A method of predicting the folded stable state of a parabolic bistable deployable composite stretchable arm

Abstract

A method for predicting the folded stable state of a parabolic bistable deployable composite stretchable arm, the method having three steps: step one, defining the geometry and size of the parabolic bistable deployable composite stretchable arm, and determining the mathematical expression of the relationship between various geometric parameters; step two, determining the total strain energy of the parabolic bistable deployable composite stretchable arm in the folded stable state according to the classical laminated plate theory and the minimum energy principle; step three, determining the stress level of the parabolic bistable deployable composite stretchable arm in the folded stable state according to the Tsai-Hill criterion, and deriving the Tsai-Hill failure coefficient expression.

Description

Technical Field

[0001] This invention provides a method for predicting the folding steady state of a parabolic bistable deployable composite material extension arm, which belongs to the field of manned spaceflight. Background Technology

[0002] Due to their lightweight, high stiffness, high folding efficiency, and reliable deployment process, deployable composite material extendable arms have received widespread attention and research in the aerospace field, showing promising application prospects. Parabolic bistable deployable composite material extendable arms are typically made of carbon fiber resin-based composite materials and are thin-walled tubular rod structures capable of folding and unfolding. Folding steady-state is a key mechanical performance indicator for parabolic bistable deployable composite material extendable arms, making its analysis necessary. Directly measuring the folding steady-state of parabolic bistable deployable composite material extendable arms experimentally is costly and susceptible to many accidental factors during testing. Finite element numerical simulation methods require complex finite element models, resulting in complex calculations, low computational efficiency, and difficulty in guaranteeing computational accuracy. Therefore, this paper establishes an effective method for predicting the folding steady-state of parabolic bistable deployable composite material extendable arms. This invention requires only a small number of component material performance parameters and geometric parameters to quickly and accurately predict the folding steady-state of parabolic bistable deployable composite material extendable arms, demonstrating its significant academic value and broad engineering application prospects. Summary of the Invention

[0003] This invention establishes a method for predicting the folding steady state of a parabolic bistable deployable composite material extension arm. This method has the advantages of simple calculation and high accuracy. The technical solution is as follows:

[0004] Step 1: Define the geometry and dimensions of the parabolic bistable deployable composite material extension arm, and determine the mathematical expression for the relationship between the various geometric parameters.

[0005] A parabolic bistable deployable composite material extender can achieve the interconversion between a deployed steady state and a folded steady state by storing and releasing elastic strain energy. The geometric configuration of the parabolic bistable deployable composite material extender in the deployed steady state is determined by the length L, thickness t, the coefficient J of the parabolic function of the cross-section, and the boundary conditions K. Figure 1 As shown. To characterize the geometric properties of the parabolic bistable deployable composite extension arm in both steady states, this paper makes the following basic assumptions:

[0006] (1) The centerline of the longitudinal section of the parabolic bistable deployable composite material extension arm under folded steady state is an Archimedean spiral, and any two loops are in close contact, such as Figure 2 As shown.

[0007] (2) Ignoring the wall thickness change of the parabolic bistable deployable composite extension arm during the bistable deformation process, the overall deformation can be described by the change in the shape of the neutral surface, which is not stretched.

[0008] (3) When the parabolic bistable deployable composite material extension arm is in a folded steady state, the curvature of the y-direction and the xy-plane is 0.

[0009] The cross-sectional configuration of the parabolic bistable deployable composite extension arm is parabolic, satisfying the functional relationship:

[0010] z = Jy 2 , y∈-K,K (1)

[0011] Taking the derivative and second derivative of equation (1) yields...

[0012]

[0013] The curvature in the y-direction at any position of the parabolic bistable deployable composite material extender arm in its deployed steady state can be expressed as:

[0014]

[0015] Substituting equation (2) into equation (3) yields

[0016]

[0017] Based on the basic assumptions (1) and Figure 2 The geometry of the cross-section of a parabolic bistable deployable composite articulated arm in its folded steady state can be described by a polynomial shape function in polar coordinates (ρ, α).

[0018] ρ=aα+b,α∈α0,α1 (5)

[0019] Where 'a' controls the distance between two adjacent loops, 'b' is the distance from the starting point to the origin of the polar coordinates, and 'α0' and 'α1' are the polar angles of the starting and ending points of the parabolic bistable deployable composite material extension arm under folded steady state, respectively.

[0020] Based on basic assumption (1), we can obtain

[0021]

[0022] According to geometric relationships, the boundary conditions that equation (5) needs to satisfy are:

[0023]

[0024] Where r0 and r1 are the extreme radii of the starting and ending points of the parabolic bistable deployable composite material extension arm under folded steady state, respectively.

[0025] Typically, α0 = 0, then equation (7) can be simplified to

[0026]

[0027] Substituting equation (6) into equation (8), we get

[0028]

[0029] Based on the basic assumption (2), we can obtain

[0030]

[0031] Where L is the length of the parabolic bistable deployable composite material extension arm.

[0032] Substituting equations (5)-(9) into equation (10) and integrating, we get

[0033]

[0034] According to the definition of radius of curvature, the curvature in the x-direction of a parabolic bistable deployable composite material extendable arm in a folded steady state can be expressed as:

[0035]

[0036] Based on the geometric configuration of the unfolded steady state, the curvature in the x-direction of the parabolic bistable deployable composite extender arm in the unfolded steady state is...

[0037] k x,0 =0 (13)

[0038] According to basic assumption (2), the curvature in the y-direction of the parabolic bistable deployable composite extension arm in the folded steady state is _____.

[0039] κ y =0 (14)

[0040] Step 2: Determine the total strain energy of the parabolic bistable deployable composite extension arm when it is in a folded steady state, based on classical laminate theory and the principle of minimum energy.

[0041] Using the ABD matrix from classical laminates, the bistable deformation process of a parabolic bistable deployable composite extension arm of an antisymmetric ply can be represented, and the relationship between internal forces and deformation can be expressed as follows:

[0042]

[0043] Based on the basic assumptions (2) and (3), we can obtain

[0044]

[0045] The formula for the bending strain energy per unit area of ​​a laminate can be expressed as:

[0046]

[0047] in,

[0048]

[0049] Substituting equations (4), (12) to (16) and (18) into equation (17) yields

[0050]

[0051] Substituting equation (16) into equation (15) and solving equation (15), we can obtain...

[0052]

[0053] in,

[0054] B * =-A -1 B (21)

[0055] Where A and B are the tensile stiffness matrix and coupling stiffness matrix of the laminate, respectively.

[0056] Substituting equations (20) and (21) into equation (19) and integrating, we can obtain the strain energy stored per unit length of the parabolic bistable deployable composite material extension arm in the x-direction when it is in a folded steady state, which is:

[0057]

[0058] Therefore, the total strain energy of the parabolic bistable deployable composite extension arm in its folded steady state can be expressed as:

[0059]

[0060] Substituting equation (22) into equation (23) yields

[0061]

[0062] Substituting equation (12) into equation (24) yields...

[0063]

[0064] By representing all α in equation (25) with ρ, equation (25) can be simplified to:

[0065]

[0066] It is known that the folding steady state of a parabolic bistable deployable composite articulated arm corresponds to an energy minimum. Therefore, applying the principle of minimum energy to equation (26) yields...

[0067]

[0068] Where H1 and H2 are intermediate variables, they can be represented as follows:

[0069]

[0070] In Equation (27), there is a differential term dr1 / dr0. The present invention uses the central difference algorithm to realize numerical differentiation.

[0071] Equations (11) and (27) each contain only two unknowns, namely r0 and r1. By combining equations (11) and (27) and solving them numerically, these two unknowns can be obtained. Substituting the obtained r0 and r1 into equation (9) yields the final polar angle α1.

[0072] Step 3: Determine the stress level of the parabolic bistable deployable composite extension arm when it is in a folded steady state according to the Tsai-Hill criterion, and derive the Tsai-Hill failure coefficient expression.

[0073] Based on fundamental assumption (1), the parabolic bistable deployable composite material extender arm, when in a folded steady state, exhibits maximum strain and stress in the x-direction at its starting position. The maximum curvature change of the parabolic bistable deployable composite material extender arm in the x-direction is...

[0074]

[0075] The maximum curvature change in the y-direction of the parabolic bistable deployable composite extension arm in its folded steady state is...

[0076] Δκ y | y＝0 =-2J (29)

[0077] The stress-strain relationship of the k-th layer at the critical point in the laminate is as follows:

[0078]

[0079] Among them, Q ij Let z be the distance of the k-th layer from the neutral plane, representing the material stiffness of the laminate after conversion.

[0080] Substituting equations (28) and (29) into equation (30) and simplifying, we get

[0081]

[0082] According to the stress component coordinate transformation equation of the k-th layer in the laminate, as follows: Figure 3 As shown, the maximum stress in the principal direction of the k-th layer of the laminate can be expressed as:

[0083]

[0084] Substituting equations (33) to (35) into equation (36) yields

[0085]

[0086] The maximum stress in the principal direction of the k-th layer is substituted into the Tsai-Hill criterion to evaluate the stress level of the parabolic bistable deployable composite extension arm in a folded steady state. When the Tsai-Hill failure factor I... f When the value reaches or exceeds 1, the parabolic bistable deployable composite material extension arm fails; otherwise, it does not fail.

[0087] Tsai-Hill Failure Factor I f The following formula can be used to calculate

[0088]

[0089] in,

[0090]

[0091] in, and These are the principal stresses in the longitudinal and transverse directions of the k-th laminate, respectively. Let X be the shear stress in the longitudinal and transverse directions of the k-th layer of the laminate. t and X c These are the longitudinal tensile strength and compressive strength of the composite material, respectively, with X1 and X2 as intermediate variables, and Y... t and Y c These are the transverse tensile strength and compressive strength of the composite material, respectively, with Y being an intermediate variable.

[0092] Figure 4 The flowchart shows the analytical model established for predicting the folding steady state of a parabolic bistable deployable composite material extension arm, which is used in this invention.

[0093] This invention provides a method for predicting the folding steady state of a parabolic bistable deployable composite material extension arm. Its key feature is that, based on the component material performance parameters and geometric parameters of the parabolic bistable deployable composite material extension arm, the folding steady state of the extension arm can be predicted conveniently and quickly. Attached Figure Description

[0094] Figure 1This is a schematic diagram of the geometric configuration of a parabolic bistable deployable composite material extension arm in its stable deployment state.

[0095] Figure 2 This is a schematic diagram of the geometric configuration of a parabolic bistable deployable composite material extension arm in its folded steady state.

[0096] Figure 3 This is a schematic diagram of stress analysis for the k-th layer of the laminate.

[0097] Figure 4 A flowchart for predicting the steady-state folding of a parabolic bistable deployable composite extension arm.

[0098] The symbols in the diagram are explained as follows:

[0099] Figure 1 In this context, L represents the length of the parabolic bistable deployable composite extension arm, t represents the thickness of the parabolic bistable deployable composite extension arm, J represents the coefficient of the cross-sectional parabolic function, K represents the boundary conditions, and x, y, and z represent the coordinate axes of the Cartesian coordinate system.

[0100] Figure 2 In the figure, r0 and r1 are the starting and ending polar radii of the parabolic bistable deployable composite material extension arm under folded steady state, respectively; ρ is the polar radii of the parabolic bistable deployable composite material extension arm under folded steady state; α is the polar angle of the parabolic bistable deployable composite material extension arm under folded steady state; and u and v are the coordinate axes of the rectangular coordinate system.

[0101] Figure 3 middle and Let be the normal stresses in the x and y directions of the k-th composite material layer in the laminate, respectively. Let denot be the shear stress of the k-th layer of composite material in the laminate, and β be the layup angle, 1. Principal direction 1, 2. Principal direction 2. Detailed implementation method:

[0102] Step 1: Define the geometry and dimensions of the parabolic bistable deployable composite material extension arm, and determine the mathematical expression for the relationship between the various geometric parameters.

[0103] A parabolic bistable deployable composite material extender can achieve the interconversion between a deployed steady state and a folded steady state by storing and releasing elastic strain energy. The geometric configuration of the parabolic bistable deployable composite material extender in the deployed steady state is determined by the length L, thickness t, the coefficient J of the parabolic function of the cross-section, and the boundary conditions K. Figure 1 As shown. To characterize the geometric properties of the parabolic bistable deployable composite extension arm in both steady states, this paper makes the following basic assumptions:

[0104] (1) The centerline of the longitudinal section of the parabolic bistable deployable composite material extension arm under folded steady state is an Archimedean spiral, and any two loops are in close contact, such as Figure 2 As shown.

[0105] (2) Ignoring the wall thickness change of the parabolic bistable deployable composite extension arm during the bistable deformation process, the overall deformation can be described by the change in the shape of the neutral surface, which is not stretched.

[0106] (3) When the parabolic bistable deployable composite material extension arm is in a folded steady state, the curvature of the y-direction and the xy-plane is 0.

[0107] The cross-sectional configuration of the parabolic bistable deployable composite extension arm is parabolic, satisfying the functional relationship:

[0108] z = Jy 2 , y∈-K,K (1)

[0109] Taking the derivative and second derivative of equation (1) yields...

[0110]

[0111] The curvature in the y-direction at any position of the parabolic bistable deployable composite material extender arm in its deployed steady state can be expressed as:

[0112]

[0113] Substituting equation (2) into equation (3) yields

[0114]

[0115] Based on the basic assumptions (1) and Figure 2 The geometry of the cross-section of a parabolic bistable deployable composite articulated arm in its folded steady state can be described by a polynomial shape function in polar coordinates (ρ, α).

[0116] ρ=aα+b,α∈α0,α1 (5)

[0117] Where 'a' controls the distance between two adjacent loops, 'b' is the distance from the starting point to the origin of the polar coordinates, and 'α0' and 'α1' are the polar angles of the starting and ending points of the parabolic bistable deployable composite material extension arm under folded steady state, respectively.

[0118] Based on basic assumption (1), we can obtain

[0119]

[0120] According to geometric relationships, the boundary conditions that equation (5) needs to satisfy are:

[0121]

[0122] Where r0 and r1 are the extreme radii of the starting and ending points of the parabolic bistable deployable composite material extension arm under folded steady state, respectively.

[0123] Typically, α0 = 0, then equation (7) can be simplified to

[0124]

[0125] Substituting equation (6) into equation (8), we get

[0126]

[0127] Based on the basic assumption (2), we can obtain

[0128]

[0129] Where L is the length of the parabolic bistable deployable composite material extension arm.

[0130] Substituting equations (5)-(9) into equation (10) and integrating, we get

[0131]

[0132] According to the definition of radius of curvature, the curvature in the x-direction of a parabolic bistable deployable composite material extendable arm in a folded steady state can be expressed as:

[0133]

[0134] Based on the geometric configuration of the unfolded steady state, the curvature in the x-direction of the parabolic bistable deployable composite extender arm in the unfolded steady state is...

[0135] k x,0 =0 (13)

[0136] According to basic assumption (2), the curvature in the y-direction of the parabolic bistable deployable composite extension arm in the folded steady state is _____.

[0137] κ y =0 (14)

[0138] Step 2: Determine the total strain energy of the parabolic bistable deployable composite extension arm when it is in a folded steady state, based on classical laminate theory and the principle of minimum energy.

[0139] Using the ABD matrix from classical laminates, the bistable deformation process of a parabolic bistable deployable composite extension arm of an antisymmetric ply can be represented, and the relationship between internal forces and deformation can be expressed as follows:

[0140]

[0141] Based on the basic assumptions (2) and (3), we can obtain

[0142]

[0143] The formula for the bending strain energy per unit area of ​​a laminate can be expressed as:

[0144]

[0145] in,

[0146]

[0147] Substituting equations (4), (12) to (16) and (18) into equation (17) yields

[0148]

[0149] Substituting equation (16) into equation (15) and solving equation (15), we can obtain...

[0150]

[0151] in,

[0152] B * =-A -1 B (21)

[0153] Where A and B are the tensile stiffness matrix and coupling stiffness matrix of the laminate, respectively.

[0154] Substituting equations (20) and (21) into equation (19) and integrating, we can obtain the strain energy stored per unit length of the parabolic bistable deployable composite material extension arm in the x-direction when it is in a folded steady state, which is:

[0155]

[0156] Therefore, the total strain energy of the parabolic bistable deployable composite extension arm in its folded steady state can be expressed as:

[0157]

[0158] Substituting equation (22) into equation (23) yields

[0159]

[0160] Substituting equation (12) into equation (24) yields...

[0161]

[0162] By representing all α in equation (25) with ρ, equation (25) can be simplified to:

[0163]

[0164] It is known that the folding steady state of a parabolic bistable deployable composite articulated arm corresponds to an energy minimum. Therefore, applying the principle of minimum energy to equation (26) yields...

[0165]

[0166] Where H1 and H2 are intermediate variables, they can be represented as follows:

[0167]

[0168] In Equation (27), there is a differential term dr1 / dr0. The present invention uses the central difference algorithm to realize numerical differentiation.

[0169] Equations (11) and (27) each contain only two unknowns, namely r0 and r1. By combining equations (11) and (27) and solving them numerically, these two unknowns can be obtained. Substituting the obtained r0 and r1 into equation (9) yields the final polar angle α1.

[0170] Step 3: Determine the stress level of the parabolic bistable deployable composite extension arm when it is in a folded steady state according to the Tsai-Hill criterion, and derive the Tsai-Hill failure coefficient expression.

[0171] Based on fundamental assumption (1), the parabolic bistable deployable composite material extender arm, when in a folded steady state, exhibits maximum strain and stress in the x-direction at its starting position. The maximum curvature change of the parabolic bistable deployable composite material extender arm in the x-direction is...

[0172]

[0173] The maximum curvature change in the y-direction of the parabolic bistable deployable composite extension arm in its folded steady state is...

[0174] Δκ y | y＝0 =-2J (29)

[0175] The stress-strain relationship of the k-th layer at the critical point in the laminate is as follows:

[0176]

[0177] Among them, Q ij Let z be the distance of the k-th layer from the neutral plane, representing the material stiffness of the laminate after conversion.

[0178] Substituting equations (28) and (29) into equation (30) and simplifying, we get

[0179]

[0180] According to the stress component coordinate transformation equation of the k-th layer in the laminate, as follows: Figure 3 As shown, the maximum stress in the principal direction of the k-th layer of the laminate can be expressed as:

[0181]

[0182] Substituting equations (33) to (35) into equation (36) yields

[0183]

[0184] The maximum stress in the principal direction of the k-th layer is substituted into the Tsai-Hill criterion to evaluate the stress level of the parabolic bistable deployable composite extension arm in a folded steady state. When the Tsai-Hill failure factor I... f When the value reaches or exceeds 1, the parabolic bistable deployable composite material extension arm fails; otherwise, it does not fail.

[0185] Tsai-Hill Failure Factor I f The following formula can be used to calculate

[0186]

[0187] in,

[0188]

[0189] in, and These are the principal stresses in the longitudinal and transverse directions of the k-th laminate, respectively. Let X be the shear stress in the longitudinal and transverse directions of the k-th layer of the laminate. t and X c These are the longitudinal tensile strength and compressive strength of the composite material, respectively, with X1 and X2 as intermediate variables, and Y... t and Y c These are the transverse tensile strength and compressive strength of the composite material, respectively, with Y being an intermediate variable.

[0190] Figure 4 The flowchart shows the analytical model established for predicting the folding steady state of a parabolic bistable deployable composite material extension arm, which is used in this invention.

[0191] This invention provides a method for predicting the folding steady state of a parabolic bistable deployable composite material extension arm. Its key feature is that, based on the component material performance parameters and geometric parameters of the parabolic bistable deployable composite material extension arm, the folding steady state of the extension arm can be predicted conveniently and quickly.

Claims

1. A method for predicting the folding steady state of a parabolic bistable deployable composite material extension arm, characterized in that: The specific steps of this method are as follows: Step 1: Define the geometry and dimensions of the parabolic bistable deployable composite material extension arm, and determine the mathematical expression for the relationship between the various geometric parameters; A parabolic bistable deployable composite extender can achieve the interconversion between a deployed steady state and a folded steady state by storing and releasing elastic strain energy. The geometric configuration of the parabolic bistable deployable composite extender in the deployed steady state is determined by the length L, thickness t, coefficient J of the parabolic function of the cross section, and boundary conditions K. To characterize the geometric properties of the parabolic bistable deployable composite extender in the two steady states, this paper makes the following basic assumptions: (1) The longitudinal section centerline of the parabolic bistable deployable composite material extension arm under folding steady state is an Archimedean spiral, and any two loops are in close contact; (2) Ignoring the wall thickness change of the parabolic bistable deployable composite extension arm during the bistable deformation process, the overall deformation can be described by the change in the shape of the neutral surface, which is not stretched. (3) When the parabolic bistable deployable composite material extension arm is in a folded steady state, the curvature of the y-direction and the xy-plane is 0; The cross-sectional configuration of the parabolic bistable deployable composite extension arm is parabolic, satisfying the functional relationship: z=You 2 ,y∈-K,K (1) Taking the derivative and second derivative of equation (1) yields... The curvature in the y-direction at any position of the parabolic bistable deployable composite material extender arm in its deployed steady state can be expressed as: Substituting equation (2) into equation (3) yields Based on the fundamental assumption (1), the geometric configuration of the cross-section of the parabolic bistable deployable composite extension arm in the folded steady state can be described by a polynomial shape function in polar coordinates (ρ, α). ρ=aα+b,α∈α0,α1 (5) where a controls the distance between two adjacent circles, b is the distance from the starting point to the origin of the polar coordinates, and α0 and α1 are the polar angles of the starting point and the ending point of the parabolic bistable deployable composite material extension arm under folding steady state, respectively. Based on basic assumption (1), we can obtain According to geometric relationships, the boundary conditions that equation (5) needs to satisfy are: Where r0 and r1 are the starting and ending polar radii of the parabolic bistable deployable composite material extension arm under folded steady state, respectively. Typically, α0 = 0, then equation (7) can be simplified to Substituting equation (6) into equation (8), we get Based on the basic assumption (2), we can obtain Where L is the length of the parabolic bistable deployable composite material extension arm; Substituting equations (5)-(9) into equation (10) and integrating, we get According to the definition of radius of curvature, the curvature in the x-direction of a parabolic bistable deployable composite material extendable arm in a folded steady state can be expressed as: Based on the geometric configuration of the unfolded steady state, the curvature in the x-direction of the parabolic bistable deployable composite extender arm in the unfolded steady state is... k x,0 ＝0 (13) According to basic assumption (2), the curvature in the y-direction of the parabolic bistable deployable composite extension arm in the folded steady state is _____. k y =0 (14) Step 2: Determine the total strain energy of the parabolic bistable deployable composite extension arm when it is in a folded steady state, based on classical laminate theory and the principle of minimum energy. Using the ABD matrix from classical laminates, the bistable deformation process of a parabolic bistable deployable composite extension arm of an antisymmetric ply can be represented, and the relationship between internal forces and deformation can be expressed as follows: Based on the basic assumptions (2) and (3), we can obtain The formula for the bending strain energy per unit area of ​​a laminate can be expressed as: in, Substituting equations (4), (12) to (16) and (18) into equation (17) yields Substituting equation (16) into equation (15) and solving equation (15), we can obtain... in, B * =-A -1 B (21) Where A and B are the tensile stiffness matrix and coupling stiffness matrix of the laminate, respectively; Substituting equations (20) and (21) into equation (19) and integrating, we can obtain the strain energy stored per unit length of the parabolic bistable deployable composite material extension arm in the x-direction when it is in a folded steady state, which is: Therefore, the total strain energy of the parabolic bistable deployable composite extension arm in its folded steady state can be expressed as: Substituting equation (22) into equation (23) yields Substituting equation (12) into equation (24) yields... By representing all α in equation (25) with ρ, equation (25) can be simplified to: It is known that the folding steady state of a parabolic bistable deployable composite material extension arm corresponds to an energy minimum; therefore, applying the minimum energy principle to equation (26) yields... Where H1 and H2 are intermediate variables, they can be represented as follows: In Equation (27), there is a differential term dr1 / dr0. The present invention uses the central difference algorithm to realize numerical differentiation. Equations (11) and (27) contain only two unknowns, namely r0 and r1. By combining equations (11) and (27) and solving them numerically, these two unknowns can be obtained. Substituting the obtained r0 and r1 into equation (9) will yield the final polar angle α1. Step 3: Determine the stress level of the parabolic bistable deployable composite extension arm in the folded steady state according to the Tsai-Hill criterion, and derive the Tsai-Hill failure coefficient expression. Based on basic assumption (1), the parabolic bistable deployable composite material extension arm has the maximum strain and stress in the x-direction at the starting position when it is in a folded steady state; the maximum curvature change of the parabolic bistable deployable composite material extension arm in the x-direction is... The maximum curvature change in the y-direction of the parabolic bistable deployable composite extension arm in its folded steady state is... Dk y | y＝0 =-2J (29) The stress-strain relationship of the k-th layer at the critical point in the laminate is as follows: Among them, Q ij To convert the material stiffness of the laminate, z is the distance of the k-th layer from the neutral plane; Substituting equations (28) and (29) into equation (30) and simplifying, we get According to the stress component coordinate transformation equation of the k-th layer in a laminate, the maximum stress in the principal direction of the k-th layer can be expressed as: Substituting equations (33) to (35) into equation (36) yields The maximum stress in the principal direction of the k-th layer is substituted into the Tsai-Hill criterion to evaluate the stress level of the parabolic bistable deployable composite extension arm in a folded steady state; when the Tsai-Hill failure factor I f When the value reaches or exceeds 1, the parabolic bistable deployable composite material extension arm fails; otherwise, it does not fail. Tsai-Hill Failure Factor I f The following formula can be used to calculate in, in, and These are the principal stresses in the longitudinal and transverse directions of the k-th laminate, respectively. Let X be the shear stress in the longitudinal and transverse directions of the k-th layer of the laminate. t and X c These are the longitudinal tensile strength and compressive strength of the composite material, respectively, with X1 and X2 as intermediate variables, and Y... t and Y c These are the transverse tensile strength and compressive strength of the composite material, respectively, with Y being an intermediate variable.

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