An analytical method for the cantilever bending deformation of a fiber metal laminated beam with variable cross-section

Abstract

An analytical method for the cantilever bending deformation of a fiber metal laminated beam with variable cross-section belongs to the fields of materials engineering and mechanics, and includes the following steps: obtaining finite-segment discrete variable cross-section micro-element segments of an overall continuously variable cross-section fiber metal heterogeneous laminated beam; based on the laminated plate theory, using the bending stiffness calculation method of the fiber metal heterogeneous laminated beam micro-element segments to obtain the stiffness of the finite-segment discrete variable cross-section micro-element segments; according to the composite beam theory, based on the stiffness of the finite-segment discrete variable cross-section micro-element segments, obtaining the bending deflection at any position of the cantilever of the variable cross-section fiber metal laminated beam under a non-uniform load. This method provides a feasible method for the bending stiffness design of variable cross-section fiber metal laminated structures, provides a fast analytical calculation approach for the ply angle optimization of fiber metal laminated beams, and has reference value for the theoretical evaluation of the mechanical properties of heterogeneous laminated variable cross-section structures.

Description

Technical Field

[0001] The present invention belongs to the fields of materials engineering and mechanics, and relates to an analytical method for the cantilever bending deformation of a fiber metal laminated beam with variable cross-section. Background Art

[0002] Fiber metal laminated material is a hybrid composite material formed by alternatingly laying metal thin plates and fiber prepregs, which perfectly combines the advantages of metal materials and fiber-reinforced composite materials, and has both the elastoplasticity and impact resistance of metal materials, as well as the low density, bridging effect, and fatigue resistance of fiber-reinforced composite materials. With the gradual reduction of the cost of composite materials and the progress of the technology for suppressing the galvanic corrosion between carbon fibers and metals, fiber metal laminated materials have gradually become the key materials for improving the mechanical properties of engineering structures, reducing the unnecessary weight of structures, and thus achieving efficiency increase and energy consumption reduction.

[0003] Fully evaluating the mechanical properties of fiber metal laminated materials is an important prerequisite for promoting the development of the manufacturing and application technologies of fiber metal laminated structures. However, for the laminated structures with variable cross-sections of composite materials and metal heterogeneous materials commonly found in large engineering fields such as aviation, shipbuilding, and architecture, it is usually necessary to carry out structural design, ply angle optimization, and structural load-bearing performance evaluation. Due to the lack of an efficient and fast analytical calculation method for the mechanical properties of fiber metal laminated structures, and at the same time, experimental tests are limited by the number of samples and experimental costs and cannot study numerous parameters, currently, it is still necessary to carry out cumbersome and time-consuming finite element analysis to obtain approximate results, which seriously hinders the multi-parameter collaborative design and multi-factor evaluation of large engineering structures. Therefore, studying the analytical algorithm for the cantilever bending deformation of a fiber metal laminated beam with variable cross-section is conducive to promoting the extended application of fiber metal laminated materials and the development of fiber metal laminated manufacturing technologies. Summary of the Invention

[0004] In order to solve the above problems, the technical solution adopted by the present invention is: an analytical method for the cantilever bending deformation of a fiber metal laminated beam with variable cross-section, comprising the following steps:

[0005] Obtain a finite-segment discrete variable cross-section micro-element segment of an overall continuous variable cross-section fiber metal heterogeneous laminated beam;

[0006] Based on the laminated plate theory, use the bending stiffness calculation method of the fiber metal heterogeneous laminated beam micro-element segment to obtain the stiffness of the finite-segment discrete variable cross-section micro-element segment;

[0007] According to the composite beam theory, based on the stiffness of the finite-segment discrete variable cross-section micro-element segment, obtain the cantilever bending deflection at any position of the fiber metal laminated beam with variable cross-section under a non-uniform load.

[0008] Further: Based on the laminate theory, the bending stiffness of a finite-segment discrete variable cross-section micro-element is obtained by using the bending stiffness calculation method of a fiber-metal heterogeneous laminated beam micro-element segment, including the following steps:

[0009] Judge the material properties of the k-th layer of the fiber-metal heterogeneous laminated beam micro-element segment, and calculate the ply angle transformation matrix and flexibility matrix of the k-th layer material;

[0010] Based on the laminate theory, calculate the stiffness matrix of the k-th layer material within the micro-element segment;

[0011] Based on the laminate theory, calculate the bending stiffness matrix of the fiber-metal heterogeneous laminated beam micro-element segment.

[0012] Further: The judgment of the material properties of the k-th layer of the fiber-metal heterogeneous laminated beam micro-element segment and the calculation of the ply angle transformation matrix and flexibility matrix of the k-th layer material are as follows:

[0013] When the k-th layer material of the micro-element segment is a fiber-reinforced composite material, according to the anisotropic elasticity mechanics, calculate the ply angle transformation matrix and compliance matrix;

[0014] When the k-th layer material of the micro-element segment is a metal material, according to the isotropic elasticity mechanics, calculate the ply angle transformation matrix and compliance matrix.

[0015] Further: The stiffness matrix is used to calculate the stiffness matrix of the k-th layer material within the micro-element segment by the following formula:

[0016] Q (k) =T -1 S -1 [T -1 T (1)

[0017] where T represents the ply angle transformation matrix related to the ply angle θ k of the k-th layer, and S is a 6x6 symmetric compliance matrix related to the engineering material constants;

[0018] From the stiffness matrix of the k-th layer material within the micro-element segment, obtain the bending stiffness matrix D of the fiber-metal heterogeneous laminated beam micro-element segment. Each element D ij of the bending stiffness matrix D satisfies the following formula:

[0019]

[0020] where the subscript ij represents the i-th row and j-th column in the 6x6 stiffness matrix, n i represents the total number of layers of this micro-element segment, and z k+1 and z k represent the upper surface and lower surface thickness coordinates of the k-th layer material, respectively.

[0021] ​Furthermore: The ply angle transformation matrix T is calculated using the following formula:

[0022]

[0023] where: When the k-th layer material of the microelement segment is fiber-reinforced composite material, the ply angle θ k takes the angle between the fiber laying direction and the beam length direction; when the k-th layer material of the microelement segment is metal, its ply angle is set to θ k = 0.

[0024] Furthermore: When the k-th layer material of the microelement segment is fiber-reinforced composite material, according to the anisotropic elasticity theory, each element of the compliance matrix S is calculated using the following formula:

[0025]

[0026] where: E 1 、E 2 and E 3 are the elastic moduli of the fiber-reinforced composite material along the fiber direction, perpendicular to the fiber direction, and along the thickness direction respectively, G 12 、G 13 and G 23 are the shear moduli, S 11 、S 12 、…、S 66 are the elements of the compliance matrix S respectively, ν 12 、ν 13 and ν 23 are the Poisson ratios.

[0027] Furthermore: When the k-th layer material of the microelement segment is metal, according to the isotropic elasticity theory, each element of the compliance matrix S is calculated using the following formula:

[0028]

[0029] where, E represents the elastic modulus of the metal, ν represents the Poisson ratio of the metal, S 11 、S 12 、…、S 66 are the elements of the compliance matrix S respectively.

[0030] Furthermore: The process of obtaining the bending deflection of a variable cross-section microelement cantilever beam under a non-uniform load based on the stiffness of a finite-segment discrete variable cross-section microelement according to the composite beam theory is as follows:

[0031] For a cantilever beam with a length of L, any microelement segment i of the cantilever beam is intercepted. Within the microelement segment i, the load intensity q(x), the beam width b(x), and the beam height h(x) can be approximated as constants q i 、the beam width b iand the beam height h i ;

[0032] The load intensity q within this infinitesimal segment i At any x on the beam k Position (0 ≤ x k ≤ L), the bending moment Satisfies the following formula:

[0033]

[0034] For all infinitesimal segments of the cantilever beam at any x k Position, 0 ≤ x k ≤ L, the total bending moment Satisfies the following formula:

[0035]

[0036] Where k represents the number of any infinitesimal segment on the beam, and i represents the number of any infinitesimal segment with a number larger than k;

[0037] According to the theory of composite beams, the differential equation of the nonlinear deflection curve satisfies the following formula:

[0038]

[0039] Where ω x Represents the deflection at any x position, E xx (x) represents the homogenized elastic modulus along the beam length direction, and I yy (x) represents the moment of inertia of the beam cross-section with respect to the y-axis;

[0040] If under the small deformation condition, the deflection curve is very flat, the Term can be ignored, and the linearized approximate differential equation of the deflection curve is obtained:

[0041]

[0042] For the pure bending loading condition, the bending stiffness matrix D of the infinitesimal segment of the fiber-metal heterogeneous laminated beam and the beam stiffness E xx (x)I yy (x) satisfy the following relationship:

[0043]

[0044] Where [D] -1 (1,1) represents the element in the first row and first column of the matrix obtained after inverting the bending stiffness matrix of the infinitesimal segment of the fiber-metal heterogeneous laminated beam;

[0045] Therefore, the differential equations of the nonlinear and linear deflection curves can be further rewritten in the following formula form:

[0046]

[0047]

[0048] Integrate the differential equation of the deflection curve and substitute the boundary conditions. At x = 0, it satisfies ω 0 = 0 and Obtain the deflection ω of the fiber metal laminated cantilever beam with variable cross-section at any x position x .

[0049]

[0050] An analytical method for the cantilever bending deformation of a fiber metal laminated beam with variable cross-section provided by the present invention aims to quickly calculate the bearing stiffness performance index of the cantilever beam structure of the variable cross-section heterogeneous laminated material, provides an analytical calculation method for the design and mechanical property evaluation of the fiber metal laminated cantilever beam, considers the heterogeneous laminated material characteristics of the metal and fiber-reinforced composite material, and the structural characteristics of variable thickness and variable width. By equivalently converting the overall continuous variable cross-section characteristics into the characteristics of a finite number of discrete variable cross-section micro-element segments, the calculation difficulty of the cantilever bending deformation of the variable cross-section heterogeneous laminated structure is simplified. The method combines the isotropic and anisotropic compliance matrix S calculation methods, establishes a functional relationship between the cantilever bending deformation of the fiber metal heterogeneous laminated beam and the properties of the heterogeneous materials, and introduces a functional relationship between the cantilever bending deformation of the fiber metal heterogeneous laminated beam and the ply angles of each layer of materials through the ply angle transformation matrix T. It has the following advantages: (1) provides a feasible method for the bending stiffness design of the fiber metal laminated structure with variable cross-section; (2) provides a quick analytical calculation approach for the ply angle optimization of the fiber metal laminated beam; (3) has reference value for the theoretical evaluation of the mechanical properties of the heterogeneous laminated variable cross-section structure. BRIEF DESCRIPTION OF THE DRAWINGS

[0051] In order to more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required for the description of the embodiments or the prior art. Obviously, the drawings in the following description are some embodiments of the present invention. For those of ordinary skill in the art, other drawings can be obtained based on these drawings without creative efforts.

[0052] Figure 1 is the method flow chart of the present invention;

[0053] Figure 2 is a schematic diagram of a fiber metal laminated cantilever beam with variable cross-section;

[0054] Figure 3 is the distribution diagram of the thickness function and width function of the cantilever beam in the embodiment;

[0055] Figure 4 It is the deflection curve diagram of the cantilever beam in the embodiment. Specific embodiments

[0056] It should be noted that, without conflict, the embodiments in the present invention and the features in the embodiments can be combined with each other. The present invention will be described in detail below with reference to the drawings and in combination with the embodiments.

[0057] To make the objectives, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. The following description of at least one exemplary embodiment is actually only illustrative and in no way restricts the present invention and its application or use. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts belong to the scope of protection of the present invention.

[0058] It should be noted that the terms used herein are only for describing specific embodiments and are not intended to limit the exemplary embodiments of the present invention. As used herein, unless the context clearly indicates otherwise, the singular form is also intended to include the plural form. In addition, it should be understood that when the terms "comprise" and / or "include" are used in this specification, they indicate the presence of features, steps, operations, devices, components, and / or combinations thereof.

[0059] Unless otherwise specifically stated, the relative arrangements of the components and steps, numerical expressions and values set forth in these embodiments do not limit the scope of the present invention. At the same time, it should be clear that, for the sake of convenience of description, the dimensions of the various parts shown in the drawings are not drawn according to the actual proportional relationship. Technologies, methods and devices known to those of ordinary skill in the relevant art may not be discussed in detail, but where appropriate, the technologies, methods and devices should be regarded as part of the authorized specification. In all the examples shown and discussed herein, any specific values should be construed as merely exemplary and not as a limitation. Therefore, other examples of the exemplary embodiments may have different values. It should be noted that: like reference numerals and letters denote like items in the following drawings, and thus, once an item is defined in one drawing, it does not need to be further discussed in subsequent drawings.

[0060] In the description of the present invention, it should be understood that the orientation or positional relationship indicated by orientation terms such as "front, rear, upper, lower, left, right", "lateral, vertical, perpendicular, horizontal" and "top, bottom", etc. is usually based on the orientation or positional relationship shown in the drawings. It is only for the convenience of describing the present invention and simplifying the description. Without contrary explanations, these orientation terms do not indicate or imply that the device or element referred to must have a specific orientation or be constructed and operated in a specific orientation. Therefore, it should not be construed as limiting the protection scope of the present invention: the orientation terms "inside, outside" refer to the inside and outside relative to the contour of each component itself.

[0061] For the convenience of description, spatial relative terms can be used here, such as "above...", "over...", "on the upper surface of...", "upper...", etc., to describe the spatial positional relationship between a device or feature shown in the figure and other devices or features. It should be understood that the spatial relative terms are intended to include different orientations in use or operation in addition to the orientation described in the figure for the device. For example, if the device in the drawing is inverted, the device described as "above other devices or structures" or "over other devices or structures" will then be positioned as "below other devices or structures" or "under other devices or structures". Thus, the exemplary term "above..." can include both the orientations of "above..." and "below...". The device can also be positioned in other different ways (rotated 90 degrees or in other orientations), and corresponding interpretations are made for the spatial relative descriptions used here.

[0062] In addition, it should be noted that the use of words such as "first", "second", etc. to limit components is only for the convenience of distinguishing the corresponding components. Without additional statements, the above words have no special meanings. Therefore, it should not be construed as limiting the protection scope of the present invention.

[0063] An analytical method for the cantilever bending deformation of a fiber metal laminated beam with variable cross-section includes the following steps:

[0064] Step 1: Considering the non-uniform load-bearing characteristics and non-linear variable cross-section characteristics of the cantilever beam, the overall variable cross-section fiber metal laminated cantilever beam conforms to the non-uniform load-bearing characteristics and non-linear variable cross-section characteristics, and within the local micro-element section, it satisfies the uniform load-bearing and equal cross-section characteristics, and obtain the finite-segment discrete variable cross-section micro-element section of the overall continuous variable cross-section fiber metal heterogeneous laminated beam;

[0065] Step 2: Based on the laminated plate theory, use the bending stiffness calculation method of the fiber metal heterogeneous laminated beam micro-element section to obtain the stiffness of the finite-segment discrete variable cross-section micro-element section;

[0066] Step 3: Based on the composite beam theory, obtain the cantilever bending deflection at any position of the variable cross-section fiber metal laminated cantilever beam under non-uniform load based on the stiffness of the finite-segment discrete variable cross-section micro-elements.

[0067] Steps 1, 2, and 3 are executed in sequence;

[0068] Furthermore, based on the laminated plate theory, the stiffness of the finite-segment discrete variable cross-section micro-elements is obtained by using the bending stiffness calculation method of the fiber metal heterogeneous laminated beam micro-elements; it includes the following steps:

[0069] Judge the material property of the k-th layer of the fiber metal heterogeneous laminated beam micro-element, and calculate the layer orientation transformation matrix and flexibility matrix of the k-th layer material;

[0070] Based on the laminated plate theory, calculate the stiffness matrix of the k-th layer material within the micro-element;

[0071] Based on the laminated plate theory, calculate the bending stiffness matrix of the fiber metal heterogeneous laminated beam micro-element.

[0072] Furthermore, the judgment of the material property of the k-th layer of the fiber metal heterogeneous laminated beam micro-element and the calculation of the layer orientation transformation matrix and flexibility matrix of the k-th layer material are as follows:

[0073] When the k-th layer material of the micro-element is a fiber-reinforced composite material, calculate the layer orientation transformation matrix and compliance matrix according to the anisotropic elasticity mechanics;

[0074] When the k-th layer material of the micro-element is a metal material, calculate the layer orientation transformation matrix and compliance matrix according to the isotropic elasticity mechanics.

[0075] Furthermore, based on the laminated plate theory, the stiffness of the finite-segment discrete variable cross-section micro-elements is obtained by using the bending stiffness calculation method of the fiber metal heterogeneous laminated beam micro-elements; it includes the following steps:

[0076] Based on the laminated plate theory, use the following formula to calculate the stiffness matrix of the k-th layer material within the micro-element:

[0077] Q (k) =T -1 S -1 [T -1 T (1)

[0078] Among them, T represents the layer orientation transformation matrix related to the layer orientation angle of the k-th layer, and S is a 6x6 symmetric compliance matrix related to the engineering material constants;

[0079] Among them: The layer orientation transformation matrix T is calculated by using the following formula:

[0080] ​

[0081] Where: when the k-th layer material of the microelement segment is a fiber-reinforced composite material, the ply angle θ k is the angle between the fiber laying direction and the beam length direction; when the k-th layer material of the microelement segment is a metal, its ply angle is set to θ k = 0.

[0082] Where: when the k-th layer material of the microelement segment is a fiber-reinforced composite material, according to the anisotropic elasticity mechanics, each element of the compliance matrix S is calculated by the following formula.

[0083]

[0084] Where: E 1 、E 2 and E 3 are the elastic moduli of the fiber-reinforced composite material along the fiber direction, perpendicular to the fiber direction, and along the thickness direction, respectively, G 12 、G 13 and G 23 are the shear moduli, S 11 、S 12 、…、S 66 are the elements of the compliance matrix S respectively, ν 12 、ν 13 and ν 23 are the Poisson's ratios.

[0085] If the k-th layer material of the microelement segment is a metal, according to the isotropic elasticity mechanics, each element of the compliance matrix S is calculated by formula (4);

[0086]

[0087] where E represents the elastic modulus of the metal, ν represents the Poisson's ratio of the metal, and S 11 、S 12 、…、S 66 are the elements of the compliance matrix S respectively.

[0088] From the stiffness matrix of the k-th layer material in the microelement segment, the bending stiffness matrix D of the microelement segment of the fiber-metal heterogeneous laminated beam is obtained, and each element D ij of the bending stiffness matrix D satisfies the following formula:

[0089]

[0090] where the subscript ij represents the i-th row and j-th column in the 6x6 stiffness matrix, n i represents the total number of layers of the microelement segment, z k+1 and z k represent the thickness coordinates of the upper surface and the lower surface of the k-th layer material respectively.

[0091] The process of obtaining the bending deflection of a cantilever beam with a variable cross-section under a non-uniform load based on the stiffness of a finite-segment discrete variable cross-section micro-element segment according to the composite beam theory is as follows:

[0092] For a cantilever beam with a length of L, any micro-element segment i of the cantilever beam is intercepted. Within the micro-element segment i, the load intensity q(x), the beam width b(x), and the beam height h(x) can be approximated as constants q i , the beam width b i , and the beam height h i respectively;

[0093] The load intensity q i within this micro-element segment generates a bending moment k at any position x k (0 ≤ x ≤ L) that satisfies the following formula:

[0094]

[0095] The total bending moment k generated by all micro-element segments of the cantilever beam at any position x k (0 ≤ x ≤ L) satisfies the following formula:

[0096]

[0097] where k represents the number of any micro-element segment on the beam, and i represents the number of any micro-element segment with a number larger than k;

[0098] According to the composite beam theory, the differential equation of the non-linear deflection curve satisfies the following formula:

[0099]

[0100] where ω x represents the deflection at any position x, E xx (x) represents the homogenized elastic modulus along the length direction of the beam, and I yy (x) represents the moment of inertia of the beam cross-section with respect to the y-axis;

[0101] If under the condition of small deformation, the deflection curve is very flat, the term can be ignored to obtain the linearized approximate differential equation of the deflection curve:

[0102]

[0103] For the pure bending loading condition, the following relationship holds between the bending stiffness matrix D of the micro-element segment of the fiber-metal heterogeneous laminated beam and the beam stiffness E xx (x)I yy (x):

[0104]

[0105] Among them, [D] -1 (1, 1) represents the element at the 1st row and 1st column of the matrix obtained after inverting the bending stiffness matrix of the fiber-metal heterogeneous laminated beam microelement segment;

[0106] Therefore, the nonlinear and linear deflection curve differential equations can be further rewritten in the following formula form:

[0107]

[0108]

[0109] Integrate the deflection curve differential equation and substitute the boundary conditions. At x = 0, ω 0 = 0 and The deflection ω of the fiber-metal laminated cantilever beam with variable cross-section at any x position is obtained x .

[0110]

[0111] Example 1: As Figure 1 shown, this example takes a fiber-metal laminated beam composed of an IMS194 / 977-2 carbon fiber reinforced resin matrix composite layer and a 2A12 aluminum alloy thin layer as an example to illustrate the specific implementation manner of the present invention, including the following steps:

[0112] S1. The elastic moduli of the HF10 / EM710 carbon fiber reinforced resin matrix composite are E 1 = 165 GPa, E 2 = 8.17 GPa, and E 3 = 8.17 GPa, the Poisson's ratios are ν 12 = 0.33, ν 13 = 0.33, and ν 23 = 0.48, the shear moduli are G 12 = 4.27 GPa, G 13 = 4.27 GPa, and G 23 = 2.76 GPa, and the single-layer thickness is t c = 0.188 mm; the elastic modulus of the 2A12 aluminum alloy is E = 72 GPa, the Poisson's ratio is ν = 0.3, and the single-layer thickness is t m = 0.2 mm;

[0113] S2. As Figure 2 shown, establish an oxyz coordinate system with the middle layer position of the fixed constraint end of the fiber-metal laminated beam as the origin. The x direction is along the beam length direction, the y direction is along the beam width direction, and the z direction is along the beam thickness direction;

[0114] In S3, the length of the fiber metal laminated beam in the embodiment is L = 50 mm, the width function is b(x) = -0.001x 2 -0.006x + 3 (unit: mm), and the height function is h(x) = 0.0005x 2 -0.05x + 4.56 (unit: mm), as Figure 3 shown, the ply angles from the bottom to the top of the section with the maximum thickness on the beam are [45 / -45 / 0 / 0 / 0 / 90] 2S , the total number of plies is 24 layers, and the 4th, 10th, 15th, and 21st layers are aluminum alloy layers, and the ply angles of the aluminum alloy plies are set to 0°;

[0115] In S4, the length of the selected microelement segment is Δx = 5 mm, the total number of microelement segments obtained is L / Δx = 10, and the load intensity within the 10 microelement segments of the embodiment is q = [0.1; 0.09; 0.08; 0.07; 0.06; 0.05; 0.04; 0.03; 0.02; 0.01] (unit: MPa). The average values of the width function b(x) and the height function h(x) within each microelement segment are taken to obtain the beam width b i and the beam height h i within any microelement segment i, and the number of layers n i within the microelement segment i and the ply angles of all layers are obtained;

[0116] In S5, according to the classical laminated plate theory in this embodiment, the stiffness matrix Q of the kth layer material within the microelement segment is calculated using formula (1) (k) ;

[0117] Q (k) = T -1 S -1 [T -1 T (1)

[0118] where S is a 6x6 symmetric compliance matrix related to engineering material constants, and T represents the transformation matrix related to the ply angle θ k of the kth layer. The calculation method of the ply angle transformation matrix T is shown in formula (2);

[0119]

[0120] In S6, for the fiber-reinforced composite material layer within the microelement segment of the fiber metal heterogeneous laminated beam, according to the anisotropic elasticity mechanics, the elements of the compliance matrix S are shown in formula (3), where E 1 , E 2 and E 3 are the elastic moduli along the fiber direction, perpendicular to the fiber direction, and along the thickness direction respectively, and ν 12 , ν 13 ​and ν 23 is the Poisson's ratio, G 12 、G 13 and G 23 are the shear moduli;

[0121]

[0122] S7. For the metal layer in the micro - element segment of the fiber - metal hybrid laminated beam, set its ply angle θ k = 0. According to isotropic elasticity mechanics, use formula (4) to calculate each element of the compliance matrix S, where E represents the metal elastic modulus and ν represents the Poisson's ratio;

[0123]

[0124] S8. Based on the classical laminated plate theory, each element D ij in the bending stiffness matrix D of the micro - element segment of the fiber - metal hybrid laminated beam can be calculated by formula (5), where the subscript ij represents the element in the i - th row and j - th column of the stiffness matrix, z k +1 and z k represent the thickness coordinates of the upper surface and the lower surface of the k - th layer material respectively;

[0125]

[0126] S9. After taking the inverse matrix of the bending stiffness matrix D of the micro - element segment of the fiber - metal hybrid laminated beam, take the element in the first row and first column, denoted as [D] -1 (1,1);

[0127] S10. Convert the overall continuously variable cross - section into a finite - segment discrete variable cross - section. For any micro - element segment i with a length of Δx, the load intensity q i at any position x k on the beam (0 ≤ x k ≤ L) generates a bending moment that satisfies formula (6);

[0128]

[0129] S11. The total bending moment k generated by all micro - element segments of the cantilever beam at any position x k on the beam (0 ≤ x ≤ L) satisfies formula (7);

[0130]

[0131] S12. For the pure - bending loading condition, the bending stiffness matrix D and E xx (x)I yy (x) of the micro - element segment of the fiber - metal hybrid laminated beam satisfy the following formula:

[0132]

[0133] Among them, [D] -1 (1,1) represents the element in the first row and first column after taking the inverse matrix of the bending stiffness matrix D of the fiber-metal heterogeneous laminated beam microelement segment;

[0134] S13. This embodiment satisfies the small deformation condition, and the deflection curve is very flat and can be ignored terms. According to the classical composite beam theory, a linear approximate differential equation of the deflection curve is obtained:

[0135]

[0136] Among them, ω x represents the deflection at any x position;

[0137] S14. Integrate the differential equation of the deflection curve and substitute the boundary conditions (satisfying ω 0 = 0 and ) at x = 0), and the deflection ω x of the variable cross-section fiber-metal laminated cantilever beam at any x position is obtained, as shown in formula (10);

[0138]

[0139] After substituting the corresponding parameters of this embodiment, the deflection curve diagram of the variable cross-section fiber-metal laminated beam under the cantilever bending load is calculated, as shown in Figure 4 . The deflection values at different positions from the fixed boundary are shown in the following table:

[0140] Table 1 Analytically calculated deflection values at different distances from the fixed boundary

[0141] Distance x / mm 0 5 10 15 20 25 30 35 40 45 50 <![CDATA[Deflection ω k / μm]]> 0 0.14 0.27 0.35 0.42 0.47 0.52 0.54 0.55 0.56 0.56

[0142] Finally, it should be noted that: the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it; although the present invention has been described in detail with reference to the foregoing embodiments, those of ordinary skill in the art should understand that: they can still modify the technical solutions recorded in the foregoing embodiments, or perform equivalent replacements on some or all of the technical features; and these modifications or replacements do not make the essence of the corresponding technical solutions deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. An analytical method for the cantilever bending deformation of a fiber metal laminated beam with variable cross-section, characterized in that: It includes the following steps: Obtain the finite-segment discrete variable cross-section micro-elements of the overall continuous variable cross-section fiber metal heterogeneous laminated beam; Based on the laminated plate theory, use the bending stiffness calculation method of the fiber metal heterogeneous laminated beam micro-elements to obtain the stiffness of the finite-segment discrete variable cross-section micro-elements; The above-mentioned based on the laminated plate theory, using the bending stiffness calculation method of the fiber metal laminated beam micro-elements to obtain the stiffness of the finite-segment discrete variable cross-section micro-elements; It includes the following steps: Determine the material properties of the k layer of the fiber metal heterogeneous laminated beam microelement section, and calculate the k layer material ply angle transformation matrix and flexibility matrix; Based on the laminate theory, calculate the stiffness matrix of the k layer material within the infinitesimal segment; Based on the laminated plate theory, calculate the bending stiffness matrix of the fiber metal heterogeneous laminated beam micro-elements; According to the composite beam theory, based on the stiffness of the finite-segment discrete variable cross-section micro-elements, obtain the cantilever bending deflection at any position of the variable cross-section fiber metal laminated beam under non-uniform load; The process of obtaining the bending deflection of the variable cross-section micro-element cantilever beam under non-uniform load based on the stiffness of the finite-segment discrete variable cross-section micro-elements according to the composite beam theory is as follows: For a cantilever beam with a length of L , any infinitesimal segment of the cantilever beam is intercepted i . Within the infinitesimal segment i , the load intensity q ( x ), the beam width b ( x ), and the beam height h ( x ) can be approximated as constants q i , the beam width b i , and the beam height h i respectively; The load intensity within this infinitesimal segment q i At any arbitrary x k position ( ) the bending moment satisfies the following formula: (6) For all infinitesimal segments of the cantilever beam at any x k position, , the total bending moment generated satisfies the following formula: (7) Among them, k represents the serial number of any infinitesimal element segment on the beam, i represents that the serial number is larger than k the serial number of any infinitesimal element segment; According to the composite beam theory, the differential equation of the non-linear deflection curve satisfies the following formula: (8) Among them, represents the deflection at any x position, E xx ( x ) represents the homogenized elastic modulus along the length of the beam, I yy ( x ) represents the moment of inertia of the beam cross-section with respect to y the axis; If the deflection curve is very flat under small deformation conditions, the term can be ignored, and a linearized approximate differential equation for the deflection curve is obtained: (9) For the pure bending load condition, the bending stiffness matrix of the micro-element segment of the fiber-metal heterogeneous laminated beam D and the beam stiffness E xx ( x ) I yy ( x ) satisfy the relationship shown in the following formula: (10) Among them, represents the element in the first row and first column of the matrix obtained after inverting the bending stiffness matrix of the micro-element segment of the fiber metal heterogeneous laminated beam; Therefore, the differential equations of the non-linear and linear deflection curves can be further rewritten in the following formula form: (11) (12) Integrate the differential equation of the deflection curve and substitute the boundary conditions, x which is satisfied at and to obtain the deflection x of the variable cross-section fiber metal laminated cantilever beam at any position: (13) 。 2. The analytical method for the cantilever bending deformation of a fiber metal laminated beam with variable cross-section according to claim 1, characterized in that: The material properties of the k th layer of the fiber metal heterogeneous laminated beam microelement are judged, and the ply angle transformation matrix and flexibility matrix of the k th layer material are calculated as follows: When the k layer material of the infinitesimal segment is a fiber-reinforced composite material, the ply angle transformation matrix and the compliance matrix are calculated according to anisotropic elasticity mechanics; When the k layer material of the micro-element segment is a metallic material, the ply angle transformation matrix and the compliance matrix are calculated based on isotropic elastic mechanics.

3. The analytical method for the cantilever bending deformation of a fiber metal laminated beam with variable cross-section according to claim 1, characterized in that: The stiffness matrix is calculated using the following formula for the stiffness matrix of the k layer material within the infinitesimal segment: (1) Among them, T represents the ply angle transformation matrix related to the k ply angle of the th layer, and S is a 6 x 6 symmetric compliance matrix related to the engineering material constants; From the stiffness matrix of the k layer material within the infinitesimal segment, the bending stiffness matrix of the infinitesimal segment of the fiber-metal heterogeneous laminated beam is obtained D , and the bending stiffness matrix D each element satisfies the following formula: (2) Among them, the subscript ij represents the i th row and the j th column in the 6x6 stiffness matrix. n i represents the total number of layers of this micro-element segment. z k+1 and z k respectively represent the upper surface and lower surface thickness coordinates of the k th layer of material.

4. The analytical method for the cantilever bending deformation of a fiber metal laminated beam with variable cross-section according to claim 2, characterized in that: The ply angle transformation matrix T is calculated using the following formula: (3) Wherein: when the k layer material is a fiber-reinforced composite material, the ply angle is taken as the angle between the fiber laying direction and the beam length direction; when the k layer material is a metal, its ply angle is set to = 0.

5. The analytical method for the cantilever bending deformation of a fiber metal laminated beam with variable cross-section according to claim 2, characterized in that: When the k layer material of the microelement segment is a fiber-reinforced composite material, according to the anisotropic elasticity mechanics, the elements of the compliance matrix S are calculated using the following formulas: (4) Wherein: E 1 、 E 2 and E 3 are the elastic moduli of the fiber-reinforced composite material in the fiber direction, perpendicular to the fiber direction, and in the thickness direction, respectively, G 12 、 G 13 and G 23 are the shear moduli, S 11 、 S 12 、…、 S 66 are the elements of the compliance matrix S respectively, ν 12 、 ν 13 and ν 23 are the Poisson's ratios.

Images