failure of component FM3, and the calculation formula is: where ρ 13 is the correlation coefficient between element FM1 and element FM3, ρ 13 =α1 T α3, α3 is the direction cosine vector of element FM3; β 2|3 is the reliability index of component FM2 in the case of component FM3 failure, and the calculation formula is: where ρ 23 is the correlation coefficient between element FM2 and element FM3, ρ 23 =α2 T α3; Step 3.2 The mean matrix μ of the random vector ξ ξ System reliability index β sys The system sensitivity index matrix S s In the calculation formula, the first partial derivative parameter The calculation formula is: Step 3.2 The mean matrix μ of the random vector ξ ξ System reliability index β sys The system sensitivity index matrix S s In the calculation formula, the second partial derivative parameter The calculation formula is:

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