Method for calculating reasonable arch axis of through arch bridge

Abstract

The invention relates to the technical field of bridge engineering construction and discloses a reasonable arch axis calculation method for a through arch bridge, which comprises the following steps of: establishing a calculation model, and obtaining a control differential equation of a reasonable arch axis considering dead loads of a main arch, a bridge road system and a suspender; solving an approximate analytical solution of the control differential equation to obtain a second-order linear ordinary differential equation of the control differential equation; solving an arch axis coefficient based on an explicit approximate calculation formula of the arch axis coefficient; and substituting the arch axis coefficient into an approximate analytical solution of a second-order linear ordinary differential equation to obtain a reasonable arch axis equation. The method for calculating the reasonable arch axis of the through arch bridge has the advantages of high speed, high efficiency and high precision, the dead loads of the main arch, the bridge road system and the suspender are considered, the theoretical dead load distribution conforms to the actual situation of the bridge, the calculation precision is improved, and the essential characteristics between the linear shape of the reasonable arch axis and the dead load distribution of the arch bridge are revealed.

Description

Technical Field

[0001] The present invention relates to the technical field of bridge engineering construction, and particularly relates to a method for calculating a reasonable arch axis of a through arch bridge. Background Art

[0002] The selection of the arch axis is the key to the design of an arch bridge. The linear shape of the arch axis directly affects the internal force distribution and magnitude of the main arch section. Whether the arch axis is reasonable has an important impact on the economy, safety, and durability of the arch bridge. In the prior art, the analytical method is usually used to obtain the reasonable arch axis. Specifically, the analytical method constructs the control differential equation of the dead load pressure line according to the static equilibrium principle of a statically determinate arch, and takes the pressure line obtained by solving the differential equation as the reasonable arch axis.

[0003] However, the defect of the prior art is that: generally, the existing analytical methods for the reasonable arch axis of a through arch bridge do not give an explicit function arch axis equation, resulting in a large amount of calculation work and low efficiency. In addition, the prior art does not consider the influence of the weight of the suspenders on the reasonable arch axis, making its dead load distribution not conform to the actual situation of the bridge, affecting the calculation accuracy, and it is also difficult to reveal the essential characteristics between the linear shape of the reasonable arch axis and the dead load distribution of the arch bridge. Summary of the Invention

[0004] The technical problem to be solved by the present invention is that the existing analytical methods for the reasonable arch axis of a through arch bridge have a large amount of calculation work, low efficiency, and do not consider the influence of the weight of the suspenders on the reasonable arch axis. The purpose is to provide a method for calculating the reasonable arch axis of a through arch bridge to solve the above problems.

[0005] The present invention is achieved by the following technical solutions: A method for calculating the reasonable arch axis of a through arch bridge includes the following steps: Establish a calculation model to obtain the control differential equation of the reasonable arch axis considering the dead loads of the main arch, the bridge deck system, and the suspenders; Obtain a simplified second-order linear ordinary differential equation of the control differential equation, and obtain an approximate analytical solution of the control differential equation; Based on the explicit approximate calculation formula of the arch axis coefficient, obtain the arch axis coefficient; Substitute the arch axis coefficient into the approximate analytical solution of the second-order linear ordinary differential equation to obtain the reasonable arch axis equation.

[0006] In a possible design, obtaining the control differential equation of the reasonable arch axis considering the dead loads of the main arch, the bridge deck system, and the suspenders includes: Obtain the parameters required for the calculation and establish a calculation model; Set up the reasonable arch axis equation, and obtain the control differential equation of the reasonable arch axis through the static equilibrium condition; Based on the coordinate system adopted by the computational model, establish the boundary conditions of the governing differential equations.

[0007] In a possible design, the computational model includes: Establish the computational diagram of a through arch bridge with a constant cross-section, with the highest point of the main arch as the origin of the rectangular coordinate system O , the abscissa x is positive to the right along the bridge axis, and the ordinate y is positive downward; Correspondingly, in the rectangular coordinate system, the two arch springing points of the main arch are respectively A point and B point, the span of the main arch is L , the rise is f , and the rise-span ratio is n = f / L .

[0008] In a possible design, the computational parameters include: Let the weight per unit length of the main arch along the arch axis be p , and the weight per unit length of the bridge deck system in the horizontal direction be q ; Let the suspenders of the through arch bridge be membranes with resistance only in the vertical direction. The weight per unit length of the suspenders is φ , and the spacing between adjacent suspenders is d . Obtain the computational formula for the weight per unit area γ of the membrane simulating the suspenders: ; Let the ratio of the self-weight of the main arch to all the dead loads acting on the main arch be the main arch dead load ratio λ . Obtain the computational formula for the main arch dead load ratio λ : .

[0009] In a possible design, the governing differential equations of the reasonable arch axis include: Establish the equation of the reasonable arch axis: ; Based on the equation of the reasonable arch axis, obtain the relationship between the main arch micro-segment and the horizontal coordinate micro-segment at any point on the main arch: ; In the formula, ds is the main arch micro-segment; dx is ds the corresponding horizontal coordinate micro-segment.

[0010] In a possible design, the governing differential equations of the reasonable arch axis obtained through the static equilibrium conditions include: Based on the main arch micro-segment dsBased on the static equilibrium condition in the horizontal direction, the following equation is obtained: ; wherein, N is the pressure at any point of the main arch; Based on Equation (5), the horizontal component of the axial pressure of the main arch is obtained as a constant H , and the following equation is obtained: ; Based on the static equilibrium condition in the vertical direction of the infinitesimal segment ds of the main arch, the following equation is obtained: ; Using Equation (6) to transform the terms in the left bracket of Equation (7) and obtaining the following equation: ; Based on Equation (4) and Equation (8), substituting Equation (4) and Equation (8) into Equation (7), the control differential equation of the reasonable arch axis is obtained: ; wherein, μ ( x ) is the uniform load intensity acting on the main arch, and is expressed by the following equation μ ( x ): .

[0011] In a possible design, the boundary conditions for establishing the control differential equation include: based on the coordinate system adopted in the calculation model, the following boundary conditions are obtained: .

[0012] In a possible design, obtaining an approximate analytical solution of the control differential equation, obtaining a simplified second-order linear ordinary differential equation form of the control differential equation, and obtaining the analytical solution of the second-order linear ordinary differential equation include: Setting up a linear function of the uniform load intensity μ ( x ); based on special coordinate points, obtaining the expression of the arch axis coefficient, the undetermined parameters of the linear function, and the linear expression of the uniform load intensity μ ( x ); Based on the linear expression of the uniform load intensity μ ( x ), obtaining the simplified form of the second-order linear ordinary differential equation of the control differential equation; Based on the arch axis coefficient, obtaining the solution of the second-order linear ordinary differential equation; Correspondingly, each solution of the second-order linear ordinary differential equation corresponds to a reasonable arch axis equation.

[0013] In a possible design, the expression of the arch axis coefficient, the undetermined parameters of the linear function formula, and the dead load intensity μ ( x ) The linear expression includes: Set up the linear function formula of the dead load intensity μ ( x ): ; In the formula, a , b are undetermined parameters; Based on the static equilibrium condition, when x = 0, y = 0, dy / dx = 0, substitute (0, 0) into the expression (10) of the dead load intensity μ ( x ) to obtain μ ( 0 ) expression: ; Substitute into the linear function formula (12) of the dead load intensity μ ( x ) to obtain the expression of the parameter a : ; Based on the rectangular coordinate system, the coordinates of the arch foot B point are ( L / 2 , f ). Define the ratio of the dead load intensity of the arch foot B point to the arch crown O as the arch axis coefficient m , and obtain the expression of the arch axis coefficient: ; Based on the linear function formula of the dead load intensity μ ( x ), obtain μ ( 0 ), =a , μ ( L / 2 ) = a + bf ; Substitute μ ( 0 ) =a , μ ( L / 2 ) =a + bf into the expression (14) of the arch axis coefficient to obtain the expression of the parameter b : ; Based on the parametera Expressions and parameters b For the expression, substitute Equation (13) and Equation (15) into Equation (12) to obtain the dead load intensity μ ( x ) of the linear expression: .

[0014] In a possible design, based on the dead load intensity μ ( x ), the simplified form of the second-order linear ordinary differential equation of the control differential equation is obtained as follows: Based on Equation (16), substitute Equation (16) into the control differential equation of the reasonable arch axis to obtain the second-order linear ordinary differential form of the control differential equation: ; In the formula, K is a constant, and the expression of the constant K is as follows: .

[0015] In a possible design, based on the arch axis coefficient, the solution of the second-order linear ordinary differential equation is obtained as follows: Based on the expression of the constant K , the solution of the second-order linear ordinary differential equation depends on the relationship between the arch axis coefficient m and the constant 1. Accordingly, based on m > 1, m = 1 and m<1 , the second-order linear ordinary differential equation has three solutions; When m=1 , K=0 , combined with the boundary conditions, the solution of the second-order linear ordinary differential equation is obtained as a parabola: ; When m>1 , K>0 , let k 2 = K , and the following formula is obtained: ; Combined with the boundary conditions, when x = 0, y = 0, dy / dx = 0, the solution of the second-order linear ordinary differential equation is obtained as: ; In the formula, cosh represents the hyperbolic cosine function; Substitute the boundary condition x = L / 2, y = f into Equation (21) to obtain the following formula: ; In the formula, arch represents the inverse hyperbolic cosine function; Obtained from Equation (21) y The first derivative of is:

[0016] In the formula, sinh represents the hyperbolic sine function; When m<1 At this time, K<0 , let , the following formula is obtained: ; Based on Equation (24), the solution of the second-order linear ordinary differential equation is: ; Substitute the boundary condition x = L / 2, y = f into Equation (25), the following formula is obtained: .

[0017] In a possible design, based on the explicit approximate calculation formula of the arch axis coefficient, the arch axis coefficient includes: Based on the working conditions divided by the dead load, the analytical solution of the reasonable arch axis corresponding to the working conditions is obtained by solving the control differential equation; Based on the function fitting method, the approximate value corresponding to the working condition is obtained; the approximate value of the reasonable arch axis under the action of all dead loads is obtained by weighted average; Based on formula substitution, the explicit approximate calculation formula of the arch axis coefficient is obtained.

[0018] In a possible design, the analytical solution of the reasonable arch axis corresponding to the working condition obtained by solving the control differential equation includes: Based on in the dead load intensity expression (10) , set up: ; All dead loads are divided into two simple working conditions. Working condition 1 is to only consider the action of the main arch dead load p , and working condition 2 is to consider the action of the bridge deck system dead load q and the hanger dead load γ ; The analytical solution of the reasonable arch axis corresponding to the working condition is obtained by solving the corresponding control differential equation. Among them, the reasonable arch axis equation of working condition 1 is the following standard catenary: ; The derivative dy / dx is obtained from Equation (28) and substituted into the approximate calculation formula of the arch axis coefficient to obtain the function formula of the main arch: ; In equations (28) and (29), H are parameters to be determined; Similarly, the reasonable arch axis equation for working condition two is obtained: .

[0019] In a possible design, obtaining the analytical solution of the reasonable arch axis for the corresponding working condition includes: Based on the boundary conditions x = L / 2, y = f , the η(L / 2 of working condition one is obtained by the function fitting method β 1: ; Based on the boundary conditions x = L / 2, y = f , the η(L / 2 of working condition two is obtained by the function fitting method β 2: ; Taking the dead load intensity p and q + yf of working condition one and working condition two respectively as weights, taking β 1 and β 2's weighted average and using it as the η(L / 2) approximation of the reasonable arch axis under all dead loads: ; Obtaining the explicit approximate calculation formula for the arch axis coefficient includes: Based on equations (14) and (10), we get:

[0020] Substituting equation (33) into equation (34), the explicit approximate calculation formula for the arch axis coefficient m is obtained: .

[0021] Compared with the prior art, the present invention has the following advantages and beneficial effects: The calculation method for the reasonable arch axis of the through arch bridge has the advantages of fast speed, high efficiency and high precision, effectively solves the problem of low efficiency caused by large calculation workload in the prior art, and also improves the efficiency and quality in the design and construction process of the through arch bridge, and has great practical value. In addition, the calculation method for the reasonable arch axis of the through arch bridge considers the dead loads of the main arch, the bridge deck system and the suspenders, making the theoretical dead load distribution conform to the actual situation of the bridge, not only improving the calculation precision, but also revealing the essential characteristics between the reasonable arch axis line shape and the dead load distribution of the arch bridge. Description of the Drawings

[0022] To more clearly illustrate the technical solutions of the exemplary embodiments of the present invention, 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 invention and should not be regarded as limiting the scope. For those of ordinary skill in the art, without creative efforts, other relevant drawings can also be obtained based on these drawings. In the drawings: Figure 1 It is a schematic flowchart of a calculation method for a reasonable arch axis of a through arch bridge.

[0023] Figure 2 It is a schematic diagram of a constant cross-section of a through arch bridge.

[0024] Figure 3 It is a schematic diagram of the static equilibrium of a micro-segment. Specific embodiments

[0025] To make the objectives, technical solutions, and advantages of the present invention clearer and more understandable, the present invention will be further described in detail below in combination with the embodiments and the drawings. The illustrative embodiments of the present invention and their descriptions are only used to explain the present invention and are not intended to limit the present invention.

[0026] In the following description, a large number of specific details are set forth in order to provide a thorough understanding of the present invention. However, it is obvious to those of ordinary skill in the art that the present invention does not have to employ these specific details. In other embodiments, well-known structures, circuits, materials, or methods are not specifically described to avoid obscuring the present invention.

[0027] Throughout the specification, the reference to "an embodiment", "embodiments", "an example", or "examples" means that the specific features, structures, or characteristics described in connection with the embodiment or example are included in at least one embodiment of the present invention. Thus, the phrases "an embodiment", "embodiments", "an example", or "examples" appearing throughout the specification do not necessarily all refer to the same embodiment or example. Additionally, the specific features, structures, or characteristics can be combined in any suitable combination and / or sub-combination in one or more embodiments or examples. Furthermore, those of ordinary skill in the art should understand that the diagrams provided herein are for illustrative purposes only and are not necessarily drawn to scale. The term "and / or" used herein includes any and all combinations of one or more of the related listed items.

[0028] In the description of the present invention, the orientation or positional relationship indicated by terms such as "front", "rear", "left", "right", "upper", "lower", "vertical", "horizontal", "high", "low", "inner", "outer", etc. is based on the orientation or positional relationship shown in the drawings. It is only for the convenience of describing the present invention and simplifying the description, rather than indicating or implying that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation, and therefore cannot be construed as a limitation on the protection scope of the present invention.

[0029] Embodiment 1: As Figures 1 - 3 shown, a method for calculating the reasonable arch axis of a through arch bridge includes the following steps: S10: Establish a calculation model to obtain the control differential equation of the reasonable arch axis considering the dead loads of the main arch, the bridge deck system, and the suspenders. S20: Obtain the simplified second-order linear ordinary differential equation of the control differential equation and find the approximate analytical solution of the control differential equation. S30: Based on the explicit approximate calculation formula of the arch axis coefficient, obtain the arch axis coefficient. S40: Substitute the arch axis coefficient into the approximate analytical solution of the second-order linear ordinary differential equation to obtain the reasonable arch axis equation.

[0030] The method for calculating the reasonable arch axis of the through arch bridge is proposed for the design of the through arch bridge. The control differential equation of the reasonable arch axis considering the combined action of the dead loads of the main arch, the bridge deck system, and the suspenders is established. A reasonable approximate assumption is used to solve the control differential equation, and a high-precision approximate analytical solution of the reasonable arch axis is obtained, as well as the calculation formula of the arch axis coefficient applicable to different parameter cases. The approximate analytical solution combined with the explicit approximate calculation formula of the arch axis coefficient constitutes a practical calculation method for the reasonable arch axis of the through arch bridge.

[0031] Compared with the prior art, the method for calculating the reasonable arch axis of the through arch bridge has the advantages of fast speed, high efficiency, and high precision. It effectively solves the problem of low efficiency caused by large computational workload in the prior art, and also improves the efficiency and quality in the design and construction process of the through arch bridge, with great practical value. In addition, the method for calculating the reasonable arch axis of the through arch bridge considers the dead loads of the main arch, the bridge deck system, and the suspenders, making the theoretical dead load distribution conform to the actual situation of the bridge. It not only improves the calculation accuracy, but also reveals the essential characteristics between the reasonable arch axis shape and the dead load distribution of the arch bridge.

[0032] The following further illustrates the method for calculating the reasonable arch axis of the through arch bridge with examples: For S10, the steps for S10 to obtain the control differential equation of the reasonable arch axis considering the dead loads of the main arch, the bridge deck system, and the suspenders include the following: S11: Obtain the parameters required for calculation and establish a calculation model; S12: Set up a reasonable arch axis equation and obtain the control differential equation of the reasonable arch axis through the static equilibrium condition; S13: Based on the coordinate system adopted by the calculation model, establish the boundary conditions of the control differential equation.

[0033] Where: As Figure 1 shown, establish the calculation diagram of the through arch bridge with equal cross-section, with the highest point of the main arch as the origin of the rectangular coordinate system O , the abscissa x is positive to the right along the bridge axis, and the ordinate y is positive downward; correspondingly, in the rectangular coordinate system, the two arch springing points of the main arch are respectively A point and B point, the span of the main arch is L , the rise is f , and the rise-span ratio is n = f / L .

[0034] It should be noted that the reasonable arch axis of the main arch is the pressure line of the main arch under the action of dead load. Assuming that the centroid of each cross-section of the main arch is located on the pressure line and the influence of elastic deformation is not considered, the internal force acting on each cross-section of the main arch only has the pressure along the tangent direction of the arch axis, without bending moment, torque and shear force.

[0035] The calculation method of the reasonable arch axis of the through arch bridge described above simultaneously considers the weights of the main arch, the bridge deck system and the suspenders. Specifically: Let the weight per unit length of the main arch along the arch axis be p , and the weight per unit length of the bridge deck system along the horizontal direction be q ; Let the suspenders of the through arch bridge be membranes that only have resistance in the vertical direction. The weight per unit length of the suspenders is φ , and the spacing between adjacent suspenders is d , and obtain the calculation formula for the weight per unit area γ of the membrane simulating the suspenders: ; Let the ratio of the self-weight of the main arch to all the dead loads acting on the main arch be the main arch dead load ratio λ , and obtain the calculation formula for the main arch dead load ratio λ : .

[0036] In addition, for other data required for calculation, they can be obtained through any appropriate means such as consulting the design drawings, and are not elaborated in this invention.

[0037] In S12, set up a reasonable arch axis equation: ; Based on the reasonable arch axis equation, the relationship between the infinitesimal segment of the main arch and the infinitesimal segment of the horizontal coordinate at any point on the main arch is obtained: ; In the formula, ds is the infinitesimal segment of the main arch; dx is ds the corresponding infinitesimal segment of the horizontal coordinate.

[0038] Based on the static equilibrium condition of the infinitesimal segment of the main arch ds in the horizontal direction, the following formula is obtained: ; In the formula, as Figure 2 shown, N is the pressure at any point on the main arch; Based on Equation (5), the horizontal component of the axial pressure of the main arch is obtained as a constant H , and the following formula is obtained: ; Based on the static equilibrium condition of the infinitesimal segment of the main arch ds in the vertical direction, the following formula is obtained: ; Using Equation (6) to change the terms in the left - hand - side parentheses of Equation (7) and obtain the following formula: ; Based on Equation (4) and Equation (8), substituting Equation (4) and Equation (8) into Equation (7), the control differential equation of the reasonable arch axis is obtained: ; In the formula, μ ( x ) is the uniform load intensity acting on the main arch, and is expressed by the following formula μ ( x ): .

[0039] Based on the rectangular coordinate system established by S11, the following boundary conditions are obtained: ; Based on the boundary conditions, the control differential equation of the reasonable arch axis is verified.

[0040] Based on this, S11 completes the preparatory work related to the calculation. In S12, the basic calculation formula is obtained through reasonable assumptions, so as to obtain the required control differential equation through subsequent derivation, and then the control differential equation of the reasonable arch axis is obtained through the static equilibrium condition.

[0041] For S20, S20 obtains a simplified second-order linear ordinary differential equation form of the governing differential equation. Obtaining the analytical solution of the second-order linear ordinary differential equation includes the following steps: S21: Set up the intensity of dead load μ ( x ) linear function; Based on special coordinate points, obtain the expression of the arch axis coefficient, the undetermined parameters of the linear function, and the intensity of dead load μ ( x ) linear expression; S22: Based on the intensity of dead load μ ( x ) linear expression, obtain the simplified form of the second-order linear ordinary differential equation of the governing differential equation; S23: Based on the arch axis coefficient, obtain the solution of the second-order linear ordinary differential equation; Correspondingly, each solution of the second-order linear ordinary differential equation corresponds to a reasonable arch axis equation.

[0042] Where: The governing differential equation obtained in S10 is a second-order constant coefficient nonlinear non-homogeneous ordinary differential equation. In the general case of considering both p, q and γ acting, it is difficult to directly solve this differential equation. The reason for the governing differential equation to be nonlinear is that the μ ( x ) in the expression of the intensity of dead load is a nonlinear term.

[0043] In order to obtain the approximate analytical solution of this nonlinear differential equation, is approximately expressed as y linear function. Correspondingly, the expression of the intensity of dead load μ ( x ) can also be expressed as y linear function. Specifically: Set up the intensity of dead load μ ( x ) linear function: ; In the formula, a , b are undetermined parameters.

[0044] Based on the static equilibrium condition, when x = 0, y = 0, dy / dx = 0, substitute (0, 0) into the expression (10) of the intensity of dead load μ ( x ), and obtain μ ( 0The expression of ( ; Substitute the dead load intensity μ ( x ) into the linear function formula (12) to obtain the expression of the parameter a : ; Based on the rectangular coordinate system, the coordinates of the springing B point are ( L / 2 , f ). Define the ratio of the dead load intensity of the springing B point to that of the crown O as the arch axis coefficient m , and obtain the expression of the arch axis coefficient: ; Based on the linear function formula of the dead load intensity μ ( x ), obtain μ ( 0 ) =a , μ ( L / 2 ) = a + bf ; Substitute μ ( 0 ) =a , μ ( L / 2 ) =a + bf into the expression (14) of the arch axis coefficient to obtain the expression of the parameter b : ; Based on the expression of the parameter a and the expression of the parameter b , substitute equations (13) and (15) into equation (12) to obtain the linear expression of the dead load intensity μ ( x ): .

[0045] Based on equation (16), substitute equation (16) into the control differential equation of the reasonable arch axis to obtain the second-order linear ordinary differential form of the control differential equation: ; In the formula, K is a constant, and the expression of the constant K is as follows: .

[0046] Based on the constant KThe expression for obtaining the solution of the second-order linear ordinary differential equation depends on the arch axis coefficient m The relationship with the constant 1. Accordingly, based on m > 1, m = 1 and m<1 , the second-order linear ordinary differential equation has three solutions; When m=1 , K=0 , combined with the boundary conditions, the solution of the second-order linear ordinary differential equation is obtained as a parabola: ; When m>1 , K>0 , let k 2 = K , the following formula is obtained: ; Combined with the boundary conditions, when x = 0, y = 0, dy / dx = 0, the solution of the second-order linear ordinary differential equation is obtained as: ; In the formula, cosh represents the hyperbolic cosine function; Substitute the boundary condition x = L / 2, y = f into Equation (21), the following formula is obtained: ; In the formula, arch represents the inverse hyperbolic cosine function; The first derivative of y obtained from Equation (21) is:

[0047] In the formula, sinh represents the hyperbolic sine function; When m<1 , K<0 , let , the following formula is obtained: ; Based on Equation (24), the solution of the second-order linear ordinary differential equation is obtained as: ; Substitute the boundary condition x = L / 2, y = f into Equation (25), the following formula is obtained: .

[0048] Based on this, in S21, a linear function formula of the dead load intensity μ ( x ) is established through reasonable assumptions, and then the solution of the control differential equation is obtained. Then, combined with the rectangular coordinate system established in S10, the arch axis coefficient is reasonably introducedm , combined with special points with special coordinates, the undetermined parameters in Equation (12) are obtained to obtain the uniform load intensity μ ( x ) of the linear expression.

[0049] Through formula substitution, using the linear expression of the uniform load intensity μ ( x ) to obtain the simplified form of the second-order linear ordinary differential equation. At the same time, a constant m related to the arch axis coefficient K is introduced. Based on m > 1, m = 1 and m<1 , the second-order linear ordinary differential equation has three solutions, and then the expressions of the three solutions are obtained by using the boundary conditions.

[0050] It is easy to understand that when implementing the calculation method of the reasonable arch axis of the through arch bridge, after obtaining the arch axis coefficient m through the explicit approximate calculation formula of the arch axis coefficient, select the corresponding expression among the three expressions and substitute the arch axis coefficient to obtain the reasonable arch axis equation.

[0051] For S30, based on the explicit approximate calculation formula of the arch axis coefficient, obtaining the arch axis coefficient includes the following steps: S31: Based on the uniform load to divide the working conditions, the analytical solution of the reasonable arch axis corresponding to the working conditions is obtained by solving the control differential equation; S32: Based on the function fitting method, the approximate value corresponding to the working conditions is obtained; the approximate value of the reasonable arch axis under the action of all uniform loads is obtained through weighted average; S33: Based on formula substitution, the explicit approximate calculation formula of the arch axis coefficient is obtained.

[0052] Where: As can be seen from Equation (14), to obtain the value of the arch axis coefficient m , it is necessary to first find out μ ( L / 2 ) and μ ( 0 ); as can be seen from Equation (10), , . Because the parameters p, q, y and f are both known, the key to calculating the arch axis coefficient lies in finding the value of x = L / 2 when . However, in the general load condition considering the simultaneous action of p, q, y , it is difficult to obtain the value of , so the function fitting method is used to obtain the approximate solution.

[0053] Specifically: obtaining the analytical solution of the reasonable arch axis corresponding to the working conditions by solving the control differential equation includes the following steps: Based on the in the dead load intensity expression (10), set up: ; Divide all the dead loads into two simple working conditions. Divide all the dead loads into two simple working conditions. Working condition 1 is to only consider the dead load of the main arch p 's effect, and working condition 2 is to consider the dead load of the bridge deck system q and the dead load of the suspenders γ 's effect; Solve the corresponding control differential equation to obtain the analytical solution of the reasonable arch axis for the corresponding working condition. Among them, the equation of the reasonable arch axis for working condition 1 is the following standard catenary: ; Obtain the derivative dy / dx from equation (28), and substitute it into the approximate calculation formula of the arch axis coefficient to obtain the functional formula: ; In equations (28) and (29), H are undetermined parameters.

[0054] For the parameter H , if the boundary condition x = L / 2, y = f is substituted into equation (28), only a transcendental equation about H can be obtained, and the explicit calculation formula of η(L / 2 cannot be obtained. Therefore, the function fitting method is used to obtain the approximate value of η(L / 2 for working condition 1, that is, based on the boundary condition x = L / 2, y = f , the approximate value of η(L / 2 for working condition 1 is obtained through the function fitting method β 1: ; Similarly, obtain the equation of the reasonable arch axis for working condition 2: .

[0055] Based on the boundary condition x = L / 2, y = f , the approximate value of η(L / 2 for working condition 2 is obtained through the function fitting method β 2: ; Taking the dead load intensities p and q + yf of working condition 1 and working condition 2 respectively as weights, take β 1 and β 2's weighted average and use it as the approximate value of η(L / 2) of the reasonable arch axis under all dead loads: ; Based on Equation (14) and Equation (10), we have:

[0056] Substitute Equation (33) into Equation (34) to obtain the explicit approximate calculation formula for the arch axis coefficient m : .

[0057] Based on this, by combining the function fitting method and the boundary conditions, the explicit approximate calculation formula for the arch axis coefficient m is obtained. When implementing the reasonable arch axis calculation method for the through arch bridge, after the staff prepares the relevant data, they can substitute it into the formula for calculation, greatly improving the calculation efficiency.

[0058] It should be noted that for the derivation of Equation (34), given and the boundary conditions , it is easy to obtain from Equation (10). In addition, since has been obtained previously, the staff can use the substitution method to obtain Equation (34).

[0059] Based on this, this embodiment gives the derivation process of the reasonable arch axis calculation method for the through arch bridge. It should be noted that when implementing the reasonable arch axis calculation method for the through arch bridge, the staff can directly apply Equation (35), that is, calculate the arch axis coefficient m according to the known design parameters, and then substitute the relationship between the arch axis coefficient m and 1 into the corresponding equation to obtain the corresponding arch axis. Thus, the derivation process of the arch axis coefficient m that can be ignored during implementation can be simplified, and the design steps can be simplified to improve the design efficiency.

[0060] Embodiment 2: This embodiment provides a hardware device for implementing the reasonable arch axis calculation method for the through arch bridge described in Embodiment 1, including: The first calculation unit: establish a calculation model to obtain the control differential equation of the reasonable arch axis considering the dead loads of the main arch, the bridge deck system, and the suspenders; The second calculation unit: obtain the approximate analytical solution of the control differential equation to obtain the second-order linear ordinary differential equation of the control differential equation; The third calculation unit: based on the explicit approximate calculation formula of the arch axis coefficient, obtain the arch axis coefficient; The fourth calculation unit: substitute the arch axis coefficient into the approximate analytical solution of the second-order linear ordinary differential equation to obtain the reasonable arch axis equation.

[0061] For the working process, working details and technical effects of the device provided in this embodiment, reference may be made to Embodiment 1, which will not be elaborated herein.

[0062] This embodiment provides a device, which includes a memory, a processor and a transceiver that are communicatively connected in sequence. Among them, the memory is used to store computer programs, the transceiver is used to send and receive messages, and the processor is used to read the computer programs and execute the reasonable arch axis calculation method for the through arch bridge.

[0063] Specifically, the memory may include, but is not limited to, random access memory (RAM), read only memory (ROM), flash memory, first input first output (FIFO) and / or first in last out (FILO), etc.; specifically, the processor may include one or more processing cores, such as a 4-core processor, an 8-core processor, etc. The processor may be implemented in at least one hardware form of DSP (Digital Signal Processing), FPGA (Field-Programmable Gate Array), PLA (Programmable Logic Array). At the same time, the processor may also include a main processor and a coprocessor. The main processor is a processor used to process data in the wake state, also known as the CPU (Central Processing Unit); the coprocessor is a low-power processor used to process data in the standby state.

[0064] In some embodiments, the processor may integrate a GPU (Graphics Processing Unit), which is responsible for rendering and drawing the content to be displayed on the display screen. For example, the processor may be, but is not limited to, a microprocessor of the STM32F105 series, a reduced instruction set computer (RISC) microprocessor, an X86 architecture processor, or a processor integrated with an embedded neural-network processing unit (NPU); the transceiver may be, but is not limited to, a Wi-Fi wireless transceiver, a Bluetooth wireless transceiver, a General Packet Radio Service (GPRS) wireless transceiver, a ZigBee (low-power local area network protocol based on the IEEE802.15.4 standard) wireless transceiver, a 3G transceiver, a 4G transceiver, and / or a 5G transceiver, etc. In addition, the device may also include, but is not limited to, a power module, a display screen, and other necessary components.

[0065] For the working process, working details, and technical effects of the device provided in this embodiment, reference may be made to Embodiment 1, which will not be elaborated herein.

[0066] This embodiment provides a storage medium storing the reasonable arch axis calculation method for the through arch bridge described in Embodiment 1, that is, instructions are stored on the storage medium, and when the instructions run on a computer, the reasonable arch axis calculation method for the through arch bridge is executed.

[0067] Among them, the storage medium refers to a carrier for storing data, which may include, but is not limited to, a floppy disk, an optical disc, a hard disk, a flash memory, a USB flash drive, and / or a Memory Stick, etc. The computer may be a general-purpose computer, a special-purpose computer, a computer network, or other programmable devices.

[0068] For the working process, working details, and technical effects of the storage medium provided in this embodiment, reference may be made to Embodiment 1, which will not be elaborated herein.

[0069] This embodiment provides a computer program product containing instructions, which, when running on a computer, causes the computer to execute the reasonable arch axis calculation method for the through arch bridge. Among them, the computer may be a general-purpose computer, a special-purpose computer, a computer network, or other programmable devices.

[0070] The specific embodiments described above further elaborate on the object, technical solution and beneficial effects of the present invention. It should be understood that the above description is only the specific embodiments of the present invention and is not used to limit the protection scope of the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principle of the present invention shall be included in the protection scope of the present invention.

Claims

1. A calculation method for a reasonable arch axis of a through arch bridge, characterized in that, It includes the following steps: Establish a calculation model to obtain the control differential equation of the reasonable arch axis considering the dead loads of the main arch, the bridge deck system and the suspenders; Obtain the simplified second-order linear ordinary differential equation of the control differential equation and find the approximate analytical solution of the control differential equation; Based on the explicit approximate calculation formula of the arch axis coefficient, obtain the arch axis coefficient; Substitute the arch axis coefficient into the approximate analytical solution of the second-order linear ordinary differential equation to obtain the reasonable arch axis equation.

2. The calculation method of the reasonable arch axis of the through arch bridge according to claim 1, characterized in that, Obtaining the control differential equation of the reasonable arch axis considering the dead loads of the main arch, the bridge deck system and the suspenders includes: Obtain the parameters required for calculation and establish a calculation model; Set up the reasonable arch axis equation and obtain the control differential equation of the reasonable arch axis through the static equilibrium condition; Based on the coordinate system adopted by the calculation model, establish the boundary conditions of the control differential equation.

3. The method for calculating the reasonable arch axis of a through arch bridge according to claim 2, characterized in that, The calculation model includes: Establish a calculation diagram for a through arch bridge with a constant cross-section, taking the highest point of the main arch as the origin of the rectangular coordinate system O , the abscissa x is positive to the right along the bridge axis, and the ordinate y is positive downward; Accordingly, in the rectangular coordinate system, the two springing points of the main arch are respectively A point and B point, the span of the main arch is L , the rise is f , and the rise-to-span ratio is n = f / L .

4. The reasonable arch axis calculation method of the through arch bridge according to claim 3, characterized in that, The calculation parameters include: Let the weight per unit length of the main arch along the arch axis be p , and the weight per unit length of the bridge deck system in the horizontal direction be q ; Let the suspenders of the through arch bridge be membranes that only have resistance in the vertical direction, and the weight per unit length of the suspenders be φ , the spacing between adjacent suspenders be d , and obtain the calculation formula for the weight per unit area of the membrane simulating the suspenders γ : ； Let the ratio of the self-weight of the main arch to all the dead loads acting on the main arch be the dead load ratio of the main arch λ , and obtain the calculation formula for the dead load ratio of the main arch λ : 。 5. The calculation method of the reasonable arch axis of the through arch bridge according to claim 4, characterized in that The control differential equation of the reasonable arch axis includes: Set up the reasonable arch axis equation: ； Based on the reasonable arch axis equation, obtain the relationship between the main arch micro-segment and the horizontal coordinate micro-segment at any point on the main arch: ； In the formula, ds is the infinitesimal segment of the main arch; dx is ds the corresponding infinitesimal segment of the horizontal coordinate.

6. The calculation method of the reasonable arch axis of the through arch bridge according to claim 5, characterized in that, Obtaining the control differential equation of the reasonable arch axis through the static equilibrium condition includes: Based on the static equilibrium condition of a micro-segment of the main arch ds in the horizontal direction, the following equation is obtained: ； In the formula, N is the pressure at any point of the main arch; Based on Equation (5), the horizontal component of the axial pressure of the main arch is obtained as a constant H , and the following equation is obtained: ； Based on the static equilibrium condition of the micro-segment of the main arch ds The following formula is obtained based on the static equilibrium condition in the vertical direction: ； Use Equation (6) to change the terms in the left bracket of Equation (7) and obtain the following equation: ； Based on Equation (4) and Equation (8), substitute Equation (4) and Equation (8) into Equation (7) to obtain the control differential equation of the reasonable arch axis: ； In the formula, μ ( x ) is the dead load intensity acting on the main arch, and is expressed by the following formula μ ( x ): 。 7. The calculation method of the reasonable arch axis of the through arch bridge according to claim 6, characterized in that, Establishing the boundary conditions of the control differential equation includes: Based on the coordinate system adopted by the calculation model, obtain the following boundary conditions: 。 8. The rational arch axis calculation method of the through arch bridge according to claim 6 or 7, characterized in that Obtain the simplified second-order linear ordinary differential equation form of the control differential equation and find the analytical solution of the second-order linear ordinary differential equation includes: Set the uniform load intensity μ ( x ) linear function; Based on special coordinate points, obtain the expression of the arch axis coefficient, the undetermined parameters of the linear function, and the uniform load intensity μ ( x ) linear expression Based on the uniform load intensity μ ( x ) a simplified form of the second-order linear ordinary differential equation for the governing differential equation is obtained according to the linear expression; Based on the arch axis coefficient, find the solution of the second-order linear ordinary differential equation; Correspondingly, each solution of the second-order linear ordinary differential equation corresponds to a reasonable arch axis equation.

9. The method for calculating the reasonable arch axis of a through arch bridge according to claim 8, characterized in that, Obtain the expression of the arch axis coefficient, the undetermined parameters of the linear function formula, and the dead load intensity μ ( x ) The linear expressions include: Set the uniform load intensity μ ( x ) linear function formula: ； wherein, a and b are parameters to be determined; Based on the static equilibrium condition, when x = 0, y = 0, dy / dx = 0, substituting (0, 0) into the expression (10) of the dead load intensity μ ( x ) to obtain the expression of μ ( 0 ): ; Substitute the dead load intensity μ into the linear function formula (12) of x to obtain the expression of parameter a : ； Based on the rectangular coordinate system, the coordinates of the springing points are obtained. B The coordinates of the point are ( L / 2 , f ). The ratio of the dead load intensity of the springing B point to that of the crown O is defined as the arch axis coefficient m , and the expression of the arch axis coefficient is obtained: ； Based on the linear function of the uniform dead load intensity μ ( x ), obtain μ ( 0 ); =a , μ ( L / 2 ); = a + bf Substitute μ ( 0 ) =a , μ ( L / 2 ) = a + bf into the expression (14) of the arch axis coefficient to obtain the expression of the parameter b : ； Based on the parameter a 's expression and the parameter b 's expression, substitute Equation (13) and Equation (15) into Equation (12) to obtain the linear expression of the dead load intensity μ ( x ) 。 10. The rational arch axis calculation method of the through arch bridge according to claim 9, characterized in that Based on the uniform load intensity μ ( x ) The simplified form of the second-order linear ordinary differential equation for obtaining the control differential equation includes: Based on Equation (16), substitute Equation (16) into the control differential equation of the reasonable arch axis to obtain the second-order linear ordinary differential form of the control differential equation: ； In the formula, K is a constant, and the constant K has the following expression: 。 11. The method for calculating the reasonable arch axis of a through arch bridge according to claim 10, wherein Based on the arch axis coefficient, find the solution of the second-order linear ordinary differential equation includes: Based on a constant K , obtaining the solution of a second-order linear ordinary differential equation depends on the relationship between the arch axis coefficient m and the constant 1. Correspondingly, based on m > 1, m = 1 and m<1 , the second-order linear ordinary differential equation has three solutions; When m=1 then K=0 and combining with the boundary conditions, the solution of the second-order linear ordinary differential equation is a parabola: ； When m>1 , K>0 , let k 2 = K , and the following equation is obtained: ； Combined with the boundary conditions, when x = 0, y = 0, dy / dx = 0, the solution of the second-order linear ordinary differential equation is obtained as follows: ； Where, cosh represents the hyperbolic cosine function; Substitute the boundary conditions x = L / 2, y = f into Equation (21) to obtain the following equation: ； Where, arch represents the inverse hyperbolic cosine function; Obtained from Equation (21) y The first derivative of is as follows: Where, sinh represents the hyperbolic sine function; When m<1 time K<0 , let , the following formula is obtained: ； Based on Equation (24), the solution of the second-order linear ordinary differential equation is obtained as: ； Substitute the boundary conditions x = L / 2, y = f into Equation (25), and the following equation is obtained: 。 12. The calculation method of the reasonable arch axis of the through arch bridge according to claim 11, characterized in that, Based on the explicit approximate calculation formula of the arch axis coefficient, obtaining the arch axis coefficient includes: Based on the dead load to divide the working conditions, obtain the analytical solution of the reasonable arch axis corresponding to the working conditions by solving the control differential equation; Based on the function fitting method, obtain the approximate value corresponding to the working conditions; obtain the approximate value of the reasonable arch axis under the action of all dead loads through weighted average; Based on the formula substitution, obtain the explicit approximate calculation formula of the arch axis coefficient.

13. The calculation method of the reasonable arch axis of the through arch bridge according to claim 12, characterized in that, Obtaining the analytical solution of the reasonable arch axis corresponding to the working conditions by solving the control differential equation includes: Based on in the dead load intensity expression (10), set up: ； All the dead loads are divided into two simple working conditions. Working condition 1 only considers the dead load of the main arch p acting, and working condition 2 considers the dead load of the bridge deck system q and the dead load of the suspenders γ acting; Obtain the analytical solution of the reasonable arch axis corresponding to the working conditions by solving the corresponding control differential equation, where the reasonable arch axis equation of the first working condition is the following standard catenary: ； The derivative is obtained from Equation (28). dy / dx and substituted into the approximate calculation formula of the arch axis coefficient to obtain the function formula of the main arch: ； In Equations (28) and (29), H are parameters to be determined; Similarly, the reasonable arch axis equation for working condition 2 is obtained: 。 14. The rational arch axis calculation method of the through arch bridge according to claim 13, characterized in that, The analytical solutions for the reasonable arch axes corresponding to the working conditions include: Based on boundary conditions x = L / 2, y = f , the approximate value of η(L / 2 under working condition 1 is obtained by the function fitting method β 1: ； Based on the boundary conditions x = L / 2, y = f , the approximate value of η(L / 2 under working condition 2 is obtained by the function fitting method β 2: ； Taking the dead load intensity of each of working condition 1 and working condition 2 p and q + γf as weights, taking β 1 and β 2's weighted average value and using it as the η(L / 2) approximate value of the reasonable arch axis under all dead loads: ； The explicit approximate calculation formulas for the arch axis coefficient include: Based on Equation (14) and Equation (10), we have: Substitute Equation (33) into Equation (34) to obtain the explicit approximate calculation formula for the arch axis coefficient m : 。

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