CN106055731A - Method for non-probability reliability optimization of composite laminated plate - Google Patents

Abstract

The method discloses a method for non-probability reliability optimization of a composite laminated plate. The method comprises the steps that firstly, according to force bearing characteristics of the composite laminated plate and with consideration of uncertainty effects of material fiber strength under limited samples, a non-probability reliability assessment model of the composite laminated plate is established based on non-probability reliability theories; and a gradient optimization method is further adopted, a light weight is taken as a goal, reliability is taken as a constraint, a paving layer thickness is taken as an optimization variable, and a composite paving layer scheme which can satisfy certain reliability requirements can be obtained through iterations. The method has the advantages that the composite laminated plate can have the high reliability and the light weight under uncertainty conditions; and safety and economic efficiency are also ensured.

Description

A kind of composite laminated plate Multidisciplinary systems optimization method

Technical field

The present invention relates to the design optimizing field of laminated composite plate structures, particularly to twin shaft stand under load laminate The Multidisciplinary systems method for designing of structure, the method considers the reliability optimization problem under fibre strength condition of uncertainty, with Under certain reliability requirement, the thickness of laminate is optimized design.

Background technology

The tool that composite is made up of on a macroscopic scale the different material physics of two or more and chemical method Having the material of new capability, the performance of general composite is better than the performance of its component material, and some performance is original component Material does not has, and composite improves the mechanical properties such as the rigidity of component material, intensity.

Composite is widely used in the fields such as Aero-Space, automobile, machinery because of its good characteristic.These fields are to material The safety of material has the highest requirement.Laminated plate structure is typical composite structure, and its material properties has the biggest Randomness and discreteness.Therefore its real mechanical response can not be accurately reflected, to lamination by conventional strength computational methods Plate structure carries out fail-safe analysis and optimizes is the most necessary.

Currently, the uncertainty analysis containing initial imperfection plate structure is ground with design by Chinese scholars with engineers and technicians Study carefully and be concentrated mainly on laminated plate structure fail-safe analysis based on Probability Statistics Theory and optimize design.Above-mentioned work is to a certain degree On enrich laminate fail-safe analysis and Optimization Theory, but have ignored the random method dependency to sample information, Greatly limit the practical application process of its theory.

Owing in Practical Project, uncertain information often can not happen occasionally with the situation that Probability Forms is expressed, set up with non- Laminated composite plate structures fail-safe analysis based on probability theory framework has significant realistic meaning with optimizing design. At present, correlational study work is the most immature, for laminated composite plate structures, causes the serious wasting of resources.

Summary of the invention

The technical problem to be solved in the present invention is: overcome the deficiencies in the prior art, it is provided that a kind of for composite laminated Plate loss of weight Optimization Design.The present invention takes into full account the uncertain factor generally existed in Practical Project problem, to propose Non-probability time-varying reliability metric as the optimization aim of Optimized model, obtained design result more conforms to truly Situation, engineering adaptability is higher.

The technical solution used in the present invention is: a kind of composite laminated plate Multidisciplinary systems optimization method, its feature It is to realize step as follows:

Step one: according to the geometric properties of composite laminated plate, material properties and boundary condition, laminate is carried out The derivation of Tsai-Hill coefficient in Tsai-Hill strength theory, obtains Tsai-Hill strength factor, the wherein geometry of laminate Feature includes length a and width b in laminate face；Material properties includes elastic constant and intensive parameter, and elastic constant includes: 1 To elastic modulus E₁, 2 to elastic modulus E₂, 1-2 shear modulus G₁₂, Poisson's ratio υ, wherein 1 to for fiber axial direction, 2 to for Vertical fibers axial direction in laminate plane；Boundary condition includes x and y direction compressive load N_xAnd N_y；Intensive parameter is the most true Fixed, compare X including longitudinal tensile strength_T, longitudinal compressive strength X_C, transverse tensile strength Y_T, transverse compression intensity Y_C, cut in face Shearing stress S；Based on Tsai-Hill strength theory, Tsai-Hill strength factor isX is longitudinally Stretching or compressive strength, the X=X when X-direction pressurized_C, X=X during pressurized_T；Y is cross directional stretch or compressive strength, during horizontal pressurized Y=Y_C, during horizontal tension, Y=Y_T；

Step 2: utilize interval vector x ∈ x^I=(X_T,X_C,Y_T,Y_C, S) rationally characterize the strength of materials parameter in step one Uncertainty, obtain:

Wherein, longitudinal tensile strength X_T, longitudinal compressive strength X_C, transverse tensile strength Y_T, transverse compression intensity Y_C, in face Shear strength S can be expressed as interval variable, and subscript U represents the value upper bound of parameter, and subscript L represents under the value of parameter Boundary, subscript c represents central value, and subscript r represents radius, x^IInterval for intensive parameter；

Step 3: in application composite laminated plate macromechanics theory analysis laminate, a laying 1 is to stress σ₁, 2 to Stress σ₂With shearing stress τ₁₂；In the Tsai-Hill expression formula that strength of materials unascertained information is updated in step one, introduce Non-probability interval is theoretical, utilizes the intensive parameter interval computation Tsai-Hill coefficient in step 2 interval, strong according to composite Topology degree, when Tsai-Hill coefficient t is more than 1, this layer by layer plywood lost efficacy, during less than 1, this plywood safety layer by layer, due to compound The intensive parameter of material laminate is interval variable, and therefore laminate each layer Tsai-Hill coefficient t is also an interval, i.e. Tsai- Hill intensity interval, then be layer less than the siding-to-siding block length of 1 part with the ratio of whole siding-to-siding block length in Tsai-Hill intensity interval Reliability P of plywood_si, i=1,2 ... n, wherein n is the laminate number of plies, monolayer can spend in minima P_s=min (P_si) be The reliability of laminate；

Step 4: according to the reliability of calculating resultant layer plywood in step 3, carries out laminate being about with reliability P Bundle, laminate total quality is target, single berth layer thickness h=(h₁,h₂,…,h_n) be optimized variable optimization design；Lamination Plate quality layer thickness each with laminate is linear, and therefore, the iteration that direct application gradient method carries out thickness in monolayer h is excellent Change, h_I+1=h_I+εh_I, wherein I is current iteration number of times, ε h_IIt is iteration step length；

Step 5: in iterative process, set convergence criterion as:Can if current layer plywood reliability still meets By degree constraints, then iterations I adds increase by 1, and returns step 3, until the reliability of laminate is less than rule in step 3 Definite value, the then thickness h of I-1 step_I-1It is the minimum thickness of satisfied current Reliability Constraint.

The Multidisciplinary systems of described step one laminate is decided by that intensive parameter includes longitudinal tensile strength X_T, longitudinally pressure Contracting intensity X_C, transverse tensile strength Y_T, transverse compression intensity Y_C, the uncertainty of in-plane shear strength S.

In described step 3, reliability P_siIt is calculated as follows:

In formula, P_siFor laminate reliability, t is Tsai-Wu coefficient, and subscript U represents the upper bound of range of variables, subscript L table Showing the lower bound of range of variables, n is the laying number of plies, according to first-ply failure criterion, P_siMinima be the reliability of this laminate.

The optimization that single berth layer thickness h is optimized variable in step 4 is designed as: at fibre strength condition of uncertainty Under, with the minimum target of laminate quality, the thickness of each layer is optimized design, can column be specifically:

Wherein, h=(h₁,h₂,…,h_n) it is each thickness in monolayer of laminate；W is laminate thickness, is the function of thickness h；P_s For the reliability of laminate, it is laminate thickness h, fibre strength x, length a, width b, material elastic modulus E₁, E₂, modulus of shearing G₁₂, the function of Poisson's ratio υ；For the Design permissible value of reliability,The biggest, laminate reliability is the highest, and weight is the biggest；

In described step 5, Optimized Iterative stopping criterion is set to reliability less than reliability allowable value

Present invention advantage compared with prior art is: the invention provides the new think of of composite reliability design Road, makes up and perfect tradition limitation based on probability theory reliability design approach.The constructed non-probability of laminate can By property measurement model, on the one hand can significantly reduce the dependency to sample information, on the other hand can take into full account uncertainty Laminated plate structure strength reliability under Zuo Yong, under laminated plate structure reliability requirement, is designed by optimization, alleviates lamination Plate carries out light-weight design.

Accompanying drawing explanation

Fig. 1 is to the present invention is directed to composite laminated plate Multidisciplinary systems Optimizing Flow figure；

Fig. 2 is composite laminated plate load schematic；

Fig. 3 is Composite Laminated Panel scheme schematic diagram；

Fig. 4 is to the present invention is directed to fibre strength indeterminacy of calculation laminate monolayer reliability schematic diagram；

Fig. 5 is that the present invention optimizes process reliability to iterations course curve to laminate；

Fig. 6 is that the present invention optimizes process thickness in monolayer to iterations course curve to laminate.

Detailed description of the invention

Below in conjunction with the accompanying drawings and specific embodiment further illustrates the present invention.

As it is shown in figure 1, the present invention is a kind of for composite laminated plate Multidisciplinary systems method for designing, including following step Rapid:

(1) according to the geometric properties of composite laminated plate, material properties and boundary condition, laminate is carried out based on The Multidisciplinary systems analysis and optimization that the first floor lost efficacy.Length a and width in wherein the geometric properties of laminate includes laminate face Degree b；Lamina elastic constant includes: 1 to elastic modulus E₁, 2 to elastic modulus E₂, 1-2 shear modulus G₁₂, Poisson's ratio υ, wherein 1 to for fiber axial direction, and 2 to for vertical fibers axial direction in laminate plane；Boundary condition includes that x and y direction is compressed Load N_xAnd N_y；Intensive parameter includes that longitudinal tensile strength compares X_T, longitudinal compressive strength X_C, transverse tensile strength Y_T, transverse compression Intensity Y_C, in-plane shear strength S.Based on Tsai-Hill strength theory, Tsai-Hill strength factor isWherein: X is longitudinal stretching or compressive strength, the X=X when X-direction pressurized_C, X=during pressurized X_T；Y is cross directional stretch or compressive strength, Y=Y during horizontal pressurized_C, during horizontal tension, Y=Y_T。

(2) interval vector x ∈ xI=(X is utilized_T,X_C,Y_T,Y_C, S) rationally characterize lean information, minority according under the conditions of structure Uncertainty, then has:

Wherein, intensive parameter includes that longitudinal tensile strength compares X_T, longitudinal compressive strength X_C, transverse tensile strength Y_T, laterally press Contracting intensity Y_C, in-plane shear strength S can be expressed as interval variable, and subscript U represents the value upper bound of parameter, and subscript L represents The value lower bound of parameter, subscript c represents central value, and subscript r represents radius；

(3) application composite macromechanics theory Analysis for Composite Laminated plate ply stress σ_i1,σ_i2,τ_i12, i=1,2 ..., n, In formula, n is the laminate laying number of plies.

By laminate stress and fibre strength substitution Tsai-Hill strength theory expression formula:

When fiber axial compression, X=X_C, during axial tension, X=X_T；When fiber at laminate face inner vertical shaft to pressurized Time, Y=Y_C, Y=Y during tension_T。

By Tsai-Hill expression formula, whenTime, the upper limit of t should obtain when Y, S remove the limit, i.e. at XThe lower limit of t should obtain during Y, S capping, i.e. at X

WhenTime, the upper limit of t should obtain when Y, S remove the limit, i.e. at XThe lower limit of t should obtain during Y, S capping, i.e. at X

WhenTime, the upper limit of t should obtain when Y, S remove the limit, i.e. at X cappingThe lower limit of t should remove the limit at X, obtains, i.e. during Y, S capping

Accordingly, the interval range [t of Tsai-Hill coefficient is obtained^L,t^U], theoretical according to composite material strength, when coefficient is big When equal to 1, this layer lost efficacy, when coefficient is less than 1, and this layer of safety, therefore can get each layer reliability i.e. Tsai-Hill coefficient Probability less than 1:

In formula, n is the laminate laying number of plies, and according to first-ply failure criterion, the reliability of whole laminate is equal to laminate The reliability of middle reliability lowermost layer, it may be assumed that P_s=min (P_si)。

(4) carrying out laminate with reliability for constraint, laminate total quality is target, single berth layer thickness h= (h₁,h₂,…,h_n) be optimized variable optimization design.Owing to each layer thickness of laminate quality and laminate is linear, because of This, can directly apply gradient method to carry out the optimization of h, h_i+1=h_i-εh_i, wherein i is current iteration number of times, ε h_iIt is step Long, take ε=0.01.

(5) repeat step 3, four, until reliability is less than setting, i.e. P_s＜ | P_s| time, optimize and stop, P in formula_sFor layer Plywood reliability, | P_s| for the laminate reliability requirement pre-set.

Embodiment:

In order to understand the feature of this invention and the suitability actual to engineering thereof more fully, the present invention is directed to such as Fig. 2 institute Compressive load N in the bearing plane of the surrounding freely-supported shown_xAnd N_yLaminate carry out optimization based on reliability design.Laminate spreads Layer mode is [0/45/-45/90]_s, such as Fig. 3.Laminate face inside dimension is a × b=(20*12.5) cm², the thickness of lamina is 0.125mm, therefore laminate total thickness is 0.125mm × 8=1mm.Table gives the uncertainty of Rectangular Plate Structure in embodiment Information.Elastic constant and the load parameter of lamina are as follows:

E₁=181.0GPa, E₂=10.8GPa, G₁₂=7.17Gpa, υ=0.28, N_x=400kN/m, N_y=72kN/m

Table 1

This embodiment uses, and reliability applies Tsai-Hill interval strength reliability as shown in Figure 4 to try to achieve, shade in figure Region representation laminate safety, its ratio with whole interval is the reliability of lamina, RELIABILITY DESIGN allowable value | P_s| Being set to 0.95, Fig. 5 and Fig. 6 gives the iteration course curve of object function and constraint function, and it is reliable that Fig. 5 gives laminate Along with the variation tendency of Optimized Iterative number of times, along with the reduction of thickness, reliability is substantially linear reduction, is reduced to by 0.9939 0.9524, slightly larger than reliability allowable value 0.95, laminate thickness in monolayer is reduced to 0.1231 by 0.125 as can be seen from Figure 6, Loss of weight 1.52%.

In sum, the present invention proposes a kind of composite laminated plate Multidisciplinary systems method for designing.First, according to Laminate physical dimension, elastic parameter, the computation layer plywood stress such as laying information；Secondly, by the uncertain letter of fibre strength Breath introduces Tsai-Hill strength theory, it is achieved the calculating of Tsai-Hill coefficient interval upper and lower bound；Strong according to Tsai-Hill Topology degree, in conjunction with the reliability of non-probability interference technique computation layer plywood；Finally, turn to target with light weight, complete with laminate Reliability is constraint, for the Multidisciplinary systems optimization design that each layer thickness is variable, reaches to meet the layer of reliability requirement The target of plywood light-weight design.

Below it is only the concrete steps of the present invention, protection scope of the present invention is not constituted any limitation；Its expansible should The technical scheme replaced for the optimization design field of Defective structure, all employing equivalents or equivalence and formed, all falls Within the scope of rights protection of the present invention.

Non-elaborated part of the present invention belongs to the known technology of those skilled in the art.

Claims (5)

1. a composite laminated plate Multidisciplinary systems optimization method, it is characterised in that realize step as follows:

Step one: according to the geometric properties of composite laminated plate, material properties and boundary condition, laminate is carried out The derivation of Tsai-Hill coefficient in Tsai-Hill strength theory, obtains Tsai-Hill strength factor, the wherein geometry of laminate Feature includes length a and width b in laminate face；Material properties includes elastic constant and intensive parameter, and elastic constant includes: 1 To elastic modulus E₁, 2 to elastic modulus E₂, 1-2 shear modulus G₁₂, Poisson's ratio υ, wherein 1 to for fiber axial direction, 2 to for Vertical fibers axial direction in laminate plane；Boundary condition includes x and y direction compressive load N_xAnd N_y；Intensive parameter is the most true Fixed, compare X including longitudinal tensile strength_T, longitudinal compressive strength X_C, transverse tensile strength Y_T, transverse compression intensity Y_C, cut in face Shearing stress S；Based on Tsai-Hill strength theory, Tsai-Hill strength factor isX is longitudinally Stretching or compressive strength, the X=X when X-direction pressurized_C, X=X during pressurized_T；Y is cross directional stretch or compressive strength, during horizontal pressurized Y=Y_C, during horizontal tension, Y=Y_T；

Step 2: utilize interval vector x ∈ x^I=(X_T,X_C,Y_T,Y_C, S) rationally characterize the strength of materials parameter in step one not Definitiveness, obtains:

$x^U=(X_T^U,X_C^U,Y_T^U,Y_C^U,S^U)=(X_T^c+X_T^r,X_C^c+X_C^r,Y_T^c+Y_T^r,Y_C^c+Y_C^r,S^c+S^r)$

$x^L=(X_T^L,X_C^L,Y_T^L,Y_C^L,S^L)=(X_T^c-X_T^r,X_C^c-X_C^r,Y_T^c-Y_T^r,Y_C^c-Y_C^r,S^c-S^r)$

Wherein, longitudinal tensile strength X_T, longitudinal compressive strength X_C, transverse tensile strength Y_T, transverse compression intensity Y_C, inplane shear is strong Degree S can be expressed as interval variable, and subscript U represents the value upper bound of parameter, and subscript L represents the value lower bound of parameter, subscript c Representing central value, subscript r represents radius, x^IInterval for intensive parameter；

Step 3: in application composite laminated plate macromechanics theory analysis laminate, a laying 1 is to stress σ₁, 2 to stress σ₂ With shearing stress τ₁₂；In the Tsai-Hill expression formula being updated in step one by strength of materials unascertained information, introduce non-probability Interval theory, utilizes the intensive parameter interval computation Tsai-Hill coefficient in step 2 interval, manages according to composite material strength Opinion, when Tsai-Hill coefficient t is more than 1, this layer by layer plywood lost efficacy, during less than 1, this plywood safety layer by layer, due to composite The intensive parameter of laminate is interval variable, and therefore laminate each layer Tsai-Hill coefficient t is also an interval, i.e. Tsai-Hill Intensity interval, then be laminate less than the siding-to-siding block length of 1 part with the ratio of whole siding-to-siding block length in Tsai-Hill intensity interval Reliability P_si, i=1,2 ... n, wherein n is the laminate number of plies, monolayer can spend in minima P_s=min (P_si) it is lamination The reliability of plate；

Step 4: according to the reliability of calculating resultant layer plywood in step 3, laminate is carried out with reliability for constraint, lamination Plate total quality is target, single berth layer thickness h=(h₁,h₂,…,h_n) be optimized variable optimization design；Laminate quality with The each layer thickness of laminate is linear, and therefore, directly application gradient method carries out the iteration optimization of thickness in monolayer h, h_I+1=h_I +εh_I, ε h_IBeing iteration step length, in iterative process, monolayer reliability isLaminate reliability isWherein I is current iteration number of times；

Step 5: in iterative process, set convergence criterion as:WhereinIt is the laminate reliability of the I time iteration,For reliability allowable value, if i.e. current layer plywood reliability still meets Reliability Constraint condition, then iterations I adds increasing Add 1, and return step 3, until the reliability of laminate is less than setting in step 3, then the thickness h of I-1 step_I-1It is Meet the minimum thickness of current Reliability Constraint.

A kind of composite laminated plate Multidisciplinary systems optimization method the most according to claim 1, it is characterised in that: institute The Multidisciplinary systems stating step one laminate is decided by that intensive parameter includes longitudinal tensile strength X_T, longitudinal compressive strength X_C, horizontal To hot strength Y_T, transverse compression intensity Y_C, the uncertainty of in-plane shear strength S.

A kind of composite laminated plate Multidisciplinary systems optimization method the most according to claim 1, it is characterised in that: institute State in step 3, reliability P_siIt is calculated as follows:

$P_{si}=\frac{1-\left(t^L\right)}{\left(t^U\right)-\left(t^L\right)},i=1,2,...,n$

In formula, P_siFor laminate reliability, t is Tsai-Wu coefficient, and subscript U represents that the upper bound of range of variables, subscript L represent change The lower bound that amount is interval, n is the laying number of plies, according to first-ply failure criterion, P_siMinima be the reliability of this laminate.

A kind of composite laminated plate Multidisciplinary systems optimization method the most according to claim 1, it is characterised in that: institute The optimization that single berth layer thickness h is optimized variable stated in step 4 is designed as: under fibre strength condition of uncertainty, with layer The minimum target of plywood quality, is optimized design to the thickness of each layer, can column be specifically:

$\left\{\begin{array}{cc}find& W\left(h\right)\\ s.t.& P_s(h,x,a,b,E_1,E_2,G_{12},\mathrm{\υ})\≥P_s^{cr}\\ & h>0,a>0,b>0\end{array}\right.$

Wherein, h=(h₁,h₂,…,h_n) it is each thickness in monolayer of laminate；W is laminate weight, is the function of thickness h；P_sFor layer The reliability of plywood, is laminate thickness h, fibre strength x, length a, width b, material elastic modulus E₁, E₂, shear modulus G₁₂, The function of Poisson's ratio υ；For the Design permissible value of reliability,The biggest, laminate reliability is the highest, and weight is the biggest.

A kind of composite laminated plate Multidisciplinary systems optimization method the most according to claim 1, it is characterised in that: institute State Optimized Iterative stopping criterion in step 5 and be set to reliability less than reliability allowable value