Mechanical analysis method of low-tension cable

Abstract

The invention discloses a mechanical analysis method for a low-tension cable, and belongs to the field of cable simulation. Through static balance, the midspan sag of the cable unit is solved based on the self-weight distribution load of the cable unit, and sag distribution of the cable unit is solved according to the midspan sag of the cable unit; determining an equivalent node load of the cable unit according to the self-weight distribution load of the cable unit; integrating a cable unit load vector on a two-dimensional plane based on the equivalent node load of the cable unit; the cable unit load vectors on the two-dimensional plane are popularized to a three-dimensional space, and cable unit load vectors of the cable units in the three-dimensional space are obtained; and calculating an elastic stiffness matrix of the cable unit based on the cable unit load vector and the displacement parameter of the cable unit in the three-dimensional space, and integrating a total stiffness matrix of the cable structure according to the elastic stiffness matrix of the cable unit. The method solves the problem that the existing method cannot reflect the tension-bending-torsion coupling phenomenon and the mechanical behaviors of uniformly distributed loads such as the dead weight of the cable.

Description

Technical Field

[0001] The invention belongs to the field of cable simulation, and in particular relates to a mechanical analysis method for low-tension cables. Background Art

[0002] At present, cable elements are generally used for cable simulation, considering that the structure is only subjected to pure tension, and the tension is very large. This assumption is reasonable for most high-tension, large-span cable analyses. However, for cables with shorter spans, especially cables with lower tension, it is necessary to consider the coupled mechanical properties of tension, bending, and torsion of the cable, which are no longer small quantities relative to the axial mechanical properties. Traditional straight beam elements can simulate the tension, bending, and torsion characteristics of the structure, but cannot reflect the coupling terms between them, and require multiple elements for high-precision simulation. They also cannot reflect the initial curve shape of the cable, and the effects of uniformly distributed loads such as deadweight on the beam element. Summary of the invention

[0003] In view of the above-mentioned deficiencies in the prior art, the present invention provides a mechanical analysis method for low-tension cables, which solves the problem that the prior method cannot reflect the mechanical behavior of uniformly distributed loads such as the tension-bending-torsion coupling phenomenon and the deadweight of the cable.

[0004] In order to achieve the above-mentioned purpose of the invention, the technical solution adopted by the present invention is: a mechanical analysis method of a low-tension cable, comprising the following steps:

[0005] S1, using quintic interpolation beam element to simulate cable element;

[0006] S2. According to the catenary shape of the cable unit, the load is distributed based on the deadweight of the cable unit through static equilibrium to solve the mid-span sag of the cable unit, and according to the mid-span sag of the cable unit, the sag distribution of the cable unit is solved;

[0007] S3. Determine the equivalent node load of the cable unit according to the self-weight distribution load of the cable unit;

[0008] S4, according to the sag distribution of the cable element, integrating the cable element load vector on the two-dimensional plane based on the equivalent node load of the cable element;

[0009] S5. Extend the cable unit load vector on the two-dimensional plane to the three-dimensional space to obtain the cable unit load vector in the three-dimensional space;

[0010] S6. Based on the load vector of the cable unit in the three-dimensional space, the displacement parameters of the cable unit are solved, based on the displacement parameters of the cable unit, the elastic stiffness matrix of the cable unit is calculated, and the total stiffness matrix of the cable structure is integrated according to the elastic stiffness matrix of the cable unit.

[0011] The beneficial effects of the present invention are as follows: the present invention adopts a beam element with quintic interpolation to realize accurate simulation of mechanical behavior within the unit scale. Based on the rigid body criterion, large deformation behavior is considered, and the geometric nonlinearity of the beam element can be accurately simulated within the macroscopic scale. At the same time, the initial deformation, initial deadweight and tension load characteristics of the cable element are considered to realize efficient and accurate simulation of the cable element.

[0012] Furthermore, the expressions of the mid-span sag of the cable unit and the sag distribution of the cable unit in step S2 are respectively:

[0013]

[0014] v g =v0(1-t 2 )

[0015] w=γA

[0016] t=2x / L

[0017] -L / 2≤x≤L / 2

[0018] Where v0 is the mid-span sag of the cable unit; w is the self-weight distributed load of the cable unit; L is the length of the cable unit; P is the axial tension; v g is the sag distribution of the cable unit; t is the dimensionless length of the cable unit; γ is the weight of the cable unit; A is the cross-sectional area of ​​the cable unit; x is the position coordinate of any point on the cable unit.

[0019] The beneficial effect of the above further solution is that a true cable unit sag distribution function taking into account the deadweight can be obtained.

[0020] Furthermore, the expression of the equivalent node load of the cable unit in step S3 is:

[0021]

[0022] P EQV =0

[0023] in, is the equivalent fixed end bending moment at one end of the cable element; is the equivalent fixed end bending moment at the other end of the cable element; P EQV is the equivalent axial tension; w is the self-weight distributed load of the cable unit; L is the length of the cable unit.

[0024] The beneficial effect of the above further solution is that the equivalent node load generated by the deadweight can be considered, and the effect of the deadweight can be fed back more realistically.

[0025] Furthermore, the step S4 is specifically as follows:

[0026] S401, calculating the lateral displacement of the cable unit;

[0027] S402. According to the sag distribution of the cable element and the lateral displacement of the cable element, the total potential energy function of the beam element composed of the strain energy and the external force potential energy is obtained by quintic interpolation:

[0028]

[0029] u1=θ1

[0030] u2=θ2

[0031] u3=e

[0032] Where Π is the total potential energy function of the beam element with quintic interpolation; L is the length of the cable element; E is the Young's elastic model; A is the cross-sectional area of ​​the cable element; is the first derivative of the axial displacement at any position of the cable element with respect to the x coordinate; d is the differential symbol; x is the position coordinate of any point on the cable element; I is the second moment of area; is the second-order derivative of the lateral displacement at any position of the cable unit with respect to the x-coordinate; P is the axial tension; is the first derivative of the lateral displacement at any position of the cable element with respect to the x-coordinate; is the first derivative of the sag distribution of the cable element with respect to the x-coordinate; F i is the conjugate external load in the i-th direction on the two nodes of the cable element; u i is the conjugate displacement of the cable element in the i-th direction; i is the direction number of the node displacement and load; w is the self-weight distributed load of the cable element; v is the lateral displacement of the cable element; u1 is the conjugate displacement of the cable element in the first direction; u2 is the conjugate displacement of the cable element in the second direction; u3 is the conjugate displacement of the cable element in the third direction; θ1 is the tangent rotation angle of the node at one end of the cable element relative to the line connecting the cable element nodes; θ2 is the tangent rotation angle of the node at the other end of the cable element relative to the line connecting the cable element nodes; e is the axial expansion and contraction displacement of the cable element;

[0033] S403. According to the total potential energy function of the beam element by quintic interpolation, the secant stiffness relationship is obtained:

[0034]

[0035] q=PL 2 / EI

[0036] Among them, δ is the sign of the change; is the symbol of partial derivative; q is the substitution load variable related to the axial tension P;

[0037] S404. According to the secant stiffness relationship, the cable unit load vector on the two-dimensional plane is obtained:

[0038]

[0039]

[0040] Among them, F1 is the conjugate external load in the first direction on the two nodes of the cable unit; F2 is the conjugate external load in the second direction on the two nodes of the cable unit; F3 is the conjugate external load in the third direction on the two nodes of the cable unit; M1 is the node bending moment at one end of the cable unit; is the equivalent fixed end bending moment at one end of the cable element; c1, c2, c0, c w , b1, b2, b vs 、b vv 、b s 、b w , H1 and H2 are coefficient variables related to the replacement load variable q; v0 is the mid-span sag of the cable element; is the substitution variable for the mid-span bending moment M0 of the cable element caused by the lateral load on the beam; M2 is the node bending moment at the other end of the cable element; is the equivalent fixed end bending moment at the other end of the cable element; P EQV is the equivalent axial tension; M0 is the mid-span bending moment of the cable unit.

[0041] The beneficial effect of the above further solution is that after taking into account the effect of the deadweight load, the stress condition of the cable unit is closer to the actual stress condition.

[0042] Furthermore, the expression of the lateral displacement of the cable unit in step S401 is:

[0043] v=N1(Lθ1)+N2(Lθ2)+N0v0+N w M0V w +N s LS0

[0044]

[0045] N w =[ζ 2 -2ζ 3 +ζ 4 ]

[0046] N s =[ζ 2 -4ζ 3 +5ζ 4 -2ζ 5 ]

[0047] ζ=x / L+0.5

[0048] Where v is the lateral displacement of the cable unit; N1 and N2 are both shape functions; N0 is the shape function related to the mid-span sag; N w is the shape function related to the midpoint equivalent bending moment generated by the deadweight load; Ns is the shape function related to the shear load generated by the deadweight load; v w is the displacement caused by the equivalent bending moment at the mid-span point generated by the distributed self-weight load; S0 is the shear force at the mid-span of the cable unit; ζ is the dimensionless coordinate variable of the coordinates of any position of the cable unit; t is the dimensionless length of the cable unit.

[0049] The beneficial effect of the above further solution is that an accurate displacement function is used here, which takes into account the influence of the cable unit axial force, the sag shape, the midpoint equivalent bending moment caused by the distributed deadweight load and the displacement caused by the equivalent shear load.

[0050] Furthermore, the boundary condition of the lateral displacement of the cable unit in step S401 is:

[0051] x=±L / 2,v=0

[0052]

[0053] The beneficial effect of the above further solution is: boundary constraints are imposed on the lateral displacement of the cable unit to prepare for the calculation of the lateral displacement of the cable unit.

[0054] Furthermore, the expression of the bending moment and shear force equilibrium condition of the lateral displacement of the cable unit in step S401 is:

[0055]

[0056] Where v is the lateral displacement of the cable unit; v g is the sag distribution of the cable element; is the third-order derivative of the lateral displacement at any position of the cable element with respect to the x-coordinate; S0 is the shear force at the mid-span of the cable element.

[0057] The beneficial effect of the above further solution is that a force-displacement balance relationship is obtained which can take into account the deadweight load and sag shape of the cable unit.

[0058] Furthermore, the expression of the cable unit load vector in the three-dimensional space in step S5 is:

[0059]

[0060]

[0061] q n =PL 2 / EI n

[0062] Among them, F 1n is the conjugate external load on the two nodes of the cable element in the first direction corresponding to the n-axis; F 2nis the conjugate external load of the second direction corresponding to the n-axis on the two nodes of the cable element; F4 is the conjugate external load of the third direction on the two nodes of the cable element; M 1n is the node bending moment at one end of the cable unit corresponding to the n-axis; n is the direction, including the Y-axis and the Z-axis; is the equivalent fixed end bending moment of one end of the cable unit corresponding to the n-axis; E is Young's elastic model; I n is the second-order moment of the area corresponding to the n-axis; L is the length of the cable unit; c 1n 、c 2n 、c wn 、c 0n 、b 1n 、b 2n 、b sn 、b wn 、b vsn and b vvn are replaced by the load variable q n The coefficient variable of interest; q n is the replacement load variable corresponding to the n-axis related to the axial tension P; θ 1n θ is the tangent rotation angle of the node at one end of the cable element relative to the corresponding n-axis of the line connecting the cable element nodes; 2n is the tangent rotation angle of the other end node of the cable element relative to the corresponding n-axis of the line connecting the cable element nodes; is the mid-span bending moment M of the cable element corresponding to the n-axis caused by the transverse load on the beam 0n Substitution variable; M 0n is the mid-span bending moment of the cable unit corresponding to the n-axis; v 0n M is the mid-span sag of the cable unit corresponding to the n-axis; 2n is the node bending moment at the other end of the cable element corresponding to the n-axis; is the equivalent fixed end bending moment of the other end of the cable unit corresponding to the n-axis; P is the axial tension; A is the cross-sectional area of ​​the cable unit; e is the axial expansion and contraction displacement of the cable unit; y is the Y-axis direction; z is the Z-axis direction; I n is the second-order moment of area of ​​the cable unit cross section corresponding to the n-axis.

[0063] The beneficial effect of the above further solution is that the force-displacement balance relationship of the cable unit is expanded from a two-dimensional stress condition to a three-dimensional stress condition, which is closer to the stress condition of the cable structure in actual engineering.

[0064] Furthermore, the step S6 is specifically as follows:

[0065] S601. Based on the cable unit load vector in three-dimensional space, solve the displacement parameter (θ 1n +θ 2n ) and (θ 1n -θ 2n );θ 1nθ is the tangent rotation angle of the node at one end of the cable element relative to the corresponding n-axis of the line connecting the cable element nodes; 2n is the tangent rotation angle of the other end node of the cable element relative to the corresponding n-axis of the line connecting the cable element nodes;

[0066] S602, according to the displacement parameter (θ 1n +θ 2n ) and (θ 1n -θ 2n ), solve the elastic stiffness matrix of the cable element;

[0067] S603. According to the elastic stiffness matrix of the cable unit, the total stiffness matrix of the integrated cable structure is:

[0068]

[0069] T L =T g T off T A T R

[0070]

[0071] Among them, K t is the total stiffness matrix of the cable structure; Nele is the number of units after the cable structure is divided into units; T L is the coordinate transformation matrix; k t is the unit stiffness matrix in the local coordinate system of the cable unit; T is the transpose; T0 is the rod end force conversion matrix from 6 node forces to 12 node forces; is the elastic stiffness matrix of the cable unit; K0 is the initial tension geometric stiffness matrix; T g is the local to global transformation matrix; T off is the transformation matrix of the rigid connection; T A is the transformation matrix between the global coordinates and the intermediate reference local coordinates; T R is the standard rotation transformation matrix between the two orthogonal coordinates of the cross-section at both ends of the beam element; L is the length of the cable element.

[0072] The beneficial effect of the above further solution is that the stiffness matrix of the cable unit in the local unit coordinate system is integrated into the stiffness matrix of the entire cable structure in the global coordinate system, which facilitates the analysis of any cable structure.

[0073] Furthermore, the expression of the elastic stiffness matrix of the cable unit in step S602 is:

[0074]

[0075] n=[y,z]m=[1,2]c1′ n =2b1n c2′ n =2b 2n

[0076]

[0077]

[0078] I max =max{I z ,I y}

[0079] Where E is Young's elastic model; I max I is the second moment of area z and I y The maximum value of ξ z is the dimensionless second-order moment of area corresponding to the Z axis; c 1z and c 2z are corresponding to the Z axis and replace the load variable q z The coefficient variable of interest; q z is the replacement load variable corresponding to the Z axis related to the axial tension P; G 1z and G 2z are all substitution variables corresponding to the Z axis; H, η and λ are all coefficient variables; G 1y and G 2y are all substitution variables corresponding to the Y-axis; ξ y is the dimensionless second moment of area corresponding to the Y axis; c 1y and c 2y are corresponding to the Y-axis and replace the load variable q y The coefficient variable of interest; q y is the replacement load variable corresponding to the Y axis related to the axial tension P; G 1n and G 2n are all substitution variables corresponding to the n-axis; n is the direction, including the Y-axis and the Z-axis; y is the Y-axis direction; z is the Z-axis direction; c1′ n Substitute variable c 1n Q n The coefficient of the derivative; c2′ n Substitute variable c 2n Q n The coefficient of the derivative; c 1n 、c 2n 、c 0n 、b vsn 、b vvn 、b wn 、b sn 、b 1n and b 2n are corresponding to the n-axis and replace the load variable q nThe coefficient variable of interest; q n is the replacement load variable corresponding to the n-axis related to the axial tension P; c0′ n is the coefficient variable c 0n Q n The coefficient of the derivative; v 0n is the mid-span sag of the cable element corresponding to the n-axis; is the mid-span bending moment M of the cable element corresponding to the n-axis caused by the transverse load on the beam 0n Substitution variable; M 0n is the mid-span bending moment of the cable unit corresponding to the n-axis; m is the end point number of the cable unit; is the derivative symbol; q n is the replacement load variable corresponding to the n-axis related to the axial tension P; θ mn G is the tangent rotation angle of the mth end node of the cable element relative to the corresponding n-axis of the line connecting the cable element nodes; mn is the substitution variable of the mth end node of the cable element corresponding to the n-axis; θ t is the torsion angle; e is the axial expansion displacement of the cable unit; b1′ n is the coefficient variable b 1n Q n The coefficient of the derivative; b2′ n is the coefficient variable b 2n Q n The coefficient of the derivative; b v ' sn is the coefficient variable b vsn Q n The coefficient of the derivative; b v ' vn is the coefficient variable b vvn Q n The coefficient of the derivative; b s ' n is the coefficient variable b sn Q n The coefficient of the derivative; b′ wn is the coefficient variable b wn Q n The coefficient of the derivative; ξ n is the dimensionless second-order moment of area corresponding to the n-axis; G is the shear modulus; J is the polar moment of inertia of the cable unit section; P is the axial tension; r is the polar radius of gyration; A is the cross-sectional area of ​​the cable unit; I z is the second-order moment of area of ​​the cable element corresponding to the Z axis; I y is the second moment of area of ​​the cable element corresponding to the Y axis.

[0080] The beneficial effect of the above further solution is that the elastic stiffness matrix of the cable unit in the three-dimensional force and deformation state in the three-dimensional space is obtained. BRIEF DESCRIPTION OF THE DRAWINGS

[0081] Figure 1 The figure is a flow chart of the method of the present invention.

[0082] Figure 2 It is a force diagram of a cable unit that comprehensively considers the distributed load, cable sag, tension-bending coupling and other characteristics in an embodiment of the present invention.

[0083] Figure 3 Schematic diagram of a typical calculation case of a cable unit in an embodiment of the present invention.

[0084] Figure 4 It is an analysis diagram of the right end axial displacement of a high-efficiency straight beam unit and a classic Euler straight beam unit in an embodiment of the present invention.

[0085] Figure 5 It is an analysis diagram of the right end axial displacement of one high-efficiency straight beam unit and 100 classic Euler straight beam units in an embodiment of the present invention.

[0086] Figure 6 It is an analysis diagram of the right end axis rotation angle of one high-efficiency straight beam unit and 100 classic Euler straight beam units in an embodiment of the present invention.

[0087] Figure 7 It is an analysis diagram of the right end axis rotation angle of a high-efficiency straight beam unit and a classic Euler straight beam unit in an embodiment of the present invention.

[0088] Figure 8 It is an analysis diagram of the lateral displacement at the mid-span of one efficient straight beam unit and 100 classic Euler straight beam units in an embodiment of the present invention.

[0089] Fig. 9 It is an analysis diagram of the lateral displacement at the mid-span of a cable of a high-efficiency straight beam unit and a classic Euler straight beam unit in an embodiment of the present invention. DETAILED DESCRIPTION

[0090] The specific implementation modes of the present invention are described below so that those skilled in the art can understand the present invention. However, it should be clear that the present invention is not limited to the scope of the specific implementation modes. For those of ordinary skill in the art, as long as various changes are within the spirit and scope of the present invention as defined and determined by the attached claims, these changes are obvious, and all inventions and creations utilizing the concept of the present invention are protected.

[0091] like Figure 1 As shown, in one embodiment of the present invention, a mechanical analysis method for a low tension cable comprises the following steps:

[0092] S1, using quintic interpolation beam element to simulate cable element;

[0093] S2. According to the catenary shape of the cable unit, the load is distributed based on the deadweight of the cable unit through static equilibrium to solve the mid-span sag of the cable unit, and according to the mid-span sag of the cable unit, the sag distribution of the cable unit is solved;

[0094] S3. Determine the equivalent node load of the cable unit according to the self-weight distribution load of the cable unit;

[0095] S4, according to the sag distribution of the cable element, integrating the cable element load vector on the two-dimensional plane based on the equivalent node load of the cable element;

[0096] S5. Extend the cable unit load vector on the two-dimensional plane to the three-dimensional space to obtain the cable unit load vector in the three-dimensional space;

[0097] S6. Based on the load vector of the cable unit in the three-dimensional space, the displacement parameters of the cable unit are solved, based on the displacement parameters of the cable unit, the elastic stiffness matrix of the cable unit is calculated, and the total stiffness matrix of the cable structure is integrated according to the elastic stiffness matrix of the cable unit.

[0098] like Figure 2 As shown in the figure, a force diagram of the cable unit is established which comprehensively considers the distributed load, cable sag, tension-bending coupling and other characteristics. The cable unit generally produces catenary deformation under the action of its own weight. For engineering analysis, a parabola can generally be used to simulate the sag.

[0099] In step S2, the expressions of the mid-span sag of the cable unit and the sag distribution of the cable unit are respectively:

[0100]

[0101] v g =v0(1-t 2 )

[0102] w=γA

[0103] t=2x / L

[0104] -L / 2≤x≤L / 2

[0105] Where v0 is the mid-span sag of the cable unit; w is the self-weight distributed load of the cable unit; L is the length of the cable unit; P is the axial tension; v g is the sag distribution of the cable unit; t is the dimensionless length of the cable unit; γ is the weight of the cable unit; A is the cross-sectional area of ​​the cable unit; x is the position coordinate of any point on the cable unit.

[0106] In this embodiment, the coordinate origin is placed at the midpoint of the line connecting the two end points of the cable.

[0107] The expression of the equivalent node load of the cable unit in step S3 is:

[0108]

[0109] P EQV =0

[0110] in, is the equivalent fixed end bending moment at one end of the cable element; is the equivalent fixed end bending moment at the other end of the cable element; P EQV is the equivalent axial tension; w is the self-weight distributed load of the cable unit; L is the length of the cable unit.

[0111] The step S4 is specifically as follows:

[0112] S401, calculating the lateral displacement of the cable unit;

[0113] S402. According to the sag distribution of the cable element and the lateral displacement of the cable element, the total potential energy function of the beam element composed of the strain energy and the external force potential energy is obtained by quintic interpolation:

[0114]

[0115] u1=θ1

[0116] u2=θ2

[0117] u3=e

[0118] Where Π is the total potential energy function of the beam element with quintic interpolation; L is the length of the cable element; E is the Young's elastic model; A is the cross-sectional area of ​​the cable element; is the first derivative of the axial displacement at any position of the cable element with respect to the x coordinate; d is the differential symbol; x is the position coordinate of any point on the cable element; I is the second moment of area; is the second-order derivative of the lateral displacement at any position of the cable unit with respect to the x-coordinate; P is the axial tension; is the first derivative of the lateral displacement at any position of the cable element with respect to the x-coordinate; is the first derivative of the sag distribution of the cable element with respect to the x-coordinate; F i is the conjugate external load in the i-th direction on the two nodes of the cable element; u i is the conjugate displacement of the cable element in the i-th direction; i is the direction number of the node displacement and load; w is the self-weight distributed load of the cable element; v is the lateral displacement of the cable element; u1 is the conjugate displacement of the cable element in the first direction; u2 is the conjugate displacement of the cable element in the second direction; u3 is the conjugate displacement of the cable element in the third direction; θ1 is the tangent rotation angle of the node at one end of the cable element relative to the line connecting the cable element nodes; θ2 is the tangent rotation angle of the node at the other end of the cable element relative to the line connecting the cable element nodes; e is the axial expansion and contraction displacement of the cable element;

[0119] S403. According to the total potential energy function of the beam element by quintic interpolation, the secant stiffness relationship is obtained:

[0120]

[0121] q=PL 2 / EI

[0122] Among them, δ is the sign of the change; is the symbol of partial derivative; q is the substitution load variable related to the axial tension P;

[0123] S404. According to the secant stiffness relationship, the cable unit load vector on the two-dimensional plane is obtained:

[0124]

[0125]

[0126] H1=q+80

[0127] H2=q+48

[0128] Among them, F1 is the conjugate external load in the first direction on the two nodes of the cable unit; F2 is the conjugate external load in the second direction on the two nodes of the cable unit; F3 is the conjugate external load in the third direction on the two nodes of the cable unit; M1 is the node bending moment at one end of the cable unit; is the equivalent fixed end bending moment at one end of the cable element; c1, c2, c0, c w , b1, b2, b vs 、b vv 、b s 、b w , H1 and H2 are coefficient variables related to the replacement load variable q; v0 is the mid-span sag of the cable element; is the substitution variable for the mid-span bending moment M0 of the cable element caused by the lateral load on the beam; M2 is the node bending moment at the other end of the cable element; is the equivalent fixed end bending moment at the other end of the cable element; P EQV is the equivalent axial tension; M0 is the mid-span bending moment of the cable unit.

[0129] The expression of the lateral displacement of the cable unit in step S401 is:

[0130] v=N1(Lθ1)+N2(Lθ2)+N0v0+N w M0V w +N s LS0

[0131]

[0132] N w =[ζ2 -2ζ 3 +ζ 4 ]

[0133] N s =[ζ 2 -4ζ 3 +5ζ 4 -2ζ 5 ]

[0134] ζ=x / L+0.5

[0135] Where v is the lateral displacement of the cable unit; N1 and N2 are both shape functions; N0 is the shape function related to the mid-span sag; N w is the shape function related to the midpoint equivalent bending moment generated by the deadweight load; N s is the shape function related to the shear load generated by the deadweight load; v w is the displacement caused by the equivalent bending moment at the mid-span point generated by the distributed self-weight load; S0 is the shear force at the mid-span of the cable unit; ζ is the dimensionless coordinate variable of the coordinates of any position of the cable unit; t is the dimensionless length of the cable unit.

[0136] In this embodiment, the lateral displacement of the cable unit is the distance away from the original equilibrium position. The displacement function is an exact function. Unlike the interpolation function of a straight beam, the parameters here all take into account the influence of the axial force. The first two items are the deformation of the beam unit considering the influence of the axial force, the third item is part of the sag shape parameter of the cable unit, the fourth item is the displacement caused by the equivalent bending moment of the midpoint generated by the distributed self-weight load, and the fifth item is the mid-part displacement caused by the shear load generated by the distributed self-weight load.

[0137] The boundary condition for the lateral displacement of the cable unit in step S401 is:

[0138] x=±L / 2,v=0

[0139]

[0140] The expression of the bending moment and shear force equilibrium condition of the lateral displacement of the cable unit in step S401 is:

[0141]

[0142] Where v is the lateral displacement of the cable unit; v g is the sag distribution of the cable element; is the third-order derivative of the lateral displacement at any position of the cable element with respect to the x-coordinate; S0 is the shear force at the mid-span of the cable element.

[0143] The expression of the cable unit load vector in the three-dimensional space in step S5 is:

[0144]

[0145] q n =PL 2 / EI n

[0146] Among them, F 1n is the conjugate external load on the two nodes of the cable element in the first direction corresponding to the n-axis; F 2n is the conjugate external load of the second direction corresponding to the n-axis on the two nodes of the cable element; F4 is the conjugate external load of the third direction on the two nodes of the cable element; M 1n is the node bending moment at one end of the cable unit corresponding to the n-axis; n is the direction, including the Y-axis and the Z-axis; is the equivalent fixed end bending moment of one end of the cable unit corresponding to the n-axis; E is Young's elastic model; I n is the second-order moment of the area corresponding to the n-axis; L is the length of the cable unit; c 1n 、c 2n 、c wn 、c 0n 、b 1n 、b 2n 、b sn 、b wn 、b vsn and b vvn are replaced by the load variable q n The coefficient variable of interest; q n is the replacement load variable corresponding to the n-axis related to the axial tension P; θ 1n θ is the tangent rotation angle of the node at one end of the cable element relative to the corresponding n-axis of the line connecting the cable element nodes; 2n is the tangent rotation angle of the other end node of the cable element relative to the corresponding n-axis of the line connecting the cable element nodes; is the mid-span bending moment M of the cable element corresponding to the n-axis caused by the transverse load on the beam 0n Substitution variable; M 0n is the mid-span bending moment of the cable unit corresponding to the n-axis; v 0n M is the mid-span sag of the cable unit corresponding to the n-axis; 2n is the node bending moment at the other end of the cable element corresponding to the n-axis; is the equivalent fixed end bending moment of the other end of the cable unit corresponding to the n-axis; P is the axial tension; A is the cross-sectional area of ​​the cable unit; e is the axial expansion and contraction displacement of the cable unit; y is the Y-axis direction; z is the Z-axis direction; I n is the second-order moment of area of ​​the cable unit cross section corresponding to the n-axis.

[0147] The step S6 is specifically as follows:

[0148] S601. Based on the cable unit load vector in three-dimensional space, solve the displacement parameter (θ 1n +θ2n ) and (θ 1n -θ 2n );θ 1n θ is the tangent rotation angle of the node at one end of the cable element relative to the corresponding n-axis of the line connecting the cable element nodes; 2n is the tangent rotation angle of the other end node of the cable element relative to the corresponding n-axis of the line connecting the cable element nodes;

[0149] S602, according to the displacement parameter (θ 1n +θ 2n ) and (θ 1n -θ 2n ), solve the elastic stiffness matrix of the cable element;

[0150] S603. According to the elastic stiffness matrix of the cable unit, the total stiffness matrix of the integrated cable structure is:

[0151]

[0152] T L =T g T off T A T R

[0153]

[0154] Among them, K t is the total stiffness matrix of the cable structure; Nele is the number of units after the cable structure is divided into units; T L is the coordinate transformation matrix; k t is the unit stiffness matrix in the local coordinate system of the cable unit; T is the transpose; T0 is the rod end force conversion matrix from 6 node forces to 12 node forces; is the elastic stiffness matrix of the cable unit; K0 is the initial tension geometric stiffness matrix; T g is the local to global transformation matrix; T off is the transformation matrix of the rigid connection; T A is the transformation matrix between the global coordinates and the intermediate reference local coordinates; T R is the standard rotation transformation matrix between the two orthogonal coordinates of the cross-section at both ends of the beam element; L is the length of the cable element.

[0155] The expression of the elastic stiffness matrix of the cable unit in step S602 is:

[0156]

[0157]

[0158] n=[y,z]

[0159] m=[1,2]

[0160] c1′ n =2b 1n

[0161] c2′ n =2b 2n

[0162]

[0163] I max =max{I z ,I y}

[0164] Where E is Young's elastic model; I max I is the second moment of area z and I y The maximum value of ξ z is the dimensionless second-order moment of area corresponding to the Z axis; c 1z and c 2z are corresponding to the Z axis and replace the load variable q z The coefficient variable of interest; q z is the replacement load variable corresponding to the Z axis related to the axial tension P; G 1z and G 2z are all substitution variables corresponding to the Z axis; H, η and λ are all coefficient variables; G 1y and G 2y are all substitution variables corresponding to the Y-axis; ξ y is the dimensionless second moment of area corresponding to the Y axis; c 1y and c 2y are corresponding to the Y-axis and replace the load variable q y The coefficient variable of interest; q y is the replacement load variable corresponding to the Y axis related to the axial tension P; G 1n and G 2n are all substitution variables corresponding to the n-axis; n is the direction, including the Y-axis and the Z-axis; y is the Y-axis direction; z is the Z-axis direction; c1′ n Substitute variable c 1n Q n The coefficient of the derivative; c2′ n Substitute variable c 2n Q n The coefficient of the derivative; c 1n 、c 2n 、c 0n 、b vsn 、b vvn 、b wn 、b sn 、b 1n and b2n are corresponding to the n-axis and replace the load variable q n The coefficient variable of interest; q n is the replacement load variable corresponding to the n-axis related to the axial tension P; c0′ n is the coefficient variable c 0n Q n The coefficient of the derivative; v 0n is the mid-span sag of the cable element corresponding to the n-axis; is the mid-span bending moment M of the cable element corresponding to the n-axis caused by the transverse load on the beam 0n Substitution variable; M 0n is the mid-span bending moment of the cable unit corresponding to the n-axis; m is the end point number of the cable unit; is the derivative symbol; q n is the replacement load variable corresponding to the n-axis related to the axial tension P; θ mn G is the tangent rotation angle of the mth end node of the cable element relative to the corresponding n-axis of the line connecting the cable element nodes; mn is the substitution variable of the mth end node of the cable element corresponding to the n-axis; θ t is the torsion angle; e is the axial expansion displacement of the cable unit; b1′ n is the coefficient variable b 1n Q n The coefficient of the derivative; b2′ n is the coefficient variable b 2n Q n The coefficient of the derivative; b v ' sn is the coefficient variable b vsn Q n The coefficient of the derivative; b v ' vn is the coefficient variable b vvn Q n The coefficient of the derivative; b s ' n is the coefficient variable b sn Q n The coefficient of the derivative; b′ wn is the coefficient variable b wn Q n The coefficient of the derivative; ξ n is the dimensionless second-order moment of area corresponding to the n-axis; G is the shear modulus; J is the polar moment of inertia of the cable unit section; P is the axial tension; r is the polar radius of gyration; A is the cross-sectional area of ​​the cable unit; I z is the second-order moment of area of ​​the cable element corresponding to the Z axis; I y is the second moment of area of ​​the cable element corresponding to the Y axis.

[0165] In this embodiment, the classic case is Figure 3As shown: a cable, hinged at both ends, one end axially free, elastic modulus E = 2e5Mpa; A = 1.0mm 2 ;I y =5.0e5mm 4 ;I z =5.0e3mm 4 ; G = 7.6923e2mm 4 ; L=10, uniformly distributed load at the mid-span w=P / L, the initial disturbance can be determined by the equilibrium condition.

[0166] The analysis results using one efficient straight beam element, one classic Euler straight beam element and 100 traditional straight beam elements are as follows Figure 4-Figure 9 As shown. It can be seen that using a high-efficiency straight beam unit and a classic Euler straight beam unit to calculate a classic Euler straight beam unit respectively cannot get the correct answer, but as the number of unit divisions increases, the traditional classic Euler straight beam unit can also achieve accurate simulation of the cable unit. In comparison, 100 traditional classic Euler straight beam units can achieve the same accuracy as the high-efficiency straight beam unit of the present invention. The high-efficiency straight beam unit proposed by the present invention has higher accuracy and higher efficiency.

Claims

1. A mechanical analysis method for low tension cables, characterized in that: The following steps are involved: S1, using quintic interpolation beam element to simulate cable element; S2. According to the catenary shape of the cable unit, the load is distributed based on the deadweight of the cable unit through static equilibrium to solve the mid-span sag of the cable unit, and according to the mid-span sag of the cable unit, the sag distribution of the cable unit is solved; S3. Determine the equivalent node load of the cable unit according to the self-weight distribution load of the cable unit; S4, according to the sag distribution of the cable element, integrating the cable element load vector on the two-dimensional plane based on the equivalent node load of the cable element; S5. Extend the cable unit load vector on the two-dimensional plane to the three-dimensional space to obtain the cable unit load vector in the three-dimensional space; S6. Based on the load vector of the cable unit in the three-dimensional space, the displacement parameters of the cable unit are solved, based on the displacement parameters of the cable unit, the elastic stiffness matrix of the cable unit is calculated, and the total stiffness matrix of the cable structure is integrated according to the elastic stiffness matrix of the cable unit.

2. The mechanical analysis method of low tension cable according to claim 1, characterized in that: In step S2, the expressions of the mid-span sag of the cable unit and the sag distribution of the cable unit are respectively: v g =v0(1-t 2 ) w=γA t=2x / L -L / 2≤x≤L / 2 Where v0 is the mid-span sag of the cable unit; w is the self-weight distributed load of the cable unit; L is the length of the cable unit; P is the axial tension; v g is the sag distribution of the cable unit; t is the dimensionless length of the cable unit; γ is the weight of the cable unit; A is the cross-sectional area of ​​the cable unit; x is the position coordinate of any point on the cable unit.

3. The mechanical analysis method of low tension cable according to claim 1, characterized in that: The expression of the equivalent node load of the cable unit in step S3 is: P EQV ＝0 in, is the equivalent fixed end bending moment at one end of the cable element; is the equivalent fixed end bending moment at the other end of the cable element; P EQV is the equivalent axial tension; w is the self-weight distributed load of the cable unit; L is the length of the cable unit.

4. The mechanical analysis method of low tension cable according to claim 1, characterized in that: The step S4 is specifically as follows: S401, calculating the lateral displacement of the cable unit; S402. According to the sag distribution of the cable element and the lateral displacement of the cable element, the total potential energy function of the beam element composed of the strain energy and the external force potential energy is obtained by quintic interpolation: u1=θ1 u2=θ2 u3=e Where Π is the total potential energy function of the beam element with quintic interpolation; L is the length of the cable element; E is the Young's elastic model; A is the cross-sectional area of ​​the cable element; is the first derivative of the axial displacement at any position of the cable element with respect to the x coordinate; d is the differential symbol; x is the position coordinate of any point on the cable element; I is the second moment of area; is the second-order derivative of the lateral displacement at any position of the cable unit with respect to the x-coordinate; P is the axial tension; is the first derivative of the lateral displacement at any position of the cable element with respect to the x-coordinate; is the first derivative of the sag distribution of the cable element with respect to the x-coordinate; F i is the conjugate external load in the i-th direction on the two nodes of the cable element; u i is the conjugate displacement of the cable element in the i-th direction; i is the direction number of the node displacement and load; w is the self-weight distributed load of the cable element; v is the lateral displacement of the cable element; u1 is the conjugate displacement of the cable element in the first direction; u2 is the conjugate displacement of the cable element in the second direction; u3 is the conjugate displacement of the cable element in the third direction; θ1 is the tangent rotation angle of the node at one end of the cable element relative to the line connecting the cable element nodes; θ2 is the tangent rotation angle of the node at the other end of the cable element relative to the line connecting the cable element nodes; e is the axial expansion and contraction displacement of the cable element; S403. According to the total potential energy function of the beam element by quintic interpolation, the secant stiffness relationship is obtained: q=PL 2 / I Among them, δ is the sign of the change; is the symbol of partial derivative; q is the substitution load variable related to the axial tension P; S404. According to the secant stiffness relationship, the cable unit load vector on the two-dimensional plane is obtained: H1=q+80 H2=q+48 Among them, F1 is the conjugate external load in the first direction on the two nodes of the cable unit; F2 is the conjugate external load in the second direction on the two nodes of the cable unit; F3 is the conjugate external load in the third direction on the two nodes of the cable unit; M1 is the node bending moment at one end of the cable unit; is the equivalent fixed end bending moment at one end of the cable element; c1, c2, c0, c w , b1, b2, b vs , b vv , b s , b w , H1 and H2 are coefficient variables related to the replacement load variable q; v0 is the mid-span sag of the cable element; is the substitution variable for the mid-span bending moment M0 of the cable element caused by the lateral load on the beam; M2 is the node bending moment at the other end of the cable element; is the equivalent fixed end bending moment at the other end of the cable element; P EQV is the equivalent axial tension; M0 is the mid-span bending moment of the cable unit.

5. The mechanical analysis method of low tension cable according to claim 4, characterized in that: The steps The expression of the lateral displacement of the cable element in S401 is: v=N1(Lθ1)+N2(Lθ2)+N0v0+N w M0Lv w +N s LS0 N w =[ζ 2 -2g 3 +g 4 ] N s =[ζ 2 -4g 3 +5g 4 -2g 5 ] ζ=x / L+0.5 Where v is the lateral displacement of the cable unit; N1 and N2 are both shape functions; N0 is the shape function related to the mid-span sag; N w is the shape function related to the midpoint equivalent bending moment generated by the deadweight load; N s is the shape function related to the shear load generated by the deadweight load; v w is the displacement caused by the equivalent bending moment at the mid-span point generated by the distributed self-weight load; S0 is the shear force at the mid-span of the cable unit; ζ is the dimensionless coordinate variable of the coordinates of any position of the cable unit; t is the dimensionless length of the cable unit.

6. The mechanical analysis method of low tension cable according to claim 4, characterized in that: The boundary condition for the lateral displacement of the cable unit in step S401 is: x=±L / 2,v=0 x=-L / 2, x=+L / 2, 7. The mechanical analysis method of low tension cable according to claim 4, characterized in that: The expression of the bending moment and shear force equilibrium condition of the lateral displacement of the cable unit in step S401 is: Where v is the lateral displacement of the cable unit; v g is the sag distribution of the cable element; is the third-order derivative of the lateral displacement at any position of the cable element with respect to the x-coordinate; S0 is the shear force at the mid-span of the cable element.

8. The mechanical analysis method of low tension cable according to claim 1, characterized in that: The expression of the cable unit load vector in the three-dimensional space in step S5 is: what n =PL 2 / I n Among them, F 1n is the conjugate external load on the two nodes of the cable element in the first direction corresponding to the n-axis; F 2n is the conjugate external load of the second direction corresponding to the n-axis on the two nodes of the cable element; F4 is the conjugate external load of the third direction on the two nodes of the cable element; M 1n is the node bending moment at one end of the cable unit corresponding to the n-axis; n is the direction, including the Y-axis and the Z-axis; is the equivalent fixed end bending moment of one end of the cable unit corresponding to the n-axis; E is Young's elastic model; I n is the second-order moment of the area corresponding to the n-axis; L is the length of the cable unit; c 1n 、c 2n 、c wn 、c 0n , b 1n , b 2n , b sn , b wn , b vsn and b vvn are replaced by the load variable q n The coefficient variable of interest; q n is the replacement load variable corresponding to the n-axis related to the axial tension P; θ 1n θ is the tangent rotation angle of the node at one end of the cable element relative to the corresponding n-axis of the line connecting the cable element nodes; 2n is the tangent rotation angle of the other end node of the cable element relative to the corresponding n-axis of the line connecting the cable element nodes; is the mid-span bending moment M of the cable element corresponding to the n-axis caused by the transverse load on the beam 0n Substitution variable; M 0n is the mid-span bending moment of the cable unit corresponding to the n-axis; v 0n M is the mid-span sag of the cable unit corresponding to the n-axis; 2n is the node bending moment at the other end of the cable element corresponding to the n-axis; is the equivalent fixed end bending moment of the other end of the cable unit corresponding to the n-axis; P is the axial tension; A is the cross-sectional area of ​​the cable unit; e is the axial expansion and contraction displacement of the cable unit; y is the Y-axis direction; z is the Z-axis direction; I n is the second-order moment of area of ​​the cable unit cross section corresponding to the n-axis.

9. The mechanical analysis method of low tension cable according to claim 1, characterized in that: The step S6 is specifically as follows: S601. Based on the cable unit load vector in three-dimensional space, solve the displacement parameter (θ 1n +θ 2n ) and (θ 1n -θ 2n );θ 1n is the tangent rotation angle of the node at one end of the cable element relative to the corresponding n-axis of the line connecting the cable element nodes; θ 2n is the tangent rotation angle of the other end node of the cable element relative to the corresponding n-axis of the line connecting the cable element nodes; S602, according to the displacement parameter (θ 1n +θ 2n ) and (θ 1n -θ 2n ), solve the elastic stiffness matrix of the cable element; S603. According to the elastic stiffness matrix of the cable unit, the total stiffness matrix of the integrated cable structure is: T L =T g T off T A T R Among them, K t is the total stiffness matrix of the cable structure; Nele is the number of units after the cable structure is divided into units; T L is the coordinate transformation matrix; k t is the unit stiffness matrix in the local coordinate system of the cable unit; T is the transpose; T0 is the rod end force conversion matrix from 6 node forces to 12 node forces; is the elastic stiffness matrix of the cable unit; K0 is the initial tension geometric stiffness matrix; T g is the local to global transformation matrix; T off is the transformation matrix of the rigid connection; T A is the transformation matrix between the global coordinates and the intermediate reference local coordinates; T R is the standard rotation transformation matrix between the two orthogonal coordinates of the cross-section at both ends of the beam element; L is the length of the cable element.

10. The mechanical analysis method of low tension cable according to claim 9, characterized in that: The expression of the elastic stiffness matrix of the cable unit in step S602 is: n=[y,z] m＝[1,2] c1′ n =2b 1n c2′ n =2b 2n I max =max{I z ,I y } Where E is Young's elastic model; I max I is the second moment of area z and I y The maximum value of ξ z is the dimensionless second-order moment of area corresponding to the Z axis; c 1z and c 2z are corresponding to the Z axis and replace the load variable q z The coefficient variable of interest; q z is the replacement load variable corresponding to the Z axis related to the axial tension P; G 1z and G 2z are all substitution variables corresponding to the Z axis; H, η and λ are all coefficient variables; G 1y and G 2y are all substitution variables corresponding to the Y-axis; ξ y is the dimensionless second moment of area corresponding to the Y axis; c 1y and c 2y are corresponding to the Y axis and replace the load variable q y The coefficient variable of interest; q y is the replacement load variable corresponding to the Y axis related to the axial tension P; G 1n and G 2n are all substitution variables corresponding to the n-axis; n is the direction, including the Y-axis and the Z-axis; y is the Y-axis direction; z is the Z-axis direction; c1′ n Substitute variable c 1n Q n The coefficient of the derivative; c2′ n Substitute variable c 2n Q n The coefficient of the derivative; c 1n 、c 2n 、c 0n , b vsn , b vvn , b wn , b sn , b 1n and b 2n are corresponding to the n-axis and replace the load variable q n The coefficient variable of interest; q n is the replacement load variable corresponding to the n-axis related to the axial tension P; c0′ n is the coefficient variable c 0n Q n The coefficient of the derivative; v 0n is the mid-span sag of the cable element corresponding to the n-axis; is the mid-span bending moment M of the cable element corresponding to the n-axis caused by the transverse load on the beam 0n Substitution variable; M 0n is the mid-span bending moment of the cable unit corresponding to the n-axis; m is the end point number of the cable unit; is the derivative symbol; q n is the replacement load variable corresponding to the n-axis related to the axial tension P; θ mn G is the tangent rotation angle of the mth end node of the cable element relative to the corresponding n-axis of the line connecting the cable element nodes; mn is the substitution variable of the mth end node of the cable element corresponding to the n-axis; θ t is the torsion angle; e is the axial expansion displacement of the cable unit; b1′ n is the coefficient variable b 1n Q n The coefficient of the derivative; b2′ n is the coefficient variable b 2n Q n The coefficient of the derivative; b v ' sn is the coefficient variable b vsn Q n The coefficient of the derivative; b v ' vn is the coefficient variable b vvn Q n The coefficient of the derivative; b s ' n is the coefficient variable b sn Q n The coefficient of the derivative; b′ wn is the coefficient variable b wn Q n The coefficient of the derivative; ξ n is the dimensionless second-order moment of area corresponding to the n-axis; G is the shear modulus; J is the polar moment of inertia of the cable unit section; P is the axial tension; r is the polar radius of gyration; A is the cross-sectional area of ​​the cable unit; I z is the second-order moment of area of ​​the cable element corresponding to the Z axis; I y is the second moment of area of ​​the cable element corresponding to the Y axis.

Images