CN105930618B - A kind of mixing fatigue reliability optimization method for composite laminated plate - Google Patents

Abstract

The invention discloses a hybrid fatigue reliability optimization method for composite material laminated plates. This method is firstly based on the non-probability statistical method of small samples, fully considers the multi-source uncertain factors such as material strength and applied load, and uses the gray scale theory to integrate the irregular sample data, explore its potential regularity, and give a reasonable quantification Results; secondly, by establishing the residual strength model under fatigue load, introducing uncertain parameters into the model, and developing a non-probabilistic reliability solution method; thirdly, considering the mixed form of random variables and interval variables, using the established non-probabilistic reliability The probabilistic reliability solving method is combined with the probabilistic reliability solving method to solve the mixed reliability index. Finally, the thickness optimization design of the composite laminate is carried out with the mixed reliability index as the constraint. Based on this, it is possible to realize the strength optimization design in the case of poor information and little data such as large structures, and ensure that the design itself takes into account safety and economy.

Description

A Hybrid Fatigue Reliability Optimization Method for Composite Laminates

technical field

The present invention relates to the technical field of solving reliability indexes of composite laminated plate structures, in particular to the reasonable characterization of the fatigue mixed reliability model of laminated structures, considering the interaction of random variables and interval variables in the case of incomplete probability information. The establishment and formulation of the exact solution method of the reliability.

Background technique

With its superior mechanical properties, composite materials have become a hot topic of research by scholars at home and abroad. For the typical structure of composite materials, laminates are widely used in various aspects such as aviation, aerospace, ships, and medical treatment. The depth and breadth of composite material technology research has reached It has become an important benchmark for the development of national science and technology. Therefore, the research on the mechanical characteristics analysis and design technology of laminated plate structures has important theoretical significance and engineering practical value.

However, due to its own anisotropy and complex service environment, the engineering laminate structure has the influence of various uncertain sources, such as the uncontrollability of processing technology, the inhomogeneity of material properties, the measurement of geometric structure, etc. Fuzziness, randomness of external loads, etc. will aggravate the uncertainty of plate structure damage, and the problem of incomplete probability information will appear, which is difficult to solve with a single reliability solution theory; therefore, how to solve the joint effect of random variables and interval variables, The solution of mixed reliability index becomes the key. It can be seen that a single traditional structural reliability analysis and solution method is no longer applicable. Based on the above situation, under the fatigue effect of laminated plate structure, the solution method of hybrid reliability index has more engineering application value.

At present, domestic and foreign scholars and engineers and technicians mainly focus on two aspects of uncertainty analysis and mixed reliability solution of laminate structures: (1) Structural uncertainty influence envelope based on probability statistics theory and safety factor equation ; (2) Considering the reliability solution of a single type of structure variable. The above work has certain engineering practical value, but it ignores the degree of influence of the refined measurement of uncertain factors on the structural reliability, and the influence of the joint action of random variables and interval variables, which greatly limits the engineering practical application of its theory .

Since in practical engineering, especially complex structures, often face the situation of incomplete probability information, it is of great practical significance to establish a mixed uncertainty characterization technology based on the combination of probability and non-probability theory, and a structure mixed reliability solution evaluation technology.

Contents of the invention

The technical problem to be solved by the present invention is: to overcome the deficiencies of the prior art, and to provide a hybrid reliability optimization method for the composite material laminate structure considering the fatigue failure mode. The present invention fully considers the ubiquitous uncertainties in practical engineering problems, builds a mathematical model that can reasonably characterize the structural residual strength under the action of fatigue failure modes, and proposes a mixed reliability index solution method that considers the simultaneous effects of random variables and interval variables, and the obtained The results are more in line with the real situation, and the engineering applicability is stronger.

The technical solution adopted in the present invention is: a method for optimizing the reliability of composite fatigue reliability for composite laminates, and the implementation steps of the method are as follows:

Step 1: According to the material properties of the laminated plate structure: structural strength R, external load S, number of cycles n, fatigue life N, the explicit expression of the limit state equation of the structure is deduced by introducing the residual strength model. Figure 2 shows the remaining Geometric interpretation of the strength model, namely:

Among them, R is the interval variable of structural strength, S is the random process of external load, n is the number of cycles, N is the fatigue life, and c is the parameter obtained through test data;

Step 2: Use the interval vector x∈x ^I = R to reasonably represent the structural uncertainty under the condition of poor information and less data, and use the random vector y∈y ^I = S to describe the random variable. Then:

x ^U ＝R ^U ＝R ^c +R ^r

x ^L ＝R ^L ＝R ^c -R ^r

S∈N(μ,σ)

Among them, the structural strength R can be expressed as an interval variable, the superscript U represents the upper bound of the parameter value, the superscript L represents the lower bound of the parameter value, the superscript c represents the center value, the superscript r represents the radius, and the external load S can represent is a random variable subject to a normal distribution, μ is the mean, and σ is the variance;

Step 3: Substituting the interval variables and random variables in the second step into the structural limit state equation in the first step, introducing the non-probability interval process theory, establishing the fatigue failure probability-mixed reliability limit state equation, and realizing the limit state function explicit expression; that is:

Among them, M is the fatigue failure limit state function of laminated plate structure, R is an interval variable, S is a random variable that obeys normal distribution, and n, N, c are test parameters;

Step 4: Combined with the stress-strength interference model, Figure 3 shows the geometric interpretation of the non-probabilistic reliability solution method. According to the limit state equation established in the third step, the fatigue non-probabilistic reliability is solved:

Among them, _{R'set, i} is the fatigue non-probabilistic reliability, R is the structural strength, S is the external load, n, N, c are the test parameters;

Step 5: Combining probability theory, non-probability theory and interval process model, the fatigue failure mixed reliability calculation index of laminated plate structure is proposed:

Among them, R _set represents the structural fatigue mixed reliability, η _i (s) represents the structural non-probabilistic fatigue reliability, f(S) is the probability density function of the random variable S, is the lower bound of the integral, ψ _i (R) is the upper bound of the integral, which are respectively the functions of R about interval variables, and i represents the number of events;

Step 6: Taking the structural fatigue failure mixed reliability R _set as the constraint condition, taking the laminated plate weight G as the optimization target, and taking the plate thickness d as the design variable, carry out the fatigue mixed reliability optimization design for the laminated plate structure, and use The particle swarm intelligent algorithm realizes the complete optimization iterative process.

Compared with the prior art, the present invention has the advantages that: the present invention provides a new method for solving the mixed reliability index of the composite laminate structure under the mixed action of random variables and interval variables, which makes up for and perfects the situation of incomplete probability information. Limitations of traditional probability theory and reliability design method of factor of safety method. The constructed probability-mixed reliability model provides a new way to solve the reliability problem under the joint action of random variables and interval variables. On the one hand, it reduces the dependence on sample information. Combined with the theory of probabilistic reliability solution, a reasonable mixed reliability solution model is constructed, which improves the accuracy and rationality of the structural reliability solution.

Description of drawings

Fig. 1 is the reliability optimization method flow chart that the present invention considers probability-non-probability joint action for laminated plate structure;

Fig. 2 is the schematic diagram of the fatigue residual strength model of the laminated plate structure according to the present invention;

Fig. 3 is a schematic diagram of solving reliability index based on non-probability theory proposed by the present invention, wherein Fig. 3 (a) is a two-dimensional non-probabilistic reliability model, and Fig. 3 (b) is a three-dimensional non-probabilistic reliability model;

Fig. 4 is a schematic diagram of analysis of different non-probabilistic reliability indicators for different value positions of structural residual strength according to the present invention;

Fig. 5 is a schematic diagram of the geometric model of the composite material laminate structure proposed by the present invention.

Detailed ways

The present invention will be further described below in conjunction with the accompanying drawings and specific embodiments.

As shown in Figure 1, the present invention proposes a hybrid fatigue reliability optimization method for composite material laminates, comprising the following steps:

(1) Use the interval vector x∈x ^I = R to reasonably represent the structural uncertainty under the condition of poor information and less data, and use the random vector y∈y ^I = S to describe the random variable, so we have:

x ^U ＝R ^U ＝R ^c +R ^r

x ^L ＝R ^L ＝R ^c -R ^r

S∈N(μ,σ)

Among them, the structural strength R can be expressed as an interval variable, the superscript U represents the upper bound of the parameter value, the superscript L represents the lower bound of the parameter value, the superscript c represents the center value, the superscript r represents the radius, and the external load S can represent is a random variable subject to a normal distribution, μ is the mean, σ is the variance, and the 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

Among them, e∈Ξ ² , Ξ ² is defined as a 2-dimensional vector set with all elements contained in [-1,1], and the symbol "×" is defined as an operator for multiplying the corresponding elements of two vectors, and the product is still dimensional A vector whose number is 2, for the interval variable R, x ^L ＝ ^RL , x ^U ＝ ^RU , x ^c ＝R ^c , x ^r ＝R ^r .

(2) According to the material properties of the laminated plate structure: structural strength R, external load S, cycle number n, fatigue life N, the explicit expression of the limit state equation of the structure is deduced by introducing the residual strength model. which is:

Among them, R is the interval variable of structural strength, S is the random process of external load, n is the number of cycles, N is the fatigue life, and c is the parameter obtained through test data;

(3) Substituting the interval variable and random variable in the second step into the structural limit state equation in the first step, introducing the non-probability interval process theory, establishing the fatigue failure probability-mixed reliability limit state equation, and realizing the explicit limit state function expression; that is:

Among them, M is the fatigue failure limit state function of laminated plate structure, R is an interval variable, S is a random variable subject to normal distribution, n, N, c are test parameters; the basic interval variable x _i (i=1) is used as the standard transform:

x _i = x _i ^c + x _i ^r ×e

where x _i (i=1) corresponds to the interval variable R; for the structural fatigue failure limit state equation, it is explicitly represented by interval mathematics and standardized processing methods, and the working coordinate system is converted to the standard coordinate system, where e∈(-1,1 ), the limit state equation is:

It can be seen from the standardized processing method that R=R ^c +R ^r ×e, then Substitute into the above formula, n, c, N are test parameters, so let The standardized limit state equation is obtained:

M=G(e,S,k)=(1-k)×(R ^c +R ^r ×e)-(k+1)S

(4) Aiming at the realization of the random variable S taking one, combined with the standardized limit state equation, the non-probability theory and the interval process model are combined, as shown in Figure 4, and the non-probabilistic reliability calculation of the fatigue failure mode for the composite laminate structure is proposed index:

I as a random variable hour,

R' _set,1 ＝η ₁ (S)＝η ₁ (G(e,S,k)>0)＝1

II as a random variable hour,

III as a random variable hour,

R' _set,3 ＝η ₃ (S)＝η ₃ (G(e,S,k)>0)＝0

Among them, η _i (S) represents the non-probabilistic reliability of the standardized limit state equation, G(e,S,k) is the standardized fatigue limit state equation, e represents the standardized interval variable R, and S represents the external load random variable , k represents the test parameters, R ^c is the central value of the interval variable R, and R ^r is the radius of the interval variable R;

(5) Considering the probability theory and non-probability theory comprehensively, the calculation index of fatigue failure mixed reliability of laminated plate structure is proposed:

Among them, R _set represents the structural fatigue mixed reliability, η _i (s) represents the structural non-probabilistic fatigue reliability, f(S) is the probability density function of the random variable S, is the lower bound of the integral, ψ _i (R) is the upper bound of the integral, which are respectively the functions of R about interval variables, and i represents the number of events;

It is known that the random variable S obeys the normal distribution with mean value μ and variance σ, and its probability density function is:

Substituting the probability density function and non-probabilistic reliability into the calculation formula of fatigue mixed reliability, we get:

Among them, R _set,i (i=1,2,…,n) represents the fatigue non-probabilistic reliability of the laminated plate in the i-th case;

(6) With the structural fatigue failure mixed reliability R _set as the constraint condition, the weight of the laminated plate G as the optimization target, and the plate thickness t as the design variable, the fatigue mixed reliability optimization design for the laminated plate structure is carried out, and the particle swarm The intelligent algorithm realizes the complete optimization iterative process. The optimized column formula is described as:

in, is the design allowable value of reliability, R _set is the fatigue mixed reliability, t is the plate thickness, in the iterative process, by modifying the geometric dimensions of the laminate, the fatigue mixed reliability R _set and the reliability allowable value are solved Comparison until the reliability requirements are met, and the optimal design results are given.

Particle swarm optimization algorithm is an intelligent global optimization solution technology, each particle represents a potential optimal solution, and its position represents a certain direction vector. The initial population will be randomly assigned initial positions and initial speeds, and they will accelerate updates along the previous optimal positions, and the determination of the global optimal point will rely on the following two formulas:

In the formula, i represents the i-th particle, k represents the k-th iteration process, v _i represents the update speed of the i-th particle, and x _i is the current position of the i-th particle. and represents the acceleration constant, and is a random number that satisfies a uniform distribution in the [0,1] interval, w ^* represents the weight coefficient, pbest _i and gbest _i represent the optimal position based on the individual and the overall, respectively. The completion of the above iterative process depends on the minimum error or the preset value of the number of iteration steps, which also determines the accuracy of the calculation results.

Example:

In order to fully understand the characteristics of the invention and its applicability to engineering practice, the present invention carries out the fatigue hybrid reliability solution based on the residual strength theory for the proposed laminate structure as shown in Figure 5. The laminated plate structure bears cyclic load S∈N(317,12.6), fatigue life N=118000, the static strength of the laminated plate structure R∈[876,896], plate length l=100mm, plate width w=40mm, plate thickness t= 5mm, to investigate the fatigue reliability when the number of cycles n=70000 under working conditions, considering the geometric dimensions and loading conditions, using the established method for solving the mixed fatigue reliability, the mixed fatigue reliability can be calculated as 0.9786.

According to the desired mixed fatigue reliability index, with the allowable reliability as the constraint, the optimal design is carried out, and finally when the allowable reliability is satisfied Under the constraints, the optimization result of the geometric dimension of the laminate structure is given as t=5.5mm. It can be seen that the weight of the laminate increases with the increase of the reliability.

To sum up, the present invention proposes a fatigue hybrid reliability solution method for the composite laminated plate structure considering the joint action of random variables and interval variables. Firstly, according to the specific characteristics of the laminate structure materials and loads, combined with the residual strength theory, the limit state function under the fatigue failure mode is obtained; secondly, random variables, interval variables and other information are introduced into the residual strength model to establish the fatigue mixed reliability solution equation , realize the mixed fatigue reliability solution considering the coexistence of random variables and interval variables; finally, based on the established mixed fatigue reliability solution method, with the mixed fatigue reliability as the constraint and the geometric parameters as variables, the particle swarm optimization algorithm is used to optimize design.

The above are only the specific steps of the present invention, and do not constitute any limitation to the scope of protection of the present invention; it can be extended and applied to the field of reliability solution of structural multi-failure modes, and all technical solutions formed by equivalent transformation or equivalent replacement fall within the scope of the present invention. Within the protection scope of the present invention.

Parts not described in detail in the present invention belong to the known techniques of those 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.