An approximate analytical method for the critical load of nonlinear jump buckling of asymmetric arch bridges

Abstract

The invention discloses an approximate analytical method for the critical load of nonlinear jump buckling of an asymmetric arch bridge. First, an asymmetric arch bridge under uniformly distributed load is taken as the research object. Based on the Euler-Bernoulli beam theory and the principle of virtual work, the geometric nonlinear equilibrium equation of the asymmetric arch bridge surface is deduced, and then the approximate analytical method for the geometric nonlinear displacement of the asymmetric arch bridge surface is deduced by introducing the asymmetric degree coefficient γ of the arch bridge; then, the post-buckling equilibrium differential equation of the asymmetric arch bridge is deduced based on the principle of virtual work, and the internal force conditions and the approximate analytical method for the critical load of the branch point buckling of the asymmetric arch bridge under two boundary conditions are obtained; finally, the nonlinear jump buckling critical load corresponds to the characteristics of the extreme point in the buckling behavior curve when it is unstable, and the approximate analytical method for the nonlinear jump buckling critical load of the asymmetric arch bridge is derived. The method of the invention can effectively reduce the workload of engineers in solving the geometric nonlinearity of the asymmetric arch bridge structure surface.

Description

Technical Field

[0001] The invention belongs to the technical field of in-plane geometric stability analysis of arch structures, and in particular relates to an approximate analytical method for critical load of nonlinear leap buckling of an asymmetric arch bridge. Background Art

[0002] At present, the geometric nonlinear stability analysis of the arch structure in-plane is mainly divided into two aspects: the geometric nonlinear stability analysis of the arc symmetric arch in-plane under the polar coordinate system and the geometric nonlinear stability analysis of the non-circular arc symmetric arch in-plane under the rectangular coordinate system, as follows:

[0003] 1. The in-plane geometric nonlinear stability analysis of circular arc symmetric arches in polar coordinates is based on Boussinesq's in-plane stability study of circular arc arch structures. The in-plane equilibrium differential equation of circular arc symmetric arches in polar coordinates is established, and the approximate analysis of the in-plane nonlinear buckling critical load is obtained. This method is simple and feasible in the case of circular arc symmetric arches, but it is not applicable to non-circular arc arch structures and asymmetric arch structures.

[0004] 2. The geometric nonlinear stability analysis of the non-circular arc symmetric arch surface in the rectangular coordinate system is to establish a finite element calculation model, study the force characteristics of the non-circular arc arch bridge, obtain the nonlinear equilibrium differential equation of the parabolic arch under the vertical uniformly distributed load, and solve the nonlinear buckling equilibrium equation of the parabolic arch. This method has good applicability when the arch foot is symmetrical, but the geometric nonlinear equilibrium equation of the asymmetric parabolic arch surface is a fourth-order differential equation and cannot be solved by this method.

[0005] Based on the above in-plane geometric nonlinear stability analysis of the arch structure, it is difficult to solve the nonlinear stability problem of the asymmetric arch bridge under the uniformly distributed load, which is mainly manifested in the following points:

[0006] 1. The nonlinear approximate analytical method for circular arc symmetric arch surface in polar coordinate system. Since the strain expression of arch structure in rectangular coordinate system is different from that in polar coordinate system, this method cannot be used for nonlinear approximate analysis of parabolic arch surface in rectangular coordinate system.

[0007] 2. The research methods on the geometric nonlinear stability of non-circular arc symmetric arch surfaces in the rectangular coordinate system are all based on the arch structure with symmetrical arrangement of arch feet. There is a lack of analysis on the influence of buckling behavior of asymmetric arch structures and approximate analysis of critical buckling load. Summary of the invention

[0008] The purpose of the present invention is to provide an approximate analytical method for the nonlinear leap buckling critical load of an asymmetric arch bridge in view of the deficiencies in the prior art. The asymmetric arch bridge under uniformly distributed load is taken as the analysis object. Based on the Euler-Bernoulli beam theory and the principle of virtual work, a high-precision approximate analytical method for the nonlinear leap buckling critical load is deduced, so that engineering and technical personnel can avoid using finite element software for complex solutions, thereby reducing the workload of solving the geometric nonlinearity in the asymmetric arch bridge structure surface.

[0009] In order to achieve the above object, the present invention adopts the following technical solution.

[0010] An approximate analytical method for the critical load of nonlinear leap buckling of an asymmetric arch bridge comprises the following steps:

[0011] Step S1, taking an asymmetric arch bridge under uniformly distributed load as a research object, obtaining a nonlinear strain-displacement expression of the structural plane of the asymmetric arch bridge in a rectangular coordinate system;

[0012] Step S2, based on the obtained in-plane nonlinear strain-displacement expression of the asymmetric arch bridge in the rectangular coordinate system and the principle of virtual work, deduce the in-plane geometric nonlinear equilibrium differential equation of the asymmetric arch bridge under the uniformly distributed load;

[0013] Step S3, according to the obtained geometric nonlinear equilibrium differential equation of the asymmetric arch bridge in the structural plane, the asymmetric degree coefficient γ of the arch bridge is introduced, and the variational method is used to simplify the integration process, and the approximate analysis of the vertical displacement in the nonlinear equilibrium path of the asymmetric arch bridge with no hinge arch and two hinge arches is deduced;

[0014] Step S4, further obtaining the relationship between the dimensionless load and the dimensionless load parameter of the asymmetric arch bridge with no hinge arch and the asymmetric arch bridge with two hinge arches according to the obtained vertical displacement approximate analysis in the nonlinear equilibrium path of the asymmetric arch bridge with no hinge arch and the asymmetric arch bridge with two hinge arches;

[0015] Step S5, based on the principle of constant axial force of nonlinear branch point buckling, the equilibrium differential equation of nonlinear branch point buckling of the asymmetric arch bridge is deduced, and further the approximate analysis of the critical load is obtained according to the internal force conditions of the branch point buckling of the asymmetric arch bridge;

[0016] Step S6: based on the characteristic that the nonlinear leap buckling critical load corresponds to the extreme point in the buckling behavior curve when the bridge is unstable, an approximate analysis of the nonlinear leap buckling critical load of the asymmetric arch bridge is deduced.

[0017] Specifically, the deduction in step S3 obtains an approximate analysis of the vertical displacement in the nonlinear equilibrium path of the asymmetric arch bridge with no hinge arch and two hinge arches, wherein:

[0018] The expression of the approximate solution of the vertical displacement in the nonlinear equilibrium path of an asymmetric arch bridge without hinged arches is:

[0019] ;

[0020] In formula (1), It is the vertical displacement of the asymmetric arch bridge without hinges after the arch structure is deformed; is a dimensionless load; is the axial force parameter; p is the arch function, , is the span of an asymmetric parabolic arch. f 2 is the height loss at the arch foot of the right half arch of the asymmetric arch bridge; z is the horizontal coordinate in the cross section of the main arch ring; is the dimensionless axial force parameter, , It is the span of the right half arch of the asymmetric arch bridge; and is a custom coefficient, expressed as:

[0021] ;

[0022] ;

[0023] is the vertical displacement parameter of the hingeless arch, expressed as:

[0024] ;

[0025] γ is the coefficient of asymmetry of the arch bridge, expressed as; , It is the span of the left half arch of the asymmetric arch bridge;

[0026] The expression of the approximate solution of the vertical displacement in the nonlinear equilibrium path of an asymmetric arch bridge with two hinged arches is:

[0027] ;

[0028] In formula (2), K p It is the vertical displacement of the asymmetric arch bridge with two hinged arches after deformation; and is a custom coefficient, expressed as: , .

[0029] Specifically, the relationship between the dimensionless load and the dimensionless load parameter of the asymmetric arch bridge with no hinged arch and two hinged arches in step S4 is as follows:

[0030] The relationship between the dimensionless load and dimensionless load parameters of an asymmetric arch bridge without hinged arches is:

[0031] ;

[0032] In formula (3), is the dimensionless load of the hingeless arch, , , is a dimensionless load parameter, expressed as:

[0033] ;

[0034] ;

[0035] ;

[0036] In formula (6), is the modified slenderness ratio of the asymmetric arch bridge, expressed as:

[0037] ;

[0038] In formula (7), f Loss of height for asymmetric arch bridge; is the radius of gyration of the main arch section, , I x is the section moment of inertia, , is the normal coordinate in the cross section of the main arch ring, is the cross-sectional area of ​​the main arch ring;

[0039] The relationship between the dimensionless load and dimensionless load parameters of an asymmetric arch bridge with two hinged arches is:

[0040] ;

[0041] In formula (8), is the dimensionless load of the two-hinged arch, , , is a dimensionless load parameter, expressed as:

[0042] ;

[0043] ;

[0044] .

[0045] Specifically, the nonlinear branch point buckling equilibrium differential equation of the asymmetric arch bridge in step S5 is expressed as:

[0046] ;

[0047] In formula (12), is the axial force parameter of the arch structure after buckling, expressed as: , H * is the horizontal thrust of the arch foot after flexion, is the elastic modulus of the asymmetric arch bridge; v *'' for v * The second derivative of v *iv for v * The fourth-order derivative of v * is the vertical displacement of the arch structure after buckling; is the change in compression strain when the arch structure buckles; y' for y The first derivative of y is the axis equation of the asymmetric parabolic arch;

[0048] The internal force condition for the branch point buckling of the asymmetric arch bridge is that when the asymmetric arch bridge structure undergoes antisymmetric instability, the arch bridge structure transitions from a non-instability state to a state infinitely close to instability, and the horizontal thrust of the arch foot before buckling is infinitely close to the horizontal thrust of the arch foot after buckling, that is, Substituting into formula (12), we can obtain the general solution of the critical load equation for branch point buckling of an asymmetric arch bridge: v b :

[0049] ;

[0050] In formula (13), C is a constant.

[0051] Specifically, the deduction in step S6 obtains an approximate analysis of the critical load of nonlinear leap buckling of an asymmetric arch bridge, and the process is as follows:

[0052] The critical load for nonlinear jump buckling of an asymmetric arch bridge corresponds to the extreme point in the buckling behavior curve, including the maximum and minimum points of the nonlinear equilibrium path of the asymmetric arch bridge. The extreme point is obtained by solving the stationary point of the buckling behavior curve, which is expressed as:

[0053] ;

[0054] In formula (14), is the geometrically nonlinear jump buckling load in the asymmetric arch bridge surface; is the horizontal thrust when the arch bridge is at critical load;

[0055] Based on the boundary conditions of the instability of asymmetric arch bridges with hingeless arches and two-hinged arches, the approximate analysis of the nonlinear leap buckling critical load of the corresponding asymmetric arch bridge is obtained, where:

[0056] Critical dimensionless load for buckling of branch points of hingeless asymmetric arch bridges The expression is:

[0057] ;

[0058] Critical dimensionless buckling load at branch point of an asymmetric arch bridge with two hinged arches The expression is:

[0059] ;

[0060] The approximate solution of the nonlinear jump buckling critical load of the hingeless arch asymmetric arch bridge is deduced by the characteristics of the extreme point in the buckling behavior curve corresponding to the nonlinear jump buckling critical load. It is expressed as:

[0061] ;

[0062] In formula (17), , , is a dimensionless load parameter, expressed as:

[0063] ;

[0064] In formula (18), the implicit function Expressed as , its The partial derivative of It is expressed as:

[0065] ;

[0066] In formula (18), the implicit function Expressed as , its The partial derivative of It is expressed as:

[0067] ;

[0068] In formula (19) and formula (20), ' It means to find the first-order derivative;

[0069] In formula (18), the dimensionless load factor is The partial derivative of It is expressed as:

[0070] ;

[0071] The approximate solution of the critical load of nonlinear jump buckling of an asymmetric arch bridge with two hinged arches is deduced It is expressed as:

[0072] ;

[0073] In formula (22), , , is a dimensionless load parameter, expressed as:

[0074] ;

[0075] In formula (23), the implicit function Expressed as , its The partial derivative of It is expressed as:

[0076] ;

[0077] In formula (23), the implicit function Expressed as , its The partial derivative of It is expressed as:

[0078] ;

[0079] In formula (24) and formula (25), ' It means to find the first-order derivative;

[0080] In formula (23), the dimensionless load parameter right The partial derivative of It is expressed as:

[0081] .

[0082] Compared with the prior art, the present invention has the following beneficial effects:

[0083] The method of the present invention takes an asymmetric arch bridge under uniformly distributed load as the research object, and by introducing the asymmetry coefficient γ of the arch bridge, based on the Euler-Bernoulli beam theory and the principle of virtual work, can quickly obtain a high-precision approximate analysis of the geometric nonlinearity within the surface of the asymmetric arch bridge structure. The method is simple, the mechanical concept is clear, and the goal is clear, so that engineering and technical personnel can avoid using finite element software for complex solutions, thereby reducing the workload of solving the geometric nonlinearity within the surface of the asymmetric arch bridge structure. BRIEF DESCRIPTION OF THE DRAWINGS

[0084] In order to more clearly illustrate the technical solutions of the embodiments of the present disclosure, the drawings required for use in the embodiments will be briefly introduced below. It should be understood that the following drawings only show certain embodiments of the present disclosure and therefore should not be regarded as limiting the scope. For ordinary technicians in this field, other relevant drawings can be obtained based on these drawings without paying creative work.

[0085] Figure 1 It is a flow chart of the approximate analytical method for the nonlinear leap buckling critical load of an asymmetric arch bridge of the present invention;

[0086] Figure 2 Schematic diagram of the analysis of an asymmetric arch bridge in an embodiment of the present invention.

[0087] Figure 3 It is a schematic diagram of curve comparison between the approximate analytical result and the finite element calculation result of the hingeless asymmetric arch bridge of the present invention;

[0088] Figure 4 It is a schematic diagram of curve comparison between the approximate analytical results and the finite element calculation results of the hingeless asymmetric arch bridge of the present invention, taking into account the rise-to-span ratio, the slenderness ratio and the asymmetry coefficient of the arch bridge;

[0089] Figure 5 It is a schematic diagram of curve comparison between the approximate analytical result and the finite element calculation result of the asymmetric arch bridge with two hinged arches according to the present invention;

[0090] Figure 6 It is a schematic diagram of curve comparison between the approximate analytical results and the finite element calculation results of the asymmetric arch bridge with two hinged arches in the present invention, taking into account the rise-to-span ratio, the slenderness ratio and the asymmetry coefficient of the arch bridge. DETAILED DESCRIPTION

[0091] In order to facilitate those of ordinary skill in the art to understand and implement the present invention, the following detailed description is given to each step of the method proposed by the present invention, and it should be understood that these embodiments are only used to illustrate the present invention and are not used to limit the scope of the present invention. In addition, it should be understood that after reading the content taught by the present invention, those skilled in the art can make various changes or modifications to the present invention, and these equivalent forms also fall within the scope defined by the appended claims of the present application.

[0092] Example

[0093] like Figure 1 As shown, the present invention discloses an approximate analytical method for the critical load of nonlinear leap buckling of an asymmetric arch bridge, comprising the following steps:

[0094] Step S1, taking an asymmetric arch bridge under uniformly distributed load as a research object, obtaining a nonlinear strain-displacement expression of the structural plane of the asymmetric arch bridge in a rectangular coordinate system;

[0095] Step S2, based on the obtained in-plane nonlinear strain-displacement expression of the asymmetric arch bridge in the rectangular coordinate system and the principle of virtual work, deduce the in-plane geometric nonlinear equilibrium differential equation of the asymmetric arch bridge under the uniformly distributed load;

[0096] Step S3, according to the obtained geometric nonlinear equilibrium differential equation of the asymmetric arch bridge in the structural plane, the asymmetric degree coefficient γ of the arch bridge is introduced, and the variational method is used to simplify the integration process, and the approximate analysis of the vertical displacement in the nonlinear equilibrium path of the asymmetric arch bridge with no hinge arch and two hinge arches is deduced;

[0097] Step S4, further obtaining the relationship between the dimensionless load and the dimensionless load parameter of the asymmetric arch bridge with no hinge arch and the asymmetric arch bridge with two hinge arches according to the obtained vertical displacement approximate analysis in the nonlinear equilibrium path of the asymmetric arch bridge with no hinge arch and the asymmetric arch bridge with two hinge arches;

[0098] Step S5, based on the principle of constant axial force of nonlinear branch point buckling, the equilibrium differential equation of nonlinear branch point buckling of the asymmetric arch bridge is deduced, and further the approximate analysis of the critical load is obtained according to the internal force conditions of the branch point buckling of the asymmetric arch bridge;

[0099] Step S6: based on the characteristic that the nonlinear leap buckling critical load corresponds to the extreme point in the buckling behavior curve when the bridge is unstable, an approximate analysis of the nonlinear leap buckling critical load of the asymmetric arch bridge is deduced.

[0100] Specifically, Figure 2 As shown, the structural plane nonlinear strain-displacement expression of the asymmetric arch bridge in the rectangular coordinate system in step S1 is:

[0101] ;

[0102] In the above formula, and are the compression strain and bending strain at any point on the asymmetric arch bridge, respectively; and are the horizontal displacement and vertical displacement of the arch structure after deformation; y is the axis equation of the arch structure of the asymmetric arch bridge; is the normal coordinate in the cross section of the main arch ring; w for w The first derivative of ; for v The second-order derivative of , The span of an asymmetric arch bridge is It is the span of the left half arch of the asymmetric arch bridge; It is the span of the right half arch of the asymmetric arch bridge; z The horizontal coordinate within the cross section of the main arch.

[0103] Specifically, the process of deducing the geometric nonlinear equilibrium differential equation of the structure plane of the asymmetric arch bridge under the uniformly distributed load in step S2 is as follows:

[0104] Based on the principle of virtual work, the virtual displacement of an asymmetric arch bridge under load is expressed as , the virtual strain is expressed as , then:

[0105] ;

[0106] In the above formula, is the total virtual work of the asymmetric arch bridge structure; is the total internal force virtual work of the system; is the total external force virtual work of the system; is the elastic modulus of the asymmetric arch bridge; is the total strain of the asymmetric arch bridge, ; is the total volume of the asymmetric arch bridge, , is the cross-sectional area of ​​the main arch ring; is the load concentration on the asymmetric arch bridge;

[0107] According to the calculus of variations, let the energy The first-order variation of is reduced to zero and the formula is simplified. According to the Euler-Bernoulli beam theory and the compressive member stability theory, the nonlinear equilibrium differential equation of the asymmetric arch bridge is obtained:

[0108] ;

[0109] In the above formula, , is the section moment of inertia; v iv for v The fourth derivative of ; H is the horizontal reaction force of the abutment;

[0110] Introducing axial force parameters Further simplifying the formula yields:

[0111] ;

[0112] In the above formula, is the axial force parameter, ; is a dimensionless load, expressed as:

[0113] ;

[0114] The asymmetry coefficient γ of the arch bridge is expressed as:

[0115] .

[0116] Specifically, the deduction in step S3 obtains an approximate analysis of the vertical displacement in the nonlinear equilibrium path of the asymmetric arch bridge with no hinge arch and two hinge arches, wherein:

[0117] The expression of the approximate solution of the vertical displacement in the nonlinear equilibrium path of an asymmetric arch bridge without hinged arches is:

[0118] ;

[0119] In formula (1), It is the vertical displacement of the hingeless arch structure of the asymmetric arch bridge after deformation; is a dimensionless load; is the axial force parameter; p is the arch function, , The span of an asymmetric arch bridge is f 2 is the height loss at the arch foot of the right half arch of the asymmetric arch bridge; z is the horizontal coordinate in the cross section of the main arch ring; is the dimensionless axial force parameter, , It is the span of the right half arch of the asymmetric arch bridge; and is a custom coefficient, expressed as:

[0120] ;

[0121] ;

[0122] is the vertical displacement parameter of the hingeless arch, expressed as:

[0123] ;

[0124] γ is the coefficient of asymmetry of the arch bridge, expressed as; , It is the span of the left half arch of the asymmetric arch bridge;

[0125] The expression of the approximate solution of the vertical displacement in the nonlinear equilibrium path of an asymmetric arch bridge with two hinged arches is:

[0126] ;

[0127] In formula (2), K p It is the vertical displacement of the asymmetric arch bridge with two hinged arches after deformation; and is a custom coefficient, expressed as: , .

[0128] Specifically, the relationship between the dimensionless load and the dimensionless load parameter of the asymmetric arch bridge with no hinged arch and two hinged arches in step S4 is as follows:

[0129] The relationship between the dimensionless load and dimensionless load parameters of an asymmetric arch bridge without hinged arches is:

[0130] ;

[0131] In formula (3), is the dimensionless load of the hingeless arch, , , is a dimensionless load parameter, expressed as:

[0132] ;

[0133] ;

[0134] ;

[0135] In formula (6), is the modified slenderness ratio of the asymmetric arch bridge, expressed as:

[0136] ;

[0137] In formula (7), f Loss of height for asymmetric arch bridge; is the radius of gyration of the main arch section, , I x is the section moment of inertia, , is the normal coordinate in the cross section of the main arch ring, is the cross-sectional area of ​​the main arch ring;

[0138] The relationship between the dimensionless load and dimensionless load parameters of an asymmetric arch bridge with two hinged arches is:

[0139] ;

[0140] In formula (8), is the dimensionless load of the two-hinged arch, , , is a dimensionless load parameter, expressed as:

[0141] ;

[0142] ;

[0143] .

[0144] Specifically, the nonlinear branch point buckling equilibrium differential equation of the asymmetric arch bridge in step S5 is expressed as:

[0145] ;

[0146] In formula (12), is the axial force parameter of the arch structure after buckling, expressed as: , H * is the horizontal thrust of the arch foot after flexion, is the elastic modulus of the asymmetric arch bridge; v *'' for v * The second derivative of v *iv for v * The fourth-order derivative of v * is the vertical displacement of the arch structure after buckling; is the change in compression strain when the arch structure buckles; y' for y The first derivative of y is the axis equation of the asymmetric parabolic arch;

[0147] The internal force condition for the branch point buckling of the asymmetric arch bridge is that when the asymmetric arch bridge structure undergoes antisymmetric instability, the arch bridge structure transitions from a non-instability state to a state infinitely close to instability, and the horizontal thrust of the arch foot before buckling is infinitely close to the horizontal thrust of the arch foot after buckling, that is, Substituting into formula (12), we can obtain the general solution of the critical load equation for branch point buckling of an asymmetric arch bridge: v b :

[0148] ;

[0149] In formula (13), C is a constant.

[0150] Specifically, the deduction in step S6 obtains an approximate analysis of the critical load of nonlinear leap buckling of an asymmetric arch bridge, and the process is as follows:

[0151] The critical load for nonlinear jump buckling of an asymmetric arch bridge corresponds to the extreme point in the buckling behavior curve, including the maximum and minimum points of the nonlinear equilibrium path of the asymmetric arch bridge. The extreme point is obtained by solving the stationary point of the buckling behavior curve, which is expressed as:

[0152] ;

[0153] In formula (14), is the geometrically nonlinear jump buckling load in the asymmetric arch bridge surface; is the horizontal thrust when the arch bridge is at critical load;

[0154] Based on the boundary conditions of the instability of asymmetric arch bridges with hingeless arches and two-hinged arches, the approximate analysis of the nonlinear leap buckling critical load of the corresponding asymmetric arch bridge is obtained, where:

[0155] Critical dimensionless load for buckling of branch points of hingeless asymmetric arch bridges The expression is:

[0156] ;

[0157] Critical dimensionless buckling load at branch point of an asymmetric arch bridge with two hinged arches The expression is:

[0158] ;

[0159] The approximate solution of the nonlinear jump buckling critical load of the hingeless arch asymmetric arch bridge is deduced by the characteristics of the extreme point in the buckling behavior curve corresponding to the nonlinear jump buckling critical load. It is expressed as:

[0160] ;

[0161] In formula (17), , , is a dimensionless load parameter, expressed as:

[0162] ;

[0163] In formula (18), the implicit function Expressed as , its The partial derivative of It is expressed as:

[0164] ;

[0165] In formula (18), the implicit function Expressed as , its The partial derivative of It is expressed as:

[0166] ;

[0167] In formula (19) and formula (20), 'It means to find the first-order derivative;

[0168] In formula (18), the dimensionless load factor is The partial derivative of It is expressed as:

[0169] ;

[0170] The approximate solution of the critical load of nonlinear jump buckling of an asymmetric arch bridge with two hinged arches is deduced It is expressed as:

[0171] ;

[0172] In formula (22), , , is a dimensionless load parameter, expressed as:

[0173] ;

[0174] In formula (23), the implicit function Expressed as , its The partial derivative of It is expressed as:

[0175] ;

[0176] In formula (23), the implicit function Expressed as , its The partial derivative of It is expressed as:

[0177] ;

[0178] In formula (24) and formula (25), ' It means to find the first-order derivative;

[0179] In formula (23), the dimensionless load parameter right The partial derivative of It is expressed as:

[0180] .

[0181] The nonlinear jump buckling critical load results of the asymmetric parabolic arch obtained by the method of the present invention and the traditional finite element analysis method are compared to verify the accuracy of the method of the present invention.

[0182] In this example, the ANSYS finite element software Beam4 beam element is used to establish an asymmetric parabolic arch structure model with a solid rectangular cross-section for verification. The arc length method is enabled during the analysis process, and the uniformly distributed load is simulated by adding nodal forces.

[0183] 1. Finite element verification of asymmetric arch bridge without hinged arch

[0184] The Beam4 beam element in ANSYS finite element software was used to establish an asymmetric arch bridge structure model with a solid rectangular cross-section and no hinged arch for verification, and the corresponding uniformly distributed load was simulated by adding nodal forces.

[0185] In order to prevent the asymmetric arch bridge from bending and torsional deformation outside the structural plane, the out-of-plane degrees of freedom of all nodes are constrained in the model; the relevant parameters of the hingeless asymmetric arch bridge used in the example are: the height of the rectangular section of the arch structure ,width , the asymmetry coefficient of the arch bridge , the rise-span ratio of the long half arch , corrected slenderness ratio ; Material elastic modulus , Poisson's ratio .

[0186] The results of the finite element software were compared with the approximate analytical curve obtained according to the method of the present invention. Figure 3 As shown, Figure 3 Middle: (a) is the rise-span ratio of the long half arch Comparison of load-displacement curves when (a) is the rise-span ratio of the long half arch (c) is the rise-span ratio of the long half arch Comparison of load-displacement curves when (d) is the rise-span ratio of the long half arch Comparison of the buckling behavior curves when .

[0187] from Figure 3 It can be seen that the approximate analytical results of the nonlinear leap buckling load of the hingeless asymmetric arch bridge solved by the method of the present invention are highly consistent with the finite element calculation results, and the rise and fall trends of the curves are also the same.

[0188] In order to further verify the accuracy of the method of the present invention, the approximate analytical results of the present invention are compared with the finite element calculation results under the condition of considering the rise-span ratio, slenderness ratio and asymmetric coefficient of the arch bridge, and the corresponding curves are drawn, as shown in FIG. Figure 4 As shown, Figure 4 Middle: (a) is the comparison of the rise-to-span ratio parameter curve; (b) is the comparison of the slenderness ratio parameter curve; (c) is the critical load of a given hingeless arch Comparison of asymmetry coefficient curves of arch bridges at different times.

[0189] The relevant parameters in the calculation example are: the degree of asymmetry of the arch foot , slenderness ratio , the rise-to-span ratio at the long half arch .

[0190] from Figure 4 It can be seen that, considering the changes in the rise-to-span ratio and the slenderness ratio, the approximate analysis of the nonlinear leap buckling load of the hingeless asymmetric arch bridge solved by the method of the present invention is highly consistent with the finite element calculation results, and the rise and fall trends of the curves are also the same. The curves under the influence of various parameters drawn by the approximate analysis of the nonlinear leap buckling load of the hingeless asymmetric arch bridge solved by the method of the present invention are compared with the finite element results. The two curves are basically consistent, which proves the accuracy and applicability of the approximate analytical results of the nonlinear leap buckling critical load of the hingeless asymmetric arch bridge solved by the method of the present invention.

[0191] 2. Finite Element Verification of Asymmetric Arch Bridge with Two Hinged Arches

[0192] The Beam4 beam element in ANSYS finite element software was used to establish an asymmetric arch bridge structure model with two hinged arches of solid rectangular cross-section for verification, and the corresponding uniformly distributed load was simulated by adding node forces. In order to prevent the asymmetric arch bridge from bending and torsional deformation outside the structural plane, the structural out-of-plane degrees of freedom of all nodes were constrained in the model.

[0193] The parameters of the two-hinged asymmetric arch bridge used in the example are: the height of the rectangular section ,width , material elastic modulus , the rise-span ratio of the long half arch , corrected slenderness ratio , the degree of asymmetry of the arch foot , Poisson's ratio .like Figure 5 As shown, Figure 5 Middle: (a) is the rise-span ratio of the long half arch Comparison of load-displacement curves when (a) is the rise-span ratio of the long half arch (c) is the rise-span ratio of the long half arch Comparison of load-displacement curves when (d) is the rise-span ratio of the long half arch Comparison of the buckling behavior curves when .

[0194] from Figure 5 It can be seen that the approximate analytical results of the nonlinear leap buckling load of the asymmetric arch bridge with two hinged arches solved by the method of the present invention are highly consistent with the finite element calculation results, and the rise and fall trends of the curves are also the same.

[0195] In order to further verify the accuracy of the method of the present invention, the approximate analytical results of the present invention are compared with the finite element calculation results under the condition of considering the rise-span ratio, slenderness ratio and asymmetric coefficient of the arch bridge, and the corresponding curves are drawn, as shown in FIG. Figure 6 As shown, Figure 6 Middle: (a) is the comparison of the rise-to-span ratio parameter curve; (b) is the comparison of the slenderness ratio parameter curve; (c) is the critical load of a given two-hinged arch Comparison of asymmetry coefficient curves of arch bridges at different times.

[0196] The relevant parameters in the calculation example are: arch bridge asymmetry coefficient , slenderness ratio , the rise-to-span ratio at the long half arch .

[0197] from Figure 6 It can be seen that, considering the changes in the rise-to-span ratio and the slenderness ratio, the approximate analysis of the nonlinear leap buckling load of the asymmetric arch bridge with two hinged arches solved by the method of the present invention is highly consistent with the finite element calculation results, and the rise and fall trends of the curves are also the same. The curves under the influence of various parameters drawn by the approximate analysis of the nonlinear leap buckling load of the asymmetric arch bridge with two hinged arches solved by the method of the present invention are compared with the finite element results. The two curves are basically consistent, which proves the accuracy and applicability of the approximate analytical results of the nonlinear leap buckling critical load of the asymmetric arch bridge with two hinged arches of the method of the present invention.

[0198] In summary, Figure 4 and Figure 6 It can be seen that the approximate analytical results of the nonlinear leap buckling load of the asymmetric arch bridge calculated by the method of the present invention are highly consistent with the finite element calculation results, and under various working conditions, the rise and fall trends of the approximate analytical curves obtained by the method of the present invention are generally consistent with the trends of the calculation results of the finite element software, which reflects the accuracy of the method of the present invention.

[0199] The above is only a preferred embodiment of the present invention, and does not limit the present invention in other forms. Any technician familiar with the profession may use the above disclosed technical content to change or modify it into an equivalent embodiment with equivalent changes. However, any simple modification, equivalent change and modification made to the above embodiment according to the technical essence of the present invention without departing from the technical solution of the present invention still belongs to the protection scope of the technical solution of the present invention.

Claims

1. An approximate analytical method for the critical load of nonlinear leap buckling of an asymmetric arch bridge, characterized in that: The following steps are involved: Step S1, taking an asymmetric arch bridge under uniformly distributed load as a research object, obtaining a nonlinear strain-displacement expression of the structural plane of the asymmetric arch bridge in a rectangular coordinate system; The nonlinear strain-displacement expression in the structural plane of the asymmetric arch bridge in the rectangular coordinate system is: ； In the above formula, and are the compression strain and bending strain at any point on the asymmetric arch bridge, respectively; and are the horizontal displacement and vertical displacement of the arch structure after deformation; y is the axis equation of the arch structure of the asymmetric arch bridge; is the normal coordinate in the cross section of the main arch ring; for w The first derivative of ; for v The second-order derivative of , For an asymmetric arch bridge span, It is the span of the left half arch of the asymmetric arch bridge; It is the span of the right half arch of the asymmetric arch bridge; z is the horizontal coordinate in the cross section of the main arch ring; Step S2, based on the obtained in-plane nonlinear strain-displacement expression of the asymmetric arch bridge in the rectangular coordinate system and the principle of virtual work, the in-plane geometric nonlinear equilibrium differential equation of the asymmetric arch bridge under the uniformly distributed load is deduced, and the process is as follows: Based on the principle of virtual work, the virtual displacement of an asymmetric arch bridge under load is expressed as , the virtual strain is expressed as , then: ； In the above formula, is the total virtual work of the asymmetric arch bridge structure; is the total internal force virtual work of the system; is the total external force virtual work of the system; is the elastic modulus of the asymmetric arch bridge; is the total strain of the asymmetric arch bridge, ; is the total volume of the asymmetric arch bridge, , is the cross-sectional area of ​​the main arch ring; is the load concentration on the asymmetric arch bridge; According to the calculus of variations, let the energy The first-order variation of is reduced to zero and the formula is simplified. According to the Euler-Bernoulli beam theory and the compressive member stability theory, the nonlinear equilibrium differential equation of the asymmetric arch bridge is obtained: ； In the above formula, , is the section moment of inertia; v iv for v The fourth derivative of ; H is the horizontal reaction force of the abutment; Introducing axial force parameters Further simplifying the formula yields: ； In the above formula, is the axial force parameter, ; is a dimensionless load, expressed as: ； The asymmetry coefficient γ of the arch bridge is expressed as: ； Step S3: According to the obtained geometric nonlinear equilibrium differential equation of the asymmetric arch bridge in the structural plane, the asymmetric degree coefficient γ of the arch bridge is introduced, and the variational method is used to simplify the integration process to deduce the approximate analysis of the vertical displacement in the nonlinear equilibrium path of the asymmetric arch bridge without hinged arch and two-hinged arch, where: The expression of the approximate solution of the vertical displacement in the nonlinear equilibrium path of an asymmetric arch bridge without hinged arches is: ； In formula (1), It is the vertical displacement of the asymmetric arch bridge without hinges after the arch structure is deformed; is a dimensionless load; is the axial force parameter; p is the arch function, , For an asymmetric arch bridge span, f 2 is the height loss at the arch foot of the right half arch of the asymmetric arch bridge; z is the horizontal coordinate in the cross section of the main arch ring; is the dimensionless axial force parameter, , It is the span of the right half arch of the asymmetric arch bridge; and is a custom coefficient, expressed as: ； ； is the vertical displacement parameter of the hingeless arch, expressed as: ； γ is the coefficient of asymmetry of the arch bridge, expressed as; , It is the span of the left half arch of the asymmetric arch bridge; The expression of the approximate solution of the vertical displacement in the nonlinear equilibrium path of an asymmetric arch bridge with two hinged arches is: ； In formula (2), K p It is the vertical displacement of the asymmetric arch bridge with two hinged arches after deformation; and is a custom coefficient, expressed as: , ; Step S4: Based on the obtained vertical displacement approximate analysis in the nonlinear equilibrium path of the asymmetric arch bridge with no hinge arch and two hinge arches, the relationship between the dimensionless load and the dimensionless load parameter of the asymmetric arch bridge with no hinge arch and two hinge arches is further obtained, wherein: The relationship between the dimensionless load and dimensionless load parameters of an asymmetric arch bridge without hinged arches is: ； In formula (3), is the dimensionless load of the hingeless arch, , , The dimensionless load parameter is expressed as: ； ； ； In formula (6), is the modified slenderness ratio of the asymmetric arch bridge, expressed as: ； In formula (7), f Loss of height for asymmetric arch bridge; is the radius of gyration of the main arch section, , I x is the section moment of inertia, , is the normal coordinate in the cross section of the main arch ring, is the cross-sectional area of ​​the main arch ring; The relationship between the dimensionless load and dimensionless load parameters of an asymmetric arch bridge with two hinged arches is: ； In formula (8), is the dimensionless load of the two-hinged arch, , , is a dimensionless load parameter, expressed as: ； ； ； Step S5, based on the principle of constant axial force of nonlinear branch point buckling, the equilibrium differential equation of nonlinear branch point buckling of the asymmetric arch bridge is deduced, and further the approximate analysis of the critical load is obtained according to the internal force conditions of the branch point buckling of the asymmetric arch bridge; The nonlinear branch point buckling equilibrium differential equation of the asymmetric arch bridge is expressed as: ； In formula (12), is the axial force parameter of the arch structure after buckling, expressed as: , H * is the horizontal thrust of the arch foot after flexion, is the elastic modulus of the asymmetric arch bridge; v *'' for v * The second derivative of v *iv for v * The fourth-order derivative of v * is the vertical displacement of the arch structure after buckling; is the change in compression strain when the arch structure buckles; y' for y The first derivative of y is the axis equation of the asymmetric parabolic arch; The internal force condition for the branch point buckling of the asymmetric arch bridge is that when the asymmetric arch bridge structure undergoes antisymmetric instability, the arch bridge structure transitions from a non-instability state to a state infinitely close to instability, and the horizontal thrust of the arch foot before buckling is infinitely close to the horizontal thrust of the arch foot after buckling, that is, Substituting into formula (12), we can obtain the general solution of the critical load equation for branch point buckling of an asymmetric arch bridge: v b : ； In formula (13), C is a constant; Step S6, based on the characteristic that the nonlinear leap buckling critical load corresponds to the extreme point in the buckling behavior curve when the bridge is unstable, an approximate analysis of the nonlinear leap buckling critical load of the asymmetric arch bridge is deduced; The critical load for nonlinear jump buckling of an asymmetric arch bridge corresponds to the extreme point in the buckling behavior curve, including the maximum and minimum points of the nonlinear equilibrium path of the asymmetric arch bridge. The extreme point is obtained by solving the stationary point of the buckling behavior curve, which is expressed as: ； In formula (14), is the geometrically nonlinear jump buckling load in the asymmetric arch bridge surface; is the horizontal thrust when the arch bridge is at critical load; Based on the boundary conditions of the instability of asymmetric arch bridges with hingeless arches and two-hinged arches, the approximate analysis of the nonlinear leap buckling critical load of the corresponding asymmetric arch bridge is obtained, where: Critical dimensionless load for buckling of branch points of hingeless asymmetric arch bridges The expression is: ； Critical dimensionless buckling load at branch point of an asymmetric arch bridge with two hinged arches The expression is: ； The approximate solution of the nonlinear jump buckling critical load of the hingeless arch asymmetric arch bridge is deduced by the characteristics of the extreme point in the buckling behavior curve corresponding to the nonlinear jump buckling critical load. It is expressed as: ； In formula (17), , , is a dimensionless load parameter, expressed as: ； In formula (18), the implicit function Expressed as , its The partial derivative of It is expressed as: ； In formula (18), the implicit function Expressed as , its The partial derivative of It is expressed as: ； In formula (19) and formula (20), ' It means to find the first-order derivative; In formula (18), the dimensionless load factor is The partial derivative of It is expressed as: ； The approximate solution of the critical load of nonlinear jump buckling of an asymmetric arch bridge with two hinged arches is deduced It is expressed as: ； In formula (22), , , is a dimensionless load parameter, expressed as: ； In formula (23), the implicit function Expressed as , its The partial derivative of It is expressed as: ； In formula (23), the implicit function Expressed as , its The partial derivative of It is expressed as: ； In formula (24) and formula (25), ' It means to find the first-order derivative; In formula (23), the dimensionless load parameter right The partial derivative of It is expressed as: 。

Images