CN109284574B - Non-probability reliability analysis method for series truss structure system - Google Patents
Non-probability reliability analysis method for series truss structure system Download PDFInfo
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Abstract
The invention discloses a method for analyzing the non-probability reliability of a series truss structure system, which comprises the following steps: 1. determining a function of each failure mode of a series truss structure system; 2. establishing a multi-dimensional ellipsoid model describing an uncertainty variable; 3. acquiring a multidimensional normalization equivalent ellipsoid model of an uncertainty variable; 4. acquiring a hypersphere model of an uncertainty variable; 5. calculating the volume of the unit hypersphere model; 6. obtaining a wide limit of the total volume of the failure domains of the series truss structure system; 7. obtaining a narrow limit of the total volume of the failure domains of the series truss structure system; 8. calculating the value range of the non-probability failure degree of the series truss structure system; 9. and calculating the value range of the non-probability reliability of the series truss structure system. According to the method, the workload of non-probability reliability solving in the series truss structure system is greatly reduced through the estimation of the wide and narrow limit interval of the total volume of the failure domains of the series truss structure system, and a relatively accurate and reasonable estimation value is given.
Description
Technical Field
The invention belongs to the technical field of non-probabilistic reliability analysis of trusses, and particularly relates to a non-probabilistic reliability analysis method of a series truss structure system.
Background
The truss is a truss structure composed of a plurality of rod pieces, and is widely applied to the fields of machinery, construction, civil engineering, aerospace and the like due to the advantages of uniform internal force distribution, reduction of material consumption, light structure dead weight and the like. In the truss design and manufacturing process, uncertain information related to load, material characteristics, geometric dimensions, boundary conditions and the like often exist, and scientific consideration is needed. The reliability analysis method is one of effective ways for processing the uncertain information, and the probability reliability method is widely applied. However, in many engineering practical structural problems, sample information for determining distribution parameters or probability density functions of a probability reliability model is generally lacking, and in this context, a non-probability reliability analysis method for performing security evaluation on uncertain parameters only by knowing their boundaries or variation ranges is gradually proposed. The existing structural non-probability reliability analysis mostly aims at single failure modes, such as a first approximation method and a second approximation method, a truss structure is a typical series-structure multi-failure-mode system, theoretically, monte Carlo can provide an accurate solution of the non-probability reliability analysis of the series-truss structure system, but the solution efficiency is low due to the huge calculation workload, and therefore an effective non-probability reliability analysis method of the series-truss structure system is lacked at present.
Disclosure of Invention
The technical problem to be solved by the invention is to provide a method for analyzing the non-probability reliability of the series truss structure system aiming at the defects in the prior art, greatly reduce the workload of solving the non-probability reliability of the series truss structure system by estimating the wide and narrow limit interval of the total volume of the failure domain of the series truss structure system, give a relatively accurate and reasonable estimation value, give a structure system non-probability reliability analysis result which is more in line with the actual engineering requirements, and have wide application range and wide application prospect, thereby being convenient for popularization and use.
In order to solve the technical problems, the technical scheme adopted by the invention is as follows: a method for analyzing the non-probability reliability of a series truss structural system is characterized by comprising the following steps:
step one, determining a function of each failure mode of a series truss structure system: determining function g of each failure mode of series truss structure system by adopting truss structure failure criterion i (X) wherein i is a structural architecture failure modeI, I is the number of architectural failure modes, X is the uncertainty variable vector and X = (X =) (I =1,2.,. I, I is the number of architectural failure modes) 1 ,X 2 ,...,X m ) T M is the uncertainty variable number and m is equal to the dimension of the uncertainty variable vector X,
X l is the first uncertain variable, l is a positive integer and the value range of l is 1-m,
represents the l-th uncertainty variable X l The interval of the values is set as follows, l Xas an uncertainty variable X l The lower bound of (a) is,
as an uncertainty variable X l The upper bound of (c);
step two, establishing a multi-dimensional ellipsoid model for describing uncertainty variables: establishing a multidimensional ellipsoid model for the uncertainty variable by adopting a data processor to obtain the multidimensional ellipsoid model
Wherein, the vector X 0 Does not determine the central point vector for the multidimensional ellipsoid and
is the l uncertainty variable X l Middle point of value interval of (1), omega x Is a feature matrix of the multi-dimensional ellipsoid for determining the shape and direction of the multi-dimensional ellipsoid and
Z ll for the l-th uncertainty variable X in the determination of a multidimensional ellipsoid model according to the NATAF method l And the l uncertainty variable X l Of (2) covariance, R m A real number field in m dimensions;
step three, obtaining a multidimensional normalization equivalent ellipsoid model of the uncertainty variable, wherein the process is as follows:
step 301, normalization processing of uncertainty variable vectors: according to the formula
Obtaining an uncertainty variable normalization vector U of the uncertainty variable vector X, wherein U = (U) 1 ,U 2 ,...,U m ) T ，U l Is the l uncertainty variable X l The corresponding normalized variable is set to be the corresponding normalized variable,
is the l uncertainty variable X l Radius of the interval of (1)
Step 302, constructing a multidimensional normalization equivalent ellipsoid model of the uncertainty variable: adopting a data processor to construct a multidimensional normalization equivalent ellipsoid model of the uncertainty variable for the uncertainty variable normalization vector U
Ω u Normalizing the feature matrix of the multi-dimensional ellipsoid determined in the normalization space U by the vector U for the uncertainty variable and
to be composed of
Is an m-dimensional diagonal matrix of diagonal elements;
step four, acquiring a hypersphere model of the uncertain variable, wherein the process is as follows:
step 401, determining the multidimensional normalization vector U of the uncertainty variable in the normalization space UCharacteristic matrix omega of ellipsoid u Performing Choleskey decomposition, i.e.
Wherein L is 0 A lower triangular matrix obtained for cholesky decomposition;
step 402, converting the multidimensional normalization equivalent ellipsoid model by using a data processor to obtain a unit hypersphere model E of the uncertainty variable in the standard space delta space δ ＝{δ|δ T δ≤1,δ∈R m δ is a normalized vector of the uncertainty variable normalized vector U in the normalized space δ space and
the dimension of the standard space delta space is m, delta l To normalize variable U l A normalized variable in the normalized space δ -space;
obtaining the relation between the uncertainty variable vector X and the standardized vector delta in the standardized space delta space:
function g for failure modes of series truss structure system i (X) performing deformation processing in the standard space delta space to obtain a structural function g of a failure mode of the standard space delta space i (δ)；
Step five, according to the formula
Computational unit hypersphere model E δ Volume V of General (1) Wherein Γ (·) is a Gamma function;
step six, obtaining a wide limit of the total volume of the failure domains of the series truss structure system: in the standard space, when the ith failure mode, the structural function g i (delta) curved surface and unit hypersphere model E δ Structural function g of failure mode using standard space delta space when intersecting i (delta) upper point of curved surface and unit hypersphere model E δ Origin of a documentCalculating structural function g of failure mode of standard space delta space by using minimum distance between the two i (delta) corresponding dead zone volume V i ；
According to the formula of wide-margin method
Calculating total volume V of failure domain of series truss structure system F, total of The wide limits of (a), wherein,
step seven, obtaining a narrow limit of the total volume of the failure domains of the series truss structure system: for the volume V of failure region i The failure domain volumes corresponding to the middle I failure modes are adjusted in a sequence from large to small to obtain the adjusted failure domain volume W = (W) 1 ,W 2 ,...,W I ) T Wherein W is 1 ～W I Is a V 1 ～V I The results being ordered from large to small, i.e. W 1 >W 2 >...>W I ，W 1 ＝max(V i )，W I ＝min(V i )；
According to a narrow law formula
Calculating total volume V of failure domain of series truss structure system F, total of The narrow limit of (a) of (b),
a common failure domain which is an ith failure mode and a jth failure mode;
step eight, calculating the value range eta of the non-probability failure degree of the series truss structure system according to the narrow limit of the total volume of the failure domain of the series truss structure system s,F Wherein, in the step (A),
step nine, according to a formula eta s,R ＝1-η s,F Of a system of trusses in seriesRange η of non-probability reliability s,R 。
The method for analyzing the non-probability reliability of the series truss structure system is characterized by comprising the following steps of: the uncertainty variable number m is not less than 2;
when m =2, unit hypersphere model E δ Structural function g of failure mode in unit circle, standard space delta space i (delta) a two-dimensional arcuate surface formed by intersecting the curved surface and the unit circle, wherein the volume V of the failure zone i By area of two-dimensional arcuate surfaces
Represents, where h is the height of the two-dimensional arcuate surface;
when m =3, unit hypersphere model E δ Structural function g of failure mode of unit sphere and standard space delta space i (delta) three-dimensional segment formed by intersecting curved surface and unit sphere, wherein the volume V of failure region i By volume of three-dimensional segments
Represents, wherein h' is the height of the three-dimensional spherical segment;
when m is more than or equal to 4, the unit hypersphere model E δ Structural function g of failure mode in standard space delta space as m-dimensional hypersphere i (delta) the curved surface formed by the structure is intersected with the m-dimensional hypersphere to form an m-dimensional hypersphere segment, and the volume V of the failure domain at the moment i By volume of m-dimensional super-spherical segment
Wherein h' is the height of the m-dimensional super-spherical segment.
The method for analyzing the non-probability reliability of the series truss structure system is characterized by comprising the following steps of: function g of each failure mode of the series truss structure system i (X) =0 is called failure critical plane, and when the function g of each failure mode of the series truss structure system i (X)<At 0, the series truss structure system fails; function g of each failure mode of series truss structure system i When the (X) is more than or equal to 0, the series truss structure system is safe.
The method for analyzing the non-probability reliability of the series truss structure system is characterized by comprising the following steps of: the uncertainty variables include static load, dynamic load, length, width, modulus of elasticity.
The method for analyzing the non-probability reliability of the series truss structure system is characterized by comprising the following steps of: a common failure domain of the ith failure mode and the jth failure mode
By numerical integration.
Compared with the prior art, the invention has the following advantages:
1. the multidimensional normalization equivalent ellipsoid model of the uncertain variables is obtained by performing normalization processing on the multidimensional ellipsoid model of the uncertain variables, the problem that the characteristic matrix of the multidimensional ellipsoid model is seriously ill when the magnitude difference between the variables in the uncertain variable vector is large is solved, the precision of the calculation result in the numerical calculation process is ensured, all elements in the characteristic matrix of the multidimensional normalization equivalent ellipsoid model after normalization processing are ensured to have the same magnitude, and the multidimensional normalization equivalent ellipsoid model is convenient to popularize and use.
2. According to the method, the failure domain is estimated through the wide-narrow limit interval of the total volume of the failure domains of the series truss structure system, the failure domain is estimated through the single-failure-mode failure domain volume through the wide-limit interval of the total volume of the failure domains of the series truss structure system, the narrow-limit interval estimation of the total volume of the failure domains of the series truss structure system further considers the multi-mode common failure domain to give a narrower interval estimation range, corresponding non-probability reliability measurement indexes are defined, and the method is reliable and stable and good in using effect.
3. The method provided by the invention has the advantages that the steps are simple, the actual engineering requirements are fully considered, the non-probabilistic reliability analysis result of the structural system which is more in line with the actual engineering requirements is given, the application range is wide, the application prospect is wide, the workload of calculating the failure domain volume of the series truss structural system is greatly simplified, the defect that the prior art can only carry out non-probabilistic reliability analysis on the structure under a single failure mode is effectively overcome, the range of the structural non-probabilistic reliability analysis method is expanded, the method has very important significance on the reliability analysis of the structural system, and the method is convenient to popularize and use.
In conclusion, the method greatly reduces the workload of the non-probability reliability solution of the series truss structure system through the estimation of the wide and narrow limit interval of the total volume of the failure domain of the series truss structure system, provides a relatively accurate and reasonable estimation value, provides a structure system non-probability reliability analysis result which is more in line with the actual engineering requirement, and has the advantages of wide application range, wide application prospect and convenience in popularization and use.
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 structural diagram of the tandem truss structural system in this embodiment.
FIG. 3 is a structural function g of the failure mode of the standard space delta space in this embodiment i (delta) curved surface and unit hypersphere model E δ Schematic intersection.
Detailed Description
As shown in fig. 1 to fig. 3, the method for analyzing the non-probabilistic reliability of the series truss structure system of the present invention includes the following steps:
step one, determining a function of each failure mode of a series truss structure system: function g for determining failure modes of series truss structure system by adopting truss structure failure criterion i (X), wherein I is the number of architectural failure modes and I =1, 2.. I, I is the number of architectural failure modes, X is the uncertainty variable vector and X = (X) 1 ,X 2 ,...,X m ) T M is an uncertainty variable number and m is equal to the dimension of the uncertainty variable vector X,
X l is the l uncertainty variable, l isA positive integer and a value range of l is 1-m,
representing the l-th uncertainty variable X l The interval of the values is set as follows, l Xas an uncertainty variable X l The lower bound of (a) is,
as an uncertainty variable X l An upper bound of (c);
in this embodiment, the uncertainty variables include static load, dynamic load, length, width, and modulus of elasticity.
In this embodiment, the function g of each failure mode of the series truss structure system i (X) =0 is called failure critical plane, and when the functional function g of each failure mode of the series truss structure system i (X)<At 0, the series truss structure system fails; function g of failure modes of serial truss structure system i When the (X) is more than or equal to 0, the series truss structure system is safe.
Step two, establishing a multi-dimensional ellipsoid model describing an uncertainty variable: establishing a multidimensional ellipsoid model for the uncertainty variable by adopting a data processor to obtain the multidimensional ellipsoid model
Wherein, the vector X 0 Does not determine the central point vector for the multidimensional ellipsoid and
is the l uncertainty variable X l Middle point of value interval of (1), omega x Is a feature matrix of the multi-dimensional ellipsoid for determining the shape and direction of the multi-dimensional ellipsoid and
Z ll for determining the l-th uncertainty variable X of the multidimensional ellipsoid model according to the NATAF method l And the l uncertainty variable X l Of (2) covariance, R m A real number field of m dimensions;
step three, acquiring a multidimensional normalization equivalent ellipsoid model of the uncertainty variable, wherein the process is as follows:
step 301, normalization processing of uncertainty variable vectors: according to the formula
Obtaining an uncertainty variable normalized vector U of the uncertainty variable vector X, wherein U = (U) 1 ,U 2 ,...,U m ) T ，U l Is the l uncertainty variable X l The corresponding normalization variable(s) is (are),
is the l uncertainty variable X l Has a radius of section of
Step 302, constructing a multidimensional normalization equivalent ellipsoid model of the uncertainty variable: adopting a data processor to construct a multidimensional normalization equivalent ellipsoid model of the uncertainty variable for the uncertainty variable normalization vector U
Ω u Normalizing the feature matrix of the multi-dimensional ellipsoid determined in the normalization space U by the vector U for the uncertainty variable and
to be composed of
Is an m-dimensional diagonal matrix of diagonal elements;
it should be noted that the multidimensional normalization equivalent ellipsoid model of the uncertainty variable is obtained by performing normalization processing on the multidimensional ellipsoid model of the uncertainty variable, so that the problem that the feature matrix of the multidimensional ellipsoid model is seriously ill-conditioned when magnitude differences among variables in an uncertainty variable vector are large is solved, the accuracy of a calculation result in a numerical calculation process is ensured, all elements in the feature matrix of the multidimensional normalization equivalent ellipsoid model after the normalization processing are ensured to have the same magnitude, and the method is convenient to popularize and use.
Step four, acquiring a hypersphere model of an uncertain variable, wherein the process is as follows:
step 401, determining a feature matrix omega of the multidimensional ellipsoid in the normalization space U of the normalization vector U of the uncertainty variable u Performing Choleskey decomposition, i.e.
Wherein L is 0 A lower triangular matrix obtained by Choleskey decomposition;
step 402, converting the multidimensional normalization equivalent ellipsoid model by using a data processor to obtain a unit hypersphere model E of the uncertainty variable in the standard space delta space δ ＝{δ|δ T δ≤1,δ∈R m δ is a normalized vector of the uncertainty variable normalized vector U in the normalized space δ space and
the dimension of the standard space delta space is m, delta l To normalize the variable U l A normalized variable in the normalized space δ space;
obtaining the relation between the uncertainty variable vector X and the standardized vector delta in the standardized space delta space:
function g for failure modes of series truss structure system i (X) performing deformation processing in the standard space delta space to obtain a structural function g of a failure mode of the standard space delta space i (δ)；
Step five, according to the publicIs of the formula
Computational unit hypersphere model E δ Volume V of General assembly Wherein Γ (·) is a Gamma function;
step six, obtaining a wide limit of the total volume of the failure domains of the series truss structure system: in the standard space, when the ith failure mode, the structural function g i (delta) curved surface and unit hypersphere model E δ Structural function g of failure mode using standard space delta space when intersecting i (delta) upper point of curved surface and unit hypersphere model E δ Calculating structural function g of failure mode of standard space delta space by distance minimum value between origins i (delta) corresponding dead zone volume V i ；
According to the formula of wide-margin method
Calculating total volume V of failure domain of series truss structure system F, total of The wide limits of (a), wherein,
in this embodiment, the uncertainty variable number m is not less than 2;
when m =2, unit hypersphere model E δ Structural function g of failure mode in unit circle, standard space delta space i (delta) a two-dimensional arcuate surface formed by intersecting the curved surface and the unit circle, wherein the volume V of the failure region i By area of two-dimensional arcuate surface
Where h is the height of the two-dimensional arcuate surface;
when m =3, the unit hypersphere model E δ Structural function g of failure mode of unit sphere and standard space delta space i (delta) three-dimensional segment formed by intersecting curved surface and unit sphere, wherein the volume V of failure region i By bodies of three-dimensional segmentsProduct of large quantities
Represents, wherein h' is the height of the three-dimensional spherical segment;
when m is more than or equal to 4, the unit hypersphere model E δ Structural function g of failure mode in standard space delta space as m-dimensional hypersphere i (delta) the curved surface formed by the structure is intersected with the m-dimensional hypersphere to form an m-dimensional hypersphere segment, and the volume V of the failure domain at the moment i By volume of m-dimensional super-spherical segment
Wherein h "is the height of the m-dimensional super-spherical segment.
Step seven, obtaining a narrow limit of the total volume of the failure domains of the series truss structure system: for the volume V of failure region i The failure domain volumes corresponding to the middle I failure modes are adjusted in a sequence from large to small to obtain the adjusted failure domain volume W = (W) 1 ,W 2 ,...,W I ) T Wherein W is 1 ～W I Is a V 1 ～V I The results being ordered from large to small, i.e. W 1 >W 2 >...>W I ，W 1 ＝max(V i )，W I ＝min(V i )；
According to the narrow-bound law formula
Calculating total volume V of failure domain of series truss structure system F, total of The narrow limit of (a) of (b),
a common failure domain which is an ith failure mode and a jth failure mode;
in this embodiment, the common failure domain of the ith failure mode and the jth failure mode
Obtained by numerical integration.
It should be noted that the estimation of the wide-narrow limit interval of the total volume of the failure domains of the series truss structure system is performed, wherein the estimation of the wide limit interval of the total volume of the failure domains of the series truss structure system estimates the failure domains by the volume of the single failure mode failure domain, and the estimation of the narrow limit interval of the total volume of the failure domains of the series truss structure system further considers the multi-mode common failure domain to give a narrower interval estimation range, so that a corresponding non-probability reliability measurement index is defined, and the method is reliable, stable and good in use effect.
Step eight, calculating the value range eta of the non-probability failure degree of the series truss structure system according to the narrow limit of the total volume of the failure domain of the series truss structure system s,F Wherein, in the process,
step nine, according to a formula eta s,R ＝1-η s,F Calculating the value range eta of the non-probability reliability of the series truss structure system s,R 。
As shown in fig. 2, in this embodiment, taking a planar 5-bar truss structure as an example, three uncertain variables X exist in the 5-bar truss structure 1 、X 2 And X 3 Three uncertain variables X 1 、X 2 And X 3 All are loads, the allowable stress of the No. 1 rod is 240kN, the allowable stress of the No. 2 rod is 200kN, the allowable stress of the No. 3 rod is 280kN, the allowable stress of the No. 4 rod is 280kN, the allowable stress of the No. 5 rod is 180kN, and three uncertain variables X 1 、X 2 And X 3 Correlation coefficient of
And
all take 0.2, three uncertain variables X 1 、X 2 And X 3 The table of values of (a) is shown in table 1.
TABLE 1
Determining a functional function g of a 5-bar failure mode in MATLAB using a truss structure failure criterion i (X) wherein, in the above-mentioned formula,
functional function of failure mode for stick # 1
Functional function of failure mode for number 2 rod
Functional function of failure mode for No. 3 Bar
Functional function of failure mode for number 4 rod
Functional function of failure mode for No. 5 rod
Determining a feature matrix of a multi-dimensional ellipsoid of an uncertainty variable vector X according to a NATAF method
Then multi-dimensional ellipsoid model
In order to effectively overcome the interference caused by the pathology of the characteristic matrix, the multidimensional ellipsoid model is normalized to obtain
Feature matrix omega in normalization space U for normalization vector U of uncertainty variable u Performing Choleskey decomposition, i.e.
Substituting the decomposed matrix into a multidimensional normalized equivalent ellipsoid model to obtain
Thus, the unit hypersphere model of the uncertainty variable in the normalized space δ space can be expressed as
Carrying out deformation processing on the uncertain variable normalization vector U in a standard space delta space to obtain
Function g for failure modes of series truss structure system i (X) carrying out deformation processing in the standard space delta space to obtain the structural function of the failure mode of the standard space delta space No. 1 rod
Structural function of failure mode of standard space delta space No. 2 rod
Structural function of failure mode of standard space delta space No. 3 rod
Structural function of failure mode of standard space delta space No. 4 rod
Structural function of failure mode of standard space delta space No. 5 rod
Unit hypersphere model E δ Volume of (2)
In this example, m is 3, and the unit hyper-sphere model E δ Structural function g of failure mode in standard space delta space as unit sphere i (delta) three-dimensional segment formed by intersecting unit sphere with plane formed by the unit sphere, wherein volume V of failure region i By volume of three-dimensional segments
Represents, wherein h' is the height of the three-dimensional segment;
structural function g of failure mode of standard space delta space i (delta) curved surface and unit hypersphere model E δ Performing intersection by using structural function g of failure mode in standard space delta space i (delta) upper point of curved surface and unit hypersphere model E δ Calculating structural function g of failure mode of standard space delta space by distance minimum value between origins i (delta) corresponding dead zone volume V i See table 2.
TABLE 2
| Failure mode | Radius of spherical segment r | Segment height h' | Volume of |
| g | |||
| δ1 | 1 | 0.1458 | 0.0635 |
| |
1 | 0.2648 | 0.2007 |
| |
1 | 0.5254 | 0.7150 |
| |
1 | 0.3893 | 0.4143 |
| |
1 | 0.3656 | 0.3686 |
According to the formula of wide-margin method
Calculating total volume V of failure domain of series truss structure system F, total of Wide margin of
Obtaining total volume V of failure domain F, total of The wide limits of (2) are: v is more than or equal to 0.7150 F, total of ≤1.7621。
For the volume V of failure region i The volumes of failure domains corresponding to the middle I failure modes are arranged from large to smallAdjusting the sequence to obtain the adjusted failure domain volume W = (W) 1 ,W 2 ,...,W I ) T ＝(0.7150,0.4143,0.3686,0.2007,0.0635) T According to the narrow-bound equation
Calculating total volume V of failure domain of series truss structure system F, total of By the total volume V of the dead zone F, total of The narrow limits of (c) are: 0.8654 is less than or equal to V F, total of ≤0.9115。
Calculating the value range of the non-probability failure degree of the series truss structure system according to the narrow limit of the total volume of the failure domains of the series truss structure system
Get eta more than or equal to 19.35 percent s,F Less than or equal to 20.38 percent, therefore, in the embodiment, the value range eta of the non-probability reliability of the series truss structure system s,R Comprises the following steps: eta of 79.62 percent or less s,R ≤80.65％。
When the method is used, the actual requirements of the engineering are fully considered, the non-probability reliability analysis result of the structural system which is more in line with the actual engineering requirements is given, the application range is wide, the application prospect is wide, the workload of calculating the failure domain volume of the series truss structural system is greatly simplified, the defect that the prior art can only carry out non-probability reliability analysis on the structure under the single failure mode is effectively overcome, the range of the structural non-probability reliability analysis method is expanded, the method has very important significance on the reliability analysis of the structural system, 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 (4)
1. A method for analyzing the non-probability reliability of a series truss structure system is characterized by comprising the following steps:
step one, determining a function of each failure mode of a series truss structure system: determining function g of each failure mode of series truss structure system by adopting truss structure failure criterion i (X), where I is the number of architectural failure modes and I =1, 2.. Wherein I, I is the number of architectural failure modes, X is the uncertainty variable vector and X = (X) 1 ,X 2 ,...,X m ) T M is an uncertainty variable number and m is equal to the dimension of the uncertainty variable vector X,
X l is the first uncertain variable, l is a positive integer and the value range of l is 1-m,
represents the l-th uncertainty variable X l The interval of the values is selected from the group, l Xas an uncertainty variable X l The lower bound of (a) is,
as an uncertainty variable X l An upper bound of (c);
step two, establishing a multi-dimensional ellipsoid model for describing uncertainty variables: establishing a multidimensional ellipsoid model for the uncertainty variable by adopting a data processor to obtain the multidimensional ellipsoid model
Wherein, the vector X 0 Does not determine the central point vector for the multidimensional ellipsoid and
is the l uncertainty variable X l Middle point of value interval of (1), omega x Is a feature matrix of the multi-dimensional ellipsoid for determining the shape and direction of the multi-dimensional ellipsoid and
Z ll for determining the l-th uncertainty variable X of the multidimensional ellipsoid model according to the NATAF method l And the l uncertainty variable X l Of (2) covariance, R m A real number field of m dimensions;
step three, acquiring a multidimensional normalization equivalent ellipsoid model of the uncertainty variable, wherein the process is as follows:
step 301, normalization processing of uncertainty variable vectors: according to the formula
Obtaining an uncertainty variable normalization vector U of the uncertainty variable vector X, wherein U = (U) 1 ,U 2 ,...,U m ) T ，U l Is the l uncertainty variable X l The corresponding normalization variable(s) is (are),
is the l uncertainty variable X l Radius of the interval of (1)
Step 302, constructing a multidimensional normalization equivalent ellipsoid model of the uncertainty variable: constructing a multidimensional normalization equivalent ellipsoid model of the uncertainty variable for the uncertainty variable normalization vector U by adopting a data processor
Ω u Normalizing the feature matrix of the multi-dimensional ellipsoid determined in the normalization space U by the vector U for the uncertainty variable and
to be composed of
Is an m-dimensional diagonal matrix of diagonal elements;
step four, acquiring a hypersphere model of an uncertain variable, wherein the process is as follows:
step 401, determining a feature matrix omega of the multidimensional ellipsoid in the normalization space U of the normalization vector U of the uncertainty variable u Performing Choleskey decomposition, i.e.
Wherein L is 0 A lower triangular matrix obtained by Choleskey decomposition;
step 402, converting the multidimensional normalized equivalent ellipsoid model by using a data processor to obtain a unit hypersphere model E of the uncertainty variable in a standard space delta space δ ＝{δ|δ T δ≤1,δ∈R m δ is a normalized vector of the uncertainty variable normalized vector U in the normalized space δ space and
the dimension of the standard space delta space is m, delta l To normalize the variable U l A normalized variable in the normalized space δ -space;
obtaining the relation between the uncertainty variable vector X and the standardized vector delta in the standardized space delta space:
function g for failure modes of series truss structure system i (X) performing deformation processing in the standard space delta space to obtain a structural function g of a failure mode of the standard space delta space i (δ)；
Step five, according to the formula
Computational unit hypersphere model E δ Volume V of General assembly Wherein Γ (·) is a Gamma function;
step six, obtaining a wide limit of the total volume of the failure domains of the series truss structure system: in standard space, when the structural function g of the ith failure mode i (delta) curved surface and unit hypersphere model E δ Structural function g of failure mode using standard space delta space when intersecting i (delta) upper point of curved surface and unit hypersphere model E δ Calculating structural function g of failure mode of standard space delta space by distance minimum value between origins i (delta) corresponding dead zone volume V i ；
According to the formula of wide-margin method
Calculating total volume V of failure domain of series truss structure system F, total of The wide limits of (a), wherein,
step seven, obtaining a narrow limit of the total volume of the failure domains of the series truss structure system: to the volume V of failure region i The failure domain volumes corresponding to the middle I failure modes are adjusted in a sequence from large to small to obtain the adjusted failure domain volume W = (W) 1 ,W 2 ,...,W I ) T Wherein W is 1 ～W I Is a V 1 ～V I The results being ordered from large to small, i.e. W 1 ＞W 2 ＞...＞W I ，W 1 ＝max(V i )，W I ＝min(V i )；
According to a narrow law formula
Calculating total volume V of failure domain of series truss structure system F, total of The narrow limit of (a) of (b),
the ith failure mode and the jth failure modeA co-failure domain of (a);
step eight, calculating the value range eta of the non-probability failure degree of the series truss structure system according to the narrow limit of the total volume of the failure domain of the series truss structure system s,F Wherein, in the process,
step nine, according to a formula eta s,R ＝1-η s,F Calculating the value range eta of the non-probability reliability of the series truss structure system s,R ；
The uncertainty variable number m is not less than 2;
when m =2, unit hypersphere model E δ Structural function g of failure mode of unit circle, standard space delta space i (delta) a two-dimensional arcuate surface formed by intersecting the curved surface and the unit circle, wherein the volume V of the failure region i By area of two-dimensional arcuate surface
Represents, where h is the height of the two-dimensional arcuate surface;
when m =3, the unit hypersphere model E δ Structural function g of failure mode of unit sphere and standard space delta space i (delta) three-dimensional segment formed by intersecting the curved surface formed by the curved surface and the unit sphere, wherein the volume V of the failure region i By volume of three-dimensional segments
Represents, wherein h' is the height of the three-dimensional segment;
when m is greater than or equal to 4, the unit hypersphere model E δ Structural function g of failure mode in standard space delta space as m-dimensional hypersphere i (delta) the curved surface formed by the structure is intersected with the m-dimensional hypersphere to form an m-dimensional hypersphere segment, and the volume V of the failure domain at the moment i By volume of m-dimensional super-spherical segment
Is represented by, wherein h' is mThe height of the super-globular segment is maintained.
2. The method for non-probabilistic reliability analysis of a tandem truss structural system according to claim 1, wherein: function g of each failure mode of the series truss structure system i (X) =0 is called failure critical plane, and when the functional function g of each failure mode of the series truss structure system i (X) < 0, the series truss structure system fails; function g of failure modes of serial truss structure system i When (X) is more than or equal to 0, the series truss structure system is safe.
3. The method for non-probabilistic reliability analysis of a tandem truss architecture in accordance with claim 1, wherein: the uncertainty variables include static load, dynamic load, length, width, modulus of elasticity.
4. The method for non-probabilistic reliability analysis of a tandem truss structural system according to claim 1, wherein: a common failure domain for the ith and jth failure modes
By numerical integration.
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