CN112632835B - Modeling method for critical buckling load analysis model of bean pod rod with large slenderness ratio - Google Patents

Modeling method for critical buckling load analysis model of bean pod rod with large slenderness ratio Download PDF

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CN112632835B
CN112632835B CN202011605081.XA CN202011605081A CN112632835B CN 112632835 B CN112632835 B CN 112632835B CN 202011605081 A CN202011605081 A CN 202011605081A CN 112632835 B CN112632835 B CN 112632835B
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姚志超
李东旭
朱仕尧
周利霖
刘望
吴军
毛少川
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National University of Defense Technology
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Abstract

The invention discloses a critical buckling load analysis model modeling method for a pod rod with a large slenderness ratio, and belongs to the technical field of structural mechanics modeling. The modeling method comprises the following steps of establishing a critical buckling load analysis model of the pod rod: step S1: constructing a critical buckling load analysis model of the pod rod with a large slenderness ratio based on an Euler formula, and respectively acquiring a cross-sectional moment of inertia of the pod rod and an axial equivalent elastic modulus of the pod rod according to the cross-sectional characteristics and the layering characteristics of the pod rod; then, establishing a pod rod finite element model through ABAQUS finite element software, verifying the pod rod critical buckling load analysis model by using a characteristic value buckling method, and determining the application range of the pod rod critical buckling load analysis model by comparing the difference of results obtained by respectively using the pod rod finite element model and the pod rod critical buckling load analysis model under different slenderness ratios. The invention has the advantages of small calculation amount, high calculation speed and the like.

Description

Modeling method for critical buckling load analysis model of pod rod with large slenderness ratio
Technical Field
The invention belongs to the technical field of structural mechanics modeling, and particularly relates to a critical buckling load analysis model modeling method for a bean pod rod with a large slenderness ratio.
Background
With the continuous development of space activities such as space exploration and the like, the size and the weight of a spacecraft are continuously increased, the launching task difficulty is sharply increased, and the requirements of light-weight, high-rigidity and high-storage-rate space extending arms are more and more urgent under the conditions. The pod rod is a thin-wall tubular extending arm which is made of composite materials, the cross section of the pod rod is in a lens shape after the pod rod is unfolded, and the pod rod has the advantages of simple structure, high stability and the like, and has wide application prospect in the unfolding process of the sail surface of the spacecraft. Due to the large structural size of the sail surface of the spacecraft, the length of the pod rods used as the supporting structure of the spacecraft is very long, the slenderness ratio is defined as the ratio of the rod length of the pod rods to the radius of gyration of the cross section, the slenderness ratio of the pod rods is usually more than 50 in application, and the pod rods are called large slenderness ratio pod rod structures, and are typical slender structures due to the long length and the relatively small cross section of the structures.
The pod rod structure with the large slenderness ratio plays a role in supporting the sail surface of the spacecraft in the unfolded state, buckling damage is easily caused by the action of internal and external loads, and the bearing capacity of the spacecraft is affected, so that the critical buckling load of the pod rods with the large slenderness ratio in the unfolded state needs to be calculated. The traditional pod rod critical buckling load analysis method is a finite element simulation method or an experimental method, and the method has the defects of large calculation amount, low calculation speed and the like and is not beneficial to analysis of critical buckling load parameter influence.
Disclosure of Invention
The invention mainly aims to provide a modeling method of a critical buckling load analysis model of a pod rod with a large slenderness ratio, and aims to solve the problems that the traditional method is large in calculation amount, low in calculation speed, not beneficial to analysis of critical buckling load parameter influence of the pod rod and the like.
In order to achieve the purpose, the invention provides a critical buckling load analysis model modeling method of a pod rod with a large slenderness ratio, which comprises the following steps:
step S1, constructing a critical buckling load analysis model of the great slenderness ratio pod rod based on an Euler formula, respectively acquiring a cross-sectional moment of inertia of the pod rod and an axial equivalent elastic modulus of the pod rod according to the cross-sectional characteristics and the layering characteristics of the pod rod, and inputting the acquired cross-sectional moment of inertia of the pod rod and the axial equivalent elastic modulus of the pod rod into the critical buckling load analysis model of the great slenderness ratio pod rod;
step S2, establishing a pod rod finite element model, verifying the pod rod critical buckling load analysis model with the large slenderness ratio by using a characteristic value buckling method, comparing the difference of results obtained by respectively using the pod rod finite element model and the pod rod critical buckling load analysis model under different slenderness ratios, and determining the application range of the pod rod critical buckling load analysis model with the large slenderness ratio according to the comparison result.
Further, the detailed step of step S1 includes the following substeps S1.1 to S1.4:
s1.1, establishing a coordinate system o-xy for a pod rod structure, taking an original point o of the coordinate system at the geometric center of a cross section, enabling the cross section of the pod rod to be symmetrical about an ox axis and an oy axis, dividing a quarter cross section of the pod rod into a convex arc section, a concave arc section and a straight line section, and performing variable definition;
s1.2, simplifying pod rods into a compression straight rod with one end fixed and the other end free under the boundary condition, and constructing a critical buckling load analysis model of the pod rods with a large slenderness ratio based on an Euler formula, wherein the critical buckling load analysis model of the pod rods with the large slenderness ratio comprises two types of parameters to be solved, namely pod rod cross-sectional moment of inertia and pod rod axial equivalent elastic modulus;
s1.3, acquiring the section characteristics of the pod rod according to the established coordinate system, and respectively acquiring the section inertia moment of the convex arc section, the concave arc section and the straight line section of the pod rod around an ox axis and the section inertia moment of the pod rod around an oy axis by integrating in the arc length direction to obtain the section inertia moment of the pod rod and input the section inertia moment into the critical buckling load analysis model of the pod rod with the large slenderness ratio;
s1.4, analyzing the layering characteristics of the pod rods based on a classical laminated plate theory, and obtaining the axial equivalent elastic modulus of the convex arc section, the concave arc section and the straight line section of the pod rods by constructing rigidity matrix elements among the pressure and the middle plane compressive strain of the convex arc section, the concave arc section and the straight line section of the pod rods to obtain the axial equivalent elastic modulus of the pod rods and inputting the axial equivalent elastic modulus to the pod rod critical buckling load analysis model with the large slenderness ratio.
Further, the critical buckling load analysis model of the pod rod with large slenderness ratio constructed in the substep S1.2 is as follows:
Fcr＝min(Fcr,x,Fcr,y)，
wherein,
Figure BDA0002870247140000021
Figure BDA0002870247140000022
in the above formula, FcrCritical buckling load of the pod stems, Fcr,xCritical buckling load for pod stems to buckle about the ox axis, Fcr,yCritical buckling load for pod rods buckling around the oy axis, l pod rod axial length, Ez,1The equivalent tensile and compression elastic modulus, E, of the convex arc section of the pod rod along the axial directionz,2The equivalent tensile and compression elastic modulus, E, of the concave arc section of the pod rod along the axial directionz,3Is the equivalent tensile and compression elastic modulus, I, of the pod rod straight-line segment along the axial directionx,1Is the moment of inertia, I, of the convex arc section of the pod rod about the ox axisx,2Is the moment of inertia, I, of the concave arc section of the pod rod about the ox axisx,3Moment of inertia about ox axis for straight line segment of pod rod, Iy,1Is the moment of inertia, I, of the convex arc section of the pod rod about the oy axisy,2Is the moment of inertia, I, of the concave arc section of the pod rod around the oy axisy,3Min (-) represents the minimum in the return given parameters for the moment of inertia of the pod rod straight line segment about the oy axis.
Further, the calculation formula of the second moment of inertia of the cross section of the pod rod in the substep S1.3 is:
Figure BDA0002870247140000023
Figure BDA0002870247140000024
Figure BDA0002870247140000031
Figure BDA0002870247140000032
Figure BDA0002870247140000033
Figure BDA0002870247140000034
in the above formula, Ix,1Is the moment of inertia, I, of the convex arc section of the pod rod about the ox axisx,2Is the moment of inertia, I, of the concave arc section of the pod rod about the ox axisx,3The moment of inertia of the straight line segment of the pod rod around the ox axis;
Iy,1is the moment of inertia, I, of the convex arc section of the pod rod about the oy axisy,2Is the moment of inertia, I, of the concave arc section of the pod rod around the oy axisy,3The moment of inertia of the straight line segment of the pod rod around the oy axis;
r1is the radius of the convex arc section of the bean pod rod, r2Is the radius of the concave arc section of the pod rod, alpha is the arc corner corresponding to the convex arc section and the concave arc section of the pod rod, and y1Is the longitudinal coordinate of the center of a convex arc section of the bean pod rod, w is the length of a straight section of the bean pod rod, h1Is the total thickness of the convex arc section of the bean pod rod, h2Is the total thickness of the concave arc section of the bean pod rod, h3Is the total thickness of the straight line segment of the pod rod.
Further, the calculation formula of the axial equivalent elastic modulus of the pod rod in the substep S1.4 is as follows:
Figure BDA0002870247140000035
Figure BDA0002870247140000036
Figure BDA0002870247140000037
in the above formula, Ez,1、Ez,2And Ez,3The equivalent tensile and compression elastic modulus of the convex arc section, the concave arc section and the straight line section of the pod rod along the axial direction are respectively;
A11,1is the first row and column element of the stiffness matrix between the convex arc pressure and the medial compressive strain of the pod rod, A22,1The second row and the second column of the stiffness matrix between the convex arc pressure and the middle plane compressive strain of the pod rod, A12,1A first row and a second column of elements of a rigidity matrix between the pressure of the convex arc section of the pod rod and the compressive strain of the middle surface;
A11,2the first row and column elements of the stiffness matrix between the concave arc pressure and the medial compressive strain of the pod rod, A22,2The second row and the second column of the stiffness matrix between the concave arc pressure and the medial compressive strain of the pod rod, A12,2A first row and a second column of elements of a stiffness matrix between the pressure of the concave arc section of the pod rod and the compressive strain of the middle surface;
A11,3the first row and column elements of the stiffness matrix between the linear section pressure and the medial compressive strain of the pod rod, A22,3The second row and column elements of the stiffness matrix between the linear section pressure and the medial compressive strain of the pod rod, A12,3A first row and a second column of elements of a rigidity matrix between the pressure of the pod rod straight line segment and the compression strain of the middle plane;
h1is the total thickness of the convex arc section of the bean pod rod, h2Is the total thickness of the concave arc section of the bean pod rod, h3Is the total thickness of straight line segments of the bean pod rod, t is the thickness of a composite material single-layer plate, betakIs the ply angle of the k-th ply in the laminate, N1Denotes the total number of layers of the convex arc segment, N2Denotes the total number of layers of the concave arc segments, N3Denotes the total number of layers of the straight line segment, Q11Is the first row and column element, Q, of the modulus matrix of the in-plane stress state under the positive axis22Second row and second column elements of the plane stress state modulus matrix under positive axis, Q12The first row and the second column of elements of the plane stress state modulus matrix under the positive axis,Q66third row and third column element of the plane stress state modulus matrix under positive axis, E1Is the longitudinal tensile modulus of a monolayer composite, E2Transverse tensile modulus, G, of a monolayer composite12Shear modulus, μ, of a monolayer composite12And mu21Respectively, the lateral and longitudinal poisson's ratios.
Further, the detailed step of step S2 includes the following substeps S2.1 to S2.3:
s2.1, establishing a finite element model of the pod rods by using ABAQUS software;
s2.2, a buckling analysis step is established in ABAQUS software, the first 2-order characteristic value and the buckling mode of the pod rod are obtained through characteristic value buckling analysis, and then the obtained characteristic value is multiplied by the applied load absolute value to obtain the characteristic value critical buckling load corresponding to the buckling mode;
s2.3, comparing and using the difference of results obtained by the pod rod finite element model and the pod rod critical buckling load analysis model, verifying the correctness of the pod rod critical buckling load analysis model, analyzing the difference of the results of the two models under different slenderness ratios, and determining the application range of the pod rod critical buckling load analysis model.
Further, the detailed step of the substep S2.1 includes: creating pod rods using the ABAQUS software, creating a composite layup in the attributes, and assigning the pod rods; creating a reference point in the center of the section of the free end of the pod rod for applying load, and performing motion coupling constraint on the reference point and the free end of the pod rod; and (4) selecting S4R thin shell units for grid division, and establishing to obtain the finite element model of the pod rod.
Further, the application range of the large slenderness ratio pod rod critical buckling load analysis model is a pod rod structure with a slenderness ratio larger than 64, wherein the slenderness ratio represents the ratio of the length of the pod rod to the radius of gyration of the cross section of the pod rod.
Compared with a conventional finite element method and an experimental method, the method has the advantages of small calculated amount, high calculating speed, convenience in quickly acquiring the critical buckling load of the pod rod, development of the influence analysis of the pod rod parameters on the buckling load and the like.
Drawings
In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the embodiments or the prior art will be briefly described below, it is obvious that the drawings in the following description are only some embodiments of the present invention, and for those skilled in the art, other drawings can be obtained according to the structures shown in the drawings without creative efforts.
FIG. 1 is a schematic diagram of a pod rod structure and a coordinate system according to the present invention;
FIG. 2 is a cross-sectional view taken along line A-A of FIG. 1;
FIG. 3 is a schematic view of a quarter cross-sectional shape of a pod rod;
FIG. 4 is a simplified model of pod rod flexure;
FIG. 5 is a schematic view of a composite layup for a convex arc segment of a pod stem;
FIG. 6 is a schematic view of a composite layup at a concave arc segment of a pod rod;
FIG. 7 is a schematic view of a finite element model of a pod rod;
FIG. 8 is a 1 st order flexion mode under pod stem axial compression;
FIG. 9 is a 2 nd order flexion mode under pod stem axial compression;
FIG. 10 is a graph of the effect of pod stem slenderness ratio on critical load for flexion about the ox axis;
FIG. 11 is a graph of the effect of pod stem slenderness on critical load for flexion about the oy axis;
FIG. 12 is a graph of the effect of straight line segment length on critical buckling load;
FIG. 13 shows the radius r of the convex arc section1The influence curve on the arc segment angle alpha;
FIG. 14 shows the radius r of the convex arc segment1An influence curve on critical buckling load;
FIG. 15 shows the radius r of the concave arc segment2The influence curve on the arc segment angle alpha;
FIG. 16 shows the radius r of the concave arc segment2An influence curve on critical buckling load;
FIG. 17 is the longitudinal axis y of the center of the convex arc segment1The influence curve on the arc segment angle alpha;
FIG. 18 is the longitudinal coordinate y of the center of the convex arc segment1An influence curve on critical buckling load;
FIG. 19 is a plot of the effect of single layer thickness t on critical buckling load;
FIG. 20 is a graph of the effect of ply angle β on critical buckling load;
FIG. 21 is a flowchart of a critical buckling load analysis model modeling method for a pod rod with a large slenderness ratio according to the present invention.
Detailed Description
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, 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.
The pod rod is a thin-wall tubular extension arm made of composite material and with a cross section in a lens shape after being unfolded, the ratio of the rod length of the pod rod to the turning radius of the cross section is defined as the slenderness ratio, and the pod rod with the large slenderness ratio is generally referred to as a pod rod structure with the slenderness ratio being more than 50. Aiming at the structure of the bean pod rod with large slenderness ratio, the invention provides a critical buckling load analysis model modeling method of the bean pod rod with large slenderness ratio, and the core thought of the method comprises the following two steps:
step S1, constructing a critical buckling load analysis model of the great slenderness ratio pod rod based on an Euler formula, respectively acquiring a cross-sectional moment of inertia of the pod rod and an axial equivalent elastic modulus of the pod rod according to the cross-sectional characteristics and the layering characteristics of the pod rod, and inputting the acquired cross-sectional moment of inertia of the pod rod and the axial equivalent elastic modulus of the pod rod into the critical buckling load analysis model of the great slenderness ratio pod rod;
step S2, establishing a pod rod finite element model, verifying the pod rod critical buckling load analysis model with the large slenderness ratio by using a characteristic value buckling method, comparing the difference of results obtained by respectively using the pod rod finite element model and the pod rod critical buckling load analysis model under different slenderness ratios, and determining the application range of the pod rod critical buckling load analysis model with the large slenderness ratio according to the comparison result.
The following describes in detail the critical buckling load analysis model modeling method for a pod rod with a large slenderness ratio provided by the present invention and the beneficial effects obtained by using the method with reference to fig. 1 to 21.
The specific implementation process of the invention comprises the following seven substeps S1.1-S2.3, wherein S1.1-S1.4 are detailed substeps contained in step S1, and S2.1-S2.3 are detailed substeps contained in step S2:
and a substep S1.1 of establishing a coordinate system for the pod rod structure and performing variable definition. Specifically, a coordinate system o-xy is established for the pod rod structure, the origin o of the coordinate system is taken at the geometric center of the cross section, the cross section of the pod rod is symmetrical about an ox axis and an oy axis, a quarter cross section of the pod rod is divided into a convex arc section, a concave arc section and a straight line section, and variable definition is performed.
The pod rods can be used in the unfolding process of the large-scale spacecraft structure, one end of each pod rod in the unfolded state is connected with the spacecraft body, and the other end of each pod rod is connected with the solar sail. Because the rigidity and the mass of the spacecraft body are large, and the solar sail does not provide displacement constraint for the pod rods, the boundary conditions of the pod rods can be simplified into one end fixed and one end free. The pod rod is a thin-wall tube formed by bonding two omega-shaped half pod rods at the boundary, the structural schematic diagram and the coordinate system of the pod rod in the unfolding state are shown in figure 1, the A-A section diagram is shown in figure 2, and the origin o of the coordinate system is taken at the geometric center of the section.
The pod rod cross section is symmetrical about the ox axis and the oy axis, so that the pod rod cross section can be divided into a quarter of the shape for variable definition and geometric description. As shown in FIG. 3, the quarter section consists of a convex arc section, a concave arc section and a straight line section, which are respectively adopted
Figure BDA0002870247140000061
Figure BDA0002870247140000062
And CD, and the tangent of the leftmost end of the convex arc section is perpendicular to the oy axis, the convex arc section is tangent to the concave arc section, and the concave arc section is tangent to the straight line section. r is1And r2Respectively the radius of a convex arc section and the radius of a concave arc section; alpha is alpha1And alpha2The arc section angles corresponding to the convex arc section and the concave arc section respectively; (x)1,y1) And (x)2,y2) The coordinates of the circle centers of the convex arc section and the concave arc section are respectively; h is the total thickness of the arc section, and the parameter subscripts of 1,2 and 3 respectively represent a convex arc section, a concave arc section and a straight line section; w is the length of the straight line segment. It can be determined from the geometric relationship that independent parameters of the cross-sectional shape, other than thickness, include r1、r2W and y1The dependent variables include alpha and x1、x2And y2And satisfies the relationship shown in formula (1):
Figure BDA0002870247140000071
and (S1.2) constructing a critical buckling load analysis model of the pod rod with large slenderness ratio based on an Euler formula. Specifically, pod rods are simplified into stressed straight rods with one end fixed and the other end free under the boundary condition, and a critical buckling load analysis model of the pod rods with the large slenderness ratio is constructed based on an Euler formula and comprises two types of parameters to be solved, namely pod rod section moment of inertia and pod rod axial equivalent elastic modulus.
The pod rod under the axial pressure is simplified into a pressure straight rod, and the boundary condition is that one end of the pod rod is fixed and the other end of the pod rod is free. When the applied load reaches the critical load, the pod rods buckle, which simplifies the buckling when subjected to axial loading as shown in figure 4.
The Eluer derives a critical load formula of the elongated pressure rod with one fixed end and one free end aiming at the buckling problem of the elongated pressure rod made of isotropic materials, and the formula is as follows:
Figure BDA0002870247140000072
in the formula: e is the modulus of elasticity, IminThe minimum moment of inertia of the cross section of the elongated rod, l is the rod length.
For the pod rod buckling problem, because the material attribute of the pod rod is no longer isotropic, the formula (2) cannot be directly used for calculating the critical buckling load of the pod rod structure, and because the layering modes of the convex arc section, the concave arc section and the straight line section may have differences, the elastic modulus and the minimum moment of inertia of the cross section of the pod rod are obtained in a segmented manner, and at this time, E represents the axial equivalent elastic modulus of the pod rod. Since the pod rod cross-section is symmetric about the ox and oy axes, the cross-sectional area of inertia product IxyAgain, because the origin o is the centroid of the pod rod cross-section, the ox and oy axes are the centroid principal axes of inertia, i.e., the minimum moment of inertia I for the pod rod cross-sectionminIs the minimum of the moments of inertia about the ox axis and about the oy axis. Because the section of the pod rod is symmetrical about the ox axis and the oy axis, 4 times of the quarter section moment of inertia of the pod rod is the integral section moment of inertia, and the critical loads of the pod rod bending around the ox axis and the oy axis are respectively:
Figure BDA0002870247140000073
Figure BDA0002870247140000074
in the formula: ez,1、Ez,2And Ez,3And the composite material laminated plates respectively comprising the convex arc section, the concave arc section and the straight line section have equivalent tensile and compression elastic modulus along the axial direction of the pod rod. I isx,1、Ix,2、Ix,3The inertia moments around the ox axis of the convex arc section, the concave arc section and the straight line section respectively, Iy,1、Iy,2And Iy,3The inertia moments of the convex arc section, the concave arc section and the straight line section around the oy axis are respectively. In summary, the theoretical critical buckling load of the pod rod is Fcr,xAnd Fcr,yMinimum of (d) as follows:
Fcr＝min(Fcr,x,Fcr,y) (5)
and a substep S1.3 of obtaining the second moment of inertia of the section of the pod rod. Specifically, the sectional characteristics of the pod rods are obtained according to the established coordinate system, the sectional moments of inertia of the convex arc sections, the concave arc sections and the straight line sections of the pod rods around the ox axis and the sectional moments of inertia of the pod rods around the oy axis are respectively obtained through integration in the arc length direction, and the sectional moments of inertia of the pod rods are obtained and input into the pod rod critical buckling load analysis model with the large slenderness ratio.
According to the geometrical shape of the cross section of the pod rod, the moments of inertia of the convex arc section, the concave arc section and the straight line section around the ox axis are deduced to be respectively:
Figure BDA0002870247140000081
Figure BDA0002870247140000082
Figure BDA0002870247140000083
in the same way, the inertia moments of the convex arc section, the concave arc section and the straight line section around the oy axis are obtained as follows:
Figure BDA0002870247140000084
Figure BDA0002870247140000085
Figure BDA0002870247140000086
substep S1.4, the axial equivalent elastic modulus of the pod rod is obtained. Specifically, the layering characteristics of the pod rods are analyzed based on a classical laminated plate theory, the axial equivalent elastic modulus of the convex arc section, the concave arc section and the straight line section of the pod rods is obtained by constructing a rigidity matrix element between the pressure and the middle plane compressive strain of the convex arc section, the concave arc section and the straight line section of the pod rods, and the axial equivalent elastic modulus of the pod rods is obtained and input to the pod rod critical buckling load analysis model with the great slenderness ratio.
Based on the classic laminated plate theory, the expression of the equivalent tensile and compressive elastic modulus of the composite laminated plate along the axial direction of the pod rod is obtained through derivation:
Figure BDA0002870247140000091
wherein, i is 1,2 and 3 respectively represent a convex arc section, a concave arc section and a straight line section; a. the11,i、A22,iAnd A12,iThe element of the rigidity matrix between the pressure and the compression strain of the middle plane is expressed as follows:
Figure BDA0002870247140000092
wherein t is the thickness of the single-layer composite material, betakIs the ply angle of the k-th ply in the laminate, NiDenotes the total number of layers in section i, Q11、Q22、Q12And Q66The modulus component of the plane stress state under the positive axis is expressed as follows:
Figure BDA0002870247140000093
in the formula, E1Is the longitudinal tensile modulus of a monolayer composite, E2Transverse tensile modulus, G, of a monolayer composite12Shear modulus, μ, of a monolayer composite12And mu21Respectively, the lateral and longitudinal poisson's ratios.
And a substep S2.1, establishing a finite element model of the pod rod by using ABAQUS software. In particular, the detailed steps of said substep S2.1 comprise: creating pod rods using the ABAQUS software, creating a composite layup in the attributes, and assigning the pod rods; creating a reference point in the center of the section of the free end of the pod rod for applying load, and performing motion coupling constraint on the reference point and the free end of the pod rod; and (4) selecting S4R thin shell units for grid division, and establishing to obtain the finite element model of the pod rod.
In the buckling analysis of this example, the parameters of the cross section of the pod rod are shown in table 1, the upper and lower parts of the pod rod are bonded by the resin bonding layer, the parameters of the single-layer composite material and the parameters of the material of the resin bonding layer are shown in tables 2 and 3, and the rod length l is 6.5 m. The 0-degree direction of the composite material layer is the axial direction of a pod rod, the layer angle of a convex arc section of the pod rod is [45 degrees/0 degrees/45 degrees ], and the layer angle of a concave arc section and a straight line section is [45 degrees/0 degrees/45 degrees ]. The single layer thickness of the composite material layer is 0.04 mm.
TABLE 1 pod rod section parameters
Figure BDA0002870247140000094
Figure BDA0002870247140000101
TABLE 2 Single layer composite parameters
Figure BDA0002870247140000102
TABLE 3 resin bond coat Material parameters
Parameter(s) Numerical value
Density (kg/m)3) 1250.00
Tensile modulus (GPa) 3.02
Poisson ratio mu12 0.33
ABAQUS software is selected for finite element modeling analysis, and shell units are selected for modeling according to the shape characteristics of the pod rods. The S4R thin shell unit in the ABAQUS has the characteristics of high calculation efficiency, strong robustness, good convergence and the like, and is a preferred unit in thin shell modeling. The pod rod belongs to a thin-shell structure, so S4R units are selected for grid division, and sufficiently fine grids are divided at the arc section. Schematic diagrams of the composite material layering of the convex arc section and the concave arc section of the pod rod are shown in fig. 5 and 6.
Finite element model of pod rods as shown in figure 7, a reference point is created at the center of the free end of the pod rod and constrained together with the free end cross section for applying a load. The MPC binding constraint or coupling constraint can be selected as a constraint mode of the reference point and the free end section, but simulation analysis shows that after the MPC binding constraint is introduced, the degree of freedom of the free end rotation direction can be additionally constrained, which is not suitable for relaxing the boundary condition of rotation limitation, and the condition of additional constraint degree of freedom can not occur when the kinematic coupling constraint is introduced, so that the reference point and the free end section are constrained by selecting the kinematic coupling constraint mode.
And S2.2, obtaining the characteristic value critical buckling load of the pod rod through characteristic value buckling analysis. Specifically, a buckling analysis step is established in ABAQUS software, the first 2-order characteristic value and the buckling mode of the pod rod are obtained through characteristic value buckling analysis, and then the obtained characteristic value is multiplied by the applied load absolute value to obtain the characteristic value critical buckling load corresponding to the buckling mode.
The buckling analysis step was created in the ABAQUS software, atAs shown in fig. 8 and 9, the first 2-order mode shape of the pod rod is calculated by applying 1N pressure to the free end, and it can be seen that the 1 st-order mode shape is a case of bending around the ox axis, and the 2 nd-order mode shape is a case of bending around the oy axis. The first 2 nd order characteristic values of the pod stem are 17.03 and 32.57, respectively, so the characteristic critical load F of pod stem buckling about the ox axiscr,x17.03N, characteristic critical load F of buckling about oy axiscr,y＝32.57N。
And a substep S2.3 of verifying the correctness of the pod rod critical buckling load analysis model and determining the application range. Specifically, the difference of results obtained by the pod rod finite element model and the pod rod critical buckling load analysis model is compared, the correctness of the pod rod critical buckling load analysis model is verified, the difference of results of the two models with different slenderness ratios is analyzed, and the application range of the pod rod critical buckling load analysis model is determined.
The parameters in table 1, table 2 and table 3 are substituted into the equivalent elastic modulus formula and the moment of inertia formula to obtain the equivalent elastic modulus and the moment of inertia of the pod rods as shown in table 4. The equivalent elastic modulus and the moment of inertia in table 4 are substituted into formula (3) and formula (4), and the theoretical critical load of pod rod buckling around the ox axis is obtained as Fcr,xThe theoretical critical load for buckling about the oy axis is F, 17.07Ncr,y32.58N. The theoretical critical buckling load of the pod rods is 17.07N.
TABLE 4 equivalent elastic modulus and Cross-sectional moment of inertia of pod rods
Parameter(s) Numerical value
Ez,1(GPa) 42.66
Ez,2(GPa) 26.43
Ez,3(GPa) 22.61
Ix,1(m4) 1.66×10-9
Ix,2(m4) 8.84×10-11
Ix,3(m4) 2.30×10-14
Iy,1(m4) 6.88×10-10
Iy,2(m4) 2.75×10-9
Iy,3(m4) 1.66×10-9
The buckling analysis based on the Euler formula ignores the deformation of the cross section and the initial geometrical defects during buckling, so the theoretical critical buckling load is larger; the characteristic value buckling analysis considers the section deformation, so the characteristic value critical buckling load is slightly smaller than the theoretical buckling load. The results of the two analytical methods are shown in Table 5, based on the theoretical analytical results. The analysis result shows that the two critical buckling loads are well matched, and the size relationship accords with the expected condition, so that the accuracy of the composite material pod rod critical buckling load analysis model is verified.
TABLE 5 comparison of results of two analysis methods under axial pressure conditions
Figure BDA0002870247140000111
Figure BDA0002870247140000121
Because the modeling method neglects the deformation of the section during buckling in the derivation, the application range of the pod rod critical buckling load analysis model needs to be discussed. The ratio of the length of the pod rod to the turning radius of the section is defined as the slenderness ratio. Fig. 10 and 11 are the aspect ratio versus theoretical and characteristic buckling loads, respectively. When the slenderness ratio is small, the deviation between the theoretical analysis result and the characteristic value analysis result is large. When the slenderness ratio is 40, the difference between the theoretical analysis result of buckling around the ox axis and the characteristic value analysis result is-13.13 percent; when the slenderness ratio is 56, the difference between the theoretical analysis result of buckling around the oy axis and the characteristic value analysis result is-15.00%, and at the moment, the difference between the two results is large, so that the theoretical critical buckling load is inaccurate. When the slenderness ratio is gradually increased, the deviation of the theoretical analysis result and the characteristic value analysis result is gradually reduced. When the slenderness ratio is larger than 64, the difference between a theoretical analysis result and a characteristic value analysis result is less than 1%, and the theoretical critical buckling load is accurate; therefore, when the length-to-thin ratio of the pod rod is larger than 64, the critical buckling load analysis model constructed by the invention is only applicable.
For the embodiment, the influence rule of the length of the straight line section of the pod rod, the radius of the convex arc section, the radius of the concave arc section, the circle center position of the convex arc section, the thickness of the layer laid and the angle of the layer laid on the critical buckling load can be analyzed.
On the basis of the original parameters of the model, the value of the length w of the straight line segment is only expanded from 5mm to 2.5 mm-7.5 mm, the step length is 0.1mm, and the change condition of the critical buckling load when the length w of the straight line segment is changed is shown in fig. 12. As can be seen from fig. 12, the length of the straight segment has little effect on the critical load for buckling about the ox axis; the length of the straight line segment is substantially linear with the critical load for flexion about the oy axis, which increases by 16.24% when the length of the straight line segment increases by 50%. The critical load of the pod rod buckling around the oy axis is greatly influenced by the length of the straight line segment, the critical buckling load is obviously influenced by the moment of inertia, the influence of the length of the straight line segment on the moment of inertia around the ox axis is extremely small, the moment of inertia around the oy axis is influenced to a certain extent, and the critical load of the pod rod buckling around the oy axis can be improved by increasing the length of the straight line segment.
The radius r is determined on the basis of the original parameters of the model1The value of (A) is expanded from 20mm to 10 mm-30 mm, and the step length is 0.5 mm. Arc segment angle alpha along with radius r1The variation of (2) is shown in fig. 13.
Radius r1The change of the critical buckling load at the time of change is shown in FIG. 14, and the radius r1A 3 rd power relation which is basically monotonously increased with the critical buckling load when the radius r1When the critical load is increased by 50%, the critical load of buckling around the ox axis is increased by 241.83%, and the critical load of buckling around the oy axis is increased by 150.31%; when radius r1At a 50% reduction, the critical load for buckling about the ox axis was reduced by 87.94% and the critical load for buckling about the oy axis was reduced by 71.96%. This is because the moment of inertia, which is affected by the radius r, has a significant effect on the critical buckling load1Is significant, so the radius r1The critical buckling load is greatly influenced.
The radius r is determined on the basis of the original parameters of the model2The value of (A) is expanded from 20mm to 10 mm-30 mm, and the step length is 0.5 mm. Arc segment angle alpha along with radius r2As shown in fig. 15.
Radius r2The change of the critical buckling load when changing is shown in FIG. 16, when the radius r is2When the critical load is increased by 50%, the critical load of buckling around the ox axis is reduced by 1.93%, and the critical load of buckling around the oy axis is increased by 32.93%; when radius r2At a 50% reduction, the critical load for buckling about the ox axis increased by 1.87% and the critical load for buckling about the oy axis decreased by 25.35%. Radius r2The critical load effect on buckling about the ox axis is small and both are inversely related, since the critical load effect on buckling about the ox axis is significant by the moment of inertia about the ox axis, while the radius r2The moment of inertia about the ox axis is less affected,and when the radius r2As this increases, the arc segment angle α decreases, resulting in a slight decrease in the moment of inertia about the ox axis. Radius r2The critical load effect on bending about the oy axis is greater because the critical load effect is significantly affected by the moment of inertia about the oy axis, which is the radius r2As a function of the radius r2When increased, the moment of inertia about the oy axis increases.
On the basis of original parameters of the model, the longitudinal coordinate y of the circle center of the convex arc section1The value of (A) is expanded from 0mm to-10 mm to 10mm, and the step length is 0.5 mm. Arc angle alpha along with longitudinal coordinate y of circle center of convex arc section1As shown in fig. 17.
Convex arc segment circle center longitudinal coordinate y1The change in critical buckling load when changing is shown in FIG. 18, when y1When the critical load is reduced to-10 mm, the critical load of buckling around the ox axis is reduced by 82.64 percent, and the critical load of buckling around the oy axis is reduced by 54.14 percent; when y is1Increasing to 10mm, the critical loads for buckling around the ox axis and the oy axis are 47.98N and 48.25N, respectively, which differ by only 0.56%, if y1Continuing to increase, the critical load for buckling about the ox axis will exceed the critical load for buckling about the oy axis. As is clear from the formulae (6) to (11), y1The moment of inertia about the ox axis is more affected and thus the critical load for buckling about the ox axis is more affected.
On the basis of the original parameters of the model, the value of the thickness t of the single layer is expanded from 0.04mm to 0.02 mm-0.06 mm, and the step length is 0.001 mm. The critical buckling load change condition when the thickness t changes is shown in fig. 19, the thickness t and the critical buckling load basically have a monotonically increasing linear relation, when the thickness t increases by 50%, the critical buckling load of buckling around the ox axis increases by 49.97%, and the critical buckling load of buckling around the oy axis increases by 49.94%; when the thickness t is reduced by 50%, the critical buckling load for buckling around the ox axis is reduced by 50.02%, and the critical buckling load for buckling around the oy axis is reduced by 50.03%. The thickness t has a greater influence on the critical buckling load because the equivalent elastic modulus and the moment of inertia of the pod rod are increased when the single-layer thickness t is increased, and further the critical buckling load is increased.
The ply pattern of the master model is defined as [45 °/β/0/β/-45 °/45 ° ], with the 3 rd and 5 th plies only being present in the convex arc segment region. On the basis of the original parameters of the model, the value of the layering angle beta is expanded from 0 degrees to-90 degrees, and the step length is 2 degrees. Fig. 20 shows the critical buckling load change when the ply angle β changes, in which the relationship curve between the ply angle β and the critical buckling load is symmetric about β equal to 0 °, and when the ply angle β is 90 °, the critical buckling load around the ox axis is reduced to 10.40N, and the critical buckling load around the oy axis is reduced to 29.80N; when the ply angle beta is 52 degrees, the critical buckling load is the minimum value, and the critical loads of buckling around the ox axis and buckling around the oy axis are 8.984N and 29.22N respectively; when the ply angle β takes 0 °, the critical buckling load takes a maximum value.
By combining the above analysis, a flow chart of the modeling method of the critical buckling load analysis model of the pod rod with large slenderness ratio is shown in fig. 21. The pod rod critical buckling load analysis model is deduced based on an Euler formula and a laminated plate theory, the correctness of the formula is verified through a finite element model, and the application range of the pod rod critical buckling load analysis model is determined by comparing the difference of results of two methods under different slenderness ratios. The parameter influence rule obtained according to the critical buckling load analysis model is as follows: the convex arc section radius has the most significant influence on the critical buckling load; concave arc segment radius and straight segment length pair Fcr,yGreater influence on Fcr,xThe influence is small; convex arc segment circle center longitudinal coordinate and layering angle pair Fcr,xGreater influence on Fcr,yThe influence is small; ply thickness pair Fcr,xAnd Fcr,yHave a major impact. Compared with a finite element method and an experimental method, the method has the advantages of small calculated amount, high calculating speed and convenience for parameter influence analysis.
The above description is only a preferred embodiment of the present invention, and is not intended to limit the scope of the present invention, and all modifications and equivalents of the present invention, which are made by the contents of the present specification and the accompanying drawings, or directly/indirectly applied to other related technical fields, are included in the scope of the present invention.

Claims (5)

1. A critical buckling load analysis model modeling method for a pod rod with a large slenderness ratio is characterized by comprising the following steps:
step S1, constructing a critical buckling load analysis model of the great slenderness ratio pod rod based on an Euler formula, respectively acquiring a cross-sectional moment of inertia of the pod rod and an axial equivalent elastic modulus of the pod rod according to the cross-sectional characteristics and the layering characteristics of the pod rod, and inputting the acquired cross-sectional moment of inertia of the pod rod and the axial equivalent elastic modulus of the pod rod into the critical buckling load analysis model of the great slenderness ratio pod rod;
the detailed step of step S1 includes the following substeps S1.1 to S1.4:
s1.1, establishing a coordinate system o-xy aiming at a pod rod structure, taking an original point o of the coordinate system at the geometric center of a cross section, enabling the cross section of the pod rod to be symmetrical about an ox axis and an oy axis, dividing a quarter cross section of the pod rod into a convex arc section, a concave arc section and a straight line section, and performing variable definition;
s1.2, simplifying pod rods into a stressed straight rod with a fixed end and a free end as boundary conditions, and constructing a critical buckling load analysis model of the pod rods with a large slenderness ratio based on an Euler formula, wherein the critical buckling load analysis model of the pod rods with the large slenderness ratio comprises two types of parameters to be solved, namely pod rod section moment of inertia and pod rod axial equivalent elastic modulus;
the critical buckling load analysis model of the pod rod with large slenderness ratio constructed in the substep S1.2 is as follows:
Fcr＝min(Fcr,x,Fcr,y)，
wherein,
Figure FDA0003641276510000011
Figure FDA0003641276510000012
in the above formula, FcrCritical buckling load of the pod stems, Fcr,xCritical buckling load for pod stems to buckle about the ox axis, Fcr,yFor pod stems flexing about the oy axisCritical buckling load, l is pod rod axial length, Ez,1The equivalent tensile and compression elastic modulus, E, of the convex arc section of the pod rod along the axial directionz,2The equivalent tensile and compression elastic modulus, E, of the concave arc section of the pod rod along the axial directionz,3Is the equivalent tensile and compression elastic modulus, I, of the pod rod straight-line segment along the axial directionx,1Is the moment of inertia, I, of the convex arc section of the pod rod about the ox axisx,2Is the moment of inertia, I, of the concave arc section of the pod rod about the ox axisx,3Moment of inertia about ox axis for straight line segment of pod rod, Iy,1Is the moment of inertia, I, of the convex arc section of the pod rod about the oy axisy,2Is the moment of inertia, I, of the concave arc section of the pod rod around the oy axisy,3For the moment of inertia of the pod rod straight line segment about the oy axis, min (·) represents the minimum value in the return given parameters;
s1.3, acquiring the section characteristics of the pod rod according to the established coordinate system, and respectively acquiring the section inertia moment of the convex arc section, the concave arc section and the straight line section of the pod rod around an ox axis and the section inertia moment of the pod rod around an oy axis by integrating in the arc length direction to obtain the section inertia moment of the pod rod and input the section inertia moment into the critical buckling load analysis model of the pod rod with the large slenderness ratio;
s1.4, analyzing the layering characteristics of the pod rods based on a classical laminated plate theory, and obtaining the axial equivalent elastic modulus of the convex arc section, the concave arc section and the straight line section of the pod rods by constructing a rigidity matrix element between the pressure and the middle plane compressive strain of the convex arc section, the concave arc section and the straight line section of the pod rods to obtain the axial equivalent elastic modulus of the pod rods and inputting the axial equivalent elastic modulus to the pod rods with the large slenderness ratio and the critical buckling load analysis model;
step S2, establishing a pod rod finite element model, verifying the pod rod critical buckling load analysis model with the large slenderness ratio by using a characteristic value buckling method, comparing the difference of results obtained by respectively using the pod rod finite element model and the pod rod critical buckling load analysis model under different slenderness ratios, and determining the application range of the pod rod critical buckling load analysis model with the large slenderness ratio according to the comparison result;
the application range of the large slenderness ratio pod rod critical buckling load analysis model is a pod rod structure with a slenderness ratio larger than 64, wherein the slenderness ratio represents the ratio of the length of a pod rod to the cross section turning radius of the pod rod.
2. The method of modeling a critical buckling load analysis model of a pod rod with a slenderness ratio of claim 1, wherein the equation for the moment of inertia of the pod rod cross section in substep S1.3 is:
Figure FDA0003641276510000021
Figure FDA0003641276510000022
Figure FDA0003641276510000023
Figure FDA0003641276510000024
Figure FDA0003641276510000025
Figure FDA0003641276510000026
in the above formula, Ix,1Is the moment of inertia, I, of the convex arc section of the pod rod about the ox axisx,2Is the moment of inertia, I, of the concave arc section of the pod rod about the ox axisx,3The moment of inertia of the straight line segment of the pod rod around the ox axis;
Iy,1is the moment of inertia about the oy axis of the convex arc section of the pod stem, Iy,2Is the moment of inertia, I, of the concave arc section of the pod rod around the oy axisy,3The moment of inertia of the straight line segment of the pod rod around the oy axis;
r1is the radius of the convex arc section of the bean pod rod, r2Is the radius of the concave arc section of the pod rod, alpha is the arc corner corresponding to the convex arc section and the concave arc section of the pod rod, and y1Is the longitudinal coordinate of the center of a convex arc section of the bean pod rod, w is the length of a straight section of the bean pod rod, h1Is the total thickness of the convex arc section of the pod rod h2Is the total thickness of the concave arc section of the pod rod, h3The total thickness of the straight line segment of the pod rod.
3. The method for modeling a critical buckling load analysis model of a pod rod with a large slenderness ratio of claim 1, wherein the calculation formula of the axial equivalent elastic modulus of the pod rod in the substep S1.4 is as follows:
Figure FDA0003641276510000027
Figure FDA0003641276510000031
Figure FDA0003641276510000032
in the above formula, Ez,1、Ez,2And Ez,3The equivalent tensile and compression elastic modulus of the convex arc section, the concave arc section and the straight line section of the pod rod along the axial direction are respectively;
A11,1is the first row and column element of the stiffness matrix between the convex arc pressure and the medial compressive strain of the pod rod, A22,1The second row and the second column of the stiffness matrix between the convex arc pressure and the middle plane compressive strain of the pod rod, A12,1A first row and a second column of elements of a rigidity matrix between the pressure of the convex arc section of the pod rod and the compressive strain of the middle surface;
A11,2the first row and column elements of the stiffness matrix between the concave arc pressure and the medial compressive strain of the pod rod, A22,2The stiffness matrix being the pressure of the concave arc section of the pod rod and the compressive strain of the medial surfaceSecond row and second column elements, A12,2A first row and a second column of elements of a stiffness matrix between the pressure of the concave arc section of the pod rod and the compressive strain of the middle plane;
A11,3the first row and column elements of the stiffness matrix between the linear section pressure and the medial compressive strain of the pod rod, A22,3The second row and column elements of the stiffness matrix between the linear section pressure and the medial compressive strain of the pod rod, A12,3A first row and a second column of elements of a stiffness matrix between the pressure of the straight line section of the pod rod and the compressive strain of the middle plane;
t is the thickness of the composite material single-layer plate, betakIs the lay-up angle of the k-th layer in the laminate, Q11Is the first row and column element, Q, of the plane stress state modulus matrix under the positive axis22Second row and second column element of plane stress state modulus matrix under positive axis, Q12Is the first row and the second column element, Q, of the plane stress state modulus matrix under the positive axis66Third row and third column element of the plane stress state modulus matrix under positive axis, E1Is the longitudinal tensile modulus of a monolayer composite, E2Transverse tensile modulus, G, of a monolayer composite12Shear modulus, μ, of a monolayer composite12And mu21Respectively, the lateral and longitudinal poisson's ratios.
4. The modeling method for the critical buckling load analysis model of the pod rod with the large slenderness ratio as claimed in any one of claims 1 to 3, wherein the detailed step of the step S2 comprises the following substeps S2.1-S2.3:
s2.1, establishing a finite element model of the pod rods by using ABAQUS software;
s2.2, a buckling analysis step is established in ABAQUS software, the first 2-order characteristic value and the buckling mode of the pod rod are obtained through characteristic value buckling analysis, and then the obtained characteristic value is multiplied by the applied load absolute value to obtain the characteristic value critical buckling load corresponding to the buckling mode;
s2.3, comparing and using the difference of results obtained by the pod rod finite element model and the pod rod critical buckling load analysis model, verifying the correctness of the pod rod critical buckling load analysis model, analyzing the difference of the results of the two models under different slenderness ratios, and determining the application range of the pod rod critical buckling load analysis model.
5. The method of modeling a critical buckling load analysis model of a pod rod with a large slenderness ratio of claim 4, wherein the detailed step of substep S2.1 comprises: creating pod rods using the ABAQUS software, creating a composite layup in the attributes, and assigning the pod rods; creating a reference point in the center of the section of the free end of the pod rod for applying load, and performing motion coupling constraint on the reference point and the free end of the pod rod; and (4) selecting S4R thin shell units for grid division, and establishing to obtain the finite element model of the pod rod.
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