CN110781606B - Multi-design-point non-probability reliability analysis method for beam structure - Google Patents
Multi-design-point non-probability reliability analysis method for beam structure Download PDFInfo
- Publication number
- CN110781606B CN110781606B CN201911081340.0A CN201911081340A CN110781606B CN 110781606 B CN110781606 B CN 110781606B CN 201911081340 A CN201911081340 A CN 201911081340A CN 110781606 B CN110781606 B CN 110781606B
- Authority
- CN
- China
- Prior art keywords
- uncertain
- taylor
- design point
- parameter vector
- unit
- Prior art date
- Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
- Active
Links
Images
Landscapes
- Management, Administration, Business Operations System, And Electronic Commerce (AREA)
Abstract
The invention discloses a multi-design-point non-probability reliability analysis method for a beam structure, which comprises the following steps of: 1. determining a function and an uncertainty variable of a beam structure; 2. establishing a corresponding ellipsoid model according to the uncertainty variable; 3. acquiring a unit ellipsoid model of an uncertain variable; 4. obtaining a plurality of design points; 5. performing Taylor primary expansion at the design point; 6. acquiring the non-probability failure degree of the beam structure; 7. and acquiring the non-probability reliability of the beam structure. The method has simple steps, reasonable design and convenient realization, establishes the functional function by considering multiple design points of the beam structure, realizes the non-probability reliability analysis of the beam structure, has high calculation efficiency, obtains relatively accurate reliability and is convenient for popularization and use.
Description
Technical Field
The invention belongs to the technical field of beam structure optimization, and particularly relates to a multi-design-point non-probabilistic reliability analysis method for a beam structure.
Background
The beam structure has good stress characteristics, can bear concentrated load, uniformly distributed load, bending moment and the like, is easy to manufacture, is widely applied to the fields of buildings, machinery and the like, and can be simplified into cantilever beam models for tower cranes, balconies, street lamps and the like. In the design and manufacturing process of the beam structure, uncertainty of relevant variables such as use load, geometric dimension and the like exists, and scientific consideration needs to be given. A commonly used method uses a reliability analysis method for processing, and a probabilistic reliability method based on the method is widely adopted. However, when the probabilistic reliability model is used to solve the problem, sufficient statistical data needs to be mastered, and the calculation model is accurate, so that the application range is limited. Therefore, a non-probability reliability model is needed to be used for reliability analysis of the beam structure, uncertainty can be described through a set, the requirement on data is not particularly strict, only uncertainty variables are required to meet uncertainty but have boundedness, and therefore the non-probability reliability analysis method becomes one of common methods under the condition of small samples. When the existing non-probability reliability analysis method is used, a Monte Carlo method is theoretically used for solving a solution with very high precision, but when a complex problem is solved, too long time is needed, so that a first-order second-order moment method or a second-order moment method is needed to be used for approximating the result. However, both the existing first order second order moment method and the existing second order moment method are performed for a functional function with a single design point, and for a beam structure functional function with multiple design points, considering only the single design point will cause a large error to a reliability analysis result. Therefore, a multi-design-point non-probabilistic reliability analysis method for a beam structure is required.
Disclosure of Invention
The technical problem to be solved by the invention is to provide a method for analyzing the non-probability reliability of a beam structure at multiple design points, aiming at the defects in the prior art, the method has simple steps, reasonable design and convenient implementation, the function is established by considering the multiple design points of the beam structure, the non-probability reliability analysis of the beam structure is realized, the calculation efficiency is high, the relatively accurate reliability is obtained, and the method is convenient to popularize and use.
In order to solve the technical problems, the invention adopts the technical scheme that: a multi-design-point non-probabilistic reliability analysis method for a beam structure is characterized by comprising the following steps of:
step one, determining a function and an uncertainty variable of a beam structure:
establishing a function g (X) of multiple design points by adopting the data processor to the beam structure in any failure mode; wherein X represents an uncertain parameter vector for the beam structure and is denoted as uncertain parameter vector X = (X) 1 ,X 2 ,...,X i ,...,X m ) T M is the dimension of the uncertain parameter vector X, X 1 Denotes the 1 st uncertain variable, X 2 Denotes the 2 nd uncertain variable, X i Denotes the ith uncertain variable, X m Representing the mth uncertain variable, i is the number of the uncertain variable, i is a positive integer and the value range of i is 1-m,
representing an uncertain variable X i The interval of the values is selected from the group, i Xand &>
Are respectively an uncertain variable X i M is more than or equal to 2;
step two, establishing a corresponding ellipsoid model according to the uncertainty variable:
establishing an ellipsoid model using the data processor
The following were used:
step three, obtaining a unit ellipsoid model of the uncertainty variable:
step 301, using said data processor according to a formula
Obtaining the ith uncertainty variable X i Is greater than or equal to the section radius>
Based on a formula +with the data processor>
Obtaining the ith uncertainty variable X i Normalized variable U of i ；
Step 302, according to the method in step 301, the data processor is adopted to perform normalization processing on the uncertain parameter vector X to obtain an uncertain normalization vector U = (U =) 1 ,U 2 ,...,U i ,...,U m ) T (ii) a Wherein, U 1 Denotes the 1 st uncertain variable X 1 Normalized variable of (U) 2 Denotes the 2 nd uncertain variable X 2 Normalized variable of (U) m Denotes the m-th uncertain variable X m A normalized variable of (d);
step 303, obtaining a normalized equivalent ellipsoid model of the uncertain parameter vector by adopting the data processor according to the method in the step two
I.e. is>
Wherein omega u A feature matrix representing a normalized equivalent ellipsoid model determining the uncertainty parameter vector, and ≥>
Is shown in
An m-dimensional diagonal matrix of diagonal elements;
step 304, determining a characteristic matrix omega of the normalized equivalent ellipsoid model of the uncertain parameter vector by using the data processor u Cholesky decomposition is carried out to obtain
Wherein L is c Feature matrix omega representing a normalized equivalent ellipsoid model for determining uncertain parameter vectors u Obtaining a lower triangular matrix through Cholesky decomposition;
305, using the data processor according to a formula
Obtaining a standardized vector delta of the uncertain parameter vector X in a standard space; wherein, delta 1 Denotes the 1 st uncertain variable X 1 Normalized variable of δ 2 Denotes the 2 nd uncertain variable X 2 Normalized variable of δ i Denotes the ith uncertain variable X i Normalized variable of δ m Denotes the m-th uncertain variable X m A normalized variable of (a);
step 306, substituting the normalized vector delta of the uncertain parameter vector X in the standard space into the normalized equivalent ellipsoid model of the uncertain parameter vector in the step 303 by adopting the data processor
Unit ellipsoid model E for obtaining uncertain parameter vectors δ I.e. E δ ＝{δ|δ T δ≤1,δ∈R m }；
Step four, obtaining a plurality of design points:
step 401, obtaining an uncertain parameter vector X and an uncertain parameter by using the data processorThe relationship between the normalized vectors δ of vector X in the normalized space is:
402, adopting the data processor to obtain a relation between the uncertain parameter vector X and a standardized vector delta of the uncertain parameter vector X in a standard space
Substituting the function g (X) to obtain a function g (delta) in the standard space;
step 403, recording the function g (delta) in the standard space as the initial function g in the standard space by using the data processor 0 (δ) using said data processor according to g 1 (δ)＝g 0 (δ)+B 0 (δ) obtaining a first order function g in the standard space 2 (δ); wherein, B 0 (δ) =0 indicates that the function is not modified once;
step 404, starting a search with the data processor at δ =0 until g 1 (δ) =0
Obtaining a first design point, based on the result of the evaluation>
A vector corresponding to the first design point; wherein it is present>
I | · | | represents the two-norm of the vector;
step 405, using the data processor according to g 2 (δ)＝g 1 (δ)+B 1 (δ) obtaining a quadratic function g in the standard space 2 (δ); wherein, B 1 (δ) represents the first correction function, and
b 1 represents the firstA minor corrected radius, and->
||·|| 2 The square of the two norms, s, of the representation vector 1 Represents a first modified scale factor and->
Representing the gradient of the functional function g (δ) in the standard space at the first design point; thereafter, in accordance with the method described in step 404, at
Start the search until at g 2 (δ) =0 down = 4>
A second design point is obtained, which is based on>
A vector corresponding to the second design point; wherein it is present>
Step 406, following the method of step 405, using the data processor to determine g n (δ)＝g n-1 (δ)+B n-1 (delta) obtaining a function g of degree n in the standard space n (δ); wherein n is a positive integer and is not less than 1, and B n-1 (δ) represents the correction function of the (n-1) th order and
represents the vector corresponding to the (n-1) th design point, b n-1 Represents the correction radius of the (n-1) th time, and->
||·|| 2 The square of the two norms, s, of the representation vector n-1 Represents the corrected scale factor of the (n-1) th time, and->
Representing the gradient of the functional function g (delta) in the standard space at the (n-1) th design point; thereafter, in accordance with the method set forth in step 404, at +>
Start the search until at g n (δ) =0 down = 4>
Then, the nth design point is obtained>
A vector corresponding to the nth design point; wherein it is present>
Step 407, repeating step 406 for multiple times until N design points are obtained; n is a positive integer, and N is more than or equal to 1 and less than or equal to N;
fifthly, performing Taylor primary expansion at the design point:
step 501, adopting the data processor to determine a vector corresponding to the nth design point
Record as
Wherein it is present>
To representThe vector corresponding to the nth design point ^ is greater than or equal to>
The first one of the parameters in (1),
represents the vector corresponding to the nth design point +>
Is greater than the second parameter of (4), is greater than the first parameter of (4)>
Represents the vector corresponding to the nth design point
Is greater than or equal to>
Indicates the vector corresponding to the nth design point->
The m-th parameter of (1);
step 502, performing taylor first-order expansion on the functional function g (δ) in the standard space at the nth design point by using the data processor to obtain a taylor first-order expansion formula at the nth design point, as follows:
step 503, obtaining taylor first-order expansion at the N design points according to the method in the steps 501 and 502;
step six, acquiring the non-probability failure degree of the beam structure:
step 601, adopting the data processor according to a formula
Unit ellipsoid model for obtaining uncertain parameter vectorE δ Volume V of q (ii) a Wherein Γ (·) represents a Gamma function;
step 602, obtaining the volumes of the parts surrounded by the Taylor first-order expansions at the N design points and the unit ellipsoid models of the uncertain parameter vectors respectively by using the data processor, and summing the volumes of the parts surrounded by the Taylor first-order expansions at the N design points and the unit ellipsoid models of the uncertain parameter vectors to obtain V f (ii) a When the volumes of parts enclosed by the Taylor first-order expansions at the N design points and the unit ellipsoid model of the uncertain parameter vector are summed, the volumes of the overlapped parts are removed;
step 603, using the data processor according to a formula
Obtaining the non-probability failure degree P of the beam structure f ；
Seventhly, acquiring the non-probability reliability of the beam structure:
using said data processor according to formula P r ＝1-P f Obtaining the non-probability reliability P of the beam structure r 。
The method for analyzing the non-probability reliability of the beam structure at multiple design points is characterized by comprising the following steps of: the uncertain parameter vector comprises concentrated load, uniformly distributed load, bending moment, geometric length or bending resistance section coefficient of the beam structure.
The method for analyzing the non-probability reliability of the beam structure at multiple design points is characterized by comprising the following steps of: in step 602, the method for acquiring the volumes of the parts surrounded by the taylor first-order expansion at the N design points and the unit ellipsoid model of the uncertain parameter vector by using the data processor is the same, wherein the method for acquiring the volumes of the parts surrounded by the taylor first-order expansion at the nth design point and the unit ellipsoid model of the uncertain parameter vector by using the data processor comprises the following specific processes:
6021, taylor first-order expansion g at nth design point when m =2 Ln (delta) is two-dimensional with the part enclosed by the unit ellipsoid model of the uncertain parameter vectorBow, taylor first order expansion g at nth design point Ln (delta) the volume of the portion enclosed by the unit ellipsoid model of the uncertain parameter vector is
Taylor first order expansion g at nth design point when m =3 Ln (delta) and the part surrounded by the unit ellipsoid model of the uncertain parameter vector is a three-dimensional spherical segment, and then the Taylor first-order expansion g at the nth design point Ln (delta) and the volume of the portion surrounded by the unit ellipsoid model of the uncertain parameter vector is
When m is>Taylor first order expansion g at nth design point at time 3 Ln (delta) and the unit ellipsoid model of the uncertain parameter vector form m-dimensional segment, and the Taylor first-order expansion g at the nth design point Ln (delta) the volume of the portion enclosed by the unit ellipsoid model of the uncertain parameter vector is
Wherein k is a positive integer and is more than or equal to 1;
step 6022, repeating the step 6021 for a plurality of times to obtain the Taylor first-order expansion at the N design points and the volume of the part enclosed by the unit ellipsoid model of the uncertain parameter vector, and respectively recording as V f1 ，V f2 ，...，V fn ，...，V fN (ii) a Wherein, V f1 Representing Taylor first order expansion g at design point 1 L1 (delta) volume of the part enclosed by the unit ellipsoid model of the uncertain parameter vector, V f2 Representing Taylor first order expansion g at design point 2 L2 (delta) volume of the part enclosed by the unit ellipsoid model of the uncertain parameter vector, V fN Representing Taylor first order expansion g at the Nth design point LN (δ) the volume of the portion enclosed by the unit ellipsoid model of the uncertain parameter vector.
The method for analyzing the non-probability reliability of the beam structure at multiple design points is characterized by comprising the following steps of: in step 602, the data processor is adopted to sum the Taylor first-order expansions at the N design points and the volume of a part enclosed by the unit ellipsoid model of the uncertain parameter vector to obtain V f The specific process comprises the following steps:
step A, adopting the data processor to enclose partial volume V by Taylor first-order expansion at N design points and a unit ellipsoid model of uncertain parameter vectors f1 ，V f2 ，...，V fn ，...，V fN The volume of the part enclosed by the Taylor first-order expansion and the unit ellipsoid model of the uncertain parameter vector at the middle and the first design point is denoted as V fl (ii) a Wherein l is a positive integer, l is more than or equal to 1 and less than or equal to N, and l is not equal to N;
step B, recording a vector corresponding to the ith design point as a design point by adopting the data processor
Step C, adopting the data processor to judge when
The sum of the volume of the portion surrounded by the Taylor first-order expansion at the first design point and the unit ellipsoid model of the uncertain parameter vector and the volume of the portion surrounded by the Taylor first-order expansion at the n design point and the unit ellipsoid model of the uncertain parameter vector is V fnl ＝V fn +V fl ；
Using the data processor to determine when
The sum of the volume of the portion surrounded by the Taylor first-order expansion at the first design point and the unit ellipsoid model of the uncertain parameter vector and the volume of the portion surrounded by the Taylor first-order expansion at the n design point and the unit ellipsoid model of the uncertain parameter vector is V fnl ＝V fn +V fl -V nl (ii) a Wherein, V nl A volume of an overlapping portion of a portion surrounded by the taylor first-order expansion and the unit ellipsoid model of the uncertain parameter vector at the ith design point and a portion surrounded by the taylor first-order expansion and the unit ellipsoid model of the uncertain parameter vector at the nth design point;
using the data processor to determine when
Then the sum of the volume of the portion enclosed by the Taylor first-order expansion at the first design point and the unit ellipsoid model of the uncertain parameter vector and the volume of the portion enclosed by the Taylor first-order expansion at the n design point and the unit ellipsoid model of the uncertain parameter vector is V fnl ＝max{V fn ,V fl }；
And D, repeating the step C for multiple times to complete the volume summation of the Taylor first-order expansion at the N design points and a part surrounded by the unit ellipsoid model of the uncertain parameter vector.
The method for analyzing the non-probability reliability of the beam structure at multiple design points is characterized by comprising the following steps of: in the step C, the volume of the overlapping part of the part enclosed by the Taylor first-order expansion at the nth design point and the unit ellipsoid model of the uncertain parameter vector and the part enclosed by the Taylor first-order expansion at the ith design point and the unit ellipsoid model of the uncertain parameter vector is obtained, and the specific process is as follows:
step C01, adopting the data processor to enable the vector corresponding to the nth design point
The vector corresponding to the/th design point->
The angle between them is denoted as theta nl The vector corresponding to the nth design point ^ is greater than or equal to>
The two norms of (A) are denoted as beta n The vector corresponding to the/th design point->
The two norms of (A) are denoted as beta l Using said data processor according to the formula rho nl ＝cosθ nl Obtaining an intermediate variable rho nl ；
Step C02, adopting the data processor, when m =2, the volume of the overlapping part of the part enclosed by the Taylor first-order expansion at the nth design point and the unit ellipsoid model of the uncertain parameter vector and the part enclosed by the Taylor first-order expansion at the ith design point and the unit ellipsoid model of the uncertain parameter vector is as follows:
when m =3, the volume of the overlapping part of the part enclosed by the taylor first-order expansion and the unit ellipsoid model of the uncertain parameter vector at the nth design point and the part enclosed by the taylor first-order expansion and the unit ellipsoid model of the uncertain parameter vector at the ith design point is:
wherein x represents a first integral variable, y represents a second integral variable, and the upper bound of x is
x has a lower bound of +>
y has an upper bound of->
y has a lower bound of->
When m > 3 and m is an even number, thThe volume of the overlapping part of the part enclosed by the Taylor first-order expansion and the unit ellipsoid model of the uncertain parameter vector at the n design points and the part enclosed by the Taylor first-order expansion and the unit ellipsoid model of the uncertain parameter vector at the ith design point is as follows:
wherein k ' represents a positive integer, k ' is not less than 1, q is a natural number, and q is not less than 0 and not more than k ' -1;
when m is more than 3 and m is an odd number, the volume of the overlapped part of the part enclosed by the Taylor first-order expansion and the unit ellipsoid model of the uncertain parameter vector at the nth design point and the part enclosed by the Taylor first-order expansion and the unit ellipsoid model of the uncertain parameter vector at the ith design point is as follows:
wherein k 'represents a positive integer, k' is not less than 1, r is a natural number, r is not less than 0 and not more than 1, s is a natural number, and s is not less than 0 and not more than 1.
The method for analyzing the non-probability reliability of the beam structure at multiple design points is characterized by comprising the following steps of: the data processor is a computer.
Compared with the prior art, the invention has the following advantages:
1. the method has the advantages of simple steps, reasonable design and high calculation efficiency.
2. The method is simple and convenient to operate and convenient to realize, and mainly comprises the steps of determining a functional function and an uncertainty variable of a beam structure, then establishing a corresponding ellipsoid model according to the uncertainty variable, and normalizing and standardizing the ellipsoid model established according to the uncertainty variable to obtain a unit ellipsoid model of the uncertainty variable and a functional function in a standard space; and finally, the data processor sums the Taylor first-order expansion at each design point and the volume of a part surrounded by the unit ellipsoid model of the uncertain parameter vector to obtain the non-probability failure degree of the beam structure, further obtain the non-probability reliability degree of the beam structure, realize the non-probability reliability analysis of the beam structure, and establish the function by considering the multiple design points of the beam structure to obtain relatively accurate reliability degree.
3. According to the method, the ellipsoid models of the uncertain variables are subjected to normalization processing to obtain the normalized equivalent ellipsoid model, even if the order of magnitude difference between the uncertain variables is overlarge, all elements in the feature matrix of the normalized equivalent ellipsoid model can be guaranteed to have the same order of magnitude by the uncertain variables in the standard space, the calculation precision in the calculation process is guaranteed, the serious ill-condition problem of the feature matrix is avoided, the adaptability of the method is improved, and the method is convenient to popularize.
4. The invention adopts the non-probability reliable model to describe the uncertain variables, avoids the need of a large amount of statistical data due to the adoption of the probability reliability design, has large calculation amount, solves the problem that the traditional probability reliability optimization design method is limited by insufficient sample information and cannot carry out scientific and reasonable design, utilizes the non-probability reliability analysis, has simple and convenient application and needs fewer samples.
5. The method for approximating the function by using the first-order Taylor expansion is simple and easy to understand, greatly simplifies the calculation steps of the failure domain volume of the beam structure with the problem of multiple design points, meets the actual requirements of engineering, and basically ignores the error of the obtained non-probability failure rate under the condition of low non-linearity.
6. The invention has obvious effect on the problem of multiple design points by approximating the function by the Taylor first-order expansion of the multiple design points in the standard space, and avoids overlarge error of a calculation result caused by using the Taylor first-order expansion approximation of a single design point.
7. The beam structure reliability analysis method provided by the invention fully considers the actual engineering situation and provides effective basis and reference for the design and manufacture of the beam structure.
In conclusion, the method has the advantages of simple steps, reasonable design and convenient implementation, the non-probability reliability analysis of the beam structure is realized by considering the establishment of the function at multiple design points of the beam structure, the calculation efficiency is high, the relatively accurate reliability is obtained, and the popularization and the use are convenient.
The technical solution of the present invention is further described in detail by the accompanying drawings and embodiments.
Drawings
FIG. 1 is a block diagram of the process flow of the present invention.
Fig. 2 is a schematic diagram of a simplified model of a beam structure according to an embodiment of the present invention.
FIG. 3 is a functional function g (delta) of the cantilever structure in the standard space, and a first-order Taylor expansion g of each design point according to the present invention Li (delta) and the unit ellipsoid model E δ Schematic intersection.
Detailed Description
A method for analyzing non-probabilistic reliability of a beam structure at multiple design points as shown in fig. 1 includes the following steps:
step one, determining a functional function and an uncertainty variable of a beam structure:
establishing a function g (X) with multiple design points by adopting the data processor to the beam structure under any failure mode; wherein X represents an uncertain parameter vector for the beam structure and is denoted as uncertain parameter vector X = (X) 1 ,X 2 ,...,X i ,...,X m ) T M is the dimension of the uncertain parameter vector X, X 1 Denotes the 1 st uncertain variable, X 2 Denotes the 2 nd uncertain variable, X i Denotes the ith uncertain variable, X m Representing the mth uncertain variable, i is the number of the uncertain variable, i is a positive integer and the value range of i is 1-m,
representing an uncertain variable X i The interval of the values is selected from the group, i Xand &>
Are respectively an uncertain variable X i M is more than or equal to 2;
step two, establishing a corresponding ellipsoid model according to the uncertainty variable:
establishing an ellipsoid model using the data processor
The following were used:
in this example, Z is 11 Denotes the 1 st uncertain variable X 1 And the 1 st uncertain variable X 1 Covariance between themselves, Z 12 Denotes the 1 st uncertain variable X 1 And 2 nd uncertain variable X 2 Covariance between, Z 1j Denotes the 1 st uncertain variable X 1 And the jth uncertain variable X j Covariance between, Z 1m Denotes the 1 st uncertain variable X 1 And the m-th uncertain variable X m The covariance between; z 21 Denotes the 2 nd uncertain variable X 2 And the 1 st uncertain variable X 1 Covariance between, Z 22 Denotes the 2 nd uncertain variable X 2 And 2 nd uncertain variable X 2 Covariance between themselves, Z 2j Denotes the 2 nd uncertain variable X 2 And the jth uncertain variable X j Covariance between, Z 2m Denotes the 2 nd uncertain variable X 2 And the m-th uncertain variable X m The covariance between; z i1 Denotes the ith uncertain variable X i And 1 st uncertain variable X 1 Covariance between, Z i2 Denotes the ith uncertain variable X i And 2 nd uncertain variable X 2 Covariance between themselves, Z im Denotes the ith uncertain variable X i And the m-th uncertain variable X m Covariance between; z m1 Denotes the m-th uncertain variable X m And the 1 st uncertain variable X 1 Covariance between, Z m2 Denotes the m-th uncertain variable X m And 2 nd uncertain variable X 2 Covariance between themselves, Z mj Denotes the m-th uncertain variable X m And the jth uncertain variable X j Covariance between, Z mm Denotes the m-th uncertain variable X m And the m-th uncertain variable X m Covariance between themselves.
In this embodiment, it should be noted that,
obtaining the ith uncertain variable X i And the jth uncertain variable X j The covariance of (a); where ρ is i,j For the ith uncertain variable X i And the jth uncertain variable X j Coefficient of correlation therebetween, in conjunction with a predetermined number of characteristic values>
For the jth uncertain variable X j The interval radius of (a).
The true bookIn the examples, it is to be noted that the 1 st uncertain variable X 1 And the 1 st uncertain variable X 1 Correlation coefficient between themselves, 2 nd uncertain variable X 2 And 2 nd uncertain variable X 2 Correlation coefficient between themselves, m-th uncertain variable X m And the m-th uncertain variable X m The correlation coefficient between themselves is 1.
Step three, obtaining a unit ellipsoid model of the uncertainty variable:
step 301, using said data processor according to a formula
Obtaining the ith uncertainty variable X i Is greater than or equal to the section radius>
Based on a formula->
Obtaining the ith uncertainty variable X i Normalized variable U of i ；
Step 302, according to the method in step 301, the data processor is adopted to perform normalization processing on the uncertain parameter vector X to obtain an uncertain normalization vector U = (U =) 1 ,U 2 ,...,U i ,...,U m ) T (ii) a Wherein, U 1 Denotes the 1 st uncertain variable X 1 Normalized variable of (U) 2 Denotes the 2 nd uncertain variable X 2 Normalized variable of (U) m Denotes the m-th uncertain variable X m A normalized variable of (d);
step 303, obtaining a normalized equivalent ellipsoid model of the uncertain parameter vector by adopting the data processor according to the method in the step two
I.e. based on>
Wherein omega u A feature matrix representing a normalized equivalent ellipsoid model determining the uncertainty parameter vector, and ≥>
Is shown in
An m-dimensional diagonal matrix of diagonal elements;
step 304, determining a characteristic matrix omega of the normalized equivalent ellipsoid model of the uncertain parameter vector by using the data processor u Cholesky decomposition is carried out to obtain
Wherein L is c Feature matrix omega representing a normalized equivalent ellipsoid model for determining uncertain parameter vectors u Obtaining a lower triangular matrix through Cholesky decomposition;
305, using the data processor according to a formula
Obtaining a standardized vector delta of the uncertain parameter vector X in a standard space; wherein, delta 1 Denotes the 1 st uncertain variable X 1 Normalized variable of δ 2 Denotes the 2 nd uncertain variable X 2 Normalized variable of δ i Denotes the ith uncertain variable X i Normalized variable of (d), d m Denotes the m-th uncertain variable X m A normalized variable of (a);
step 306, substituting the normalized vector delta of the uncertain parameter vector X in the standard space into the normalized equivalent ellipsoid model of the uncertain parameter vector in the step 303 by adopting the data processor
Unit ellipsoid model E for obtaining uncertain parameter vector δ I.e. E δ ＝{δ|δ T δ≤1,δ∈R m }；
Step four, obtaining a plurality of design points:
step 401, obtaining a relation between the uncertain parameter vector X and the normalized vector δ of the uncertain parameter vector X in the standard space by using the data processor, wherein the relation is as follows:
step 402, using the data processor, of relating the uncertain parameter vector X to a normalized vector delta in a standard space
Substituting the function g (X) to obtain a function g (delta) in the standard space;
step 403, recording the function g (delta) in the standard space as the initial function g in the standard space by using the data processor 0 (δ) using said data processor according to g 1 (δ)＝g 0 (δ)+B 0 (delta) obtaining a first order function g in the standard space 2 (δ); wherein, B 0 (δ) =0 indicates that the function is not modified once;
step 404, starting a search with the data processor at δ =0 until g 1 (δ) =0
The first design point is obtained, based on which the decision is made>
A vector corresponding to the first design point; wherein +>
I | · | | represents the two-norm of the vector;
step 405, using the data processor according to g 2 (δ)＝g 1 (δ)+B 1 (δ) obtaining a standard spaceSecond order function g in 2 (δ); wherein, B 1 (δ) represents the first correction function, and
b 1 indicates the first correction radius and->
||·|| 2 The square of the two norms, s, of the representation vector 1 Represents a first modified scale factor and->
Representing the gradient of the functional function g (δ) in the standard space at the first design point; thereafter, in accordance with the method described in step 404, at
Start the search until at g 2 (δ) =0 down->
A second design point is obtained, which is based on>
A vector corresponding to the second design point; wherein the content of the first and second substances,
step 406, following the method of step 405, using the data processor to determine g n (δ)＝g n-1 (δ)+B n-1 (δ) obtaining a function g of degree n in the standard space n (δ); wherein n is a positive integer, n is not less than 1, B n-1 (δ) represents the correction function of the (n-1) th order and
represents the vector corresponding to the (n-1) th design point, b n-1 Represents the correction radius of the (n-1) th time, and->
||·|| 2 The square of the two norms, s, of the representation vector n-1 Represents the corrected scale factor of the (n-1) th time, and->
Representing the gradient of the functional function g (delta) in the standard space at the (n-1) th design point; thereafter, in accordance with the method described in step 404, at
Start the search until at g n (δ) =0 down = 4>
Then, the nth design point is obtained>
A vector corresponding to the nth design point; wherein it is present>
Step 407, repeating step 406 for multiple times until N design points are obtained; n is a positive integer, and N is more than or equal to 1 and less than or equal to N;
fifthly, performing Taylor primary expansion at the design point:
step 501, adopting the data processor to convert the vector corresponding to the nth design point
Record as
Wherein it is present>
Indicates the vector corresponding to the nth design point->
The first one of the parameters in (1),
indicates the vector corresponding to the nth design point->
Is greater than the second parameter of (4), is greater than the first parameter of (4)>
Represents the vector corresponding to the nth design point
Is greater than or equal to>
Indicates the vector corresponding to the nth design point->
The m-th parameter of (1);
step 502, performing taylor first-order expansion on the functional function g (δ) in the standard space at the nth design point by using the data processor to obtain a taylor first-order expansion formula at the nth design point, as follows:
step 503, obtaining taylor first-order expansion at the N design points according to the method in the steps 501 and 502;
step six, acquiring the non-probability failure degree of the beam structure:
step 601, adopting the data processor according to a formula
Unit ellipsoid model E for obtaining uncertain parameter vector δ Volume V of q (ii) a Wherein Γ (·) represents a Gamma function;
step 602, respectively obtaining volumes of parts surrounded by Taylor first-order expansions at the N design points and the unit ellipsoid models of the uncertain parameter vectors by adopting the data processor, and summing the volumes of the parts surrounded by the Taylor first-order expansions at the N design points and the unit ellipsoid models of the uncertain parameter vectors to obtain V f (ii) a When the volumes of parts enclosed by the Taylor first-order expansions at the N design points and the unit ellipsoid model of the uncertain parameter vector are summed, the volumes of the overlapped parts are removed;
step 603, using the data processor according to a formula
Obtaining the non-probability failure degree P of the beam structure f ；
Seventhly, acquiring the non-probability reliability of the beam structure:
using said data processor according to formula P r ＝1-P f Obtaining the non-probability reliability P of the beam structure r 。
In this embodiment, the uncertain parameter vector includes a concentrated load, a uniform load, a bending moment, a geometric length, or a bending section coefficient of the beam structure.
In the first step 602, the method for acquiring the volumes of the portions surrounded by the taylor first-order expansion at the N design points and the unit ellipsoid model of the uncertain parameter vector by using the data processor is the same, wherein the method for acquiring the volumes of the portions surrounded by the taylor first-order expansion at the N design points and the unit ellipsoid model of the uncertain parameter vector by using the data processor comprises the following specific steps:
step 6021When m =2, the Taylor first-order expansion g at the nth design point Ln (delta) and the unit ellipsoid model of the uncertain parameter vector are enclosed to form a two-dimensional arc, and the Taylor first-order expansion g at the nth design point Ln (delta) the volume of the portion enclosed by the unit ellipsoid model of the uncertain parameter vector is
Taylor first-order expansion g at nth design point when m =3 Ln (delta) and the unit ellipsoid model of the uncertain parameter vector enclose a part which is a three-dimensional segment, and then the Taylor first-order expansion g at the nth design point Ln (delta) the volume of the portion enclosed by the unit ellipsoid model of the uncertain parameter vector is
When m is>Taylor first order expansion g at nth design point at time 3 Ln (delta) and the unit ellipsoid model of the uncertain parameter vector form m-dimensional segment, and the Taylor first-order expansion g at the nth design point Ln (delta) the volume of the portion enclosed by the unit ellipsoid model of the uncertain parameter vector is
Wherein k is a positive integer and is more than or equal to 1;
step 6022, repeating the step 6021 for a plurality of times to obtain the Taylor first-order expansion at the N design points and the volume of the part enclosed by the unit ellipsoid model of the uncertain parameter vector, and respectively recording as V f1 ，V f2 ，...，V fn ，...，V fN (ii) a Wherein, V f1 Representing Taylor first order expansion g at design point 1 L1 (delta) volume of the part enclosed by the unit ellipsoid model of the uncertain parameter vector, V f2 Representing Taylor first order expansion g at design point 2 L2 (delta) volume of the part enclosed by the unit ellipsoid model of the uncertain parameter vector, V fN Represents the Nth design pointTaylor first order expansion g of LN (δ) the volume of the portion enclosed by the unit ellipsoid model of the uncertain parameter vector.
In this embodiment, in step 602, the data processor sums the taylor first-order expansion at the N design points and the volume of the part surrounded by the unit ellipsoid model of the uncertain parameter vector to obtain V f The specific process is as follows:
step A, adopting the data processor to enclose partial volume V by Taylor first-order expansion at N design points and a unit ellipsoid model of uncertain parameter vectors f1 ，V f2 ，...，V fn ，...，V fN The volume enclosed by the Taylor first-order expansion and the unit ellipsoid model of the uncertain parameter vector at the design point I is denoted as V fl (ii) a Wherein l is a positive integer, l is more than or equal to 1 and less than or equal to N, and l is not equal to N;
step B, recording a vector corresponding to the ith design point as a design point by adopting the data processor
Step C, adopting the data processor to judge when
The sum of the volume of the portion surrounded by the Taylor first-order expansion at the first design point and the unit ellipsoid model of the uncertain parameter vector and the volume of the portion surrounded by the Taylor first-order expansion at the n design point and the unit ellipsoid model of the uncertain parameter vector is V fnl ＝V fn +V fl ；
Using the data processor to determine when
The sum of the volume of the portion surrounded by the Taylor first-order expansion at the first design point and the unit ellipsoid model of the uncertain parameter vector and the volume of the portion surrounded by the Taylor first-order expansion at the n design point and the unit ellipsoid model of the uncertain parameter vector is V fnl ＝V fn +V fl -V nl (ii) a Wherein, V nl A volume of an overlapping portion of a portion surrounded by the taylor first-order expansion and the unit ellipsoid model of the uncertain parameter vector at the ith design point and a portion surrounded by the taylor first-order expansion and the unit ellipsoid model of the uncertain parameter vector at the nth design point;
using the data processor to determine when
Then the sum of the volume of the portion enclosed by the Taylor first-order expansion at the first design point and the unit ellipsoid model of the uncertain parameter vector and the volume of the portion enclosed by the Taylor first-order expansion at the n design point and the unit ellipsoid model of the uncertain parameter vector is V fnl ＝max{V fn ,V fl }；
And D, repeating the step C for multiple times to complete the volume summation of the Taylor first-order expansion at the N design points and a part surrounded by the unit ellipsoid model of the uncertain parameter vector.
In this embodiment, in step C, the volume of the overlapping portion of the part enclosed by the taylor first-order expansion at the nth design point and the unit ellipsoid model of the uncertain parameter vector and the part enclosed by the taylor first-order expansion at the ith design point and the unit ellipsoid model of the uncertain parameter vector is obtained, and the specific process is as follows:
step C01, adopting the data processor to enable the vector corresponding to the nth design point
The vector corresponding to the/th design point->
The angle between them is denoted θ nl The vector corresponding to the nth design point ^ is greater than or equal to>
The two norms of (A) are denoted as beta n The vector corresponding to the/th design point->
Is denoted as beta l Using said data processor according to the formula rho nl ＝cosθ nl Obtaining an intermediate variable rho nl ；
Step C02, adopting the data processor, when m =2, the volume of the overlapping part of the part enclosed by the Taylor first-order expansion at the nth design point and the unit ellipsoid model of the uncertain parameter vector and the part enclosed by the Taylor first-order expansion at the ith design point and the unit ellipsoid model of the uncertain parameter vector is as follows:
when m =3, the volume of the overlapping part of the part enclosed by the taylor first-order expansion and the unit ellipsoid model of the uncertain parameter vector at the nth design point and the part enclosed by the taylor first-order expansion and the unit ellipsoid model of the uncertain parameter vector at the ith design point is:
wherein x represents a first integral variable, y represents a second integral variable, and the upper bound of x is
x has a lower bound of +>
y has an upper bound of->
y has a lower bound of->
When m > 3 and m is even number, the nthThe volume of the overlapping part of the part enclosed by the Taylor first-order expansion and the unit ellipsoid model of the uncertain parameter vector at the design point and the part enclosed by the Taylor first-order expansion and the unit ellipsoid model of the uncertain parameter vector at the ith design point is as follows:
wherein k ' represents a positive integer, k ' is not less than 1, q is a natural number, and q is not less than 0 and not more than k ' -1;
when m is more than 3 and m is an odd number, the volume of the overlapped part of the part enclosed by the Taylor first-order expansion and the unit ellipsoid model of the uncertain parameter vector at the nth design point and the part enclosed by the Taylor first-order expansion and the unit ellipsoid model of the uncertain parameter vector at the ith design point is as follows:
wherein k 'represents a positive integer, k' is not less than 1, r is a natural number, r is not less than 0 and not more than 1, s is a natural number, and s is not less than 0 and not more than 1.
In this embodiment, the data processor is a computer.
As shown in fig. 2, in this embodiment, the beam structure is a cantilever beam structure, and the loads borne by the cantilever beam structure are respectively 4N/mm of vertically downward uniformly distributed load, 12800N of vertically downward concentrated load, and 6400N of vertically upward concentrated load, where a distance between a position of action of the vertically downward concentrated load and a joint is 100mm, a position of action of the vertically upward concentrated load is located at the middle of the cantilever beam, and allowable stress between the cantilever beam and the joint is 100MPa.
In this embodiment, the value range of i is 1 to 2,m =2, and the uncertain parameter vector X includes an uncertain variable X 1 And an uncertain variable X 2 Then X = (X) 1 ,X 2 ) T Said uncertain variable X 1 The uncertain variable X is the bending resistance section coefficient of the cantilever beam and the joint 1 The length of the cantilever beam structure.
In this embodiment, the uncertain variable X 1 And an uncertain variable X 2 Correlation coefficient of
Take 0.2, uncertain variable X 1 And an uncertain variable X 1 Its own correlation coefficient->
Take 1, uncertain variable X 2 And an uncertain variable X 2 Self correlation coefficient
1 is taken.
In this embodiment, the uncertain variable X 1 And an uncertain variable X 2 The value range of (A) is shown in Table 1
TABLE 1 uncertain variable X 1 And an uncertain variable X 2 Value range of
In this embodiment, the failure mode is the cantilever beam failure when the bending moment at the cantilever beam connection exceeds the product of the allowable stress and the bending section coefficient, and other failure modes are not considered because the other failure modes do not reach the limit state, so that the function of multiple design points in the failure mode is determined as
Function g (X) =100X with multiple design points obtained after simplification 1 -2X 2 2 +3200X 2 -1280000；
In this embodiment, the feature matrix of the ellipsoid model is determined as
/>
In this embodiment, normalization processing is performed on the ellipsoid model in steps 301 to 303 to obtain the characteristics of the normalized equivalent ellipsoid model for determining the uncertain parameter vectorMatrix array
In this embodiment, in step 304, the data processor is used to determine the feature matrix Ω of the normalized equivalent ellipsoid model of the uncertain parameter vector u Cholesky decomposition is carried out to obtain
In this embodiment, the step 305 obtains a normalized vector of the uncertain parameter vector X in the standard space
In this embodiment, the relation between the uncertain parameter vector X and the normalized vector δ of the uncertain parameter vector X in the standard space obtained in step 401 is as follows
In step 402, the function g (δ) in the standard space is obtained as g (δ) =19595.92 δ 1 -(800δ 2 +3200) 2 +644000δ 2 +1340000。
In this embodiment, according to steps 403 to 407, two design points are obtained, where N =2, and the value of N is 1 to 2, and then a vector corresponding to the 1 st design point is obtained
The vector corresponding to the 2 nd design point ^ is greater than or equal to>
As shown in fig. 3, in this embodiment, according to step 502, the data processor is used to perform first-order taylor expansion on the functional function g (δ) in the standard space at the 1 st design point to obtain the functional functions g (δ) in the standard space
And
taylor first order expansion at design point 1 is g L1 (δ)＝19595.92δ 1 +135906.60δ 2 +114372.97, the volume of the part enclosed by the Taylor first-order expansion at the 1 st design point and the unit ellipsoid model of the uncertain parameter vector is V f1 ＝0.1254762。
According to the step 502, the data processor is adopted to perform first-order Taylor expansion on the function g (delta) in the standard space at the 2 nd design point to respectively obtain
And &>
Taylor first order expansion at design point 2 is g L2 (δ)＝19595.92δ 1 -135740.08δ 2 +121022.78, the volume of the part enclosed by the unit ellipsoid model of the Taylor first-order expansion uncertainty parameter vector at the 2 nd design point is V f2 ＝0.0746604。
In this embodiment, the data processor performs determination according to steps a to D to obtain that a volume of a part surrounded by the taylor first-order expansion at the 1 st design point and the unit ellipsoid model of the uncertain parameter vector and a part surrounded by the taylor first-order expansion at the 2 nd design point and the unit ellipsoid model of the uncertain parameter vector do not coincide, and then V is determined f12 ＝V f1 +V f2 。
In this embodiment, the unit ellipsoid model E of uncertain parameter vectors δ Volume V of q Is a V q ＝3.14。
In this embodiment, the data processor is used to convert V into V f1 =0.1254762 and V f2 Summation of =0.0746604 yields V f According to the formula
Obtaining the non-probability failure degree of the beam structure as P f ＝6.37％。
In this embodiment, the data processor is adopted according to the formula P r ＝1-P f Obtaining the non-probability reliability of the beam structure as P r ＝93.63％。
In this embodiment, when only the first design point is considered, the obtained non-probability failure degree of the beam structure is P f =3.99%, and the non-probability reliability of the beam structure is P r ＝96.01％。
In this embodiment, when only the second design point is considered, the obtained non-probability failure degree of the beam structure is P f =2.38%, and the non-probability reliability of the beam structure is P r ＝97.62％。
In this embodiment, when the monte carlo method is used to solve, the obtained non-probability failure degree of the beam structure is P f =6.44%, and the non-probability reliability of the beam structure is P r ＝93.56％。
In this embodiment, the non-probability reliability of the beam structure obtained by the present invention is close to the non-probability reliability of the beam structure solved by the monte carlo method, which indicates that the non-probability reliability of the beam structure obtained by the present invention is more accurate. In addition, the actual requirements of engineering are considered, approximation is removed through Taylor expansion of each design point of the cantilever beam structure, the method is simple and easy to understand, the calculation efficiency is improved, relatively accurate reliability is obtained, and the method is convenient to popularize and use.
The above description is only a preferred embodiment of the present invention, and is not intended to limit the present invention, and all simple modifications, changes and equivalent structural changes made to the above embodiment according to the technical spirit of the present invention still fall within the protection scope of the technical solution of the present invention.
Claims (6)
1. A multi-design-point non-probabilistic reliability analysis method for a beam structure is characterized by comprising the following steps of:
step one, determining a function and an uncertainty variable of a beam structure:
establishing a function g (X) of multiple design points by adopting the data processor to the beam structure in any failure mode; wherein X represents an uncertain parameter vector for the beam structure and is denoted as uncertain parameter vector X = (X) 1 ,X 2 ,...,X i ,...,X m ) T M is the dimension of the uncertain parameter vector X, X 1 Denotes the 1 st uncertain variable, X 2 Denotes the 2 nd uncertain variable, X i Denotes the ith uncertain variable, X m Representing the mth uncertain variable, i is the number of the uncertain variable, i is a positive integer and the value range of i is 1-m,
representing an uncertain variable X i The interval of the values is selected from the group, i Xand
are respectively an uncertain variable X i M is more than or equal to 2;
step two, establishing a corresponding ellipsoid model according to the uncertainty variable:
establishing an ellipsoid model using the data processor
The following were used:
step three, obtaining a unit ellipsoid model of the uncertainty variable:
step 301, using said data processor according to a formula
Obtaining the ith uncertainty variable X i Radius of interval (2)
Using said data processor according to a formula
Obtaining the ith uncertainty variable X i Normalized variable U of i ；
Step 302, according to the method in step 301, the data processor is adopted to perform normalization processing on the uncertain parameter vector X to obtain an uncertain normalization vector U = (U =) 1 ,U 2 ,...,U i ,...,U m ) T (ii) a Wherein, U 1 Denotes the 1 st uncertain variable X 1 Normalized variable of (U) 2 Denotes the 2 nd uncertain variable X 2 Normalized variable of (U) m Denotes the m-th uncertain variable X m A normalized variable of (a);
step 303, obtaining the normalized equivalent of the uncertain parameter vector by adopting the data processor according to the method in the step twoEllipsoid model
Namely, it is
Wherein omega u A feature matrix representing a normalized equivalent ellipsoid model determining uncertain parameter vectors, an
Is shown in
An m-dimensional diagonal matrix of diagonal elements;
step 304, determining a characteristic matrix omega of the normalized equivalent ellipsoid model of the uncertain parameter vector by using the data processor u Cholesky decomposition is carried out to obtain
Wherein L is c Feature matrix omega representing a normalized equivalent ellipsoid model for determining uncertain parameter vectors u Obtaining a lower triangular matrix through Cholesky decomposition;
305, using the data processor according to a formula
Obtaining a standardized vector delta of the uncertain parameter vector X in a standard space; wherein, delta 1 Denotes the 1 st uncertain variable X 1 Normalized variable of (d), d 2 Denotes the 2 nd uncertain variable X 2 Normalized variable of δ i Denotes the ith uncertain variable X i Normalized variable of δ m Denotes the m-th uncertain variable X m A normalized variable of (a);
step 306, adoptSubstituting a normalized vector delta of the uncertain parameter vector X in the standard space into the normalized equivalent ellipsoid model of the uncertain parameter vector in step 303 with the data processor
Unit ellipsoid model E for obtaining uncertain parameter vector δ I.e. E δ ＝{δ|δ T δ≤1,δ∈R m }；
Step four, obtaining a plurality of design points:
step 401, obtaining a relation between the uncertain parameter vector X and the normalized vector δ of the uncertain parameter vector X in the standard space by using the data processor, wherein the relation is as follows:
402, adopting the data processor to obtain a relation between the uncertain parameter vector X and a standardized vector delta of the uncertain parameter vector X in a standard space
Substituting the function g (X) to obtain a function g (delta) in the standard space;
step 403, recording the function g (delta) in the standard space as the initial function g in the standard space by using the data processor 0 (δ) using said data processor according to g 1 (δ)＝g 0 (δ)+B 0 (δ) obtaining a first order function g in the standard space 2 (δ); wherein, B 0 (δ) =0 indicates that the function is not modified once;
step 404, starting a search with the data processor at δ =0 until g 1 (δ) =0
A first design point is obtained for the first time,
a vector corresponding to the first design point; wherein the content of the first and second substances,
i | · | | represents the two-norm of the vector;
step 405, using the data processor according to g 2 (δ)＝g 1 (δ)+B 1 (delta) obtaining a quadratic function g in the standard space 2 (δ); wherein, B 1 (δ) represents the first correction function, and
b 1 indicates the first correction radius, and
||·|| 2 the square of the two norms, s, of the representation vector 1 Represents the first modified scale factor, and
representing the gradient of the functional function g (δ) in the standard space at the first design point; thereafter, in accordance with the method described in step 404, at
Start the search until at g 2 (δ) =0 or less
A second design point is obtained in that,
a vector corresponding to the second design point; wherein the content of the first and second substances,
step 406, following the method of step 405, using the data processor to determine g n (δ)＝g n-1 (δ)+B n-1 (δ) obtaining a function g of degree n in the standard space n (δ); wherein n is a positive integer and is not less than 1, and B n-1 (delta) represents the correction function of degree n-1 and
represents the vector corresponding to the (n-1) th design point, b n-1 Represents the correction radius of the n-1 th order, and
||·|| 2 the square of the two norms, s, of the representation vector n-1 Represents the modified scale factor of degree n-1, and
representing the gradient of the functional function g (delta) in the standard space at the (n-1) th design point; thereafter, the method described in step 404 is followed
Start the search until at g n (δ) =0
Then, the nth design point is obtained,
a vector corresponding to the nth design point; wherein the content of the first and second substances,
step 407, repeating step 406 for multiple times until N design points are obtained; n is a positive integer, and N is more than or equal to 1 and less than or equal to N;
fifthly, performing Taylor primary expansion at the design point:
step 501, adopting the data processor to convert the vector corresponding to the nth design point
Record as
Wherein the content of the first and second substances,
represents the vector corresponding to the nth design point
The first one of the parameters in (1),
represents the vector corresponding to the nth design point
The second one of the parameters (c) is,
represents the vector corresponding to the nth design point
The (c) th parameter of (a),
represents the vector corresponding to the nth design point
The m-th parameter of (1);
step 502, performing taylor first-order expansion on the functional function g (δ) in the standard space at the nth design point by using the data processor to obtain a taylor first-order expansion formula at the nth design point, as follows:
step 503, obtaining taylor first-order expansion at the N design points according to the methods in step 501 and step 502;
step six, acquiring the non-probability failure degree of the beam structure:
step 601, adopting the data processor according to a formula
Unit ellipsoid model E for obtaining uncertain parameter vector δ Volume V of q (ii) a Wherein Γ (·) represents a Gamma function;
step 602, obtaining the volumes of the parts surrounded by the Taylor first-order expansions at the N design points and the unit ellipsoid models of the uncertain parameter vectors respectively by using the data processor, and summing the volumes of the parts surrounded by the Taylor first-order expansions at the N design points and the unit ellipsoid models of the uncertain parameter vectors to obtain V f (ii) a When the volumes of parts enclosed by the Taylor first-order expansions at the N design points and the unit ellipsoid model of the uncertain parameter vector are summed, the volumes of the overlapped parts are removed;
step 603, using the data processor according to a formula
Obtaining the non-probability failure degree P of the beam structure f ；
Seventhly, acquiring the non-probability reliability of the beam structure:
using said data processor according to formula P r ＝1-P f Obtaining the non-probability of the beam structureDegree of reliability P r 。
2. The method for analyzing the non-probabilistic reliability of the multi-design point for the beam structure according to claim 1, wherein: the uncertain parameter vector comprises concentrated load, uniform load, bending moment, geometric length or bending resistance section coefficient of the beam structure.
3. The method for analyzing the non-probabilistic reliability of the multi-design point for the beam structure according to claim 1, wherein: in step 602, the method for acquiring the volumes of the parts surrounded by the taylor first-order expansion at the N design points and the unit ellipsoid model of the uncertain parameter vector by using the data processor is the same, wherein the method for acquiring the volumes of the parts surrounded by the taylor first-order expansion at the nth design point and the unit ellipsoid model of the uncertain parameter vector by using the data processor comprises the following specific processes:
step 6021, taylor first-order expansion g at nth design point when m =2 Ln (delta) and the unit ellipsoid model of the uncertain parameter vector are enclosed into a two-dimensional arc, then the Taylor first-order expansion g at the nth design point Ln (delta) the volume of the portion enclosed by the unit ellipsoid model of the uncertain parameter vector is
Taylor first order expansion g at nth design point when m =3 Ln (delta) and the unit ellipsoid model of the uncertain parameter vector enclose a part which is a three-dimensional segment, and then the Taylor first-order expansion g at the nth design point Ln (delta) the volume of the portion enclosed by the unit ellipsoid model of the uncertain parameter vector is
When m is>Taylor first order expansion g at nth design point at time 3 Ln The unit ellipsoid model of the (delta) and uncertain parameter vectorThe resultant part is m-dimensional spherical segment, the Taylor first-order expansion g at the nth design point Ln (delta) the volume of the portion enclosed by the unit ellipsoid model of the uncertain parameter vector is
Wherein k is a positive integer and is more than or equal to 1;
step 6022, repeating the step 6021 for a plurality of times to obtain the Taylor first-order expansion at the N design points and the volume of the part enclosed by the unit ellipsoid model of the uncertain parameter vector, and respectively recording as V f1 ，V f2 ，...，V fn ，...，V fN (ii) a Wherein, V f1 Representing Taylor first order expansion g at design point 1 L1 (delta) volume of the part enclosed by the unit ellipsoid model of the uncertain parameter vector, V f2 Representing Taylor first order expansion g at design point 2 L2 (delta) volume of the part enclosed by the unit ellipsoid model of the uncertain parameter vector, V fN Representing Taylor first order expansion g at the Nth design point LN (δ) the volume of the portion enclosed by the unit ellipsoid model of the uncertain parameter vector.
4. The method for analyzing the non-probabilistic reliability of the multi-design point for the beam structure according to claim 1, wherein: in step 602, the data processor is adopted to sum the Taylor first-order expansions at the N design points and the volume of a part enclosed by the unit ellipsoid model of the uncertain parameter vector to obtain V f The specific process is as follows:
step A, adopting the data processor to enclose partial volume V by Taylor first-order expansion at N design points and a unit ellipsoid model of uncertain parameter vectors f1 ，V f2 ，...，V fn ，...，V fN The volume enclosed by the Taylor first-order expansion and the unit ellipsoid model of the uncertain parameter vector at the design point I is denoted as V fl (ii) a Wherein l is a positive integer, l is more than or equal to 1 and less than or equal to N, and l is not equal to N;
step B, using the data processor to process the first stepThe vectors corresponding to the design points are recorded as
Step C, adopting the data processor to judge when
The sum of the volume of the portion surrounded by the Taylor first-order expansion at the first design point and the unit ellipsoid model of the uncertain parameter vector and the volume of the portion surrounded by the Taylor first-order expansion at the n design point and the unit ellipsoid model of the uncertain parameter vector is V fnl ＝V fn +V fl ；
Using the data processor to determine when
The sum of the volume of the portion surrounded by the Taylor first-order expansion at the first design point and the unit ellipsoid model of the uncertain parameter vector and the volume of the portion surrounded by the Taylor first-order expansion at the n design point and the unit ellipsoid model of the uncertain parameter vector is V fnl ＝V fn +V fl -V nl (ii) a Wherein, V nl A volume of an overlapping portion of a portion surrounded by the taylor first-order expansion and the unit ellipsoid model of the uncertain parameter vector at the ith design point and a portion surrounded by the taylor first-order expansion and the unit ellipsoid model of the uncertain parameter vector at the nth design point;
using the data processor to determine when
Then the sum of the volume of the portion enclosed by the Taylor first-order expansion at the first design point and the unit ellipsoid model of the uncertain parameter vector and the volume of the portion enclosed by the Taylor first-order expansion at the n design point and the unit ellipsoid model of the uncertain parameter vector is V fnl ＝max{V fn ,V fl }；
And D, repeating the step C for multiple times to complete the volume summation of the Taylor first-order expansion at the N design points and a part surrounded by the unit ellipsoid model of the uncertain parameter vector.
5. The method for analyzing the non-probabilistic reliability of the multi-design point for the beam structure according to claim 4, wherein: in the step C, the volume of the overlapping part of the part enclosed by the Taylor first-order expansion at the nth design point and the unit ellipsoid model of the uncertain parameter vector and the part enclosed by the Taylor first-order expansion at the ith design point and the unit ellipsoid model of the uncertain parameter vector is obtained, and the specific process is as follows:
step C01, adopting the data processor to enable the vector corresponding to the nth design point
Vector corresponding to the l-th design point
The angle between them is denoted as theta nl Vector corresponding to nth design point
The two norms of (A) are denoted as beta n Vector corresponding to the l-th design point
The two norms of (A) are denoted as beta l Using said data processor according to the formula rho nl ＝cosθ nl Obtaining an intermediate variable ρ nl ；
Step C02, adopting the data processor, when m =2, the volume of the overlapping part of the part enclosed by the Taylor first-order expansion at the nth design point and the unit ellipsoid model of the uncertain parameter vector and the part enclosed by the Taylor first-order expansion at the ith design point and the unit ellipsoid model of the uncertain parameter vector is as follows:
when m =3, the volume of the overlapping part of the part enclosed by the taylor first-order expansion and the unit ellipsoid model of the uncertain parameter vector at the nth design point and the part enclosed by the taylor first-order expansion and the unit ellipsoid model of the uncertain parameter vector at the ith design point is:
wherein x represents a first integral variable, y represents a second integral variable, and the upper bound of x is
The lower bound of x is
Upper bound of y is
Lower bound of y is
When m is more than 3 and m is even number, the volume of the overlapped part of the part enclosed by the Taylor first-order expansion and the unit ellipsoid model of the uncertain parameter vector at the nth design point and the part enclosed by the Taylor first-order expansion and the unit ellipsoid model of the uncertain parameter vector at the ith design point is as follows:
wherein k ' represents a positive integer, k ' is not less than 1, q is a natural number, and q is not less than 0 and not more than k ' -1;
when m is more than 3 and m is an odd number, the volume of the overlapped part of the part enclosed by the Taylor first-order expansion and the unit ellipsoid model of the uncertain parameter vector at the nth design point and the part enclosed by the Taylor first-order expansion and the unit ellipsoid model of the uncertain parameter vector at the ith design point is as follows:
wherein k 'represents a positive integer, k' is not less than 1, r is a natural number, r is not less than 0 and not more than 1, s is a natural number, and s is not less than 0 and not more than 1.
6. A multi-design-point non-probabilistic reliability analysis method for a beam structure according to any one of claims 1 to 5, wherein: the data processor is a computer.
Priority Applications (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN201911081340.0A CN110781606B (en) | 2019-11-07 | 2019-11-07 | Multi-design-point non-probability reliability analysis method for beam structure |
Applications Claiming Priority (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN201911081340.0A CN110781606B (en) | 2019-11-07 | 2019-11-07 | Multi-design-point non-probability reliability analysis method for beam structure |
Publications (2)
| Publication Number | Publication Date |
|---|---|
| CN110781606A CN110781606A (en) | 2020-02-11 |
| CN110781606B true CN110781606B (en) | 2023-03-24 |
Family
ID=69390134
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CN201911081340.0A Active CN110781606B (en) | 2019-11-07 | 2019-11-07 | Multi-design-point non-probability reliability analysis method for beam structure |
Country Status (1)
| Country | Link |
|---|---|
| CN (1) | CN110781606B (en) |
Families Citing this family (1)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN117556642A (en) * | 2024-01-12 | 2024-02-13 | 南昌矿机集团股份有限公司 | Cone crusher productivity reliability analysis method considering uncertainty |
Citations (3)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN105022888A (en) * | 2015-08-01 | 2015-11-04 | 西安科技大学 | Reliability evaluation method for top beam of hydraulic bracket |
| CN105930647A (en) * | 2016-04-18 | 2016-09-07 | 北京航空航天大学 | Beam structure non-probabilistic reliability solving method capable of considering multi-failure modes |
| CN110135063A (en) * | 2019-05-15 | 2019-08-16 | 西安科技大学 | A kind of non-probability failure degree calculation method of series connection truss structure system |
Family Cites Families (1)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN107526898B (en) * | 2017-09-13 | 2019-12-27 | 大连理工大学 | Variable-stiffness composite material plate-shell structure modeling analysis and reliability optimization design method |
-
2019
- 2019-11-07 CN CN201911081340.0A patent/CN110781606B/en active Active
Patent Citations (3)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN105022888A (en) * | 2015-08-01 | 2015-11-04 | 西安科技大学 | Reliability evaluation method for top beam of hydraulic bracket |
| CN105930647A (en) * | 2016-04-18 | 2016-09-07 | 北京航空航天大学 | Beam structure non-probabilistic reliability solving method capable of considering multi-failure modes |
| CN110135063A (en) * | 2019-05-15 | 2019-08-16 | 西安科技大学 | A kind of non-probability failure degree calculation method of series connection truss structure system |
Non-Patent Citations (2)
| Title |
|---|
| 基于区间有限元的吊梁非概率可靠性研究及敏感性分析;韩志杰等;《中南大学学报(自然科学版)》;20120526(第05期);全文 * |
| 基于非概率区间模型的可靠性分析与优化;韩志杰等;《中国机械工程》;20110325(第06期);全文 * |
Also Published As
| Publication number | Publication date |
|---|---|
| CN110781606A (en) | 2020-02-11 |
Similar Documents
| Publication | Publication Date | Title |
|---|---|---|
| CN105608263B (en) | A kind of adaptive processing method towards turbine blade structural life-time probability analysis | |
| Tian et al. | Low-rise gable roof buildings pressure prediction using deep neural networks | |
| CN109255173A (en) | Consider the structural realism interval computation method of bounded-but-unknown uncertainty | |
| CN110243560A (en) | A kind of temperature effect separation method in bridge deflection monitoring | |
| Lee et al. | Probabilistic fatigue life updating for railway bridges based on local inspection and repair | |
| CN110781606B (en) | Multi-design-point non-probability reliability analysis method for beam structure | |
| Swiler et al. | Calibration of a fuel relocation model in BISON | |
| CN105895089A (en) | Speech recognition method and device | |
| CN110162895A (en) | A kind of two stage high energy efficiency ship form optimization design method | |
| CN107609320B (en) | A kind of truss Multidisciplinary systems Structural shape optimization | |
| CN115455793A (en) | High-rise structure complex component stress analysis method based on multi-scale model correction | |
| Tak et al. | System-level seismic risk assessment of bridge transportation networks employing probabilistic seismic hazard analysis | |
| CN116681153A (en) | TEC forecasting method based on LSTM neural network and GNSS historical observation data | |
| Ferreira et al. | Genetic assisted selection of RBF model structures for greenhouse inside air temperature prediction | |
| CN112949944B (en) | Intelligent groundwater level prediction method and system based on space-time characteristics | |
| CN112733724B (en) | Relativity relationship verification method and device based on discrimination sample meta-digger | |
| CN112685825A (en) | Optimization method of stepwise equivalent plane method | |
| CN111798141A (en) | Method and device for evaluating technology maturity | |
| CN116757048A (en) | Structure reliability optimization method and device based on local parameterized model | |
| CN116956737A (en) | Method for inverting electron density of low ionosphere based on neural network and genetic algorithm | |
| CN114819107B (en) | Mixed data assimilation method based on deep learning | |
| CN115099066A (en) | Kriging-based power transmission tower structure wind disaster vulnerability modeling method | |
| Xu et al. | Damage identification of single-layer cylindrical latticed shells based on the model updating technique | |
| Van Biesbroeck et al. | Reference prior for Bayesian estimation of seismic fragility curves | |
| Froli et al. | Experimental static and dynamic tests on a large-scale free-form Voronoi grid shell mock-up in comparison with finite-element method results |
Legal Events
| Date | Code | Title | Description |
|---|---|---|---|
| PB01 | Publication | ||
| PB01 | Publication | ||
| SE01 | Entry into force of request for substantive examination | ||
| SE01 | Entry into force of request for substantive examination | ||
| GR01 | Patent grant | ||
| GR01 | Patent grant |