CN115831288A - Explicit rapid analysis method for buckling and post-buckling under mechanical load - Google Patents
Explicit rapid analysis method for buckling and post-buckling under mechanical load Download PDFInfo
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Abstract
The invention relates to the technical field of buckling and post-buckling analysis, in particular to an explicit rapid buckling and post-buckling analysis method under the action of mechanical load, which simultaneously considers the influences of front buckling nonlinear deformation, large post-buckling deflection, initial geometric defects and reinforcement space positions. Based on a high-order shear deformation theory, considering the influence of an accurate curvature relation, and establishing a buckling problem control equation of a reinforced composite material cylindrical shell structure, wherein the buckling problem control equation comprises a pull-bend coupling effect, a pull-twist coupling effect, a bend-twist coupling effect and a heat effect.
Description
Technical Field
The invention relates to the technical field of buckling and post-buckling analysis, in particular to an explicit rapid analysis method for buckling and post-buckling under the action of mechanical load.
Background
The composite material structure has the advantages of high specific strength and specific modulus (rigidity), good fatigue resistance, creep deformation, impact resistance, fracture toughness and the like. With the improvement of the manufacturing process, various performance indexes of the composite material are continuously improved, more and more main stressed members in aerospace and marine engineering structures are made of the composite material, and particularly, the composite material reinforced cylindrical shell structure is often used as a bearing part in the engineering structure.
With the increasing application of composite materials in engineering structures, the composite materials can be buckled under the action of complex internal and external loads to cause the situation of reduced bearing capacity and even damage. In general, in the design process of structures and materials, the buckling and post-buckling behaviors of the anisotropic composite material reinforced cylindrical shell need to be discussed in a more general sense by considering not only the rigidity difference caused by different layering conditions of the materials, but also the influence of the stretching-bending, bending-twisting and stretching-twisting coupling deformation, the transverse shearing effect and the thermal effect of the structures. As a result, the problem of buckling of composite structures has been a significant concern for engineering designers.
The composite material cylindrical shell is reinforced to improve the strength and rigidity, the integral structure is simple, the bearing capacity is greatly enhanced, and the composite material cylindrical shell is widely applied as a basic structure in engineering. However, the design practice shows that great difference exists between the classic theoretical prediction result and the experimental result under the action of the temperature field; the main reason is that designers have not studied the buckling failure mechanism of the composite material structure thoroughly.
Furthermore, for various reasons, this type of structure inevitably produces geometric defects, the presence of which seriously affects the load-bearing characteristics of the shell structure. Therefore, the buckling and post-buckling characteristics of the cylindrical shell under the conditions of different boundaries and different initial geometric defects can be researched, and a beneficial reference can be provided for the engineering application of the cylindrical shell.
In order to solve the above problems, it is necessary to provide an explicit rapid analysis method for buckling and post-buckling under the action of mechanical load, which is used to solve the buckling problem of a composite shell structure, especially a reinforced shell structure, so as to overcome the defect of the existing qualitative knowledge of the buckling behavior near the critical point, and provide a reliable basis for engineering design.
Disclosure of Invention
The invention aims to provide an explicit rapid analysis method for buckling and post-buckling under the action of mechanical load, so as to solve the problems in the background technology.
In order to achieve the purpose, the invention provides the following technical scheme:
an explicit rapid analysis method for buckling and post-buckling under the action of mechanical load is used for a composite material reinforced shell containing accurate curvature expression, and comprises the following steps:
step 1: based on a high-order shear deformation theory, setting the transverse shear displacement-strain relation of the cylindrical shell to be distributed according to a parabolic rule along the shell thickness direction;
step 2: the stiffened plate is equivalent to a variable-rigidity plate structure, namely the influence of the plate structure rigidity corresponding to the local stiffened part on the superimposed ribs is obtained, and the rigidity increment of the ribs is considered;
and step 3: the height h of the local neutral plane can be determined according to the condition of pure bending, namely that the stress on the bending neutral plane of the plate structure is zero 0 ；
And 4, step 4: combining the rigidity coefficients of the rib-free area and the reinforced area into a variable rigidity function;
meanwhile, in order to ensure the conductivity of the stiffness coefficient matrix with respect to the position coordinate, a hyperbolic tangent function is introduced to carry out smooth transition on the variable stiffness coefficient matrix;
and 5: obtaining an expression of internal force and bending moment of the reinforced cylindrical shell structure under common laying conditions according to the equivalent constitutive relation of the composite material reinforced plate;
and 6: according to the Hamilton principle, obtaining a composite material reinforced cylindrical shell balance differential equation by using a Euler-Lagrange equation;
and 7: introducing general form initial defect expression, carrying out non-dimensionalization treatment on a balance differential equation, and introducing a small parameter epsilon with obvious physical significance, namely that the small parameter epsilon is in inverse proportion to the geometric parameter Z of the equivalent length of the shell;
when epsilon is less than 1, the composite material reinforced cylindrical shell balance differential equation is a boundary layer equation;
and step 8: solving by using a singular perturbation method, and dividing the solution of an equation into a regular solution and a boundary layer solution;
and step 9: processing the initial defect numerical value of the completely anisotropic reinforced cylindrical shell to form deflection numerical values generally distributed at different positions of the shell;
meanwhile, considering the influence of a heat effect, generating a heat bending moment and initial deflection, and substituting the heat bending moment and the initial deflection into a composite material reinforced cylindrical shell balance differential equation for calculation;
step 10: obtaining perturbation equation sets of each order according to the equilibrium differential equation of the reinforced cylindrical shell made of the discrete composite material with the same power of epsilon, solving the perturbation equation sets of each order step by step and synthesizing a regular solution and a boundary layer solution to obtain a large-deflection asymptotic solution which strictly meets the boundary condition of the fixed support in the asymptotic sense;
on the basis, obtaining a quantitative relational expression of deflection and corner and a boundary layer width expression of shell buckling, and obtaining shell structure equivalent stress sigma by using constitutive relation ij An expression;
step 11: obtaining the dimensionless deflection w and the pressure lambda T And shear stress lambda s A post-flexion equilibrium pathway of expression; wherein:
and (3) axial compression buckling:
and:
external pressure buckling:
and:
and (3) torsional buckling:
and:
and relative torsion angle:
step 12: converting the parameters in the formulas (1 a) to (3 d) by using secondary perturbation parameter conversion
Conversion to dimensionless maximum deflection, i.e.:
wherein, W m In the flexibility expression, the point (x, y) = (pi/2 m, pi/2 n) is the maximum flexibility without dimension, and the point (x, y) = (pi/2 m, pi/2 n) comprises the following points:
step 13: and obtaining a post-buckling balance path of the shearing cylindrical shell with dimensionless maximum deflection as a perturbation parameter under the action of torque load.
As a further scheme of the invention: in step 2, rectangular section ribs are adopted, and the equivalent stress balance and strain displacement coordination relation of a reinforced area is obtained according to the coordinate position of the reinforced structure and the material or geometric parameters of the reinforced structure:
in the formula: sigma 1 、σ 2 、σ 6 The upper right corner marks p and s respectively mark stress variables corresponding to the plate and the rib for the stress related to the in-plane bending, and the lower right corner marks represent in-plane bending deformation;
material rigidity coefficient A of plate obtained based on composite laminated plate theory p 、B p 、D p …H p Is composed of
The ribs are simplified into a flexible beam structure, a rectangular section laminated beam is taken as an example, and a corresponding rigidity coefficient A can be obtained through similar derivation s 、B s 、D s …H s 。
As a further scheme of the invention: in step 3, the equivalence of the local region is further derivedCoefficient of stiffness
The following conditions are satisfied:
wherein, A p ～H p 、A s ～H s The rigidity coefficient matrixes of the flat plate and the rib to the neutral surface are respectively.
As a further scheme of the invention: in step 4, considering the oblique reinforcement condition, the stiffness coefficient matrix of equation (8) may be rewritten as:
82309: (9) wherein a i ，b i ，c 2i And c 2i-1 A geometric equation of the ith rib parallel edge line is obtained; t is i A local-integral coordinate transformation matrix is set for the ith rib; lambda is a transition region smoothing coefficient;
thereby obtaining an equivalent variable stiffness coefficient function established by the hyperbolic tangent function, wherein the stiffness coefficient considers the influence of the pull-twist, pull-bend and bend-twist coupling stiffness.
As a further scheme of the invention: in step 8, the boundary layer is decomposed into
Magnitude, order of different buckling boundary layer effect parameters epsilon, wherein the axial pressure boundary layer effect is epsilon 1 Step, external pressure buckling boundary layer effect is epsilon 3/2 Order, torsional boundary layer effect of epsilon 5/4 Under the action of mechanical loads such as axial pressure, external pressure, torque and the like, the small-deflection classical solution and the initial geometric defect of the completely anisotropic reinforced cylindrical shell can be taken as follows:
and (3) axial compression buckling:
external pressure buckling:
and (3) torsional buckling:
wherein the content of the first and second substances,
is a defect parameter.
As a further scheme of the invention: in step 9, the numerical value of deflection is subjected to fourier expansion to generate terms corresponding to coefficients in equations (10), (12) and (14).
As a further scheme of the invention: in step 13, the post-flexion equilibrium path is obtained by substituting formula (5) for formula (1 a) to formula (3 d).
As a further scheme of the invention: in step 13, the shell is completed
Or μ =0, order
Usually W m ≠0；
For general initial defects
Or μ i Not equal to 0, terms corresponding to the coefficients in the equations (11), (13) and (15) are generated by a fourier expansion method, and the minimum buckling load and the corresponding buckling mode (m, n) are easily obtained by comparison.
Compared with the prior art, the invention has the beneficial effects that:
(1) According to the method, accurate curvature expression, geometric nonlinear relation and the relation between a reinforcement structure and a shell coordinate position are introduced into buckling and post-buckling analysis of axial pressure, external pressure and torsion working conditions of the fiber reinforced anisotropic reinforced laminated cylindrical shell, and different combinations and distribution forms of ribs can be considered;
(2) The method simultaneously considers the influence of front buckling nonlinear deformation, rear buckling large deflection and initial geometric defects and the influence of transverse shear deformation and coupling rigidity, adopts a singular perturbation method to provide asymptotic solution of the rear buckling large deflection of the completely anisotropic reinforced laminated cylindrical shell, which not only meets a control equation under the action of axial pressure, external pressure and torsional load, but also strictly meets boundary conditions in asymptotic sense, and uses the singular perturbation method to obtain buckling load and rear buckling balance path of the sheared cylindrical shell under the action of the axial pressure;
(3) Studies by this method showed that: different paving modes, paving sequences, geometric parameters, rib rigidity and distribution forms have obvious influence on the axial pressure, external pressure, torsional buckling critical load and post-buckling path of the anisotropic cylindrical shell with the medium thickness, and can reflect the local buckling phenomenon of the structure;
(4) The results obtained by this method confirm that: (a) Under the action of axial compression, the post-buckling path of the completely anisotropic reinforced composite material laminated cylindrical shell is unstable, the cylindrical shell is sensitive to initial geometric defects, and shear stress and torsion are generated along with the completely anisotropic cylindrical shell under the action of axial compression; (b) Under the action of external pressure, a post-buckling path of the medium-length completely anisotropic reinforced composite material laminated cylindrical shell is stable, the cylindrical shell is insensitive to the initial geometric defect, and shear stress and torsion are generated along with the completely anisotropic cylindrical shell under the action of external pressure; (c) The results obtained by the method prove that under the action of torque, the post-buckling path of the completely anisotropic reinforced composite material laminated cylindrical shell is weak and stable, the cylindrical shell is insensitive to the initial geometric defect, and the completely anisotropic cylindrical shell is accompanied by compressive stress besides shear stress when being subjected to the torque action;
(5) The quantitative relation between the deflection and the corner is obtained by the method, and the method has important significance for predicting a modal coarsening energy transfer mechanism and buckling propagation.
Drawings
FIG. 1 is a schematic view of a composite cylindrical shell under axial compression and its coordinate system in an embodiment of the present invention.
FIG. 2 is a schematic diagram of a composite cylindrical shell and its coordinate system under external pressure in an embodiment of the present invention.
FIG. 3 is a schematic diagram of a composite cylindrical shell and its coordinate system under the action of torque in an embodiment of the present invention.
Fig. 4 is a schematic view of the geometric relationship between the shell and the reinforcement structure in the embodiment of the invention.
Fig. 5 is a schematic flow chart of the analysis of buckling and post-buckling of the anisotropic laminated composite cylindrical stiffened shell in the embodiment of the invention.
FIG. 6 is a schematic diagram of an exemplary initial geometric distortion defect in an embodiment of the present invention; wherein (a) is integral corrugation deformation; and (b) local convex-concave deformation.
Fig. 7 is a schematic diagram of the post-axial buckling equilibrium path of a orthotropic laminated composite cylindrical shell in an embodiment of the invention.
Fig. 8 is a schematic diagram of the equilibrium path of buckling after external pressure for a orthotropic laminated composite cylindrical shell in an embodiment of the present invention.
Fig. 9 is a schematic diagram of the post-twist buckling equilibrium path of the anisotropic laminated composite cylindrical shell in an embodiment of the invention.
FIG. 10 is a schematic diagram illustrating the effect of the anisotropic effect on the buckling equilibrium path after shearing a cylindrical shell in an embodiment of the present invention. Wherein, (a) is a torque-end shortening relationship diagram; and (b) is a torque-rotation angle relation diagram.
Detailed Description
The technical solutions in the embodiments of the present invention will be described clearly and completely with reference to the accompanying drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
Specific implementations of the present invention are described in detail below with reference to specific embodiments.
An explicit rapid analysis method for buckling and post-buckling under the action of mechanical load is used for a composite material reinforced shell containing accurate curvature expression, and comprises the following steps:
step 1: based on a high-order shear deformation theory, setting the transverse shear displacement-strain relation of the cylindrical shell to be distributed according to a parabolic rule along the shell thickness direction;
step 2: the stiffened plate is equivalent to a variable-rigidity plate structure, namely the influence of the plate structure rigidity superposed ribs corresponding to the local stiffened part is taken into consideration;
and step 3: according to the pure bending condition, the stress on the bending neutral plane of the plate structure is zero, and the height h of the local neutral plane can be determined 0 ；
And 4, step 4: combining the rigidity coefficients of the rib-free area and the reinforced area into a variable rigidity function;
meanwhile, in order to ensure the conductivity of the stiffness coefficient matrix with respect to the position coordinate, a hyperbolic tangent function is introduced to carry out smooth transition on the variable stiffness coefficient matrix;
and 5: obtaining an expression of internal force and bending moment of the reinforced cylindrical shell structure under common laying conditions according to the equivalent constitutive relation of the composite material reinforced plates;
step 6: according to the Hamilton principle, obtaining a balance differential equation of the composite material reinforced cylindrical shell by using an Euler-Lagrange equation;
and 7: introducing general form initial defect expression, carrying out non-dimensionalization treatment on a balance differential equation, and introducing a small parameter epsilon with obvious physical significance, namely that the small parameter epsilon is in inverse proportion to the geometric parameter Z of the equivalent length of the shell;
when epsilon is less than 1, the composite material reinforced cylindrical shell balance differential equation is a boundary layer equation;
and 8: solving by using a singular perturbation method, and dividing the solution of an equation into a regular solution and a boundary layer solution;
and step 9: processing the initial defect numerical value of the completely anisotropic reinforced cylindrical shell to form deflection numerical values generally distributed at different positions of the shell;
meanwhile, considering the influence of a heat effect, generating a heat bending moment and initial deflection, and substituting the heat bending moment and the initial deflection into a composite material reinforced cylindrical shell balance differential equation for calculation;
step 10: obtaining perturbation equation sets of each order according to the equilibrium differential equation of the reinforced cylindrical shell made of the discrete composite material with the same power of epsilon, solving the perturbation equation sets of each order step by step and synthesizing a regular solution and a boundary layer solution to obtain a large-deflection asymptotic solution which strictly meets the boundary condition of the fixed support in the asymptotic sense;
on the basis, obtaining a quantitative relational expression of deflection and corner and a boundary layer width expression of shell buckling, and obtaining shell structure equivalent stress sigma by using constitutive relation ij An expression;
step 11: obtaining the dimensionless deflection w and the pressure lambda T And shear stress lambda s A post-flexion equilibrium pathway of expression; wherein:
and (3) axial compression buckling:
and:
external pressure buckling:
and:
and (3) torsional buckling:
and:
and relative torsion angle:
step 12: converting the parameters (A) in the formulas (1 a) to (3 d) by using the second perturbation parameter 1 ( 1 2) Epsilon) into a dimensionless maximum deflection, i.e.:
wherein, W m In the flexibility expression, the point (x, y) = (pi/2 m, pi/2 n) is the maximum flexibility without dimension, and the point (x, y) = (pi/2 m, pi/2 n) comprises the following points:
step 13: and obtaining a post-buckling balance path of the shearing cylindrical shell with dimensionless maximum deflection as perturbation parameters under the action of torque load.
In step 2, rectangular section ribs are adopted, and the equivalent stress balance and strain displacement coordination relation of a reinforced area is obtained according to the coordinate position of the reinforced structure and the material or geometric parameters of the reinforced structure:
in the formula: sigma 1 、σ 2 、σ 6 The upper right corner marks p and s respectively mark stress variables corresponding to the plate and the rib for the stress related to the in-plane bending, and the lower right corner marks represent in-plane bending deformation;
material rigidity coefficient A of plate obtained based on composite laminated plate theory p 、B p 、D p …H p Is composed of
The ribs can be simplified into a flexible beam structure, and the corresponding rigidity coefficient A can be obtained by similar derivation by taking a laminated beam with a rectangular section as an example s 、B s 、D s …H s 。
In step 3, the equivalent stiffness coefficient of the local area is further derived
The following conditions are satisfied:
wherein A is p ～H p 、A s ～H s The rigidity coefficient matrixes of the flat plate and the rib to the neutral surface are respectively.
In step 4, considering the oblique reinforcement condition, the stiffness coefficient matrix of equation (8) may be rewritten as:
82309: (9) wherein a i ，b i ，c 2i And c 2i-1 A geometric equation of the ith rib parallel edge line is obtained; t is i Converting a matrix for the ith rib local-overall coordinate; lambda is a transition region smoothing coefficient;
thereby obtaining an equivalent variable stiffness coefficient function established by the hyperbolic tangent function, wherein the stiffness coefficient considers the influence of the pull-twist, pull-bend and bend-twist coupling stiffness.
In step 7, for most composites,
thus, when
When we have epsilon<1. In particular, for isotropic cylindrical shells, there are
Wherein
As a Batdorf shell parameter, for the classical cylindrical shell linear buckling analysis,
in actual engineering, it is common
There is always epsilon < 1.
In step 8, the boundary layer is decomposed into
Magnitude, order of different buckling boundary layer effect parameters epsilon, wherein the axial pressure boundary layer effect is epsilon 1 Step, external pressure buckling boundary layer effect is epsilon 3/2 Order, torsional boundary layer effect of epsilon 5/4 Under the action of mechanical loads such as axial pressure, external pressure, torque and the like, the small-deflection classical solution and the initial geometric defect of the completely anisotropic reinforced cylindrical shell can be taken as follows:
and (3) axial compression buckling:
external pressure buckling:
and (3) torsional buckling:
wherein the content of the first and second substances,
is a defect parameter.
In step 9, the numerical value of deflection is subjected to fourier expansion to generate terms corresponding to coefficients in equations (10), (12) and (14).
In step 13, the post-flexion equilibrium path is obtained by substituting formula (5) for formula (1 a) to formula (3 d).
In step 13, the shell is completed
Or μ =0, order
Usually W m ≠0；
For general initial defects
Or mu i Not equal to 0, terms corresponding to the coefficients in the equations (11), (13) and (15) are generated by a fourier expansion method, and the minimum buckling load and the corresponding buckling mode (m, n) are easily obtained by comparison.
Application example:
the invention relates to a composite material reinforced shell containing accurate curvature expression, which is divided into a skin and ribs, wherein the length of the reinforced shell is L, the radius of the reinforced shell is R, the skin is formed by N layers of orthogonal single layers with the thickness of t, and the ribs are uniformly distributed on the skin for local reinforcement;
taking a reinforced cylindrical structure as a variable-rigidity medium-thick shell, and considering the action of initial geometric defects and non-uniform temperature load delta T (x, y, z) so as to
And
respectively represents displacement components of the shell middle surface and an arbitrary point along X, Y and Z directions in a right-hand coordinate system,
and
representing the rotation angles of the mid-plane normal relative to the Y-axis and the X-axis, respectively, the displacement field of the anisotropic cylindrical shell is related as follows:
the method has a similar calculation process (as shown in fig. 6) for the buckling and post-buckling analysis of the cylindrical shell under the action of axial pressure, external pressure and torsion, and in order to compare the method with the existing test results conveniently, the three buckling conditions are respectively subjected to example analysis and are introduced in a crossed manner. Firstly, the buckling load and the minimum post-buckling load of the cylindrical shell under the action of axial pressure, external pressure and torsion are respectively given, and compared with the literature structure, the results are detailed in tables 1 to 3 below.
TABLE 1 comparison result of buckling critical load of additional rib axial compression cylindrical shell
1 Suppose thatIs a simple boundary condition; 2 assuming a solidus boundary condition; 3 the numerical values in brackets are respectively the axial half wave number and the circumferential wave number of the buckling mode of the shell; 4 the method is used for fixing and supporting buckling load under boundary conditions; 5 the method calculates a minimum post-buckling load theoretical value; 6 Singer,J.and Abramovich,H.Vibration techniques for definition of practical boundary conditions in stiffened shells.AIAA Journal,17:762–769,1979。
TABLE 2 static water external pressure without reinforcement (0/90) 12T Comparison result (E) of buckling critical load (MPa) of cylindrical shell 11 ＝162.0GPa,E 22 ＝9.6GPa,G 12 ＝G 13 ＝6.1GPa,G 23 ＝3.5GPa,ν 12 ＝0.298)
1 Hur,S.H.,Son,H.J.,Kweon,J.H.and Choi,J.H.,2008,“Postbuckling of composite cylinders under external hydrostatic pressure,”Composite Structures,86,pp.114-124.
2 The numerical values in parentheses are the shell buckling mode axial half wave number and the circumferential wave number, respectively.
TABLE 3 Critical load of buckling of cylindrical shell of composite material under torque action (N) xy ) cr (lbf/in) comparison (R =7.5in, L = 15in)
1 The numerical values in parentheses are the shell buckling mode circumferential wave numbers respectively.
2 Wilkins,D.J.,Love,T.S.Compression-torsion buckling test of laminated composite cylindrical shells.AIAA paper,pp.74-379,1974.
3 Simitses G.J.,Shaw,D.,Sheinman,I.Stability of imperfect laminated cylinders:a comparison between theory and experiment.AIAA Journal,1985,23:1086-1092。
The specific process is as follows:
the real boundary conditions in the experiment are close to the solidus boundary.
Calculating the rigidity coefficient A according to the geometric and material parameters of the cylindrical shell structure p ～H p And the rigidity coefficient A of the reinforced part s ～H s . The calculation method of the rigidity coefficient of each material is similar, and A is used here p For example, the following steps are carried out:
axle pressure cases (Table 1-AB6, R =120.1mm, ns =85, al 7075-76E =75.0kg/mm 2 ，v＝0.3)：
External pressure cases (Table 2-CTM3, E11=162.0GPa, E22=9.6GPa, G12= G13=6.1GPa, G23=3.5GPa, v) 12 ＝0.298)：
Torsion cases (table 3, boron/epoxy, E11=30.0 × 106lbf/in2, E22=2.7 × 106lbf/in2, G12= G13= G23=0.65 × 106lbf/in2, v 12 ＝0.21，(45/-45)S，R＝7.5in，L＝15in)：
Calculating strain according to the displacement-strain relation, and providing an expression of the section force N and the moment M and a calculation result;
calculating the reduced stiffness coefficient of the cylindrical shell
If the longitudinal or circumferential reinforcement structure exists, the geometric (such as the height and thickness of the ribs and the distance d between the longitudinal ribs and the ribs) are required to be determined according to the geometric characteristics 1 Eccentricity of longitudinal rib e 1 Outside, inThe ribbed sign is negative; or the circumferential rib inter-rib spacing d of the circumferential rib 2 Eccentricity e of circumferential rib 2 The sign of the additional rib is negative) and material parameters (e.g. modulus of elasticity E of the longitudinal rib S1 Longitudinal rib shear modulus G 1 Or modulus of elasticity E of circumferential rib S2 Shear modulus G of circumferential rib 2 ) Calculating the section area A of the longitudinal rib at the position of the reinforced coordinate 1 Sectional area A of circumferential rib 2 Longitudinal rib moment of inertia I 1 Circular rib moment of inertia I 2 Longitudinal rib torque J 1 Torque J of circumferential rib 2 Calculating the reduced stiffness coefficient of the reinforcing rib
If the geodesic line reinforcement structure exists, the geometric (such as the height and the thickness of the ribs and the inter-rib spacing d of the geodesic line ribs) are required to be determined 3 And eccentricity e of the rib of the geodesic line 3 The sign of the additional rib is negative, the number Ng of the circumferential geodesic ribs of the geodesic line rib and the axial included angle T g ) With material parameters (e.g. geodesic rib modulus of elasticity E) S3 Shear modulus G of geodesic rib 3 ) Calculating the section area A of the geodesic rib at the position of the reinforced coordinate 3 And the inertia moment I of the earth wire rib 3 And the torque J of the earth line rib 3 Calculating the reduced stiffness coefficient of the reinforcing rib
According to the Hamilton principle, obtaining a composite material reinforced cylindrical shell balance differential equation by using a Euler-Lagrange equation;
introducing general form initial defect expression, carrying out non-dimensionalization treatment on the equilibrium differential equation, and introducing small parameter epsilon
Has obvious physical significance, namely equivalent length geometric parameters of the shell and the shell
In inverse proportion.
For the most part of the composite material,
thus, when
When we have epsilon<1。
In particular, for isotropic cylindrical shells, there are
Wherein
As a Batdorf shell parameter, for the classical cylindrical shell linear buckling analysis,
in actual engineering, it is common
Always has epsilon<<1. When epsilon<1, the balance differential equation of the composite material reinforced cylindrical shell is a boundary layer equation.
The solution is performed using a singular perturbation method, the solution to the equation is divided into an "external" solution (canonical solution) and a boundary layer solution, the boundary layer solution being
Magnitude, axial buckling boundary layer effect of epsilon 1 Step, external pressure buckling boundary layer effect is epsilon 3/2 Order, torsional boundary layer effect of epsilon 5/4 And (5) step. The dimensionless small-deflection classical solution of the completely anisotropic reinforced cylindrical shell can be as follows:
buckling after axial compression:
buckling after external pressure:
buckling after torsion:
let the initial geometric defect of the shell have the following form:
buckling after axial compression:
buckling after external pressure:
buckling after torsion:
wherein the content of the first and second substances,
is a defect parameter;
the initial defect value of the completely anisotropic reinforced cylindrical shell is processed to form deflection values (for example, the integral waveform defect, shown in fig. 6 (a)) generally distributed at different positions of the shell, and a Fourier expansion method is adopted to generate terms corresponding to coefficients in an initial geometric defect expression.
For example, with local dimple defects (as shown in FIG. 6 (b)), the mathematical model can be a bi-directional exponential decay function tableSign
As a defect amplitude, C 1 And C 2 Half the characteristic length of the axial and circumferential directions of the dimple-type defect.
Further, a mathematical expression of the unquantized local pit defect can be obtained:
wherein the content of the first and second substances,
mu is a defect parameter;
meanwhile, if the influence of a thermal effect is considered, generating a thermal bending moment and initial deflection, and substituting the thermal bending moment and the initial deflection into a composite material reinforced cylindrical shell balance differential equation for calculation;
according to the equilibrium differential equation of the reinforced cylindrical shell made of the discrete composite material with the same power of epsilon, perturbation equation sets of each order can be obtained, solution is carried out step by step to synthesize a regular solution and a boundary layer solution, and a large-deflection asymptotic solution which strictly meets the boundary condition of the fixed support in the asymptotic sense is obtained (the analysis flow is shown in figure 5).
The asymptotic solution of buckling after axial compression is as follows:
the asymptotic solution of external pressure post-flexion is:
the asymptotic solution for post-torsion buckling is:
all coefficients in the above solution can be expressed as
In the form of (a). From the deflection expression, it can be seen that the forward buckling deformation is non-linear.
On the basis, obtaining a quantitative relational expression of deflection and corner under different types of loads, namely:
the axial compression corner and deflection are respectively as follows:
the external pressure corner and deflection are respectively as follows:
the torsion angle and the deflection are respectively as follows:
under the condition of axial compression: boundary layer width expression of AB6 buckling of shell
Wherein the housing is given for specific geometrical and material parameters
The values of (A) are all constant values.
Under external pressure: boundary layer width expression of CTM3 buckling of shell
Wherein the housing is given for specific geometrical and material parameters
The values of (A) are all constant values.
In the case of twisting: boron fiber resin based composite material (45/-45) S Boundary layer width expression for shell buckling
Wherein the housing is given for specific geometrical and material parameters
All values of (A) areA constant value.
Further obtain the dimensionless deflection w and the pressure lambda p And shear stress lambda s The post flexion equilibrium pathway of expression.
Bending balance path after axial compression:
and:
the equilibrium path of buckling after external pressure:
and is
Post-twist flexion equilibrium path:
accordingly, the method can be used for solving the problems that,
and:
and relative torsion angle:
it is thus possible to obtain axial compressive stresses of a completely anisotropic cylindrical shell under the action of torque of
Wherein the variable k (in this example, k is 1.9062 at the maximum value of deflection after the occurrence of buckling) can be determined by the following formula:
it should be noted here that before the occurrence of the torsional buckling, the value of the variable k is zero, and the previous small-deflection solution satisfies the boundary condition.
Converting the parameters in the expression by using secondary perturbation parameter
Conversion to dimensionless maximum deflection, i.e.:
wherein, W m Maximum deflection without dimension, theta 1 ＝6.6017×10 -2 (axial and external pressures), theta 1 ＝1.9138×10 -2 (torsion).
In the flexibility expression, the point (x, y) = (pi/2 m, pi/2 n) includes:
wherein, C 3 ＝0.8339，Θ 2 =0.4157 (axial and external pressure); c 3 ＝0.7755，Θ 2 =0.2541 (twist).
Substituting the maximum deflection expression into the post-deflection balance path expression to obtain a post-deflection balance path of the shearing cylindrical shell with dimensionless maximum deflection as perturbation parameters under the action of axial pressure load.
For perfecting the shell
(or μ) i =0,i =1,2,3,4), order
(usually W) m Not equal to 0); for general initial defects
(or μ) i Not equal to 0), generating items corresponding to coefficients in the initial defect expression by adopting a Fourier expansion method, and easily obtaining the minimum buckling load and the corresponding buckling modes (m, n) through comparison.
Analysis of axial pressure case results:
the above calculations are shown in table 1, and it can be seen that the results of the method (especially the post-buckling minimum load, including the buckling mode) are closer to the experimental results. To further verify the correctness of the method, an axially depressed inner stiffened lattice shell (0/90) was calculated S And (-45/45/90/0) S And compared to results of analysis by Gerhard et al (Gerhard, C.S., gurdal, Z., kapania, R.K.,1996.Finite element analysis of geological and vertebral composite shells using a layerwise theory. NASA CR 1471, 1996) as shown in tables 4 and 5 below:
TABLE 4 inner reinforcement of geodesic line under axial pressure orthometric symmetrically laid cylindrical shell (0/90) S Buckling load comparison results (R =85 inches, L =100 inches, t =0.2 inches, 1 × 12grid shell，
)
6 FEA-finite element analysis abbreviation; 7 the shell layering is similar to theory, and Table 5 has the same meaning;
Gerhard,C.S.,Gurdal,Z.,Kapania,R.K.,1996.Finite element analysis of geodesically stiffened cylindrical composite shells using a layerwise theory.NASACR 1471。
TABLE 5 inner reinforcement of geodesic line under axial pressure orthometric symmetrically laid cylindrical shell (-45/45/90/0) S Buckling load comparison results (R =85 inches, L =100 inches, t =0.2 inches, 1 x 12grid shell,
)
Gerhard,C.S.,Gurdal,Z.,Kapania,R.K.,1996.Finite element analysis of geodesically stiffened cylindrical composite shells using a layerwise theory.NASACR 1471。
the geometric parameters were taken as R =85 inches (1 inch =25.4 mm), L =100 inches, t =0.2 inches, 1 x 12 geodesic bars,
and the rib height is 0.5,1.0,1.5 inches respectively, and the thickness is 0.2 inches. It can be seen that the current results are reasonably consistent with those of Gerhard et al. (1996).
FIG. 7 shows a comparison of (0/90) S Post-buckling load shortening curve and farhadini (fahadini, mahood. Finish element analysis and experimental) of laminated cylindrical shell under axial compressionevaluation of packaging phenol in laboratory composite tubes and plates. PhD. Thesis, university of Missouri-Rolla, 1992). Calculated data were used of L =12.0in (1in = 254mm), R/t =74.5, t =0.02in, young's modulus and shear modulus respectively: e 11 ＝6.56×106psi(1psi＝6.895kPa)，E 22 ＝1.167×106psi，G 12 ＝G 13 ＝0.886×106psi，v 12 =0.28. The limit point load for the method results was 407.046lb/in (1lb =0.454 kg) and the finite element solution result was 449.7lb/in when compared to 394.2 lb/in.
The calculation result of the method is well matched with the experimental result.
External pressure case result analysis:
the above calculation results are shown in table 2, and it can be seen that the results of the method (especially the post-buckling minimum load, including the buckling mode) are closer to the experimental results. To further verify the correctness of the method, an externally pressed and internally stiffened grid shell (0/90) was calculated S And (-45/45/90/0) S And compared to results of analysis by Gerhard et al (Gerhard, C.S., gurdal, Z., kapania, R.K. Finite element analysis of geological and saline composite shells using a layerwise theory. NASA CR 1471, 1996), as shown in tables 6 and 7 below:
TABLE 6 inner reinforcement of geodesic line under external pressure orthometric symmetrically laid cylindrical shell (0/90) S Buckling load comparison results (R =85 inches, L =100 inches, t =0.2 inches, 1 × 12grid shell,
)
3 FEA-finite element analysis abbreviation; 4 the shell layering is similar to theory, and Table 7 has the same meaning;
Gerhard,C.S.,Gurdal,Z.,Kapania,R.K.,1996.Finite element analysis of geodesically stiffened cylindrical composite shells using a layerwise theory.NASA CR 1471。
TABLE 7 inner reinforcement of geodesic line under external pressure orthometric symmetrically laid cylindrical shell (-45/45/90/0) S Buckling load comparison results (R =85 inches, L =100 inches, t =0.2 inches, 1 x 12grid shell,
)
Gerhard,C.S.,Gurdal,Z.,Kapania,R.K.,1996.Finite element analysis of geodesically stiffened cylindrical composite shells using a layerwise theory.NASA CR 1471。
the geometric parameters were taken as R =85 inches (1 inch =25.4 mm), L =100 inches, t =0.2 inches, 1 x 12 geodesic bars,
and the rib heights are respectively 0.5,1.0,1.5 and 2.0 inches, and the thickness is 0.2 inches. It can be seen that the current results are reasonably consistent with the results of Gerhard et al (Gerhard, C.S., gurdal, Z., kapania, R.K. Finite element analysis of geodesic contaminated cylindral composition shells used a layerwise theory. NASA CR 1471, 1996).
FIG. 8 compares (0/90) numbered CTM1 12T And (3) a post-buckling load shortening curve of the laminated cylindrical shell under the action of hydrostatic external pressure under a solid-branch condition and experimental results of Hur (2008). The calculation data used were L =600.0mm, R/t =62.7, t =2.52mm, E11=162.0GPa, E22=9.6GPa, G12= G13=6.1GPa, G23=3.5GPa, v 12 =0.298. The result of the method is taken at the initial defect as
And compared with the experimental results.
The calculation result of the method is well matched with the experimental result.
Torsion case results analysis:
the above calculations are shown in Table 3, from which it can be seen that the results herein are very close to the experimental results, since the solution of Simtses et al (Simtses G.J., shaw, D., sheinman, I.Stabilty of experimental laboratory proven cyclinders: a complex between the results and the experimental. AIAA Journal, 23. It can be seen that the results of this method (especially post-buckling minimum loads, including the buckling mode) are closer to the experimental results.
In order to further verify the correctness of the method, the method is compared with the buckling load of the cylindrical shell of the multilayer composite material obtained by adopting an energy method and a finite difference method during the high pressure cycling and the high pressure cycling. Table 8 shows the experimental results of this method with high efficiency and high efficiency, tennyson (Tennyson, R.C. Buckling of laminated composite cyclines: a review. Composites, 6-24, 1975,) and Tennyson for comparison with Koiter's theoretical results. The geometric parameters are given in table 8, the test piece is a glass fiber/epoxy column shell, and the material constants are: e 11 ＝5.5×106psi，E 22 ＝2.6×106psi，G 12 ＝G 13 ＝G 23 ＝0.7×106psi，ν 12 =0.37. The results show that the results are generally closer to the experimental results than the analysis results of high frequency and high frequency, and the influence of the initial geometric defects should be considered when performing the torsional buckling analysis.
TABLE 8 shear cylindrical Shell buckling load under Torque Effect (M) S ) cr (lbf in) comparison (R =6.26in, t =0.027in, L = 12.5in)
Tennyson,R.C.Buckling oflaminated composite cylinders:areview.Composites,1975,6:17-24；
High pressure cycling, non-linear instability calculation of the cylindrical shell of multi-layer composite material, mathematics and mechanics, 1986,7 (1): 17-23.
FIG. 4 compares (15/0/10/-10/0/-15) S Post-flexion shear stress-shear strain curves of cylindrical shells under torque and results of experiments with Derstine (Derstine, m.s., pindera, m.j., bowles, d.e. combined mechanical loading of composite tubes, NASA CR-183012, 1988). The calculated data adopted are that L =10.0 inches (1 inch =254 mm), R/t =17.17, t =0.02 inch, and P75/934 resin-based graphite fiber reinforced material E is adopted as the material 11 ＝35.25×106psi(1psi＝6.895kPa)，E 22 ＝1.04×106psi，G 12 ＝G 13 ＝0.570×106psi，v 12 ＝0.331，v 23 =0.49. When accounting for initial defect effects
The calculation result of the method is well matched with the experimental result.
FIG. 5 shows the same length of the housing under torque
Shear cylinder shell (+/-45) 4T 、(±45) 2S And (-452/-302/602/152) T Torque-tip shortening and torque-turn angle curves for the three lay-down modes. Assuming that all the pavements are equal in thickness and the total thickness is t =4.0mm, R/t =40, the material constants are taken as E for the graphite/epoxy composite material 11 ＝138.0GPa，E 22 ＝8.9GPa，G 12 ＝G 13 ＝5.17GPa，G 23 ＝2.89GPa，ν 12 =0.30, and is shown in all figures
Representing the initial geometric defect value of the shell. As can be seen in FIG. 5, for the three lay down shear shells, (-452/-302/602/152) T The torsional stiffness of the shell is maximized; (+/-45) 4T The shell has the greatest buckling load and post-buckling strength, but its path is relatively flat.
To further examineAccording to the method, the buckling load of the inner longitudinal reinforcement cylindrical shell under the action of torque is calculated, and the geometric material parameter results of SCT1 (non-reinforced) and SCT2 (11 longitudinal reinforced) samples are reported by Kuenzi and Norris (Kuenzi, E.W., norris, C.B.,1962. Torque bundling of longitudinal reinforcement, thin-walled, complex cylindrical connectors, no.1563, united States Department of agricultural purpose Products Laboratory, madison, wisconsin, in collaboration with the University of Wisconsin, 1962), and are shown In the following table 9. The geometric parameters were taken to be R =9.15 inches (1 inch =25.4 mm), L =29 inches, t =0.037 inches, and 0.029 inches rib height and 0.246 inches rib thickness of the inner evenly spaced ribs. The SCT1 material parameters are as follows: e 11 ＝1.189×106psi 2，E 22 ＝1.189×106psi，G 12 ＝G 13 ＝G 23 ＝0.871×106psi，ν 12 =0.39; the SCT2 material parameters are as follows: e 11 ＝1.171×106psi，E 22 ＝1.171×106psi，G 12 ＝G 13 ＝G 23 ＝0.871×106psi，ν 12 =0.37, rib material parameters are: e 11 ＝2.026×106psi，G 12 ＝0.74×106psi。
TABLE 9 buckling load tau of longitudinal inner stiffened cylindrical shell under torque effect cr (psi) comparative results (R =9.15 inches, L =29 inches, t =0.037 inches)
4 Kuenzi,E.W.,Norris,C.B.,1962.Torsional buckling of longitudinally stiffened,thin-walled,plywood cylinders.No.1563,United States Department of Agriculture Forest Service Forest Products Laboratory,Madison,Wisconsin,In Cooperation with the University of Wisconsin；
5 The numbers in parentheses are the torsional buckling circumferential wavenumber.
Torsional buckling critical load tau given by the method cr (psi) is well matched with the experimental results.
Further comparing the calculation time lengths of the different calculation methods in the above case (as shown in table 10 below), compared with the commercial finite element method, the calculation time length of the method is greatly reduced, and the great efficiency advantage of the semi-analytic explicit solution to the problem is reflected.
TABLE 10 average calculation time of buckling of axial compression, external compression and torsion reinforced cylindrical shell
The two calculation methods run on the same workstation (InterXeon CPU E5-26962.20GHz processor, 256 GB).
It should be noted that, in the present invention, although the description is made according to the embodiments, not every embodiment includes only one independent technical solution, and such description of the description is only for clarity, and those skilled in the art should integrate the description, and the technical solutions in the embodiments can also be combined appropriately to form other embodiments understood by those skilled in the art.
Claims (8)
1. An explicit rapid analysis method for buckling and post-buckling under the action of mechanical load, which is used for a composite material reinforced shell containing accurate curvature expression, is characterized by comprising the following steps:
step 1: based on a high-order shear deformation theory, setting the transverse shear displacement-strain relation of the cylindrical shell to be distributed according to a parabolic rule along the shell thickness direction;
step 2: the stiffened plate is equivalent to a variable-rigidity plate structure, namely the influence of the plate structure rigidity corresponding to the local stiffened part on the superimposed ribs is obtained, and the rigidity increment of the ribs is considered;
and step 3: according to the pure bending condition, the stress on the bending neutral plane of the plate structure is zero, and the height h of the local neutral plane can be determined 0 ；
And 4, step 4: combining the rigidity coefficients of the rib-free area and the reinforced area into a variable rigidity function;
meanwhile, in order to ensure the conductivity of the stiffness coefficient matrix with respect to the position coordinate, a hyperbolic tangent function is introduced to carry out smooth transition on the variable stiffness coefficient matrix;
and 5: obtaining an expression of internal force and bending moment of the reinforced cylindrical shell structure under common laying conditions according to the equivalent constitutive relation of the composite material reinforced plates;
step 6: according to the Hamilton principle, obtaining a composite material reinforced cylindrical shell balance differential equation by using a Euler-Lagrange equation;
and 7: introducing general form initial defect expression, carrying out non-dimensionalization treatment on the equilibrium differential equation, and introducing small parameter epsilon with obvious physical significance, namely equivalent length geometric parameter of the shell
In inverse proportion;
when epsilon is less than 1, the composite material reinforced cylindrical shell balance differential equation is a boundary layer equation;
and 8: solving by using a singular perturbation method, and dividing the solution of an equation into a regular solution and a boundary layer solution;
and step 9: processing the initial defect numerical value of the completely anisotropic reinforced cylindrical shell to form deflection numerical values generally distributed at different positions of the shell;
meanwhile, considering the influence of a heat effect, generating a heat bending moment and initial deflection, and substituting the heat bending moment and the initial deflection into a composite material reinforced cylindrical shell balance differential equation for calculation;
step 10: obtaining perturbation equation sets of each order according to the equilibrium differential equation of the reinforced cylindrical shell made of the discrete composite material with the same power of epsilon, solving the perturbation equation sets of each order step by step and synthesizing a regular solution and a boundary layer solution to obtain a large-deflection asymptotic solution which strictly meets the boundary condition of the fixed support in the asymptotic sense;
on the basis, obtaining a quantitative relation expression of deflection and corner and a boundary layer width expression of shell deflection, and obtaining the shell structure equivalent stress sigma by using constitutive relation ij An expression;
step 11: obtaining the dimensionless deflection w and the pressure lambda T And shear stressλ s A post-flexion equilibrium pathway of expression; wherein:
and (3) axial compression buckling:
and:
external pressure buckling:
and:
and (3) torsional buckling:
and:
and relative torsion angle:
step 12: converting the parameters in the formulas (1 a) to (3 d) by using secondary perturbation parameter conversion
Conversion to dimensionless maximum deflection, i.e.:
wherein, W m In the flexibility expression, the point (x, y) = (pi/2 m, pi/2 n) is the maximum flexibility without dimension, and the point (x, y) = (pi/2 m, pi/2 n) comprises the following points:
step 13: and obtaining a post-buckling balance path of the shearing cylindrical shell with dimensionless maximum deflection as perturbation parameters under the action of torque load.
2. The explicit rapid analysis method for buckling and post-buckling under the action of mechanical load according to claim 1, characterized in that in step 2, rectangular section ribs are adopted, and the coordination relationship between equivalent stress balance and strain displacement of a reinforced area is obtained according to the coordinate position of a reinforced structure and the material or geometric parameters of the reinforced structure:
in the formula: sigma 1 、σ 2 、σ 6 The upper right corner marks p and s respectively mark stress variables corresponding to the plate and the rib for the stress related to the in-plane bending, and the lower right corner marks represent in-plane bending deformation;
material rigidity coefficient A of plate obtained based on composite laminated plate theory p 、B p 、D p …H p Is composed of
The ribs are simplified into a flexible beam structure, a rectangular section laminated beam is taken as an example, and a corresponding rigidity coefficient A can be obtained through similar derivation s 、B s 、D s …H s 。
3. The explicit rapid analysis method for buckling and post-buckling under mechanical load as claimed in claim 2, wherein in step 3, the equivalent stiffness coefficient of the local region is further derived
The following conditions are satisfied:
wherein A is p ～H p 、A s ～H s The rigidity coefficient matrixes of the flat plate and the rib to the neutral surface are respectively.
4. The explicit rapid analysis method for buckling and post-buckling under mechanical load according to claim 3, wherein in step 4, considering the oblique reinforcement condition, the stiffness coefficient matrix of equation (8) is rewritten as:
82309: (9) wherein a i ，b i ，c 2i And c 2i-1 A geometric equation of the ith rib parallel edge line is obtained; t is i Converting a matrix for the ith rib local-overall coordinate; lambda is a transition region smoothing coefficient;
thereby obtaining an equivalent variable stiffness coefficient function established by the hyperbolic tangent function, wherein the stiffness coefficient considers the influence of the pull-twist, pull-bend and bend-twist coupling stiffness.
5. Mechanical load bearing according to any of claims 1 to 4Explicit rapid analysis of down-buckling and post-buckling characterized by the fact that in step 8 the boundary layer is solved to
Magnitude, order of different buckling boundary layer effect parameters epsilon, wherein the axial pressure boundary layer effect is epsilon 1 Step, external pressure buckling boundary layer effect is epsilon 3/2 Order, torsional boundary layer effect of epsilon 5/4 Under the action of mechanical loads such as axial pressure, external pressure, torque and the like, the small-deflection classical solution and the initial geometric defect of the completely anisotropic reinforced cylindrical shell can be taken as follows:
and (3) axial compression buckling:
external pressure buckling:
and (3) torsional buckling:
wherein the content of the first and second substances,
is a defect parameter.
6. The explicit rapid analysis method for buckling and post-buckling under mechanical load according to claim 5, wherein in step 9, the bending numerical value is generated by using a Fourier expansion method to generate terms corresponding to coefficients in formula (10), (12) and (14).
7. The explicit rapid analysis method for buckling and post-buckling under mechanical load according to claim 6, wherein in step 13, the post-buckling equilibrium path is obtained by substituting formula (5) for formula (1 a) to formula (3 d).
8. Method for the explicit rapid analysis of buckling and post-buckling under mechanical load according to claim 7, characterised in that in step 13, the shell is completed for
Or μ =0, order
Usually W m ≠0；
For general initial defects
Or mu i Note that 0, terms corresponding to the coefficients in the equations (11), (13), and (15) are generated by a fourier expansion method, and the minimum buckling load and the corresponding buckling mode (m, n) are easily obtained by comparison.
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