CN112685825A - Optimization method of stepwise equivalent plane method - Google Patents

Optimization method of stepwise equivalent plane method Download PDF

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CN112685825A
CN112685825A CN202110091094.8A CN202110091094A CN112685825A CN 112685825 A CN112685825 A CN 112685825A CN 202110091094 A CN202110091094 A CN 202110091094A CN 112685825 A CN112685825 A CN 112685825A
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CN112685825B (en
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李鹏伟
司李南
李蚩行
闫琳
宋攀
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Xian Aeronautical Polytechnic Institute
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Abstract

The invention discloses an optimization method of a step-by-step equivalent plane method, which comprises the following steps: determining a normal random variable and establishing a function of each failure mode in a portal frame structure; step two, acquiring a reliability index of a single failure mode; step three, acquiring a linear structure function with a unit coefficient vector; step four, sequencing the failure modes in the order of the phase relation number from large to small; and step five, calculating the failure probability of the serial portal frame structure system. According to the method, one failure mode is used for replacing two failure modes successively, the problem of solving the reliability indexes of the multi-failure-mode serial portal frame structure system at one time is converted into the problem of solving a series of reliability indexes of the two failure modes, the complexity of the solving process is reduced, and the solving efficiency is high on the premise that the sufficient accuracy is kept.

Description

Optimization method of stepwise equivalent plane method
Technical Field
The invention belongs to the technical field of reliability analysis of a serial portal frame structure system, and particularly relates to an optimization method of a gradual equivalent plane method.
Background
The portal frame structure serving as a traditional structural system has the characteristics of simple stress, clear force transmission path, quick component manufacturing, obvious economic benefit, convenience for industrial processing, short construction period and the like, and is widely applied to industrial and civil buildings such as single-story factory buildings, civil supermarkets, exhibition halls, storehouses, cultural and entertainment public facilities and the like. In the process of processing, manufacturing and service of the structure, the door-type frame structure is often inevitably subjected to uncertainty of material characteristics, geometric parameters and the like caused by factors such as manufacturing environment, technical conditions and the like, and uncertainty of load caused by factors such as operating conditions, natural conditions and service environment and the like. These uncertain factors can have a great influence on the performance of the structure, so that the structure has certain potential safety hazard in the service process, and therefore, the structure needs to be considered scientifically and faithfully. The mathematical models of uncertain factors in the existing measurement structure comprise a probability model, a fuzzy model and a convex set model, and the corresponding reliability analysis methods are called probability reliability analysis, fuzzy reliability analysis and non-probability reliability analysis. The probability reliability analysis technology has longer development time, a theoretical system is more perfect, and the probability reliability analysis technology is more widely applied to practical problems. In view of this, probabilistic reliability analysis becomes an effective way to deal with uncertainty in the gantry framework structure. For a multi-failure-mode portal frame structure system with structural failure possibly caused by multiple damage mechanisms, two approximate ideas of interval estimation and point estimation are mainly adopted in the existing reliability analysis. The narrow-boundary method based on the interval estimation idea has poor precision and the error of the narrow-boundary method can be rapidly increased along with the increase of the number of failure modes, and the stepwise equivalent plane method based on the point estimation idea has low solving efficiency because each equivalent needs to calculate the correlation coefficient between every two failure modes and needs to differentiate all variables.
Disclosure of Invention
The invention aims to provide an optimization method of a gradual equivalent plane method to solve the problems.
In order to achieve the purpose, the invention adopts the following technical scheme:
a method for optimizing a stepwise equivalent plane method comprises the following steps:
step 1, determining a normal random variable and establishing a function of each failure mode in a portal frame structure;
step 2, judging whether normal random variables in a portal frame structure system are mutually independent or not, and obtaining the reliability index of a single failure mode;
step 3, acquiring a linear structure function with a unit coefficient vector, and determining the reliability index of the structure function;
step 4, calculating the correlation coefficient between every two failure modes, and sequencing the failure modes according to the sequence of the phase relation numbers from large to small;
and 5, calculating the reliability index of the serial portal frame structure system according to the sequence of the step 4.
Further, the step 1 specifically includes:
step 101, determining uncertain variables in the structure, and recording a random vector formed by all normal random variables in the structure as X ═ X (X)1,X2,…,Xn)TN is the total number of normal random variables in the structure; establishing a normal distribution probability density function of each random variable according to sample points given in practical problems, and recording the normal distribution obeyed by the ith random variable as
Figure BDA0002912503780000021
Wherein i is 1,2, …, n;
step 102, determining a function g of each failure mode of the serial portal frame structure system according to the structural failure criterion based on the random variable in the step 101j(X)＝gj(X1,X2,…,Xn) Where j is the number of the failure mode in the portal frame architecture and j is 1,2, …, m, m is the total number of failure modes in the architecture.
Further, step 2 specifically includes:
step 201, judging whether normal random variables in a portal frame structure system are mutually independent, if so, executing step 202, and if so, executing step 203;
step 202, reliability analysis of the independent normal random variable failure mode: if X is ═ X1,X2,…,Xn)TIs a random vector composed of n mutually independent normal random variables, expressed as
Figure BDA0002912503780000022
Converting random variable vector X into standard normal random variable Y ═ Y (Y)1,Y2,…,Yn)T. Substituting the normalized normal random variable into the jth structure function gj(X)＝gj(X1,X2,…,Xn) The structural function in the standard normal space is obtained as gj(Y)＝gj(Y1,Y2,…,Yn). If the structure function gj(Y)＝gj(Y1,Y2,…,Yn) If the function is a linear function, go to step 2021; if the structure function gj(Y)＝gj(Y1,Y2,…,Yn) If the function is a non-linear function, go to step 2022;
step 203, reliability analysis of the related normal random variable failure mode: if X is ═ X1,X2,…,Xn)TIs a random vector composed of n related normal random variables, and for the jth structure function gj(X)＝gj(X1,X2,…,Xn) Then there are X to N (mu)X,CX) Wherein
Figure BDA0002912503780000031
Is the mean vector of a normal random variable vector X, CXIs a covariance matrix of a normal random variable vector X,
and is
Figure BDA0002912503780000032
Wherein k isIs also the number of normal random variables, and k is 1,2, …, n; the probability density function of X is
Figure BDA0002912503780000033
Wherein | CXI represents the covariance matrix CXDeterminant of (4);
according to a random variable XiAnd XkCorrelation coefficient of
Figure BDA0002912503780000034
With its covariance Cov (X)i,Xk) In relation to (2)
Figure BDA0002912503780000035
Determining a matrix of correlation coefficients for a random variable vector X
Figure BDA0002912503780000036
Obtaining rho by Cholesky decompositionX＝AATWherein, A is a lower triangular matrix obtained by Cholesky decomposition; with matrix a, the associated random variable X can be represented by an independent standard normal random variable vector Y ═ Y (Y)1,Y2,…,Yn)TExpressed as X ═ σXAY+μXWherein, in the step (A),
Figure BDA0002912503780000037
make X ═ sigmaXAY+μXSubstituting into a structural function gj(X)＝gj(X1,X2,…,Xn) Obtaining a structural function g in a standard normal spacej(Y)＝gj(Y1,Y2,…,Yn) Then, the reliability index of the structure function is calculated according to step 202.
Further, step 202 specifically includes:
step 2021, calculating the reliability index of the linear structure function: the expression of a linear structure function in the X space of the original variable space is assumed as
Figure BDA0002912503780000041
Wherein a isj0Is a constant term in the jth structure function, aji(i-1, 2, …, n) is the ith random variable X in the jth structure functioniThe coefficient of (a); the structural function converted into the standard normal space is
Figure BDA0002912503780000042
Then the structure function gj(Y) has a mean value and a standard deviation of
Figure BDA0002912503780000043
And
Figure BDA0002912503780000044
obtaining the reliability index of the jth structure function
Figure BDA0002912503780000045
Step 2022, calculating the reliability index of the nonlinear structure function: when the structural function is nonlinear, the nonlinear structural function is expanded into linearity at a check point; assuming a checking point of
Figure BDA0002912503780000046
The structural function is then expanded once to
Figure BDA0002912503780000047
The mean and standard deviation thereof are respectively
Figure BDA0002912503780000048
And
Figure BDA0002912503780000049
the reliability index of the jth nonlinear structure function is obtained as
Figure BDA00029125037800000410
The process of calculating the reliability index by an iterative method comprises the following steps: first assume an initial check point
Figure BDA00029125037800000411
Substitute it into
Figure BDA00029125037800000412
Calculating a reliability index betaj(ii) a Then beta is mixedjSubstitution into
Figure BDA0002912503780000051
Calculate out
Figure BDA0002912503780000052
Then will be
Figure BDA0002912503780000053
βjAnd
Figure BDA0002912503780000054
substitution into
Figure BDA0002912503780000055
Calculating new checking points
Figure BDA0002912503780000056
Finally judging | | X*(1)-X*(0)Whether or not | l < ε holds, where | l | · | represents the 2-norm of the vector, ε is a specified tolerance and is typically taken to be 10-3If the judgment result is 'yes', the iteration is stopped, and if the judgment result is 'no', the method is used
Figure BDA0002912503780000057
Replacement of
Figure BDA0002912503780000058
And re-executing the solving process until the final judgment result is yes to obtain the reliability index of the nonlinear structure function, wherein the check point obtained by iteration is the check point of the nonlinear structure function
Figure BDA0002912503780000059
Further, the step 3 specifically comprises:
if the original structure function is a linear structure function, according to step 2021, the jth structure function in the Y space of the standard normal variable space is expressed as
Figure BDA00029125037800000510
Then its corresponding linear structure function with unit coefficient vector is
Figure BDA00029125037800000511
And the constant term in the structure function is the reliability index beta of the structure functionj(ii) a If the original structure function is a non-linear structure function, the checking point of the structure function is obtained according to step 2022
Figure BDA00029125037800000512
And at check point to structure function gj(X) performing Taylor first-order expansion to obtain linear approximate structure function
Figure BDA00029125037800000513
And by variable substitution
Figure BDA00029125037800000514
Converting it into standard normal space Y space to obtain
Figure BDA00029125037800000515
It has the structural function of the unit coefficient vector expressed as
Figure BDA0002912503780000061
And the constant term in the structure function is also the reliability index of the structure function, namely the reliability index is
Figure BDA0002912503780000062
The ith random variable Y of the jth linear structure function with the unit coefficient vector in the standard normal variable spaceiIs uniformly recorded as alphajiThen, the jth structural function with unit coefficient vector in the normal variable space is uniformly expressed as
Figure BDA0002912503780000063
Further, the step 4 specifically comprises:
step 401, calculating a correlation coefficient between every two failure modes: taking the j-th and h-th (h is 2,3, …, m) (h > j) structural function functions of the m linear structural function functions with unit coefficient vectors obtained in the third step and recording the functions as
Figure BDA0002912503780000064
The mean value of the structural function is
Figure BDA0002912503780000065
The variance of the structural function is
Figure BDA0002912503780000066
The standard normal random variables in the structural function are all independent of each other, if
Figure BDA0002912503780000067
The variance of the structural function is reduced to
Figure BDA0002912503780000068
Then the correlation coefficient of the jth and the h structural function is obtained as
Figure BDA0002912503780000069
Substituting any two structure function functions into the formula to obtain m structure function correlation coefficient matrixes
Figure BDA0002912503780000071
And step 402, sequencing the failure modes from large to small according to the number of the relations.
Further, step 402 specifically includes:
step 4021, selecting the largest correlation coefficient from the correlation coefficient matrix rho, and numbering two corresponding structure function functions as (j → h);
step 4022, selecting all function functions g related to the structure from the correlation coefficient matrix rhoj(Y) and gh(Y) the relevant correlation coefficients and selecting the largest correlation coefficient therefrom, the number of the corresponding further structure function being denoted q; if the correlation coefficient is the structural function gq(Y) and gj(Y), the ordering of the three structure function functions is (q → j → h); if the correlation coefficient is the structural function gh(Y) and gq(Y), the ordering of the three structure function functions is (j → h → q);
step 4023, if the sequence of the three structural function functions is (q → j → h), removing the structural function g from the correlation coefficient matrix ρj(Y) all relevant correlation coefficients, and judging whether the number of elements in the correlation coefficient matrix rho is zero or not; if the judgment result is 'yes', ending the sequencing process; if the judgment result is 'no', the structural function g is judgedq(Y) and gh(Y) repeating the step 4022 until the judgment result is YES; if the three structure function functions are ordered as (j → h → q), the structure function g in the correlation coefficient matrix ρ is removedh(Y) all relevant correlation coefficients, judging whether the number of elements in the correlation coefficient matrix rho is zero or not, if the judgment result is yes, finishing the sorting process, if the judgment result is no, and then performing structural function gq(Y) and gh(Y) repeating the step 4022 until the judgment result is YES;
step 4024, numbering the m structural function functions again according to the sequence obtained in the step 4023 according to the sequence numbers 1,2, … and m; and updating the correlation coefficient matrix of the structural function according to the renumbering.
Further, step 5 specifically includes:
step 501, according to the sequence of step 4024, selecting the first three structural function functions g1(Y)，g2(Y) and g3(Y), calculating the joint failure probability of every two structural function functions and the joint failure probability of the three structural function functions;
step 502, calculating the failure probability of each of the three selected structure function functions: calculating three selected structure function g according to formula P ═ phi (-beta)1(Y)、g2(Y) and g3(Y) failure probabilities, respectively denoted as P1＝Φ(-β1)、P2＝Φ(-β2) And P3＝Φ(-β3) Where Φ (·) is a standard normal distribution function;
step 503, calculating the joint failure probability of every two structure function functions:
structural function (g)1(Y),g2(Y))，(g1(Y),g3(Y)) and (g)2(Y),g3(Y)) a joint probability density function of
Figure BDA0002912503780000081
The joint failure probability of the two structural function functions is
Figure BDA0002912503780000082
Wherein
Figure BDA0002912503780000083
Is an integral domain; at a place far away from the origin of coordinates, the joint probability density function tends to be 0, and the main failure area near the origin of coordinates is taken as the integral domain, namely the integral domain is respectively taken as
Figure BDA0002912503780000084
Wherein epsilon is used to control the accuracy of the approximation calculation, and epsilon should satisfy
Figure BDA0002912503780000085
Of two structural function functionsThe joint failure probability is approximately expressed as
Figure BDA0002912503780000086
Then obtaining the joint failure probability of two structure function functions by a numerical method
Figure BDA0002912503780000091
Wherein N represents the number of the divided intervals on each one-dimensional coordinate,
Figure BDA0002912503780000092
and
Figure BDA0002912503780000093
separately representing the integral domain
Figure BDA0002912503780000094
The (r, t) th divided section, and
Figure BDA0002912503780000095
respectively represent
Figure BDA0002912503780000096
And
Figure BDA0002912503780000097
a center point of (a), and
Figure BDA0002912503780000098
step 504, calculating a structure function g1(Y)，g2(Y) and g3(Y) joint failure probability: structural function g1(Y)，g2(Y) and g3(Y) a joint probability density function of
Figure BDA0002912503780000099
Wherein [ C]Representing a structural function g1(Y)，g2(Y) and g3A variance matrix of (Y), and
Figure BDA00029125037800000910
det[C]representation matrix [ C]Is of determinant
Figure BDA00029125037800000911
[Λjh]＝[C]-1(ii) a The joint failure probability of the three structural function functions is obtained
Figure BDA00029125037800000912
Wherein G is123＝{(g1(Y),g2(Y),g3(Y))(g1(Y)＜0)∩(g2(Y)＜0)∩(g3(Y) < 0) } denotes an integration domain,
Figure BDA0002912503780000101
represents G123A primary failure zone closer to the origin of coordinates; then g is1(Y)，g2(Y) and g3(Y) the joint failure probability is expressed as
Figure BDA0002912503780000102
Wherein Δ VrtlRepresenting the integral domain
Figure BDA0002912503780000103
Volume of the upper (r, t, l) th region, and
Figure BDA0002912503780000107
(g1r(Y),g2t(Y),g3l(Y)) represents the center point of the (r, t, l) th region, and
Figure BDA0002912503780000104
step 505, equivalence of two failure modes: the equivalent process needs to satisfy three conditions: (1) the failure domain corresponding to the equivalent limit state surface and the failure domain surrounded by the two limit state surfaces have the same failure probability or (generalized) reliability index; (2) the normal vector of the equivalent limit state surface and the normal vectors of the two limit state surfaces are in the same plane; (3) the failure probability of the region enclosed by the equivalent limit state surface and the third limit state surface is equal to the failure probability of the region enclosed by the corresponding three limit state surfaces; the method comprises the following steps that a condition (1) is used for determining a reliability index of an equivalent structure function, and conditions (2) and (3) are used for determining a normal vector of an equivalent limit state surface;
step 506, judging whether the total number of the structural function functions is equal to 3, if so, numbering the three structural function functions from 1 again, and respectively calculating the failure probability P of a single structural function of the three structural function functions according to the steps 502, 503 and 5041、P2And P3Combined failure probability P of two-by-two structure function12、P13、P23And joint failure probability P of three structural function123Obtaining the reliability index beta of the serial portal frame structure systemseriesIs betaseries＝-Φ-1(P1+P2+P3-P12-P13-P23+P123) (ii) a If the judgment result is 'no', the fifth step is executed again until the judgment result is 'yes'.
Further, step 505 specifically includes:
step 5051, solving the reliability index of the equivalent structure function: for structural function
Figure BDA0002912503780000105
Assuming that its equivalent structure function is
Figure BDA0002912503780000106
Phi (-beta) can be obtained according to the condition (1)e)＝P12Where phi (-) is a normal distribution function, the equivalent reliability index is betae＝-Φ-1(P12)；
Step 5052, solving the normal vector of the corresponding failure plane of the equivalent structure function: memory structure function g1(Y)，g2(Y) normal vectors corresponding to the failure planes are respectively
Figure BDA0002912503780000111
Equivalence ofStructural function of function
Figure BDA0002912503780000112
The normal vector corresponding to the failure surface is a column vector
Figure BDA0002912503780000113
According to the condition (2), there is a unique real number λ (λ > 0) such that
Figure BDA0002912503780000114
Assuming equivalent structural function
Figure BDA0002912503780000115
And g3(Y) a joint failure probability of
Figure BDA0002912503780000116
Structural function of function
Figure BDA0002912503780000117
And g3The correlation coefficient of (Y) is ρ 12,3, depending on the condition (3)
Figure BDA0002912503780000118
Wherein
Figure BDA0002912503780000119
Is simple and easy to obtain
Figure BDA00029125037800001110
And wherein betaeAnd beta3Known as phi (-beta)e,-β3,ρ12,3)＝f(ρ12,3)＝P13+P23-P123Get rho12,3＝f-1(P13+P23-P123)；
Expression based on correlation coefficients of two structural function functions
Figure BDA00029125037800001111
Is provided with
Figure BDA00029125037800001112
Obtain a quadratic equation of unity with respect to λ
Figure BDA00029125037800001113
Solving the equation to obtain the positive root of lambda
Figure BDA00029125037800001114
Substituting the obtained lambda into
Figure BDA00029125037800001115
Obtaining the normal vector of the equivalent structure function
Figure BDA00029125037800001116
Then unitizing the vector to obtain a unit normal vector alpha of the equivalent structure functioneIs composed of
Figure BDA00029125037800001117
At this point, the structural function g is uniquely determined1(Y)，g2Equivalent structural function of (Y)
Figure BDA00029125037800001118
Structural function g1(Y) and g2(Y) replacement by
Figure BDA00029125037800001119
Compared with the prior art, the invention has the following technical effects:
the invention adopts the traditional probability model to describe the uncertain variables in the structure, provides an approximate numerical calculation method for calculating the joint failure probability of two failure modes and three failure modes, and can effectively improve the efficiency of calculating the joint failure probability in the reliability analysis of a structural system.
The invention provides a strategy of sequencing according to the correlation coefficients between the structural function functions from large to small, only the equivalent structural function is needed to replace the two equivalent structural function functions after each equivalent, the next equivalent can be carried out in sequence, on the premise of ensuring the sufficient precision of the equivalent process, the step of calculating the correlation coefficients of the equivalent structural function and other structural function functions every equivalent time in the traditional gradual equivalent plane method is avoided, and the efficiency of the gradual equivalent process can be effectively improved.
The method for determining the failure plane normal vector corresponding to the equivalent structure function avoids the complex solving process of each component in the vector through a derivation calculation method in the traditional equivalent plane method, and obviously improves the efficiency of the equivalent process of two failure modes.
In the step-by-step equivalent plane method provided by the invention, three structural function functions are considered in each step of equivalent process, and the reliability analysis error between the equivalent process of each step of equivalent process of the first two structural function functions and the equivalent process of each step of equivalent process of the third structural function is zero, so that the precision of the step-by-step equivalent plane method is greatly improved.
In conclusion, the problem that the reliability indexes of a plurality of failure modes are solved at one time in a complex way is converted into the problem that the reliability indexes of a series of two failure modes are solved by successively adopting one equivalent failure mode to replace two failure modes, so that the complexity of reliability analysis of the serial portal frame structure system is greatly simplified. The equivalent sequence sorted according to the correlation coefficients between the structural function functions from large to small and the normal vector solving method in the equivalent process of the two failure modes, provided by the invention, realize higher solving efficiency on the premise of ensuring that the method has enough precision, and the method has the advantages of wide application range, wide application prospect and convenience in popularization and use.
Drawings
FIG. 1 is a block diagram of the process flow of the present invention.
Fig. 2 is a schematic structural diagram of a serial portal frame architecture in this embodiment.
Detailed Description
As shown in fig. 1 and fig. 2, the high-precision step-by-step equivalent plane method for reliability analysis of a serial portal frame structure system of the present invention comprises the following steps:
the method comprises the following steps: determining a normal random variable and establishing a function of each failure mode in the portal frame structure, wherein the specific process is as follows:
step 101, analyzing sources of uncertain factors in a portal frame structure system according to practical problems, determining uncertain variables in a structure, and recording a random vector formed by all normal random variables in the structure as X ═ X (X is recorded as1,X2,…,Xn)TAnd n is the total number of normal random variables in the structure. Establishing a normal distribution probability density function of each random variable according to sample points given in practical problems, and recording the normal distribution obeyed by the ith random variable as
Figure BDA0002912503780000131
Wherein i is 1,2, …, n;
in this embodiment, the uncertainty factors include material properties, geometry, boundary conditions, and loading parameters of the serial portal frame architecture.
In this embodiment, the material properties of the serial portal frame structure system include elastic modulus, poisson's ratio, tensile and compressive strength, and mass density; the geometric dimension of the serial portal frame structure system comprises the cross-sectional area, bending moment, thickness, inertia moment and the like of each unit of the frame.
In this embodiment, taking the portal frame structure shown in fig. 2 as an example, the height of the portal frame structure is 6m, and the span is 2l 12 m. Random variables in the structure include the cross-sectional resistance bending moment X at cross-section 11 Section 2 against bending moment X2Section 3 against bending moment X3Section 4 against bending moment X4Section 5 against bending moment X5And the load X at the section 36The corresponding normal random variable vector is X ═ X (X)1,X2,X3,X4,X5,X6)T. All random variables in the structure are subject to mutually independent normal distribution, and the section resists bending moment X1Is normally distributed as X1～N(75,11.252) Cross section resisting bending moment X2Is normally distributed as X2～N(55,8.252) Cross sectional resistanceBending moment X3Is normally distributed as X3～N(80,122) Cross section resisting bending moment X4Is normally distributed as X4～N(55,8.252) Cross section resisting bending moment X5Is normally distributed as X5～N(75,11.252) And the load X at the section 36Is normally distributed as X6～N(20,62)。
Step 102, determining a function g of each failure mode of the serial portal frame structure system according to the structural failure criterion based on the random variable in the step 101j(X)＝gj(X1,X2,…,Xn) Where j is the number of the failure mode in the portal frame architecture and j is 1,2, …, m, m is the total number of failure modes in the architecture.
In this embodiment, the structural function functions corresponding to the four destruction mechanisms of the portal frame structure determined according to the structural failure criterion are:
g1(X)＝X1+2X3+X4-6X6；
g2(X)＝X1+2X3+X5-6X6；
g3(X)＝X2+2X3+X4-6X6；
g4(X)＝X2+2X3+X5-6X6；
step two: obtaining the reliability index of the single failure mode, wherein the process is as follows:
step 201, judging whether normal random variables in a portal frame structure system are mutually independent, if so, executing step 202, and if so, executing step 203;
in this embodiment, the normal random variables in the structure are all independent of each other.
Step 202, reliability analysis of the independent normal random variable failure mode: if X is ═ X1,X2,…,Xn)TIs a random vector composed of n mutually independent normal random variables, expressed as
Figure BDA0002912503780000141
Converting random variable vector X into standard normal random variable Y ═ Y (Y)1,Y2,…,Yn)T. Substituting the normalized normal random variable into the jth structure function gj(X)＝gj(X1,X2,…,Xn) The structural function in the standard normal space is obtained as gj(Y)＝gj(Y1,Y2,…,Yn). If the structure function gj(Y)＝gj(Y1,Y2,…,Yn) If the function is a linear function, go to step 2021; if the structure function gj(Y)＝gj(Y1,Y2,…,Yn) If the function is a non-linear function, go to step 2022;
in this embodiment, a normal random variable vector X is defined as (X)1,X2,X3,X4,X5,X6) Converted into a corresponding standard normal distribution having
Figure BDA0002912503780000142
And
Figure BDA0002912503780000143
the corresponding structural function is transformed into the standard normal space as follows:
g1(Y)＝11.25Y1+24Y3+8.25Y4-36Y6+170；
g2(Y)＝11.25Y1+24Y3+11.25Y5-36Y6+190；
g3(Y)＝8.25Y2+24Y3+8.25Y4-36Y6+150；
g4(Y)＝8.25Y2+24Y3+11.25Y5-36Y6+170；
step 2021, calculating the reliability index of the linear structure function: the expression of a linear structure function in the X space of the original variable space is assumed as
Figure BDA0002912503780000144
Wherein a isj0Is a constant term in the jth structure function, aji(i-1, 2, …, n) is the ith random variable X in the jth structure functioniThe coefficient of (a). The structural function converted into the standard normal space is
Figure BDA0002912503780000145
Then the structure function gj(Y) has a mean value and a standard deviation of
Figure BDA0002912503780000146
And
Figure BDA0002912503780000147
the reliability index of the jth structure function can be obtained
Figure BDA0002912503780000148
In this embodiment, the four structural function functions are all linear structural function functions, and the corresponding reliability indexes thereof are respectively:
Figure BDA0002912503780000151
Figure BDA0002912503780000152
Figure BDA0002912503780000153
Figure BDA0002912503780000154
step 2022, calculating the reliability index of the nonlinear structure function: when the structural function is non-linear, it will be non-linearThe structural function expands linearly at the check point. Assuming a checking point of
Figure BDA0002912503780000155
The structural function is then expanded once to
Figure BDA0002912503780000156
The mean and standard deviation thereof are respectively
Figure BDA0002912503780000157
And
Figure BDA0002912503780000158
the reliability index of the jth nonlinear structure function can be obtained as
Figure BDA0002912503780000159
The process of calculating the reliability index by an iterative method comprises the following steps: first assume an initial check point
Figure BDA00029125037800001510
(the mean of the normal random variables can be taken as the initial check point in general), and substituted into
Figure BDA00029125037800001511
Calculating a reliability index betaj(ii) a Then beta is mixedjSubstitution into
Figure BDA00029125037800001512
Calculate out
Figure BDA00029125037800001513
Then will be
Figure BDA00029125037800001514
βjAnd
Figure BDA00029125037800001515
substitution into
Figure BDA00029125037800001516
Calculating new checking points
Figure BDA00029125037800001517
Finally judging | | X*(1)-X*(0)Whether or not | l < ε holds, where | l | · | represents the 2-norm of the vector, ε is a specified tolerance and is typically taken to be 10-3If the judgment result is 'yes', the iteration is stopped, and if the judgment result is 'no', the method is used
Figure BDA00029125037800001518
Replacement of
Figure BDA00029125037800001519
And re-executing the solving process until the final judgment result is yes, so as to obtain the reliability index of the nonlinear structure function, wherein the check point obtained by iteration is the check point of the nonlinear structure function
Figure BDA0002912503780000161
Step 203, reliability analysis of the related normal random variable failure mode: if X is ═ X1,X2,…,Xn)TIs a random vector composed of n related normal random variables, and for the jth structure function gj(X)＝gj(X1,X2,…,Xn) Then there are X to N (mu)X,CX) Wherein
Figure BDA0002912503780000162
Is the mean vector of a normal random variable vector X, CXIs a covariance matrix of a normal random variable vector X, and
Figure BDA0002912503780000163
where k is also the number of the normal random variable, and k is 1,2, …, n. The probability density function of X is
Figure BDA0002912503780000164
Wherein | CXI represents the covariance matrix CXDeterminant (c). According to a random variable XiAnd XkCorrelation coefficient of
Figure BDA0002912503780000165
With its covariance Cov (X)i,Xk) In relation to (2)
Figure BDA0002912503780000166
Matrix of correlation coefficients that can determine a random variable vector X
Figure BDA0002912503780000167
Matrix ρ of correlation coefficients due to normal random variablesXIs a positive definite matrix, and obtains rho through Cholesky decompositionX＝AATWherein, A is a lower triangular matrix obtained by Cholesky decomposition. With matrix a, the associated random variable X can be represented by an independent standard normal random variable vector Y ═ Y (Y)1,Y2,…,Yn)TExpressed as X ═ σXAY+μXWherein, in the step (A),
Figure BDA0002912503780000168
make X ═ sigmaXAY+μXSubstituting into a structural function gj(X)＝gj(X1,X2,…,Xn) Obtaining a structural function g in a standard normal spacej(Y)＝gj(Y1,Y2,…,Yn) Then, the reliability index of the structure function is calculated according to step 202.
Step three: obtaining a linear structure function with unit coefficient vectors: if the original structure function is a linear structure function, according to step 2021, the jth structure function in the Y space of the normal variable space can be expressed as
Figure BDA0002912503780000171
Then its corresponding linear structure function with unit coefficient vector is
Figure BDA0002912503780000172
And it can be seen that the constant term in the structural function at this time is the reliability index beta of the structural functionj(ii) a If the original structure function is a non-linear structure function, the checking point of the structure function is obtained according to step 2022
Figure BDA0002912503780000173
And at check point to structure function gj(X) performing Taylor first-order expansion to obtain linear approximate structure function
Figure BDA0002912503780000174
And by variable substitution
Figure BDA0002912503780000175
Converting it into standard normal space Y space to obtain
Figure BDA0002912503780000176
Its structural function with a unit coefficient vector can be expressed as
Figure BDA0002912503780000177
And the constant term in the structure function is also the reliability index of the structure function, namely the reliability index is
Figure BDA0002912503780000178
For convenience of description, the ith random variable Y of the jth linear (or linearized) structure function with unit coefficient vector in the standard normal variable space isiIs uniformly recorded as alphajiThen, the jth structural function with unit coefficient vector in the normal variable space can be uniformly expressed as
Figure BDA0002912503780000179
In this embodiment, unitizing the coefficient vector of the linear structure function in the standard normal random variable space to obtain a structure function with a unit coefficient vector is:
g1(Y)＝0.24747Y1+0.52794Y3+0.18148Y4-0.79190Y6+3.73954；
g2(Y)＝0.24404Y1+0.52062Y3+0.24404Y5-0.78093Y6+4.12156；
g3(Y)＝0.18410Y2+0.53557Y3+0.18410Y4-0.80335Y6+3.34731；
g4(Y)＝0.18148Y2+0.52794Y3+0.24747Y5-0.79190Y6+3.73954。
step four: and sequencing the failure modes in the order of the relative numbers from large to small:
step 401, calculating a correlation coefficient between every two failure modes: taking the j-th and h-th (h is 2,3, …, m) (h > j) structural function functions of the m linear structural function functions with unit coefficient vectors obtained in the third step and recording the functions as
Figure BDA0002912503780000181
The mean value of the structural function is
Figure BDA0002912503780000182
The variance of the structural function is
Figure BDA0002912503780000183
It is noted that the standard normal random variables in the structural function are all independent of each other, and then
Figure BDA0002912503780000184
The variance of the structure function can then be reduced to
Figure BDA0002912503780000185
Then the correlation coefficient of the jth and the h structural function can be obtained as
Figure BDA0002912503780000186
Substituting any two structure function functions into the formula to obtain m structure function correlation coefficient matrixes
Figure BDA0002912503780000187
In this embodiment, the correlation coefficient matrix of the four structural function functions is
Figure BDA0002912503780000188
Step 402, sorting the failure modes from large to small according to the number of relations:
step 4021, selecting the largest correlation coefficient from the correlation coefficient matrix rho, and numbering two corresponding structure function functions as (j → h);
step 4022, selecting all function functions g related to the structure from the correlation coefficient matrix rhoj(Y) and gh(Y) the correlation coefficients of interest and selecting the largest correlation coefficient therefrom, the corresponding further structure function being numbered q. If the correlation coefficient is the structural function gq(Y) and gj(Y), the ordering of the three structure function functions is (q → j → h); if the correlation coefficient is the structural function gh(Y) and gq(Y), the ordering of the three structure function functions is (j → h → q);
step 4023, if the sequence of the three structural function functions is (q → j → h), removing the structural function g from the correlation coefficient matrix ρj(Y) all the related correlation coefficients, and judging whether the number of elements in the correlation coefficient matrix rho is zero or not. If the judgment result is 'yes', ending the sequencing process; if the judgment result is 'no', the structural function g is judgedq(Y) and gh(Y) repeating the step 4022 until the judgment result is YES; if the three structure function functions are ordered as (j → h → q), the structure function g in the correlation coefficient matrix ρ is removedh(Y) all relevant correlation coefficients, judging whether the number of elements in the correlation coefficient matrix rho is zero, if so, ending the sorting process, and if so, judgingNo, then function g for structural functionq(Y) and gh(Y) repeating the step 4022 until the judgment result is YES;
step 4024, numbering the m structural function functions again according to the sequence obtained in step 4023 with the sequence numbers 1,2, …, m. And updating the correlation coefficient matrix of the structural function according to the renumbering.
In this embodiment, the results of the structural function sorted from large to small according to the correlation coefficient are: 2 → 1 → 3 → 4, i.e. according to
g2(Y)→g1(Y)→g3(Y)→g4The sequence of (Y) performs a stepwise equivalent planar method. Renumbering it as follows:
g1(Y)＝0.24404Y1+0.52062Y3+0.24404Y5-0.78093Y6+4.12156；
g2(Y)＝0.24747Y1+0.52794Y3+0.18148Y4-0.79190Y6+3.73954；
g3(Y)＝0.18410Y2+0.53557Y3+0.18410Y4-0.80335Y6+3.34731；
g4(Y)＝0.18148Y2+0.52794Y3+0.24747Y5-0.79190Y6+3.73954。
the matrix of correlation coefficients of the structural function is also updated according to the renumbering. Obtain the updated correlation coefficient matrix of
Figure BDA0002912503780000191
Step five: calculating the reliability index of the serial portal frame structure system:
step 501, according to the sequence of step 4024, selecting the first three structural function functions g1(Y)，g2(Y) and g3(Y), calculating the joint failure probability of every two structural function functions and the joint failure probability of the three structural function functions;
in this embodiment, the failure modes in step four are sorted according to the selection orderThe first three structural function functions are g1(Y),g2(Y),g3(Y)。
Step 502, calculating the failure probability of each of the three selected structure function functions: calculating three selected structure function g according to formula P ═ phi (-beta)1(Y)、g2(Y) and g3(Y) failure probabilities, respectively denoted as P1＝Φ(-β1)、P2＝Φ(-β2) And P3＝Φ(-β3) Where Φ (·) is a standard normal distribution function.
In this embodiment, the failure probabilities of the three structural function functions are respectively:
P1＝Φ(-β1)＝0.0000188158；
P2＝Φ(-β2)＝0.0000922117；
P3＝Φ(-β3)＝0.000408；
step 503, calculating the joint failure probability of every two structure function functions: structural function (g)1(Y),g2(Y))，(g1(Y),g3(Y)) and (g)2(Y),g3(Y)) a joint probability density function of
Figure BDA0002912503780000201
The joint failure probability of the two structural function functions is
Figure BDA0002912503780000202
Wherein
Figure BDA0002912503780000203
Is the integral domain. It is noted that the joint probability density function tends to 0 at a distance from the origin of coordinates, and for the sake of simplifying the calculation, the main failure region closer to the origin of coordinates is taken as the integral domain, i.e., the integral domain is respectively taken as
Figure BDA0002912503780000204
Wherein epsilon is used to control the accuracy of the approximation calculation, and epsilon should satisfy
Figure BDA0002912503780000211
The joint failure probability of the two structural function functions can be approximated as
Figure BDA0002912503780000212
Then, by a numerical method, the joint failure probability of two structure function functions can be obtained
Figure BDA0002912503780000213
Wherein N represents the number of the divided intervals on each one-dimensional coordinate,
Figure BDA0002912503780000214
Figure BDA0002912503780000215
and
Figure BDA0002912503780000216
separately representing the integral domain
Figure BDA0002912503780000217
Divided into (r, t) th cell, and
Figure BDA0002912503780000218
respectively represent
Figure BDA0002912503780000219
And
Figure BDA00029125037800002110
a center point of (a), and
Figure BDA00029125037800002111
in this embodiment, the joint failure probability between every two three structure function functions in the first equivalent process is calculated as: p12＝0.00001640978,P13＝0.00001691644,P23＝0.00008194422。
Step 504, calculating a structure function g1(Y)，g2(Y) and g3(Y) joint failure probability: structural function g1(Y)，g2(Y) and g3(Y) a joint probability density function of
Figure BDA00029125037800002112
Wherein [ C]Representing a structural function g1(Y)，g2(Y) and g3A variance matrix of (Y), and
Figure BDA00029125037800002113
det[C]representation matrix [ C]Is of determinant
Figure BDA0002912503780000221
[Λjh]＝[C]-1. Thus, the joint failure probability of three structural function functions can be obtained as
Figure BDA0002912503780000222
Wherein G is123＝{(g1(Y),g2(Y),g3(Y))(g1(Y)＜0)∩(g2(Y)＜0)∩(g3(Y) < 0) } denotes an integration domain,
Figure BDA0002912503780000223
represents G123The central primary failure zone being closer to the origin of coordinates. Then g is1(Y)，g2(Y) and g3(Y) the joint failure probability can be expressed by a numerical calculation method as
Figure BDA0002912503780000224
Wherein Δ VrtlRepresenting the integral domain
Figure BDA0002912503780000225
The volume of the upper (r, t, l) th small region, and
Figure BDA0002912503780000226
(g1r(Y),g2t(Y),g3l(Y)) represents the center point of the (r, t, l) th small region, and
Figure BDA0002912503780000227
in this embodiment, the joint failure probability of simultaneous failure of three structural function in the first equivalent process is P123＝0.00001331033。
Step 505, equivalence of two failure modes: the equivalent process needs to satisfy three conditions: (1) the failure domain corresponding to the equivalent limit state surface and the failure domain surrounded by the two limit state surfaces have the same failure probability or (generalized) reliability index; (2) the normal vector of the equivalent limit state surface and the normal vectors of the two limit state surfaces are in the same plane; (3) the failure probability of the region enclosed by the equivalent limit state surface and the third limit state surface is equal to the failure probability of the region enclosed by the corresponding three limit state surfaces. The condition (1) is used for determining the reliability index of the equivalent structure function, and the conditions (2) and (3) are used for determining the normal vector of the equivalent limit state surface.
Step 5051, solving the reliability index of the equivalent structure function: for structural function
Figure BDA0002912503780000228
Assuming that its equivalent structure function is
Figure BDA0002912503780000229
Phi (-beta) can be obtained according to the condition (1)e)＝P1+P2-P12Where phi (-) is a normal distribution function, the equivalent reliability index is betae＝-Φ-1(P1+P2-P12)。
In this embodiment, in the first equivalent process, the structure function g1(Y) and g2(Y) an equivalent reliability index of the equivalent structure function of
Figure BDA00029125037800002210
Step 5052, solving the normal vector of the corresponding failure plane of the equivalent structure function: memory structure function g1(Y)，g2(Y) method for aligning failure planeThe vectors are respectively
Figure BDA0002912503780000231
Function of equivalent structure
Figure BDA0002912503780000232
The normal vector corresponding to the failure surface is a column vector
Figure BDA0002912503780000233
According to the condition (2), there is a unique real number λ (λ > 0) such that
Figure BDA0002912503780000234
Assuming equivalent structural function
Figure BDA0002912503780000235
And g3(Y) a joint failure probability of
Figure BDA0002912503780000236
Structural function of function
Figure BDA0002912503780000237
And g3The correlation coefficient of (Y) is ρ12And 3 according to the condition (3) are
Figure BDA0002912503780000238
Wherein
Figure BDA0002912503780000239
Is simple and easy to obtain
Figure BDA00029125037800002310
It is noted that
Figure BDA00029125037800002311
And wherein betaeAnd beta3Known as phi (-beta)e,-β3,ρ12,3)＝f(ρ12,3)＝P13+P23-P123Available rho12,3＝f-1(P13+P23-P123). According toExpression of correlation coefficients of two structural function functions
Figure BDA00029125037800002312
Is provided with
Figure BDA00029125037800002313
One-dimensional quadratic equation for λ can be obtained
Figure BDA00029125037800002314
Solving the equation to obtain the positive root of lambda
Figure BDA00029125037800002315
Substituting the obtained lambda into
Figure BDA00029125037800002316
The normal vector of the equivalent structure function can be obtained
Figure BDA00029125037800002317
Then unitizing the vector to obtain a unit normal vector alpha e of the equivalent structure function
Figure BDA00029125037800002318
So far, the structure function g can be uniquely obtained1(Y)，g2Equivalent structural function of (Y)
Figure BDA00029125037800002319
Structural function g1(Y) and g2(Y) replacement by
Figure BDA00029125037800002320
In this example, according to the formula Φ (- β)e,-β3,ρ12,3)＝f(ρ12,3)＝P13+P23-P123Finding rho12,30.95685935, then according to formula
Figure BDA0002912503780000241
And λ is 1.250173. Then the equivalent structural workThe energy function is given by the normal vector
Figure BDA0002912503780000242
The unit vector is converted into alphae＝[0.2488082,0,0.5307937,0.102002,0.1097162,-0.7961849]T. Then the structure function g1(Y) and g2The equivalent structural function of (Y) can be expressed as:
Figure BDA0002912503780000243
step 506, judging whether the total number of the structural function functions is equal to 3, if so, numbering the three structural function functions from 1 again, and respectively calculating the failure probability P of a single structural function of the three structural function functions according to the steps 502, 503 and 5041、P2And P3Combined failure probability P of two-by-two structure function12、P13、P23And joint failure probability P of three structural function123Obtaining the reliability index beta of the serial portal frame structure systemseriesIs betaseries＝-Φ-1(P1+P2+P3-P12-P13-P23+P123) (ii) a If the judgment result is 'no', the fifth step is executed again until the judgment result is 'yes'.
In this embodiment, the number of the remaining structure function functions after the equivalence is completed is 3, and the remaining structure function functions are renumbered to obtain:
g1(Y)＝0.24881Y1+0.53079Y3+0.10200Y4+0.10972Y5-0.79619Y6+3.73297；
g2(Y)＝0.18410Y2+0.53557Y3+0.18410Y4-0.80335Y6+3.34731；
g3(Y)＝0.18148Y2+0.52794Y3+0.24747Y5-0.79190Y6+3.73954。
calculating the failure probability P of single structure function of three structure function1、P2And P3Combined failure probability P of two-by-two structure function12、P13、P23And joint failure probability P of three structural function123Respectively as follows: p1＝0.000094618、P2＝0.000408、P3＝0.000092179、P12＝0.000080013、P13＝0.000043898、P230.000092087 and P1230.000039873. The reliability index of the serial portal frame structure system is betaseries＝-Φ-1(0.00041867)＝3.34015。

Claims (9)

1. A method for optimizing a stepwise equivalent planar method is characterized by comprising the following steps:
step 1, determining a normal random variable and establishing a function of each failure mode in a portal frame structure;
step 2, judging whether normal random variables in a portal frame structure system are mutually independent or not, and obtaining the reliability index of a single failure mode;
step 3, acquiring a linear structure function with a unit coefficient vector, and determining the reliability index of the structure function;
step 4, calculating the correlation coefficient between every two failure modes, and sequencing the failure modes according to the sequence of the phase relation numbers from large to small;
and 5, calculating the reliability index of the serial portal frame structure system according to the sequence of the step 4.
2. The method for optimizing a stepwise equivalent planar method according to claim 1, wherein the step 1 specifically comprises:
step 101, determining uncertain variables in the structure, and recording a random vector formed by all normal random variables in the structure as X ═ X (X)1,X2,…,Xn)TN is the total number of normal random variables in the structure; establishing a normal distribution probability density function of each random variable according to sample points given in practical problems, and recording the normal distribution obeyed by the ith random variable as
Figure FDA0002912503770000011
Wherein i is 1,2, …, n;
step 102, determining a function g of each failure mode of the serial portal frame structure system according to the structural failure criterion based on the random variable in the step 101j(X)＝gj(X1,X2,…,Xn) Where j is the number of the failure mode in the portal frame architecture and j is 1,2, …, m, m is the total number of failure modes in the architecture.
3. The method for optimizing a stepwise equivalent planar method according to claim 1, wherein the step 2 specifically comprises:
step 201, judging whether normal random variables in a portal frame structure system are mutually independent, if so, executing step 202, and if so, executing step 203;
step 202, reliability analysis of the independent normal random variable failure mode: if X is ═ X1,X2,…,Xn)TIs a random vector composed of n mutually independent normal random variables, expressed as
Figure FDA0002912503770000012
Converting random variable vector X into standard normal random variable Y ═ Y (Y)1,Y2,…,Yn)T(ii) a Substituting the normalized normal random variable into the jth structure function gj(X)＝gj(X1,X2,…,Xn) The structural function in the standard normal space is obtained as gj(Y)＝gj(Y1,Y2,…,Yn) (ii) a If the structure function gj(Y)＝gj(Y1,Y2,…,Yn) If the function is a linear function, go to step 2021; if the structure function gj(Y)＝gj(Y1,Y2,…,Yn) If the function is a non-linear function, go to step 2022;
step 203, reliability analysis of the related normal random variable failure mode: if X is ═ X1,X2,…,Xn)TIs a random vector composed of n related normal random variables, and for the jth structure function gj(X)＝gj(X1,X2,…,Xn) Then there are X to N (mu)X,CX) Wherein
Figure FDA0002912503770000021
Is the mean vector of a normal random variable vector X, CXIs a covariance matrix of a normal random variable vector X,
and is
Figure FDA0002912503770000022
Where k is also the number of normal random variables, and k is 1,2, …, n; the probability density function of X is
Figure FDA0002912503770000023
Wherein | CXI represents the covariance matrix CXDeterminant of (4);
according to a random variable XiAnd XkCorrelation coefficient of
Figure FDA0002912503770000024
With its covariance Cov (X)i,Xk) In relation to (2)
Figure FDA0002912503770000025
Determining a matrix of correlation coefficients for a random variable vector X
Figure FDA0002912503770000026
Obtaining rho by Cholesky decompositionX＝AATWherein, A is a lower triangular matrix obtained by Cholesky decomposition; with matrix a, the associated random variable X can be represented by an independent standard normal random variable vector Y ═ Y (Y)1,Y2,…,Yn)TExpressed as X ═ σXAY+μXWherein, in the step (A),
Figure FDA0002912503770000027
make X ═ sigmaXAY+μXSubstituting into a structural function gj(X)＝gj(X1,X2,…,Xn) Obtaining a structural function g in a standard normal spacej(Y)＝gj(Y1,Y2,…,Yn) Then, the reliability index of the structure function is calculated according to step 202.
4. The method according to claim 3, wherein the step 202 specifically includes:
step 2021, calculating the reliability index of the linear structure function: the expression of a linear structure function in the X space of the original variable space is assumed as
Figure FDA0002912503770000031
Wherein a isj0Is a constant term in the jth structure function, aji(i-1, 2, …, n) is the ith random variable X in the jth structure functioniThe coefficient of (a); the structural function converted into the standard normal space is
Figure FDA0002912503770000032
Then the structure function gj(Y) has a mean value and a standard deviation of
Figure FDA0002912503770000033
And
Figure FDA0002912503770000034
obtaining the reliability index of the jth structure function
Figure FDA0002912503770000035
Step 2022, calculating the nonlinear structure workReliability index of energy function: when the structural function is nonlinear, the nonlinear structural function is expanded into linearity at a check point; assuming a checking point of
Figure FDA0002912503770000036
The structural function is then expanded once to
Figure FDA0002912503770000037
The mean and standard deviation thereof are respectively
Figure FDA0002912503770000038
And
Figure FDA0002912503770000039
the reliability index of the jth nonlinear structure function is obtained as
Figure FDA00029125037700000310
The process of calculating the reliability index by an iterative method comprises the following steps: first assume an initial check point
Figure FDA00029125037700000311
Substitute it into
Figure FDA00029125037700000312
Calculating a reliability index betaj(ii) a Then beta is mixedjSubstitution into
Figure FDA0002912503770000041
Calculate out
Figure FDA0002912503770000042
Then will be
Figure FDA0002912503770000043
βjAnd
Figure FDA0002912503770000044
substitution into
Figure FDA0002912503770000045
Calculating new checking points
Figure FDA0002912503770000046
Finally judging | | X*(1)-X*(0)Whether or not | l < ε holds, where | l | · | represents the 2-norm of the vector, ε is a specified tolerance and is typically taken to be 10-3If the judgment result is 'yes', the iteration is stopped, and if the judgment result is 'no', the method is used
Figure FDA0002912503770000047
Replacement of
Figure FDA0002912503770000048
And re-executing the solving process until the final judgment result is yes to obtain the reliability index of the nonlinear structure function, wherein the check point obtained by iteration is the check point of the nonlinear structure function
Figure FDA0002912503770000049
5. The method for optimizing a stepwise equivalent planar method according to claim 4, wherein the step 3 specifically comprises:
if the original structure function is a linear structure function, according to step 2021, the jth structure function in the Y space of the standard normal variable space is expressed as
Figure FDA00029125037700000410
Then its corresponding linear structure function with unit coefficient vector is
Figure FDA00029125037700000411
And now in the structure functionThe constant term is the reliability index beta of the structure functionj(ii) a If the original structure function is a non-linear structure function, the checking point of the structure function is obtained according to step 2022
Figure FDA00029125037700000412
And at check point to structure function gj(X) performing Taylor first-order expansion to obtain linear approximate structure function
Figure FDA00029125037700000413
And by variable substitution
Figure FDA00029125037700000414
Converting it into standard normal space Y space to obtain
Figure FDA00029125037700000415
It has the structural function of the unit coefficient vector expressed as
Figure FDA0002912503770000051
And the constant term in the structure function is also the reliability index of the structure function, namely the reliability index is
Figure FDA0002912503770000052
The ith random variable Y of the jth linear structure function with the unit coefficient vector in the standard normal variable spaceiIs uniformly recorded as alphajiThen, the jth structural function with unit coefficient vector in the normal variable space is uniformly expressed as
Figure FDA0002912503770000053
6. The method for optimizing a stepwise equivalent planar method according to claim 1, wherein the step 4 specifically comprises:
step 401, calculating a correlation coefficient between every two failure modes: taking the j-th and h-th (h is 2,3, …, m) (h > j) structural function functions of the m linear structural function functions with unit coefficient vectors obtained in the third step and recording the functions as
Figure FDA0002912503770000054
The mean value of the structural function is
Figure FDA0002912503770000055
The variance of the structural function is
Figure FDA0002912503770000056
The standard normal random variables in the structural function are all independent of each other, if
Figure FDA0002912503770000057
The variance of the structural function is reduced to
Figure FDA0002912503770000058
Then the correlation coefficient of the jth and the h structural function is obtained as
Figure FDA0002912503770000059
Substituting any two structure function functions into the formula to obtain m structure function correlation coefficient matrixes
Figure FDA0002912503770000061
And step 402, sequencing the failure modes from large to small according to the number of the relations.
7. The method according to claim 1, wherein the step 402 specifically includes:
step 4021, selecting the largest correlation coefficient from the correlation coefficient matrix rho, and numbering two corresponding structure function functions as (j → h);
step 4022, selecting all function functions g related to the structure from the correlation coefficient matrix rhoj(Y) and gh(Y) the relevant correlation coefficients and selecting the largest correlation coefficient therefrom, the number of the corresponding further structure function being denoted q; if the correlation coefficient is the structural function gq(Y) and gj(Y), the ordering of the three structure function functions is (q → j → h); if the correlation coefficient is the structural function gh(Y) and gq(Y), the ordering of the three structure function functions is (j → h → q);
step 4023, if the sequence of the three structural function functions is (q → j → h), removing the structural function g from the correlation coefficient matrix ρj(Y) all relevant correlation coefficients, and judging whether the number of elements in the correlation coefficient matrix rho is zero or not; if the judgment result is 'yes', ending the sequencing process; if the judgment result is 'no', the structural function g is judgedq(Y) and gh(Y) repeating the step 4022 until the judgment result is YES; if the three structure function functions are ordered as (j → h → q), the structure function g in the correlation coefficient matrix ρ is removedh(Y) all relevant correlation coefficients, judging whether the number of elements in the correlation coefficient matrix rho is zero or not, if the judgment result is yes, finishing the sorting process, if the judgment result is no, and then performing structural function gq(Y) and gh(Y) repeating the step 4022 until the judgment result is YES;
step 4024, numbering the m structural function functions again according to the sequence obtained in the step 4023 according to the sequence numbers 1,2, … and m; and updating the correlation coefficient matrix of the structural function according to the renumbering.
8. The method for optimizing a stepwise equivalent planar method according to claim 7, wherein the step 5 specifically comprises:
step 501, according to the sequence of step 4024, selecting the first three structural function functions g1(Y)，g2(Y) and g3(Y), calculating the joint failure probability of every two structural function functions and the joint failure probability of the three structural function functions;
step 502, calculating the failure probability of each of the three selected structure function functions: calculating three selected structure function g according to formula P ═ phi (-beta)1(Y)、g2(Y) and g3(Y) failure probabilities, respectively denoted as P1＝Φ(-β1)、P2＝Φ(-β2) And P3＝Φ(-β3) Where Φ (·) is a standard normal distribution function;
step 503, calculating the joint failure probability of every two structure function functions:
structural function (g)1(Y),g2(Y))，(g1(Y),g3(Y)) and (g)2(Y),g3(Y)) a joint probability density function of
Figure FDA0002912503770000071
The joint failure probability of the two structural function functions is
Figure FDA0002912503770000072
Wherein
Figure FDA0002912503770000073
Is an integral domain; at a place far away from the origin of coordinates, the joint probability density function tends to be 0, and the main failure area near the origin of coordinates is taken as the integral domain, namely the integral domain is respectively taken as
Figure FDA0002912503770000074
Wherein epsilon is used to control the accuracy of the approximation calculation, and epsilon should satisfy
Figure FDA0002912503770000075
Then the joint failure probability approximation table of the two structure functionShown as
Figure FDA0002912503770000076
Then obtaining the joint failure probability of two structure function functions by a numerical method
Figure FDA0002912503770000081
Wherein N represents the number of the divided intervals on each one-dimensional coordinate,
Figure FDA0002912503770000082
and
Figure FDA0002912503770000083
separately representing the integral domain
Figure FDA0002912503770000084
The (r, t) th divided section, and
Figure FDA0002912503770000085
respectively represent
Figure FDA0002912503770000086
And
Figure FDA0002912503770000087
a center point of (a), and
Figure FDA0002912503770000088
step 504, calculating a structure function g1(Y)，g2(Y) and g3(Y) joint failure probability: structural function g1(Y)，g2(Y) and g3(Y) a joint probability density function of
Figure FDA0002912503770000089
Wherein [ C]Representing a structural function g1(Y)，g2(Y) and g3A variance matrix of (Y), and
Figure FDA00029125037700000810
det[C]representation matrix [ C]Is of determinant
Figure FDA00029125037700000811
[Λjh]＝[C]-1(ii) a The joint failure probability of the three structural function functions is obtained
Figure FDA00029125037700000812
Wherein G is123＝{(g1(Y),g2(Y),g3(Y))(g1(Y)＜0)∩(g2(Y)＜0)∩(g3(Y) < 0) } denotes an integration domain,
Figure FDA0002912503770000091
represents G123A primary failure zone closer to the origin of coordinates; then g is1(Y)，g2(Y) and g3(Y) the joint failure probability is expressed as
Figure FDA0002912503770000092
Wherein Δ VrtlRepresenting the integral domain
Figure FDA0002912503770000093
Volume of the upper (r, t, l) th region, and
Figure FDA0002912503770000094
(g1r(Y),g2t(Y),g3l(Y)) represents the center point of the (r, t, l) th region, and
Figure FDA0002912503770000095
step 505, equivalence of two failure modes: the equivalent process needs to satisfy three conditions: (1) the failure domain corresponding to the equivalent limit state surface and the failure domain surrounded by the two limit state surfaces have the same failure probability or (generalized) reliability index; (2) the normal vector of the equivalent limit state surface and the normal vectors of the two limit state surfaces are in the same plane; (3) the failure probability of the region enclosed by the equivalent limit state surface and the third limit state surface is equal to the failure probability of the region enclosed by the corresponding three limit state surfaces; the method comprises the following steps that a condition (1) is used for determining a reliability index of an equivalent structure function, and conditions (2) and (3) are used for determining a normal vector of an equivalent limit state surface;
step 506, judging whether the total number of the structural function functions is equal to 3, if so, numbering the three structural function functions from 1 again, and respectively calculating the failure probability P of a single structural function of the three structural function functions according to the steps 502, 503 and 5041、P2And P3Combined failure probability P of two-by-two structure function12、P13、P23And joint failure probability P of three structural function123Obtaining the reliability index beta of the serial portal frame structure systemseriesIs betaseries＝-Φ-1(P1+P2+P3-P12-P13-P23+P123) (ii) a If the judgment result is 'no', the fifth step is executed again until the judgment result is 'yes'.
9. The method for optimizing a stepwise equivalent planar method according to claim 8, wherein the step 505 specifically includes:
step 5051, solving the reliability index of the equivalent structure function: for structural function
Figure FDA0002912503770000096
Assuming that its equivalent structure function is
Figure FDA0002912503770000097
Phi (-beta) can be obtained according to the condition (1)e)＝P12Where phi (-) is a normal distribution function, the equivalent reliability index is betae＝-Φ-1(P12)；
Step 5052, solving the normal vector of the corresponding failure plane of the equivalent structure function: memory structure function g1(Y)，g2(Y) normal vectors corresponding to the failure planes are respectively
Figure FDA0002912503770000101
Function of equivalent structure
Figure FDA0002912503770000102
The normal vector corresponding to the failure surface is a column vector
Figure FDA0002912503770000103
According to the condition (2), there is a unique real number λ (λ > 0) such that
Figure FDA0002912503770000104
Assuming equivalent structural function
Figure FDA0002912503770000105
And g3(Y) a joint failure probability of
Figure FDA0002912503770000106
Structural function of function
Figure FDA0002912503770000107
And g3The correlation coefficient of (Y) is ρ12,3According to the condition (3) have
Figure FDA0002912503770000108
Wherein
Figure FDA0002912503770000109
Is simple and easy to obtain
Figure FDA00029125037700001010
And wherein betaeAnd beta3Known as phi (-beta)e,-β3,ρ12,3)＝f(ρ12,3)＝P13+P23-P123Get rho12,3＝f-1(P13+P23-P123)；
Expression based on correlation coefficients of two structural function functions
Figure FDA00029125037700001011
Is provided with
Figure FDA00029125037700001012
Obtain a quadratic equation of unity with respect to λ
Figure FDA00029125037700001013
Solving the equation to obtain the positive root of lambda
Figure FDA00029125037700001014
Substituting the obtained lambda into
Figure FDA00029125037700001015
Obtaining the normal vector of the equivalent structure function
Figure FDA00029125037700001016
Then unitizing the vector to obtain a unit normal vector alpha of the equivalent structure functioneIs composed of
Figure FDA00029125037700001017
At this point, the structural function g is uniquely determined1(Y)，g2Equivalent structural function of (Y)
Figure FDA00029125037700001018
Structural function g1(Y) and g2(Y) replacement by
Figure FDA00029125037700001019
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