An approximate analytical method for out-of-plane stability of tied arch bridges considering the effect of non-directional forces

Abstract

The present invention discloses an approximate analytical method for the out-of-plane stability of a tied arch bridge considering the effect of non-preserving forces. First, the positional relationship between the main arch ring and the bridge deck system when the tied arch structure subjected to non-preserving forces is analyzed when the structure is out-of-plane unstable, and the restoring force model is deduced; and the expressions of the work done by the non-preserving forces and the bending strain energy of the main beam are further deduced; then, the work done by the non-preserving forces and the bending strain energy of the main beam are introduced into the total energy equation of the tied arch; then, based on the boundary conditions of the tied arch, the out-of-plane buckling displacement function of the tied arch structure considering the effect of non-preserving forces is obtained; finally, the approximate analytical method for the out-of-plane buckling load of the tied arch considering the effect of non-preserving forces is obtained according to the Rayleigh-Ritz method. The present invention breaks through the limitation that the conventional out-of-plane stability analysis method of the arch structure cannot consider the effect of non-preserving forces, and can significantly improve the calculation efficiency of the out-of-plane buckling load of the tied arch compared with the finite element analysis method.

Description

Technical Field

[0001] The present invention mainly relates to the field of bridge engineering, and in particular to an approximate analytical method for out-of-plane stability of a tied arch bridge taking into account the effect of non-directional forces. Background Art

[0002] The bottom-supported tied arch bridge has a beautiful appearance and good mechanical properties, excellent stiffness and strong bearing capacity, and is widely used in highway and railway bridge projects in my country. The main arch ring of the tied arch bridge mainly bears axial pressure, has a large span and a narrow bridge deck, and there is a very high risk of out-of-plane instability.

[0003] The load applied to the arch structure is called a directional force if the direction of the load does not change when the arch structure loses stability out of the plane; it is called a non-directional force if the direction of the load changes when the arch structure loses stability out of the plane. When the directional force acts on the arch structure, when the arch structure has out-of-plane displacement, the directional force with unchanged load direction will increase the out-of-plane displacement, thereby causing the arch structure to lose stability out of the plane. On the contrary, when the non-directional force acts on the arch structure, when the arch structure has out-of-plane displacement, the direction of the non-directional force will change, thereby reducing the out-of-plane displacement of the arch structure, reducing the possibility of out-of-plane instability of the arch structure, and improving the out-of-plane instability bearing capacity of the arch structure.

[0004] The bottom-supported tied arch bridge consists of the main arch ring, the bridge deck system and the hangers. The load applied to the bridge deck system is transmitted to the main arch through the hangers. The main arch ring has large in-plane stiffness and small out-of-plane stiffness, while the bridge deck system has small in-plane stiffness and large out-of-plane stiffness. The two are connected to each other through the hangers to form a bridge structure with superior performance. When the out-of-plane stiffness of the main arch ring is insufficient and out-of-plane displacement occurs, the bridge deck system has basically no out-of-plane displacement or very small out-of-plane displacement due to its large out-of-plane stiffness. At this time, the upper end of the hanger that transfers the load moves out-of-plane with the main arch ring, while the lower end of the hanger connected to the bridge deck system basically does not move, which is a typical non-preserving force.

[0005] At present, the out-of-plane stability analysis of through arch bridges basically does not consider the effect of non-directional forces. There are mainly the following methods:

[0006] (1) Theory of out-of-plane stability calculation of circular arch in polar coordinate system. This theory uses balanced differential equations or energy method based on polar coordinate system to analyze the out-of-plane stability of circular arch and obtains analytical solution. Since this method does not consider the effect of non-directional force, it cannot be applied to the out-of-plane stability analysis of tied arch considering the effect of non-directional force.

[0007] (2) Calculation theory of out-of-plane stability of arch structures in rectangular coordinate system. Arch structures in rectangular coordinate system, represented by parabolas, catenaries, and cable lines, are widely used in bridge engineering. The calculation theory of out-of-plane stability of arch structures in rectangular coordinate system uses a method similar to the out-of-plane stability theory of circular arches in polar coordinate system to obtain their out-of-plane stability bearing capacity. However, it does not consider the effect of non-directional forces, and cannot solve the problem of out-of-plane stability analysis of tie arches considering the effect of non-directional forces.

[0008] In summary, the current out-of-plane stability analysis methods for bottom-supported tied-arch bridges do not consider the effect of non-directional forces, and cannot solve the problem of out-of-plane stability analysis of tied arch bridges considering the effect of non-directional forces. Summary of the invention

[0009] The purpose of the present invention is to provide an approximate analytical method for the out-of-plane stability of a tied arch bridge considering the effect of non-preserving forces in view of the deficiencies in the prior art. Based on the positional relationship between the out-of-plane deformation of the main arch ring and the bridge deck system when the tied arch structure is out-of-plane unstable, a restoring force model caused by the non-preserving force effect when the tied arch structure is out-of-plane unstable is deduced; based on the restoring force model of the non-preserving force effect, the expressions of the work done by the non-preserving force and the bending strain energy of the main beam are deduced; the work done by the non-preserving force and the bending strain energy of the main beam are introduced into the energy equation containing the lateral bending strain energy of the arch rib, the warping and torsional strain energy of the arch rib and the work done by the deadweight load; based on the Rayleigh-Ritz method, an approximate analytical method for the out-of-plane buckling load of the tied arch considering the effect of non-preserving forces is obtained, which breaks through the limitation that the out-of-plane stability analysis of conventional arch structures cannot consider the effect of non-preserving forces. The method has clear overall logic, clear concepts and simple solutions, and can help engineering and technical personnel quickly judge the safety state of the bottom-supported arch bridge structure and improve the calculation efficiency of the out-of-plane buckling load of the tied arch.

[0010] In order to achieve the above object, the present invention adopts the following technical solution.

[0011] An approximate analytical method for out-of-plane stability of a tied arch bridge considering the effect of non-directional forces comprises the following steps:

[0012] Step S1, by analyzing the position relationship between the main arch ring and the bridge deck when the tied arch structure is subjected to the non-preserving force and buckling out of the plane, a restoring force model caused by the non-preserving force effect when the tied arch structure buckles out of the plane is deduced;

[0013] Step S2, based on the restoring force model of the non-alignment force effect obtained in step S1, deriving expressions for the work done by the non-alignment force and the lateral bending strain energy of the main beam;

[0014] Step S3, introducing the non-alignment force work obtained in step S2 and the lateral bending strain energy of the main beam into the energy equation including the lateral bending strain energy of the main arch ring, the out-of-plane torsional strain energy of the main arch ring and the work done by the deadweight load, so as to obtain the total energy equation of the tied arch considering the non-alignment force effect;

[0015] Step S4, based on the tied arch boundary condition, obtain the out-of-plane buckling displacement function of the tied arch structure considering the non-alignment-preserving force effect, substitute it into the out-of-plane buckling energy equation of the tied arch structure, and obtain the specific expressions of each out-of-plane buckling energy equation of the tied arch structure considering the non-alignment-preserving force effect;

[0016] Step S5, substituting each energy equation of the out-of-plane buckling of the tied arch structure considering the effect of non-alignment force obtained in step S4 into the total energy equation of the tied arch considering the effect of non-alignment force obtained in step S3, and simplifying to obtain the total energy equation of the out-of-plane buckling of the tied arch structure;

[0017] Step S6, based on the Rayleigh-Ritz method and the principle of stationary value of potential energy, solve the conditional variational extreme value when the out-of-plane total energy of the tied arch structure considering the non-preserving force effect is constant, and then obtain the approximate analysis of the out-of-plane buckling load of the tied arch considering the non-preserving force effect.

[0018] Specifically, the restoring force model caused by the non-maintaining force effect in step S1 is F H The expression is:

[0019] ;

[0020] In the above formula, is the in-plane vertical load concentration; z is the vertical coordinate of the arch structure; u , u b They are the horizontal displacement of the main arch and the horizontal displacement of the main beam respectively; is the equation of the tied arch axis.

[0021] Specifically, the expression for the work done by the non-alignment-preserving force in step S2 is:

[0022] ;

[0023] In the above formula, W H The work done by the non-preserving force; L is the span of the arch.

[0024] The expression of the lateral bending strain energy of the main beam is:

[0025] ;

[0026] In the above formula, U B,b is the lateral bending strain energy of the main beam; E b is the Young's modulus of the main beam material; I by is the out-of-plane bending moment of inertia of the main beam section.

[0027] Specifically, the energy equation including the lateral bending strain energy of the main arch ring, the out-of-plane torsional strain energy of the main arch ring and the work done by the deadweight load described in step S3, wherein the expression of the lateral bending strain energy of the main arch ring is:

[0028] ;

[0029] In the above formula, U B is the lateral bending strain energy of the main arch ring; E is the Young’s modulus of the main arch material; is the lateral bending strain of the arch; V is the volume of the main arch;

[0030] The expression of the out-of-plane torsional strain energy of the main arch ring is:

[0031] ;

[0032] In the above formula, U T is the out-of-plane torsional strain energy of the main arch ring; T a is the total torque of the arch section; θ a is the torsion angle per unit length of the arch section;

[0033] The expression for the work done by the deadweight load is:

[0034] ;

[0035] In the above formula, W Work done for the tied arch deadweight load; g c is the load concentration at the vault position; u c is the out-of-plane linear displacement at the mid-span of the tied arch; , k is the coefficient of the tied arch equation, , m is the tie arch axis curve coefficient, f is the rise of the tied arch structure; is the displacement function, for The first derivative of ;

[0036] After introducing the work done by the non-directional force and the bending strain energy of the main beam, the expression of the total energy equation of the tied arch considering the effect of the non-directional force is obtained:

[0037] ;

[0038] In the above formula, ΠNori is the total energy of the tied arch considering the effect of non-retaining forces.

[0039] Specifically, the expression of the out-of-plane buckling displacement function of the tied arch structure considering the non-preserving force effect in step S4 is:

[0040] ;

[0041] In the above formula, u , φ and u b They are the horizontal displacement of the main arch, the angular deformation of the main arch and the horizontal displacement of the main beam; u c , φ c and u b，c They are the maximum out-of-plane displacement in the main arch mid-span, the maximum section rotation in the main arch mid-span and the maximum out-of-plane displacement in the main beam mid-span; is the displacement function, and its expression is , z is the vertical coordinate of the arch structure, L is the span of the arch;

[0042] After substituting the out-of-plane buckling energy equation of the tied arch structure, the specific expressions of the out-of-plane buckling energy equations of the tied arch structure considering the non-directional force effect are obtained:

[0043] Work done by non-alignment forces:

[0044] ;

[0045] In the above formula, To consider the calculation coefficient of the non-directional force work of the tied arch under the action of non-directional force, ,in, α is the rise-to-span ratio of the tied arch, , f is the rise of the tied arch structure, L is the span of the arch; sh (*) is the hyperbolic sine function, ch (*) is the hyperbolic cosine function;

[0046] Lateral bending strain energy of main beam:

[0047] ;

[0048] In the above formula, is the calculation coefficient of lateral bending strain energy of the tied arch main beam, ;

[0049] Lateral bending strain energy of main arch ring:

[0050] ;

[0051] In the above formula, , , are the calculation parameters related to lateral bending strain energy,

[0052] ;

[0053] ;

[0054] ;

[0055] Out-of-plane torsional strain energy of the main arch ring:

[0056] ;

[0057] In the above formula, G is the shear modulus of the material; I t is the torsional moment of inertia of the arch section; , , , , , , , , is the calculation coefficient related to torsional strain energy,

[0058] ;

[0059] ;

[0060] ;

[0061] ;

[0062] ;

[0063] ;

[0064] ;

[0065] ;

[0066] ;

[0067] In the above formula, , A is the cross-sectional area of ​​the tied arch rib, I x is the in-plane bending moment of inertia of the arch section, is the out-of-plane bending moment of inertia of the arch section;

[0068] Work done by deadweight load:

[0069] ;

[0070] In the above formula, is the calculation coefficient of work done by the tied arch self-weight load, .

[0071] Specifically, the expression of the total out-of-plane buckling energy equation of the tied arch structure obtained after simplification in step S5 is:

[0072] ;

[0073] In the above formula, A , B , C , D , E is the coefficient of the energy equation outside the tied arch surface, and its expressions are:

[0074] ;

[0075] In the above formula, is the out-of-plane bending moment of inertia of the arch section; I w is the warping moment of inertia of the arch section.

[0076] Specifically, the conditional variational extreme value of the tied arch structure out of plane considering the non-directional force effect solved in step S6 when the total energy remains unchanged is expressed as:

[0077] ;

[0078] The out-of-plane buckling load of the tied arch considering the effect of non-directional force is approximately analyzed as follows:

[0079] ;

[0080] In the above formula, g cr The out-of-plane buckling load of the tied arch considering the effect of non-retaining force; a , b , c. d. is the coefficient of the out-of-plane buckling load equation of the tied arch considering the effect of non-directional force,

[0081] ;

[0082] ;

[0083] ;

[0084] .

[0085] Compared with the prior art, the present invention has the following beneficial effects:

[0086] According to the characteristic that out-of-plane instability of a supported arch bridge has non-preserving forces, the present invention proposes a method for calculating the out-of-plane stability of a tied arch structure taking into account the effects of non-preserving forces, which breaks through the limitation that the out-of-plane stability analysis of conventional arch structures cannot take into account the effects of non-preserving forces. The method has clear overall logic, clear concepts, and simple solutions. It can help engineering and technical personnel quickly judge the safety status of the supported arch bridge structure and improve the calculation efficiency of the out-of-plane buckling load of the tied arch. BRIEF DESCRIPTION OF THE DRAWINGS

[0087] Figure 1 It is a flow chart of the approximate analytical method for the out-of-plane stability of the tied arch considering the effect of non-alignment force of the present invention;

[0088] Figure 2 Schematic diagram of a tie arch structure in an embodiment of the present invention;

[0089] Figure 3 It is a cross-sectional view of the out-of-plane buckling of the tied arch structure in the embodiment of the present invention;

[0090] Figure 4 It is a top view of the out-of-plane buckling of the tied arch structure in the embodiment of the present invention;

[0091] Figure 5 Schematic diagram of the dimensions of three types of arch rib cross sections in an embodiment of the present invention;

[0092] Figure 6 The figure is a comparison result and relative error diagram of the out-of-plane buckling load of the tie arch obtained by the method of the present invention and the finite element analysis method.

[0093] In the figure, 1. In-plane vertical uniformly distributed load; 2. Arch axis; 3. Tie-arch bridge hanger; 4. Hingeless boundary constraint; 5. Horizontal tie; 6. Global coordinate system; 7. Local coordinate system; 8. Arch axis after tie failure. DETAILED DESCRIPTION

[0094] 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.

[0095] Example

[0096] like Figure 1As shown, the present invention discloses an approximate analytical method for out-of-plane stability of a tied arch bridge considering the effect of non-directional force, comprising the following steps:

[0097] Step S1: Figure 2 As shown in the figure, by analyzing the position relationship between the main arch ring and the bridge deck when the tied arch structure is subjected to the non-preserving force and buckling out of the plane, the restoring force model caused by the non-preserving force effect when the tied arch structure is buckling out of the plane is deduced;

[0098] Step S2, based on the restoring force model of the non-alignment force effect obtained in step S1, deriving expressions for the work done by the non-alignment force and the lateral bending strain energy of the main beam;

[0099] Step S3, introducing the non-alignment force work obtained in step S2 and the lateral bending strain energy of the main beam into the energy equation including the lateral bending strain energy of the main arch ring, the out-of-plane torsional strain energy of the main arch ring and the work done by the deadweight load, so as to obtain the total energy equation of the tied arch considering the non-alignment force effect;

[0100] Step S4, based on the tied arch boundary condition, obtain the out-of-plane buckling displacement function of the tied arch structure considering the non-alignment-preserving force effect, substitute it into the out-of-plane buckling energy equation of the tied arch structure, and obtain the specific expressions of each out-of-plane buckling energy equation of the tied arch structure considering the non-alignment-preserving force effect;

[0101] Step S5, substituting each energy equation of the out-of-plane buckling of the tied arch structure considering the effect of non-alignment force obtained in step S4 into the total energy equation of the tied arch considering the effect of non-alignment force obtained in step S3, and simplifying to obtain the total energy equation of the out-of-plane buckling of the tied arch structure;

[0102] Step S6, based on the Rayleigh-Ritz method and the principle of stationary value of potential energy, solve the conditional variational extreme value when the out-of-plane total energy of the tied arch structure considering the non-preserving force effect is constant, and then obtain the approximate analysis of the out-of-plane buckling load of the tied arch considering the non-preserving force effect.

[0103] like Figure 3 As shown in the figure, by analyzing the position relationship between the main arch ring and the bridge deck when the tied arch structure is subjected to non-directional force and buckling out of the plane, the restoring force model caused by the non-directional force effect when the tied arch structure is buckled out of the plane is deduced. F H The expression is:

[0104] ;

[0105] In the above formula, is the in-plane vertical load concentration; z is the vertical coordinate of the arch structure; u , u b They are the horizontal displacement of the main arch and the horizontal displacement of the main beam respectively; is the equation of the tied arch axis.

[0106] Furthermore, based on the restoring force model of the out-of-plane buckling non-preserving force effect of the tied arch structure, the expression of the work done by the non-preserving force is obtained as follows:

[0107] ;

[0108] In the above formula, W H The work done by the non-preserving force; L is the span of the arch.

[0109] Furthermore, if Figure 4 As shown in the figure, based on the Euler-Bernoulli beam theory, when the tied arch structure buckles out of plane, the expression of the lateral bending strain energy of the main beam is:

[0110] ;

[0111] In the above formula, U B,b is the lateral bending strain energy of the main beam; E b is the Young's modulus of the main beam material; I by is the out-of-plane bending moment of inertia of the main beam section; For Find the second derivative.

[0112] The energy equation including the lateral bending strain energy of the main arch ring, the out-of-plane torsional strain energy of the main arch ring and the work done by the deadweight load in step S3 is obtained based on the out-of-plane stability theory of the tied arch:

[0113] The expression of the lateral bending strain energy of the main arch ring is:

[0114] ;

[0115] In the above formula, U B is the lateral bending strain energy of the main arch ring; E is the Young’s modulus of the main arch material; is the lateral bending strain of the arch; V is the volume of the main arch;

[0116] The expression of the out-of-plane torsional strain energy of the main arch ring is:

[0117] ;

[0118] In the above formula, U T is the out-of-plane torsional strain energy of the main arch ring; T ais the total torque of the arch section; θ a is the torsion angle per unit length of the arch section;

[0119] The expression for the work done by the deadweight load is:

[0120] ;

[0121] In the above formula, W Work done for the tied arch deadweight load; g c is the load concentration at the vault position; u c is the out-of-plane linear displacement at the mid-span of the tied arch; , k is the coefficient of the tied arch equation, , m is the arch axis curve coefficient of the tied arch, f is the rise of the tied arch structure; is the displacement function, for The first derivative of ;

[0122] After introducing the work done by the non-directional force and the bending strain energy of the main beam, the expression of the total energy equation of the tied arch considering the effect of the non-directional force is obtained:

[0123] ;

[0124] In the above formula, Π Nori is the total energy of the tied arch considering the effect of non-retaining forces.

[0125] Specifically, the expression of the out-of-plane buckling displacement function of the tied arch structure considering the non-preserving force effect in step S4 is:

[0126] ;

[0127] In the above formula, u , φ and u b They are the horizontal displacement of the main arch, the angular deformation of the main arch and the horizontal displacement of the main beam; u c , φ c and u b，c They are the maximum out-of-plane displacement in the main arch mid-span, the maximum section rotation in the main arch mid-span and the maximum out-of-plane displacement in the main beam mid-span; is the displacement function, and its expression is , z is the vertical coordinate of the arch structure, L is the span of the arch;

[0128] After substituting the out-of-plane buckling energy equation of the tied arch structure, the specific expressions of the out-of-plane buckling energy equations of the tied arch structure considering the non-directional force effect are obtained:

[0129] Work done by non-alignment forces:

[0130] ;

[0131] In the above formula, To consider the calculation coefficient of the work done by the non-directional force of the tied arch under the action of the non-directional force, ,in, α is the rise-to-span ratio of the tied arch, , f is the rise of the tied arch structure, L is the span of the arch; sh (*) is the hyperbolic sine function, ch (*) is the hyperbolic cosine function;

[0132] Lateral bending strain energy of main beam:

[0133] ;

[0134] In the above formula, is the calculation coefficient of lateral bending strain energy of the tied arch main beam, ;

[0135] Lateral bending strain energy of main arch ring:

[0136] ;

[0137] In the above formula, , , are the calculation parameters related to lateral bending strain energy,

[0138] ;

[0139] ;

[0140] ;

[0141] Out-of-plane torsional strain energy of the main arch ring:

[0142] ;

[0143] In the above formula, G is the shear modulus of the material; I t is the torsional moment of inertia of the arch section; , , , , , , , , is the calculation coefficient related to torsional strain energy,

[0144] ;

[0145] ;

[0146] ;

[0147] ;

[0148] ;

[0149] ;

[0150] ;

[0151] ;

[0152] ;

[0153] In the above formula, , A is the cross-sectional area of ​​the tied arch rib, I x The in-plane bending moment of inertia of the arch section, is the out-of-plane bending moment of inertia of the arch section;

[0154] Work done by deadweight load:

[0155] ;

[0156] In the above formula, is the calculation coefficient of work done by the tied arch self-weight load, .

[0157] Specifically, the expression of the total out-of-plane buckling energy equation of the tied arch structure obtained after simplification in step S5 is:

[0158] ;

[0159] In the above formula, A , B , C , D , E is the coefficient of the energy equation outside the tied arch surface, and its expressions are:

[0160] ;

[0161] In the above formula, is the out-of-plane bending moment of inertia of the arch section; I w is the warping moment of inertia of the arch section.

[0162] Specifically, based on the Rayleigh-Ritz method and the principle of stationary value of potential energy, the conditional variational extreme value of the tied arch structure with the non-directional force effect considered when the out-of-plane total energy remains unchanged is expressed as:

[0163] ;

[0164] The out-of-plane buckling load of the tied arch considering the effect of non-directional force is approximately analyzed as follows:

[0165] ;

[0166] In the above formula, g cr The out-of-plane buckling load of the tied arch considering the effect of non-retaining force; a , b , c. d. is the coefficient of the out-of-plane buckling load equation of the tied arch considering the effect of non-directional force,

[0167] ;

[0168] ;

[0169] ;

[0170] .

[0171] The following is a calculation example to compare the results of the approximate analytical method for the out-of-plane stability of the tie arch considering the non-directional force effect obtained by the method of the present invention with those obtained by the traditional finite element analysis method, so as to verify the accuracy of the method of the present invention.

[0172] The finite element analysis software ANSYS is used to model and calculate the tied arch structure. The specific parameters of the tied arch are: the rise-span ratio is 1 / 10~1 / 5, the tied arch axis coefficient m=1.988, and the out-of-plane slenderness ratio L / i y =120, the Young's modulus of the main arch and main beam materials are E and E b=210GPa, Poisson's ratio υ=0.3. The arch rib sections of the finite element model are I-shaped sections that consider the warping torque, rectangular sections that consider the warping normal stress but do not consider the warping torque, and circular sections that do not consider the warping at all; the radius of the circular section is 0.5m, and the dimensions of the other two types of sections are determined according to the principle of equal cross-sectional area and equal bending moment of inertia of the cross-sectional surface. The dimensions of each section are as follows: Figure 5 The main beam section adopts a rectangular section with a height × width of 1m × 2.4m; the suspension rod adopts an area of ​​0.02m 2 The cross section of the bar unit. The three-dimensional finite strain beam unit BEAM188 is selected as the main beam and main arch unit of the model, and the WARP warping degree of freedom option is turned on. 500 BEAM188 beam units are used for both the main beam and the main arch; the three-dimensional finite strain bar unit LINK180 is selected for the hanger. Each node of this unit has only three degrees of freedom, UX, UY, and UZ, which can simulate the hinge relationship between the hanger and the main beam and the main arch. The finite element model of the tied arch constrains all seven degrees of freedom at the nodes at the arch foot position, and the remaining nodes are not constrained. At the same time, the load is equivalent to the node load and applied to each node of the finite element model accordingly.

[0173] The ANSYS eigenvalue buckling module is used for calculation, and the first-order buckling load is extracted as the numerical solution of the verification load. The same calculation parameters as the finite element method are used and substituted into the analytical solution of the buckling load outside the tie arch surface considering the non-preserving force effect of the present invention to obtain the calculation results of the method of the present invention. Since the numerical result of the buckling load is more significant, it is dimensionless and the buckling load value is multiplied by ( represents the span, is the elastic modulus, is the out-of-plane bending moment of inertia of the arch section), This makes the values ​​of the buckling load more intuitive and comparable. Figure 6 The figure shows the dimensionless numerical comparison of buckling load between the method of the present invention and the finite element method, wherein the relative error is calculated as follows: relative error = |the method of the present invention - the finite element method| / the finite element method.

[0174] from Figure 6 (a) and Figure 6 It can be seen from (b) in the figure that, for circular arch rib sections with a rise-to-span ratio ranging from 1 / 10 to 1 / 5, the out-of-plane buckling load considering the non-directional force obtained by the method of the present invention has a smaller relative error than the buckling load obtained by the finite element method, with the maximum relative error being 7.29%, the minimum relative error being 0.11%, and the average relative error being 4.33%.

[0175] from Figure 6 (c) and Figure 6It can be seen from (d) in the figure that, for rectangular arch rib sections with a rise-to-span ratio ranging from 1 / 10 to 1 / 5, the out-of-plane buckling load considering the non-directional force obtained by the method of the present invention has a maximum relative error of 7.54% and a minimum relative error of 0.68% with an average relative error of 4.46% compared with the out-of-plane buckling load obtained by the finite element method.

[0176] from Figure 6 (e) and Figure 6 It can be seen from (f) in the figure that, for the I-shaped arch rib section with a rise-to-span ratio ranging from 1 / 10 to 1 / 5, the out-of-plane buckling load considering the non-directional force obtained by the method of the present invention has an average relative error of 6.84% compared with the buckling load obtained by the finite element method. In particular, in the range of the rise-to-span ratio from 1 / 6 to 1 / 7, the relative error is less than 1%.

[0177] Based on the above comparative analysis, it is shown that the relative error between the out-of-plane buckling load of the tie arch considering the non-preserving force obtained by the method of the present invention and the finite element result is small, and the approximate analytical result of the out-of-plane buckling load of the tie arch obtained by the method of the present invention can meet the needs of actual engineering calculations. Therefore, the method of the present invention can be applied to the calculation of the out-of-plane buckling load of the tie arch considering the effect of non-preserving force of different cross-section types.

[0178] 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 out-of-plane stability of a tied arch bridge considering the effect of non-alignment forces, characterized in that: The following steps are involved: Step S1, by analyzing the position relationship between the main arch ring and the bridge deck when the tied arch structure is subjected to the non-preserving force and buckling out of the plane, a restoring force model caused by the non-preserving force effect when the tied arch structure buckles out of the plane is deduced; Step S2, based on the restoring force model of the non-alignment force effect obtained in step S1, deriving expressions for the work done by the non-alignment force and the lateral bending strain energy of the main beam; Step S3, introducing the non-alignment force work obtained in step S2 and the lateral bending strain energy of the main beam into the energy equation including the lateral bending strain energy of the main arch ring, the out-of-plane torsional strain energy of the main arch ring and the work done by the deadweight load, so as to obtain the total energy equation of the tied arch considering the non-alignment force effect; Step S4, based on the tied arch boundary condition, obtain the out-of-plane buckling displacement function of the tied arch structure considering the non-alignment-preserving force effect, substitute it into the out-of-plane buckling energy equation of the tied arch structure, and obtain the specific expressions of each out-of-plane buckling energy equation of the tied arch structure considering the non-alignment-preserving force effect; Step S5, substituting each energy equation of the out-of-plane buckling of the tied arch structure considering the effect of non-alignment force obtained in step S4 into the total energy equation of the tied arch considering the effect of non-alignment force obtained in step S3, and simplifying to obtain the total energy equation of the out-of-plane buckling of the tied arch structure; Step S6, based on the Rayleigh-Ritz method and the principle of stationary value of potential energy, solve the conditional variational extreme value when the out-of-plane total energy of the tied arch structure considering the non-preserving force effect is constant, and then obtain the approximate analysis of the out-of-plane buckling load of the tied arch considering the non-preserving force effect.

2. The approximate analytical method for out-of-plane stability of a tied arch bridge considering the effect of non-alignment forces according to claim 1 is characterized in that: The restoring force model caused by the non-alignment force effect described in step S1 F H The expression is: ； In the above formula, is the in-plane vertical load concentration; z is the vertical coordinate of the arch structure; u , u b They are the horizontal displacement of the main arch and the horizontal displacement of the main beam respectively; is the axis equation of the tied arch.

3. The approximate analytical method for out-of-plane stability of a tied arch bridge considering the effect of non-alignment forces according to claim 1 is characterized in that: The expression of the work done by the non-alignment-preserving force in step S2 is: ； In the above formula, W H The work done by the non-preserving force; L is the span of the arch; The expression of the lateral bending strain energy of the main beam is: ； In the above formula, U B,b is the lateral bending strain energy of the main beam; E b is the Young's modulus of the main beam material; I by is the out-of-plane bending moment of inertia of the main beam section; For Find the second derivative.

4. The approximate analytical method for out-of-plane stability of a tied arch bridge considering the effect of non-alignment forces according to claim 1 is characterized in that: The energy equation in step S3 includes the lateral bending strain energy of the main arch ring, the out-of-plane torsional strain energy of the main arch ring, and the work done by the deadweight load, wherein the expression of the lateral bending strain energy of the main arch ring is: ； In the above formula, U B is the lateral bending strain energy of the main arch ring; E is the Young’s modulus of the main arch material; is the lateral bending strain of the arch; V is the volume of the main arch; The expression of the out-of-plane torsional strain energy of the main arch ring is: ； In the above formula, U T is the out-of-plane torsional strain energy of the main arch ring; T a is the total torque of the arch section; θ a is the torsion angle per unit length of the arch section; The expression for the work done by the deadweight load is: ； In the above formula, W Work done for the tied arch deadweight load; g c is the load concentration at the vault position; u c is the out-of-plane linear displacement at the mid-span of the tied arch; , k is the coefficient of the tied arch equation, , m is the arch axis curve coefficient of the tied arch, f is the rise of the tied arch structure; is the displacement function, for The first derivative of ; After introducing the work done by the non-directional force and the bending strain energy of the main beam, the expression of the total energy equation of the tied arch considering the effect of the non-directional force is obtained: ； In the above formula, Π Nori is the total energy of the tied arch considering the effect of non-retaining forces.

5. The approximate analytical method for out-of-plane stability of a tied arch bridge considering the effect of non-alignment forces according to claim 1 is characterized in that: The expression of the out-of-plane buckling displacement function of the tied arch structure considering the non-preserving force effect in step S4 is: ； In the above formula, u , φ and u b They are the horizontal displacement of the main arch, the angular deformation of the main arch and the horizontal displacement of the main beam; u c , φ c and u b，c They are the maximum out-of-plane displacement in the main arch mid-span, the maximum section rotation in the main arch mid-span and the maximum out-of-plane displacement in the main beam mid-span; is the displacement function, and its expression is , z is the vertical coordinate of the arch structure, L is the span of the arch; After substituting the out-of-plane buckling energy equation of the tied arch structure, the specific expressions of the out-of-plane buckling energy equations of the tied arch structure considering the non-directional force effect are obtained: Work done by non-alignment forces: ； In the above formula, To consider the calculation coefficient of the work done by the non-directional force of the tied arch under the action of the non-directional force, ,in, α is the rise-to-span ratio of the tied arch, , f is the rise of the tied arch structure, L is the span of the arch; sh (*) is the hyperbolic sine function, ch (*) is the hyperbolic cosine function; Lateral bending strain energy of main beam: ； In the above formula, is the calculation coefficient of lateral bending strain energy of the tied arch main beam, ; Lateral bending strain energy of main arch ring: ； In the above formula, , , are the calculation parameters related to lateral bending strain energy, ； ； ； Out-of-plane torsional strain energy of the main arch ring: ； In the above formula, G is the shear modulus of the material; I t is the torsional moment of inertia of the arch section; , , , , , , , , is the calculation coefficient related to torsional strain energy, ； ； ； ； ； ； ； ； ； In the above formula, , A is the cross-sectional area of ​​the tied arch rib, I x is the in-plane bending moment of inertia of the arch section, is the out-of-plane bending moment of inertia of the arch section; Work done by deadweight load: ； In the above formula, is the calculation coefficient of work done by the tied arch self-weight load, .

6. The approximate analytical method for out-of-plane stability of a tied arch bridge considering the effect of non-alignment forces according to claim 1 is characterized in that: The expression of the total energy equation of out-of-plane buckling of the tied arch structure obtained after simplification in step S5 is: ； In the above formula, A , B , C , D , E is the coefficient of the energy equation outside the tied arch surface, and its expressions are: ； In the above formula, is the out-of-plane bending moment of inertia of the arch section; I w is the warping moment of inertia of the arch section.

7. The approximate analytical method for out-of-plane stability of a tied arch bridge considering the effect of non-alignment forces according to claim 1 is characterized in that: The conditional variational extreme value of the tied arch structure out of plane total energy unchanged considering the non-directional force effect solved in step S6 is expressed as: ； The out-of-plane buckling load of the tied arch considering the effect of non-directional force is approximately analyzed as follows: ； In the above formula, g cr The out-of-plane buckling load of the tied arch considering the effect of non-retaining force; a , b , c. d. is the coefficient of the out-of-plane buckling load equation of the tied arch considering the effect of non-directional force, ； ； ； 。

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