CN105930618B - A kind of mixing fatigue reliability optimization method for composite laminated plate - Google Patents
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Abstract
The invention discloses a kind of mixing fatigue reliability optimization method for composite laminated plate.This method is primarily based on the non-probabilistic method of small sample, takes into full account multi-source uncertain factor existing for the strength of materials, used load etc., integrates irregular sample data using grey topology degree, probes into its potential rule, and provide reasonable quantized result；Secondly, by establishing the Residual Strength Model under fatigue load effect, uncertain parameter is incorporated into model, develops Multidisciplinary systems method for solving；Again, consider the mixed form of stochastic variable and interval variable, mixing reliability index is solved using the Multidisciplinary systems method for solving and join probability reliability method for solving established, finally, composite laminated plate thickness optimization design is carried out as constraint to mix reliability index.Based on this, the poor information such as large scale structure, a small number of strength optimum designs in the case of can be achieved, it is ensured that design compromise between security itself and economy.
Description
Technical Field
The invention relates to the technical field of reliability index solving of a composite laminated plate structure, in particular to establishment and formulation of a method for reasonably characterizing a fatigue mixed reliability model of a laminated structure and accurately solving the mixed reliability by simultaneously considering the combined action of a random variable and an interval variable under the condition of incomplete probability information.
Background
The composite material has excellent mechanical properties, becomes a hot problem for the research of scholars at home and abroad, and aiming at the typical structure of the composite material, the laminated plate is widely applied to various aspects of aviation, aerospace, ships, medical treatment and the like, and the depth and the breadth of the technical research of the composite material become important benchmarks of the national scientific and technological development, so that the mechanical property analysis and the design technical research of the laminated plate structure have important theoretical significance and engineering practical value.
However, due to the characteristics of anisotropy and the like of the engineering laminated plate structure, and in a complex service environment, the influence of various uncertain sources exists, the uncertainty of plate structure damage is aggravated by the uncontrollable property of a processing technology, the nonuniformity of material properties, the measurement ambiguity of a geometric structure, the randomness of external load and the like, the problem of incomplete probability information occurs, and the problem of solving by using a single reliability solving theory is difficult; therefore, how to solve the problem of solving the mixed reliability index under the combined action of the random variable and the interval variable becomes a key. Therefore, the single traditional structure reliability analysis and solution method is no longer applicable. By combining the situations, the mixed reliability index solving method has engineering application value under the fatigue action of the laminated plate structure.
Currently, uncertainty analysis and mixed reliability solving research on laminated plate structures by scholars and engineering technicians at home and abroad mainly focuses on two aspects: (1) Structural uncertainty influence envelopment based on a probability statistic theory and a safety coefficient equation; and (2) solving by considering the reliability of the structural single type variable. The work has a certain engineering practical value, but the influence degree of the refined measurement of uncertain factors on the structural reliability and the influence of the combined action of random variables and interval variables are ignored, so that the theoretical engineering practical process is greatly limited.
In actual engineering, especially in the case that a complex structure is always under the condition of incomplete probability information, the establishment of a mixed uncertainty characterization technology and a structure mixed reliability solving and evaluating technology based on the combination of probability-non-probability theory has obvious practical significance.
Disclosure of Invention
The technical problem to be solved by the invention is as follows: the method overcomes the defects of the prior art, and provides the method for optimizing the hybrid reliability under the fatigue failure mode aiming at the composite laminated plate structure. The invention fully considers the ubiquitous uncertainty factor in the practical engineering problem, constructs the mathematical model capable of reasonably representing the structure residual strength under the action of the fatigue failure mode, and provides the mixed reliability index solving method considering the simultaneous action of the random variable and the interval variable, so that the obtained result is more consistent with the real situation, and the engineering applicability is stronger.
The technical scheme adopted by the invention is as follows: a hybrid fatigue reliability optimization method for composite material laminated plates comprises the following implementation steps:
the first step is as follows: according to the material properties of the laminate structure: structural strength R, external load S, cycle number N, fatigue life N, and an explicit expression of the extreme state equation for introducing the residual strength model to deduce the structure, and FIG. 2 gives a geometric explanation of the residual strength model, that is:
wherein R is a structural strength interval variable, S is an external load random process, N is a cycle number, N is a fatigue life, and c is a parameter obtained through test data;
the second step is that: using the interval vector x ∈ x I = R reasonable characterization poor information, structural uncertainty under the condition of few data, and random vector y ∈ y I = S describes the random variable then:
x U ＝R U ＝R c +R r
x L ＝R L ＝R c -R r
S∈N(μ,σ)
the structural strength R can be expressed as an interval variable, an upper standard U represents a value upper bound of a parameter, an upper standard L represents a value lower bound of the parameter, an upper standard c represents a central value, an upper standard R represents a radius, the external load S can be expressed as a random variable obeying normal distribution, mu is a mean value, and sigma is a variance;
the third step: substituting the interval variable and the random variable in the second step into the structure extreme state equation in the first step, introducing a non-probability interval process theory, establishing a fatigue failure probability-mixed reliability extreme state equation, and realizing the explicit expression of an extreme state function; namely:
wherein M is a laminated plate structure fatigue failure limit state function, R is an interval variable, S is a random variable obeying normal distribution, and N, N and c are test parameters;
the fourth step: combining the stress-intensity interference model, fig. 3 gives a geometric explanation of the non-probabilistic reliability solving method, and according to the extreme state equation established in the third step, the fatigue non-probabilistic reliability is solved:
wherein R' set，i For fatigue non-probability reliability, R is structural strength, S is external load, and N, N and c are test parameters;
the fifth step: combining the probability theory, the non-probability theory and the interval process model, and providing the fatigue failure mixed reliability calculation index of the laminated plate structure:
wherein R is set Representing the mixed reliability, eta, of structural fatigue i (S) represents the non-probabilistic fatigue reliability of the structure, f (S) is a probability density function of a random variable S,for lower bound of integration,. Psi i (R) is the upper integral bound, each being a function of R for an interval variable, i representing the number of events;
and a sixth step: mixed reliability R based on structural fatigue failure set And as constraint conditions, taking the weight G of the laminated plate as an optimization target, taking the thickness d of the laminated plate as a design variable, developing fatigue hybrid reliability optimization design facing the laminated plate structure, and realizing a complete optimization iterative process by a particle swarm intelligent algorithm.
Compared with the prior art, the invention has the advantages that: the invention provides a novel method for solving the mixed reliability index of the composite laminated plate structure under the mixed action of random variables and interval variables, and makes up and perfects the limitations of the reliability design method of the traditional probability theory and safety coefficient method under the condition of incomplete probability information. The established probability-mixed reliability model provides a new solution for the reliability solving problem under the combined action of random variables and interval variables, on one hand, the dependence on sample information is reduced, on the other hand, the probability and non-probability reliability solving theory are combined, a reasonable mixed reliability solving model is established, and the structural reliability solving precision and the rationality are improved.
Drawings
FIG. 1 is a flow chart of a reliability optimization method of the present invention for laminate structure considering probabilistic-non-probabilistic co-action;
FIG. 2 is a schematic diagram of a fatigue residual strength model for a laminated plate structure according to the present invention;
FIG. 3 is a schematic diagram of solving a reliability index based on a non-probability theory, wherein FIG. 3 (a) is a two-dimensional non-probability reliability model, and FIG. 3 (b) is a three-dimensional non-probability reliability model;
FIG. 4 is a schematic diagram illustrating different analyses of the non-probabilistic reliability index for different values of the remaining structural strength according to the present invention;
fig. 5 is a schematic diagram of a geometric model of a proposed composite material laminated plate structure.
Detailed Description
The invention is further described with reference to the following figures and detailed description.
As shown in fig. 1, the present invention provides a hybrid fatigue reliability optimization method for a composite laminate, including the steps of:
(1) Using the interval vector x ∈ x I = R reasonable representation of poor information, structural uncertainty under the condition of few data, and random vector y belongs to y I = S describes the random variable then:
x U ＝R U ＝R c +R r
x L ＝R L ＝R c -R r
S∈N(μ,σ)
wherein, structural strength R can be expressed as interval variable, superscript U represents the value upper bound of the parameter, superscript L represents the value lower bound of the parameter, superscript c represents the central value, superscript R represents the radius, external load S can be expressed as the random variable obeying normal distribution, mu is the mean value, sigma is the variance, uncertainty parameter vector x can be expressed as:
x＝[x L ,x U ]＝[x c -x r ,x c +x r ]
＝x c +x r [-1,1]
＝x c +x r ×e
wherein e ∈ xi- 2 ，Ξ 2 Is defined as all elements contained in [ -1,1 [ ]]The 2-dimensional vector set in the interior, the symbol "x" is defined as the operator for multiplying the corresponding elements of two vectors, the product is still a vector with dimension 2, for the interval variable R, x L ＝R L ,x U ＝R U ,x c ＝R c ,x r ＝R r 。
(2) Depending on the material properties of the laminate structure: structural strength R, external load S, cycle number N, fatigue life N, and introducing an explicit expression of a residual strength model to deduce the extreme state equation of the structure. Namely:
wherein R is a structural strength interval variable, S is an external load random process, N is a cycle number, N is a fatigue life, and c is a parameter obtained through test data;
(3) Substituting the interval variable and the random variable in the second step into the structure extreme state equation in the first step, introducing a non-probability interval process theory, establishing a fatigue failure probability-mixed reliability extreme state equation, and realizing the explicit expression of an extreme state function; namely:
wherein M is a laminated plate structure fatigue failure limit state function, R is an interval variable, S is a random variable obeying normal distribution, and N, N and c are test parameters; for the basic interval variable x i (i = 1) standard transformation:
x i ＝x i c +x i r ×e
wherein x i (i = 1) corresponding to the interval variable R; for a structural fatigue failure limit state equation, applying interval mathematics and standardization processing means to explicitly characterize, and converting a working coordinate system into a standard coordinate system, wherein e belongs to (-1, 1), and the limit state equation is as follows:
by standardizationThe processing method can know that R = R c +R r X e, thenSubstituting the formula, N, c, N are test parameters, so thatObtaining a normalized limit state equation:
M＝G(e,S,k)＝(1-k)×(R c +R r ×e)-(k+1)S
(4) One is taken for realizing the random variable S, a non-probability theory is combined with an interval process model by combining a normalized extreme state equation, and as shown in figure 4, a fatigue failure mode non-probability reliability calculation index for a composite laminated plate structure is provided:
i when random variableWhen the utility model is used, the water is discharged,
R' set,1 ＝η 1 (S)＝η 1 (G(e,S,k)>0)＝1
II when random variableWhen the temperature of the water is higher than the set temperature,
III when random variableWhen the temperature of the water is higher than the set temperature,
R' set,3 ＝η 3 (S)＝η 3 (G(e,S,k)>0)＝0
wherein eta i (S) represents the non-probabilistic reliability of the normalized extreme state equation, G (e, S, k) is the normalized fatigue limit state equation, e represents the normalized interval variable R, S represents the random variable of the external load, k represents the test parameterNumber, R c Is the central value of the interval variable R, R r Is the radius of the interval variable R;
(5) And (3) comprehensively considering probability theory and non-probability theory, and providing a fatigue failure mixed reliability calculation index of the laminated plate structure:
wherein R is set Indicating the mixed reliability, η, of structural fatigue i (S) represents the non-probabilistic fatigue reliability of the structure, f (S) is a probability density function of a random variable S,for lower bound of integration,. Psi i (R) is the upper integral bound, each being a function of R for the interval variable, i representing the number of events;
it is known that the random variable S follows a normal distribution with mean μ and variance σ, and its probability density function is:
substituting the probability density function and the non-probability reliability into a fatigue mixed reliability calculation formula to obtain:
wherein R is set,i (i =1,2, \8230;, n) represents the non-probabilistic reliability of laminate fatigue for the i-th case;
(6) Mixed reliability R based on structural fatigue failure set And as constraint conditions, taking the weight G of the laminated plate as an optimization target, taking the plate thickness t as a design variable, developing fatigue hybrid reliability optimization design for the laminated plate structure, and realizing a complete optimization iterative process by a particle swarm intelligent algorithm. The optimized formula is described as:
wherein,design allowance, R, for reliability set In order to obtain the fatigue mixed reliability, t is the plate thickness, and in the iterative process, the geometric dimension of the laminated plate is modified to obtain the fatigue mixed reliability R set And reliability allowable valueAnd comparing until the reliability requirement is met, and giving an optimized design result.
The particle swarm optimization is an intelligent global optimization solution technology, each particle represents a potential optimization solution, and the position of each particle represents a certain direction vector. Initially the population will be randomly assigned initial positions and initial velocities that will accelerate the update along the previous optimal position, while the determination of the global optimal point will rely on the following two equations:
wherein i represents the ith particle, k represents the kth iteration process, and v i Indicates the update speed, x, of the ith particle i Is the current position of the ith particle.Andwhich is indicative of an acceleration constant, is,andis in [0,1 ]]Random numbers, w, that are uniformly distributed within the interval * Representing the weight coefficient, pbest i And gbest i Respectively, represent individual and population based optimal locations. The iteration process is completed depending on a preset value of a minimum error or the number of iteration steps, which determines the accuracy of the calculation result.
Example (b):
in order to more fully understand the features of the invention and its practical applicability to engineering, the invention performs a fatigue hybrid reliability solution based on residual strength theory for a laminate structure constructed as shown in fig. 5. The laminated plate structure bears cyclic load S epsilon N (317, 12.6), the fatigue life N =118000, the static strength R epsilon [876,896] of the laminated plate structure, the plate length l =100mm, the plate width w =40mm and the plate thickness t =5mm, the fatigue reliability, the comprehensive geometric dimension and the load condition when the cyclic frequency N =70000 under the working condition are considered, and the hybrid fatigue reliability can be found to be 0.9786 by utilizing the established hybrid fatigue reliability solving method.
Based on the obtained mixed fatigue reliability index, the allowable reliability is taken as the constraint to develop the optimization design, and finally the allowable reliability is metGiven the optimisation of the geometry of the laminate structure under the constraints t =5.5mm, it can be seen that the weight of the laminate increases with increasing reliability.
In summary, the invention provides a method for solving fatigue hybrid reliability under the combined action of random variables and interval variables in consideration of a composite laminated plate structure. Firstly, according to the specific characteristics of the laminated plate structure material, the load and the like, and by combining with the residual strength theory, obtaining the extreme state function under the fatigue failure mode; secondly, introducing information such as random variables, interval variables and the like into a residual strength model to establish a fatigue mixed reliability solving equation, and realizing mixed fatigue reliability solving considering the coexistence of the random variables and the interval variables; and finally, based on the established mixed fatigue reliability solving method, the mixed fatigue reliability is used as constraint, the geometric parameters are used as variables, and the particle swarm optimization algorithm is used for optimization design.
The above are only the specific steps of the present invention, and the protection scope of the present invention is not limited in any way; the method can be expanded and applied to the field of reliability solving of multiple failure modes of the structure, and all technical schemes formed by adopting equivalent transformation or equivalent replacement fall within the protection scope of the invention.
The invention has not been described in detail and is within the knowledge of a person skilled in the art.
Claims (6)
1. A hybrid fatigue reliability optimization method for composite laminated plates is characterized by comprising the following implementation steps:
the first step is as follows: according to the material properties of the laminate structure: structural strength R, external load S, cycle number N, fatigue life N, and an explicit expression of the extreme state equation of the derived structure by introducing a residual strength model, namely:
wherein R is a structural strength interval variable, S is an external load random process, N is a cycle number, N is a fatigue life, and c is a parameter obtained through test data;
the second step is that: using the interval vector x ∈ x I = R reasonable representation of poor information, structural uncertainty under the condition of few data, and random vector y belongs to y I = S describes a random variable, then:
x U ＝R U ＝R c +R r
x L ＝R L ＝R c -R r
S∈N(μ,σ)
the structural strength R can be expressed as an interval variable, an upper standard U represents a value upper bound of a parameter, an upper standard L represents a value lower bound of the parameter, an upper standard c represents a central value, an upper standard R represents a radius, the external load S can be expressed as a random variable obeying normal distribution, mu is a mean value, and sigma is a variance;
the third step: substituting the interval variable and the random variable in the second step into the structure extreme state equation in the first step, introducing a non-probability interval process theory, establishing a fatigue failure probability-mixed reliability extreme state equation, and realizing the explicit expression of an extreme state function; namely:
wherein M is a laminated plate structure fatigue failure limit state function, R is an interval variable, S is a random variable obeying normal distribution, and N, N and c are test parameters;
the fourth step: and (3) solving the fatigue non-probability reliability by combining a stress-intensity interference model and applying a non-probability reliability solving method according to the extreme state equation established in the third step:
wherein R' set,i The fatigue non-probability reliability is shown, R is the structural strength, S is the external load, and N, N and c are test parameters;
the fifth step: combining probability theory, non-probability theory and interval process model, providing fatigue failure mixed reliability calculation index of the laminated plate structure:
wherein R is set Indicating the mixed reliability, η, of structural fatigue i (s) represents a junctionConstructing a non-probability fatigue reliability, f (S) is a probability density function of a random variable S,to integrate the lower bound, psi i (R) is the upper integral bound, each being a function of R for the interval variable, i representing the number of events;
and a sixth step: mixed reliability R based on structural fatigue failure set And as constraint conditions, taking the weight G of the laminated plate as an optimization target, taking the thickness t of the laminated plate as a design variable, developing fatigue mixed reliability optimization design for the laminated plate structure, and realizing a complete optimization iterative process by using a particle swarm intelligent algorithm.
2. The method of claim 1, wherein the method comprises the steps of: the interval variable x in the second step can be expressed by a standardization means as follows:
x＝[x L ,x U ]＝[x c -x r ,x c +x r ]
＝x c +x r [-1,1]
＝x c +x r ×e
wherein e ∈ xi- 2 ，Ξ 2 Is defined as all elements contained in [ -1,1]The 2-dimensional vector set in the interior, the symbol "x" is defined as the operator for multiplying the corresponding elements of two vectors, the product is still a vector with dimension 2, for the interval variable R, x L ＝R L ,x U ＝R U ,x c ＝R c ,x r ＝R r 。
3. The method of claim 1, wherein the method comprises the steps of: in the third step, material parameters and external loads are quantized in an interval process model, and explicit representation is realized by applying a standardization means to a limit state equation of a fatigue failure mode of the laminated plate structure; and converting the working coordinate system to a standard coordinate system, wherein e belongs to (-1, 1), and the extreme state equation is as follows:
4. the hybrid fatigue reliability optimization method for composite laminated plates according to claim 1, wherein: structural reliability R 'in four steps' set The calculation of (2) needs to use a stress-intensity interference model, and transform the model through a standardization means, and then a method for calculating an area ratio and a volume ratio by using an interval interference model is used for providing a structural reliability index, so that the method comprises the following steps:
wherein R' set,i Representing the non-probabilistic reliability of the laminate structure fatigue.
5. The hybrid fatigue reliability optimization method for composite laminated plates according to claim 1, wherein: in the fifth step, the solution of the fatigue mixed reliability of the laminated plate structure needs to be combined with a probability reliability solution theory, and the normalized fatigue mixed reliability display representation is given through a standardization means, namely:
wherein R is set Mixed reliability, η, for structural fatigue i (S) is the non-probability reliability of structural fatigue, f (S) is the probability density function of the random variable S,for lower bound of integration,. Psi i (R c +R r Xe) is an upper integral bound, which is a function of R, respectively, for interval variables c Is the central value of the interval variable R, R r I represents the number of events, which is the radius of the interval variable R.
6. The hybrid fatigue reliability optimization method for composite laminated plates according to claim 1, wherein: the optimized formula in the step six is described as follows:
wherein,design allowance, R, for reliability set For the fatigue mixing reliability, t is the sheet thickness.
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