A calculation method for the reasonable arch axis of through arch bridges

Abstract

The present invention relates to the technical field of bridge engineering construction and discloses a method for calculating the reasonable arch axis of a bottom-supported arch bridge. The method comprises the following steps: establishing a calculation model to obtain a control differential equation of a reasonable arch axis that considers the dead loads of the main arch, the bridge system, and the hanger; obtaining an approximate analytical solution of the control differential equation to obtain a second-order linear ordinary differential equation of the control differential equation; obtaining the arch axis coefficient based on an explicit approximate calculation formula of the arch axis coefficient; substituting the arch axis coefficient into the approximate analytical solution of the second-order linear ordinary differential equation to obtain a reasonable arch axis equation. The method for calculating the reasonable arch axis of a bottom-supported arch bridge has the advantages of high speed, high efficiency, and high precision. It considers the dead loads of the main arch, the bridge system, and the hanger, so that the theoretical dead load distribution conforms to the actual situation of the bridge, improves the accuracy of the calculation, and reveals the essential characteristics between the reasonable arch axis line shape and the dead load distribution of the arch bridge.

Description

Technical Field

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

[0002] The selection of the arch axis is crucial in arch bridge design. Its linear shape directly influences the distribution and magnitude of internal forces within the main arch section. The rationality of the arch axis significantly impacts the economic efficiency, safety, and durability of the arch bridge. Analytical methods are commonly used to determine the optimal arch axis. Specifically, based on the principle of static equilibrium of a statically determinate arch, the governing differential equation for the constant-load pressure line is constructed. The resulting pressure line is then used as the optimal arch axis.

[0003] However, existing techniques suffer from the following drawbacks: Existing analytical methods for determining the ideal arch axis for through-arch bridges generally lack explicit function arch axis equations, resulting in high computational workload and low efficiency. Furthermore, existing techniques fail to consider the effect of hanger weight on the ideal arch axis, resulting in dead load distribution that is inconsistent with actual bridge conditions, affecting calculation accuracy and making it difficult to reveal the essential relationship between the ideal arch axis shape and the arch bridge's dead load distribution. Summary of the Invention

[0004] The technical problem to be solved by the present invention is that the existing analytical method for the reasonable arch axis of a bottom-supported arch bridge has a large calculation workload, low efficiency and does not consider the influence of the weight of the hanger on the reasonable arch axis. The purpose is to provide a method for calculating the reasonable arch axis of a bottom-supported arch bridge to solve the above problems.

[0005] The present invention is achieved through the following technical solutions:

[0006] A method for calculating a reasonable arch axis of a through arch bridge comprises the following steps:

[0007] A computational model was established to obtain the governing differential equation of the reasonable arch axis taking into account the dead loads of the main arch, the bridge system, and the suspenders.

[0008] Obtain a simplified second-order linear ordinary differential equation of the governing differential equation and obtain an approximate analytical solution of the governing differential equation;

[0009] Based on the explicit approximate calculation formula of the arch axis coefficient, the arch axis coefficient is obtained;

[0010] The arch axis coefficient is substituted into the approximate analytical solution of the second-order linear ordinary differential equation to obtain the reasonable arch axis equation.

[0011] In one possible design, the governing differential equations for obtaining a reasonable arch axis considering the dead loads of the main arch, the bridge system, and the suspenders include:

[0012] Obtain the parameters required for calculation and establish a calculation model;

[0013] Establish a reasonable arch axis equation and obtain the governing differential equation of the reasonable arch axis through static equilibrium conditions;

[0014] Based on the coordinate system adopted by the computational model, the boundary conditions of the governing differential equations are established.

[0015] In one possible design, the computational model includes:

[0016] Establish a calculation diagram for a uniform cross-section through-arch bridge, taking the highest point of the main arch as the origin of the rectangular coordinate system. O , the horizontal axis x Taking the right direction along the bridge as positive, the vertical coordinate y downward is positive;

[0017] Correspondingly, in the rectangular coordinate system, the arch foot points at both ends of the main arch are A Dot and B point, the main arch span is L , Yadaka is f , the span ratio is n=f / L .

[0018] In one possible design, the calculated parameters include:

[0019] Let the weight per unit length of the main arch along the arch axis be p , the unit length weight of the bridge system in the horizontal direction is q ;

[0020] The hanger of the through arch bridge is assumed to be a membrane with resistance only in the vertical direction, and the unit length weight of the hanger is f The spacing between adjacent booms is d , obtain the unit area weight of the membrane simulating the hanger c The calculation formula is:

[0021] ;

[0022] The ratio of the main arch dead weight to the total dead load acting on the main arch is the main arch dead load ratio. l , obtain the main arch dead load ratio l The calculation formula is:

[0023] .

[0024] In one possible design, the governing differential equations for the reasonable arch axis include:

[0025] Establish a reasonable arch axis equation:

[0026] ;

[0027] Based on the reasonable arch axis equation, the relationship between the main arch micro-segment and the horizontal coordinate micro-segment at any point on the main arch is obtained:

[0028] ;

[0029] Where, ds It is the main arch micro segment; dx for ds The corresponding horizontal coordinate micro segment.

[0030] In one possible design, the governing differential equations for obtaining a reasonable arch axis through static equilibrium conditions include:

[0031] Based on the main arch micro segment ds Under the static equilibrium condition in the horizontal direction, the following formula is obtained:

[0032] ;

[0033] Where, N is the pressure at any point of the main arch;

[0034] Based on formula (5), the horizontal component of the main arch axial pressure is obtained as a constant H , and obtain the following formula:

[0035] ;

[0036] Based on the main arch micro segment ds Under the static equilibrium condition in the vertical direction, the following formula is obtained:

[0037] ;

[0038] Using formula (6), we can change the brackets on the left side of formula (7) and obtain the following formula:

[0039] ;

[0040] Based on equations (4) and (8), substituting equations (4) and (8) into equation (7), we can obtain the governing differential equation of the reasonable arch axis:

[0041] ;

[0042] Where, m ( x ) is the concentration of dead load acting on the main arch and is expressed as follows: m ( x ):

[0043] .

[0044] In one possible design, establishing the boundary conditions for the governing differential equations includes: based on the coordinate system used in the computational model, obtaining the boundary conditions as follows:

[0045] .

[0046] In one possible design, an approximate analytical solution to the governing differential equation is obtained to obtain a simplified second-order linear ordinary differential equation form of the governing differential equation. Obtaining an analytical solution to the second-order linear ordinary differential equation includes:

[0047] Establishing a dead load concentration m ( x ) linear function; based on special coordinate points, the expression of the arch axis coefficient, the unknown parameters of the linear function and the concentration of the dead load are obtained. m ( x )’s linear expression;

[0048] Based on the concentration of dead load m ( x ) to obtain the simplified form of the second-order linear ordinary differential equation governing the differential equation;

[0049] Based on the arch axis coefficient, the solution of the second-order linear ordinary differential equation is obtained;

[0050] Accordingly, each solution of the second-order linear ordinary differential equation corresponds to a reasonable arch axis equation.

[0051] In a possible design, the expression of the arch axis coefficient, the unknown parameters of the linear function and the concentration of the dead load are obtained. m ( x ) include:

[0052] Establishing a dead load concentration m ( x ) is a linear function:

[0053] ;

[0054] Where, a 、 b is a parameter to be determined;

[0055] Based on the static equilibrium condition, when x =0, y =0, dy / dx =0, substitute (0,0) into the constant load concentration m ( x ) expression (10), we get m ( 0 ) expression: ;

[0056] Substitute the constant load concentration m ( x ) of the linear function (12), and obtain the parameters a The expression:

[0057] ;

[0058] Based on the rectangular coordinate system, obtain the arch foot B The coordinates of the point are ( L / 2 , f ), the arch foot B Points and Vaults O The ratio of the dead load concentration is defined as the arch axis coefficient m , we can get the expression of arch axis coefficient:

[0059] ;

[0060] Based on the concentration of dead load m ( x ), we can obtain m ( 0 ) =a , m ( L / 2 ) =a+bf ;Will m ( 0 ) =a , m ( L / 2 ) =a +bf Substituting into the expression (14) of the arch axis coefficient, we obtain the parameter b The expression:

[0061] ;

[0062] Parameter-based a Expressions and parameters b Substituting equations (13) and (15) into equation (12), we can obtain the constant load concentration m ( x ) is the linear expression:

[0063] .

[0064] In one possible design, based on the dead load concentration m ( x ), the simplified form of the second-order linear ordinary differential equation of the governing differential equation is obtained as follows:

[0065] Based on formula (16), substitute formula (16) into the governing differential equation of the reasonable arch axis to obtain the second-order linear ordinary differential form of the governing differential equation:

[0066] ;

[0067] Where, K is a constant, and the constant K The expression is as follows:

[0068] .

[0069] In one possible design, the solution to the second-order linear ordinary differential equation based on the arch axis coefficient includes:

[0070] Based on constant K The expression of the second-order linear ordinary differential equation depends on the arch axis coefficient m The relationship with the constant 1, accordingly, is based on m>1、m=1 and m<1 , the second-order linear ordinary differential equation has three solutions;

[0071] when m=1 hour, K=0 , combined with the boundary conditions, the solution of the second-order linear ordinary differential equation is a parabola:

[0072] ;

[0073] when m>1 hour, K>0 ,make k 2 = K , and obtain the following formula:

[0074] ;

[0075] Combined with the boundary conditions, when x =0, y =0, dy / dx =0, the solution of the second-order linear ordinary differential equation is:

[0076] ;

[0077] Where cosh represents the hyperbolic cosine function;

[0078] The boundary conditions x=L / 2,y=f Substituting into formula (21), we obtain the following formula:

[0079] ;

[0080] Where, arch represents the inverse hyperbolic cosine function;

[0081] From formula (21), we can get y The first-order derivative of is:

[0082]

[0083] Where sinh represents the hyperbolic sine function;

[0084] when m<1 hour, K<0 ,make , and obtain the following formula:

[0085] ;

[0086] Based on formula (24), the solution of the second-order linear ordinary differential equation is obtained as:

[0087] ;

[0088] The boundary conditions x=L / 2,y=f Substituting into formula (25), we obtain the following formula:

[0089] .

[0090] In one possible design, based on the explicit approximate calculation formula of the arch axis coefficient, the arch axis coefficient is obtained as follows:

[0091] Based on the dead load division working condition, the analytical solution of the reasonable arch axis corresponding to the working condition is obtained by solving the control differential equation;

[0092] Based on the function fitting method, the approximate value of the corresponding working condition is obtained; the approximate value of the reasonable arch axis under the action of all dead loads is obtained by weighted average;

[0093] Based on formula substitution, the explicit approximate calculation formula of the arch axis coefficient is obtained.

[0094] In one possible design, the analytical solution of the reasonable arch axis corresponding to the working condition is obtained by solving the governing differential equations, including:

[0095] Based on the constant load concentration expression (10) ,set up:

[0096] ;

[0097] The entire dead load is divided into two simple working conditions. Working condition 1 is to consider only the dead load of the main arch. p The second working condition is to consider the dead load of the bridge system. q and boom dead load c The role of;

[0098] The analytical solution of the reasonable arch axis of the corresponding 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:

[0099] ;

[0100] The derivative is obtained from formula (28): dy / dx , and substitute it into the approximate calculation formula of the arch axis coefficient to obtain the main arch Functional:

[0101] ;

[0102] In formula (28) and formula (29), H is a parameter to be determined;

[0103] Similarly, the reasonable arch axis equation of working condition 2 is obtained:

[0104] .

[0105] In one possible design, the analytical solution for obtaining the reasonable arch axis corresponding to the working condition includes:

[0106] Based on boundary conditions x=L / 2,y=f , the working condition 1 is obtained by function fitting method the (L / 2) ) β 1:

[0107] ;

[0108] Based on boundary conditions x=L / 2,y=f , the second working condition is obtained by function fitting method the (L / 2) ) β 2:

[0109] ;

[0110] The constant load concentration of working condition 1 and working condition 2 p and q+yf As weight, take β 1 and β 2 and take the weighted average value as the reasonable arch axis under all dead loads the (L / 2) Approximate value of :

[0111] ;

[0112] Explicit approximate calculation formulas for obtaining the arch axis coefficient include:

[0113] Based on formula (14) and formula (10), we can get:

[0114]

[0115] Substituting equation (33) into equation (34), we can obtain the arch axis coefficient m The explicit approximate calculation formula of is:

[0116] .

[0117] Compared with the prior art, the present invention has the following advantages and beneficial effects:

[0118] The proposed method for calculating the reasonable arch axis of a through-arch bridge boasts the advantages of speed, efficiency, and precision. It effectively addresses the inefficiency inherent in existing techniques due to the high computational workload. It also improves the efficiency and quality of through-arch bridge design and construction, thus possessing significant practical value. Furthermore, the proposed method considers the dead loads of the main arch, the bridgeway system, and the suspenders, ensuring that the theoretical dead load distribution aligns with the actual bridge conditions. This not only improves calculation accuracy but also reveals the essential relationship between the reasonable arch axis shape and the dead load distribution of arch bridges. BRIEF DESCRIPTION OF THE DRAWINGS

[0119] In order to more clearly illustrate the technical solutions of the exemplary embodiments of the present invention, the following briefly introduces the drawings required for use in the examples. It should be understood that the following drawings only illustrate certain embodiments of the present invention and should not be considered as limiting the scope. A person of ordinary skill in the art can also derive other relevant drawings based on these drawings without inventive effort. In the drawings:

[0120] Figure 1 The figure is a flow chart of a method for calculating the reasonable arch axis of a through arch bridge.

[0121] Figure 2 This is a schematic diagram of the equal section of a through arch bridge.

[0122] Figure 3 Schematic diagram of static equilibrium of a micro segment. DETAILED DESCRIPTION

[0123] In order to make the objectives, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with examples and drawings. The exemplary 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.

[0124] In the following description, numerous specific details are set forth to provide a thorough understanding of the present invention. However, it will be apparent to one skilled in the art that these specific details are not necessarily required to practice the present invention. In other embodiments, well-known structures, circuits, materials, or methods are not described in detail to avoid obscuring the present invention.

[0125] Throughout this specification, references to "one embodiment," "an embodiment," "an example," or "an example" mean that a particular feature, structure, or characteristic described in connection with the embodiment or example is included in at least one embodiment of the present invention. Therefore, appearances of the phrases "one embodiment," "an embodiment," "an example," or "an example" in various places throughout this specification are not necessarily all referring to the same embodiment or example. Furthermore, the particular features, structures, or characteristics may be combined in one or more embodiments or examples in any suitable combinations and / or subcombinations. Furthermore, it will be understood by those of ordinary skill in the art that the figures provided herein are for illustrative purposes only and are not necessarily drawn to scale. As used herein, the term "and / or" includes any and all combinations of one or more of the associated listed items.

[0126] In the description of the present invention, the terms "front", "back", "left", "right", "up", "down", "vertical", "horizontal", "high", "low", "inside", "outside", etc., indicating directions or positional relationships, are based on the directions or positional relationships shown in the accompanying drawings. They are only for the convenience of describing the present invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific direction, be constructed and operated in a specific direction. Therefore, they should not be understood as limiting the scope of protection of the present invention.

[0127] Example 1:

[0128] like Figure 1-Figure 3 As shown, a method for calculating the reasonable arch axis of a through arch bridge includes the following steps:

[0129] S10: Establish a computational model to obtain the governing differential equation of the reasonable arch axis taking into account the dead loads of the main arch, the bridge system, and the suspenders;

[0130] S20: Obtain a simplified second-order linear ordinary differential equation of the governing differential equation and obtain an approximate analytical solution of the governing differential equation;

[0131] S30: Based on the explicit approximate calculation formula of the arch axis coefficient, the arch axis coefficient is obtained;

[0132] S40: Substitute the arch axis coefficient into the approximate analytical solution of the second-order linear ordinary differential equation to obtain a reasonable arch axis equation.

[0133] The method for calculating the reasonable arch axis of a bottom-supported arch bridge is proposed for the design of bottom-supported arch bridges. A governing differential equation of the reasonable arch axis is established considering the combined effects of the constant loads of the main arch, the bridgeway system and the hanger. The governing differential equation is solved using reasonable approximate assumptions, and a high-precision approximate analytical solution of the reasonable arch axis and a calculation formula for the arch axis coefficient applicable to different parameter conditions are obtained. 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 a bottom-supported arch bridge.

[0134] Compared to existing technologies, the proposed method for calculating the reasonable arch axis for through-the-arch bridges offers the advantages of speed, efficiency, and precision. It effectively addresses the inefficiency inherent in existing technologies due to the high computational workload. It also improves the efficiency and quality of through-the-arch bridge design and construction, thus possessing significant practical value. Furthermore, the proposed method considers the dead loads of the main arch, the bridgeway system, and the suspenders, ensuring that the theoretical dead load distribution aligns with the actual bridge conditions. This not only improves calculation accuracy but also reveals the essential relationship between the reasonable arch axis shape and the dead load distribution of arch bridges.

[0135] The following is a further explanation of the calculation method of the reasonable arch axis of the through arch bridge with reference to an example:

[0136] For S10, the control differential equation of the reasonable arch axis considering the dead load of the main arch, the bridge system and the suspender includes the following steps:

[0137] S11: Obtain the parameters required for calculation and establish a calculation model;

[0138] S12: Establish a reasonable arch axis equation and obtain the governing differential equation of the reasonable arch axis through static equilibrium conditions;

[0139] S13: Based on the coordinate system used in the computational model, establish the boundary conditions of the governing differential equations.

[0140] in:

[0141] like Figure 1 As shown in the figure, the calculation diagram of the uniform cross-section through-arch bridge is established, with the highest point of the main arch as the origin of the rectangular coordinate system. O , the horizontal axis x Taking the right direction along the bridge as positive, the vertical coordinate y Downward is positive; correspondingly, in the rectangular coordinate system, the arch foot points at both ends of the main arch are A Dot and B point, the main arch span is L , Yadaka is f , the span ratio is n=f / L .

[0142] It is worth noting that the rational arch axis of the main arch is the pressure line of the main arch under dead load. Assuming that the centroid of each section of the main arch lies on this pressure line and that elastic deformation is not considered, the internal force acting on each section of the main arch is only pressure along the tangent direction of the arch axis, with no bending moment, torque, or shear force.

[0143] The method for calculating the reasonable arch axis of a through arch bridge considers the weight of the main arch, the bridge system, and the suspenders. Specifically:

[0144] Let the weight per unit length of the main arch along the arch axis be p , the unit length weight of the bridge system in the horizontal direction is q ;

[0145] The hanger of the through arch bridge is assumed to be a membrane with resistance only in the vertical direction, and the unit length weight of the hanger is f The spacing between adjacent booms is d , obtain the unit area weight of the membrane simulating the hanger c The calculation formula is:

[0146] ;

[0147] The ratio of the main arch dead weight to the total dead load acting on the main arch is the main arch dead load ratio. l , obtain the main arch dead load ratio l The calculation formula is:

[0148] .

[0149] In addition, other data required for calculation can be obtained by consulting design drawings or any other suitable method, and the present invention will not elaborate on them here.

[0150] In S12, a reasonable arch axis equation is established:

[0151] ;

[0152] Based on the reasonable arch axis equation, the relationship between the main arch micro-segment and the horizontal coordinate micro-segment at any point on the main arch is obtained:

[0153] ;

[0154] Where, ds It is the main arch micro segment; dx for ds The corresponding horizontal coordinate micro segment.

[0155] Based on the main arch micro segment ds Under the static equilibrium condition in the horizontal direction, the following formula is obtained:

[0156] ;

[0157] In the formula, Figure 2 As shown, N is the pressure at any point of the main arch;

[0158] Based on formula (5), the horizontal component of the main arch axial pressure is obtained as a constant H , and obtain the following formula:

[0159] ;

[0160] Based on the main arch micro segment ds Under the static equilibrium condition in the vertical direction, the following formula is obtained:

[0161] ;

[0162] Using formula (6), we can change the brackets on the left side of formula (7) and obtain the following formula:

[0163] ;

[0164] Based on equations (4) and (8), substituting equations (4) and (8) into equation (7), we can obtain the governing differential equation of the reasonable arch axis:

[0165] ;

[0166] Where, m ( x ) is the concentration of dead load acting on the main arch and is expressed as follows: m ( x ):

[0167] .

[0168] Based on the rectangular coordinate system established by S11, the boundary conditions are as follows:

[0169] ;

[0170] Based on the boundary conditions, the governing differential equations of the reasonable arch axis are verified.

[0171] Based on this, S11 completes the calculation-related preparatory work. In S12, the basic calculation formula is obtained through reasonable assumptions, so that the required control differential equation can be obtained through subsequent derivation, and then the control differential equation of the reasonable arch axis is obtained through static equilibrium conditions.

[0172] For S20, S20 obtains the 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:

[0173] S21: Establishing Dead Load Concentration m ( x ) linear function; based on special coordinate points, the expression of the arch axis coefficient, the unknown parameters of the linear function and the concentration of the dead load are obtained. m ( x )’s linear expression;

[0174] S22: Based on the concentration of dead load m ( x ) to obtain the simplified form of the second-order linear ordinary differential equation governing the differential equation;

[0175] S23: Obtain solutions to second-order linear ordinary differential equations based on the arch axis coefficient;

[0176] Accordingly, each solution of the second-order linear ordinary differential equation corresponds to a reasonable arch axis equation.

[0177] in:

[0178] The control differential equation obtained in S10 is a second-order nonlinear nonhomogeneous ordinary differential equation with constant coefficients. p、q and c In general, the differential equation is difficult to solve directly. The reason why the control differential equation becomes nonlinear is that the constant load concentration m ( x ) in the expression is a nonlinear term.

[0179] In order to obtain an approximate analytical solution to the nonlinear differential equation, Approximately expressed as y A linear function of m ( x ) can also be expressed as y A linear function of . Specifically:

[0180] Establishing a dead load concentration m ( x ) is a linear function:

[0181] ;

[0182] Where, a 、 b To be determined parameters.

[0183] Based on the static equilibrium condition, when x =0, y =0, dy / dx =0, substitute (0,0) into the constant load concentration m ( x ) expression (10), we get m ( 0 ) expression: ;

[0184] Substitute the constant load concentration m ( x ) of the linear function (12), and obtain the parameters a The expression:

[0185] ;

[0186] Based on the rectangular coordinate system, obtain the arch footB The coordinates of the point are ( L / 2 , f ), will arch the foot B Points and Vaults O The ratio of the dead load concentration is defined as the arch axis coefficient m , we can get the expression of arch axis coefficient:

[0187] ;

[0188] Based on the concentration of dead load m ( x ), we can obtain m ( 0 ) =a , m ( L / 2 ) =a+bf ;Will m ( 0 ) =a , m ( L / 2 ) =a +bf Substituting into the expression (14) of the arch axis coefficient, we obtain the parameter b The expression:

[0189] ;

[0190] Parameter-based a Expressions and parameters b Substituting equations (13) and (15) into equation (12), we can obtain the constant load concentration m ( x ) is the linear expression:

[0191] .

[0192] Based on formula (16), substitute formula (16) into the governing differential equation of the reasonable arch axis to obtain the second-order linear ordinary differential form of the governing differential equation:

[0193] ;

[0194] Where, K is a constant, and the constant K The expression is as follows:

[0195] .

[0196] Based on constant K The expression of the second-order linear ordinary differential equation depends on the arch axis coefficient m The relationship with the constant 1, accordingly, is based on m>1、m=1 and m<1, the second-order linear ordinary differential equation has three solutions;

[0197] when m=1 hour, K=0 , combined with the boundary conditions, the solution of the second-order linear ordinary differential equation is a parabola:

[0198] ;

[0199] when m>1 hour, K>0 ,make k 2 = K , and obtain the following formula:

[0200] ;

[0201] Combined with the boundary conditions, when x =0, y =0, dy / dx =0, the solution of the second-order linear ordinary differential equation is:

[0202] ;

[0203] Where cosh represents the hyperbolic cosine function;

[0204] The boundary conditions x=L / 2,y=f Substituting into formula (21), we obtain the following formula:

[0205] ;

[0206] Where, arch represents the inverse hyperbolic cosine function;

[0207] From formula (21), we can get y The first-order derivative of is:

[0208]

[0209] Where sinh represents the hyperbolic sine function;

[0210] when m<1 hour, K<0 ,make , and obtain the following formula:

[0211] ;

[0212] Based on formula (24), the solution of the second-order linear ordinary differential equation is obtained as:

[0213] ;

[0214] The boundary conditions x=L / 2,y=f Substituting into formula (25), we obtain the following formula:

[0215] .

[0216] Based on this, S21 establishes the dead load concentration through reasonable assumptions. m ( x ) linear function, and then find the solution of the control differential equation. Combined with the rectangular coordinate system established in S10, the arch axis coefficient is reasonably introduced m , combined with the special points with special coordinates to obtain the unknown parameters in formula (12), and obtain the constant load concentration m ( x ) is a linear expression of .

[0217] By replacing the formula, using the constant load concentration m ( x ) is obtained by using the linear expression of the second-order linear ordinary differential equation. At the same time, the arch axis coefficient is introduced m Related constants K ,based on m>1、m=1 and m<1 , the second-order linear ordinary differential equation has three solutions, and then the boundary conditions are used to obtain the expressions of the three solutions.

[0218] It is easy to understand that when the method for calculating the reasonable arch axis of a through arch bridge is implemented, the arch axis coefficient is obtained by the explicit approximate calculation formula of the arch axis coefficient. m Finally, select the corresponding expression from the three expressions and substitute it into the arch axis coefficient to obtain the reasonable arch axis equation.

[0219] For S30, S30 is based on an explicit approximate calculation formula for the arch axis coefficient. The following steps are involved in obtaining the arch axis coefficient:

[0220] S31: Based on the dead load division working condition, the analytical solution of the reasonable arch axis corresponding to the working condition is obtained by solving the control differential equation;

[0221] S32: Based on the function fitting method, the approximate value of the corresponding working condition is obtained; the approximate value of the reasonable arch axis under the action of all dead loads is obtained through weighted average;

[0222] S33: Based on formula substitution, an explicit approximate calculation formula for the arch axis coefficient is obtained.

[0223] in:

[0224] From formula (14), we can know that to obtain the arch axis coefficient m The value of m ( L / 2 )and m ( 0 ); From formula (10), we can see that , Because the parameter p、q、y and f are all known, so the key to calculating the arch axis coefficient is to find x=L / 2 hour But at the same time, p、q、y Under normal load conditions, it is difficult to obtain Therefore, the function fitting method is used to obtain the approximate solution.

[0225] Specifically, the analytical solution of the reasonable arch axis corresponding to the working condition by solving the governing differential equation includes the following steps:

[0226] Based on the constant load concentration expression (10) ,set up:

[0227] ;

[0228] The whole dead load is divided into two simple working conditions. The whole dead load is divided into two simple working conditions. Working condition one is to consider only the dead load of the main arch. p The second working condition is to consider the dead load of the bridge system. q and boom dead load c The role of;

[0229] The analytical solution of the reasonable arch axis of the corresponding 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:

[0230] ;

[0231] The derivative is obtained from formula (28): dy / dx , and substitute it into the approximate calculation formula of the arch axis coefficient to obtain the main arch Functional:

[0232] ;

[0233] In formula (28) and formula (29), H To be determined parameters.

[0234] For parameters H , if the boundary conditions x=L / 2,y=f Substituting into formula (28) we can only get a H The transcendental equation of the (L / 2) ). Therefore, the function fitting method is used to obtain the working condition 1 the (L / 2) ), that is, based on the boundary conditions x=L / 2,y=f , the working condition 1 is obtained by function fitting method the (L / 2) ) β 1:

[0235] ;

[0236] Similarly, the reasonable arch axis equation of working condition 2 is obtained:

[0237] .

[0238] Based on boundary conditions x=L / 2,y=f , the second working condition is obtained by function fitting method the (L / 2) ) β 2:

[0239] ;

[0240] The constant load concentration of working condition 1 and working condition 2 p and q+yf As weight, take β 1 and β 2 and take the weighted average value as the reasonable arch axis under all dead loads the (L / 2) Approximate value of :

[0241] ;

[0242] Based on formula (14) and formula (10), we can get:

[0243]

[0244] Substituting equation (33) into equation (34), we can obtain the arch axis coefficient m The explicit approximate calculation formula of is:

[0245] .

[0246] Based on this, the arch axis coefficient is obtained by combining the function fitting method with the boundary conditions. m When implementing the above-mentioned calculation method for the reasonable arch axis of a through arch bridge, the staff can prepare the relevant data and then insert it into the formula for calculation, which greatly improves the calculation efficiency.

[0247] It is worth noting that for the solution of formula (34), it is known that and boundary conditions , it is easy to get from formula (10) In addition, as mentioned above , the staff can use the substitution method to obtain formula (34).

[0248] Based on this, this embodiment provides the derivation process of the method for calculating the reasonable arch axis of the bottom-supported arch bridge. It is worth noting that when the method for calculating the reasonable arch axis of the bottom-supported arch bridge is implemented, the staff can directly apply formula (35), that is, calculate the arch axis coefficient based on the known design parameters. m , and then according to the arch axis coefficientm Substitute the relationship with 1 into the corresponding equation to obtain the corresponding arch axis. Therefore, the arch axis coefficient can be ignored during implementation. m The derivation process simplifies the design steps and improves the design efficiency.

[0249] Example 2:

[0250] This embodiment provides a hardware device for implementing the method for calculating the reasonable arch axis of a through arch bridge described in Example 1, including:

[0251] The first calculation unit: establish a calculation model to obtain the governing differential equation of the reasonable arch axis considering the dead load of the main arch, the bridge system and the suspenders;

[0252] The second computational unit: obtains the approximate analytical solution of the governing differential equation and obtains the second-order linear ordinary differential equation of the governing differential equation;

[0253] The third calculation unit: based on the explicit approximate calculation formula of the arch axis coefficient, the arch axis coefficient is obtained;

[0254] The fourth calculation unit: Substitute the arch axis coefficient into the approximate analytical solution of the second-order linear ordinary differential equation to obtain a reasonable arch axis equation.

[0255] The working process, working details and technical effects of the device provided in this embodiment can be found in Example 1 and will not be described in detail here.

[0256] This embodiment provides a device including a memory, a processor and a transceiver that are communicatively connected in sequence, wherein 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 program and execute the method for calculating the reasonable arch axis of a bottom-supported arch bridge.

[0257] For example, the memory may include, but is not limited to, random access memory (RAM), read-only memory (ROM), flash memory, first-in first-out memory (FIFO), and / or first-in last-out memory (FILO). Specifically, the processor may include one or more processing cores, such as a quad-core processor or an octa-core processor. The processor may be implemented in at least one of the following hardware forms: a DSP (Digital Signal Processing), an FPGA (Field-Programmable Gate Array), or a PLA (Programmable Logic Array). Furthermore, the processor may include a main processor and a coprocessor. The main processor is a processor for processing data in an awake state, also known as a CPU (Central Processing Unit); the coprocessor is a low-power processor for processing data in a standby state.

[0258] In some embodiments, the processor may be integrated with a GPU (Graphics Processing Unit), which is responsible for rendering and drawing the content required 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 with an integrated embedded neural network processing unit (NPU); the transceiver may be, but is not limited to, a Wireless Fidelity (WIFI) wireless transceiver, a Bluetooth wireless transceiver, a General Packet Radio Service (GPRS) wireless transceiver, a ZigBee protocol (a low-power local area network protocol based on the IEEE802.15.4 standard, ZigBee) wireless transceiver, a 3G transceiver, a 4G transceiver, and / or a 5G transceiver. In addition, the device may also include, but is not limited to, a power module, a display screen, and other necessary components.

[0259] The working process, working details and technical effects of the equipment provided in this embodiment can be found in Example 1 and will not be described in detail here.

[0260] This embodiment provides a storage medium that stores the method for calculating the reasonable arch axis of a bottom-supported arch bridge described in Example 1, that is, the storage medium stores instructions, and when the instructions are run on a computer, the method for calculating the reasonable arch axis of a bottom-supported arch bridge is executed.

[0261] The storage medium refers to a carrier for storing data, which may include but is not limited to a floppy disk, an optical disk, a hard disk, a flash memory, a USB flash drive and / or a memory stick, and the computer may be a general-purpose computer, a special-purpose computer, a computer network, or other programmable device.

[0262] The working process, working details and technical effects of the storage medium provided in this embodiment can be found in Example 1 and will not be described in detail here.

[0263] This embodiment provides a computer program product containing instructions that, when executed on a computer, cause the computer to execute the method for calculating a reasonable arch axis of a through arch bridge. The computer may be a general-purpose computer, a special-purpose computer, a computer network, or other programmable device.

[0264] The specific implementation methods described above further illustrate the objectives, technical solutions and beneficial effects of the present invention in detail. It should be understood that the above description is only a specific implementation method of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A method for calculating the reasonable arch axis of a through arch bridge, characterized in that: The following steps are involved: A computational model was established to obtain the governing differential equation of the reasonable arch axis taking into account the dead loads of the main arch, the bridge system, and the suspenders. Obtain a simplified second-order linear ordinary differential equation of the governing differential equation and obtain an approximate analytical solution of the governing differential equation; Based on the explicit approximate calculation formula of the arch axis coefficient, the arch axis coefficient is obtained; Substituting the arch axis coefficient into the approximate analytical solution of the second-order linear ordinary differential equation, we can obtain the reasonable arch axis equation. The governing differential equations for obtaining the reasonable arch axis considering the dead loads of the main arch, the bridge system, and the suspenders include: Obtain the parameters required for calculation and establish a calculation model; Establish a reasonable arch axis equation and obtain the governing differential equation of the reasonable arch axis through static equilibrium conditions; Establish the boundary conditions of the governing differential equations based on the coordinate system used in the computational model; The computational model includes: Establish a calculation diagram for a uniform cross-section through-arch bridge, taking the highest point of the main arch as the origin of the rectangular coordinate system. O , the horizontal axis x Taking the right direction along the bridge as positive, the vertical coordinate y downward is positive; Correspondingly, in the rectangular coordinate system, the arch foot points at both ends of the main arch are A Dot and B point, the main arch span is L , Yadaka is f , the span ratio is n=f / L ; Calculation parameters include: Let the weight per unit length of the main arch along the arch axis be p , the unit length weight of the bridge system in the horizontal direction is q ; The hanger of the through arch bridge is assumed to be a membrane with resistance only in the vertical direction, and the unit length weight of the hanger is φ The spacing between adjacent booms is d , obtain the unit area weight of the membrane simulating the hanger γ The calculation formula is: ； The ratio of the main arch dead weight to the total dead load acting on the main arch is the main arch dead load ratio. λ , obtain the main arch dead load ratio λ The calculation formula is: ； The governing differential equations of the reasonable arch axis include: Establish a reasonable arch axis equation: ； Based on the reasonable arch axis equation, the relationship between the main arch micro-segment and the horizontal coordinate micro-segment at any point on the main arch is obtained: ； Where, ds It is the main arch micro segment; dx for ds The corresponding horizontal coordinate micro segment; The governing differential equations for obtaining a reasonable arch axis through static equilibrium conditions include: Based on the main arch micro segment ds Under the static equilibrium condition in the horizontal direction, the following formula is obtained: ； Where, N is the pressure at any point of the main arch; Based on formula (5), the horizontal component of the main arch axial pressure is obtained as a constant H , and obtain the following formula: ； Based on the main arch micro segment ds Under the static equilibrium condition in the vertical direction, the following formula is obtained: ； Using formula (6), we can change the brackets on the left side of formula (7) and obtain the following formula: ； Based on equations (4) and (8), substituting equations (4) and (8) into equation (7), we can obtain the governing differential equation of the reasonable arch axis: ； Where, μ ( x ) is the concentration of dead load acting on the main arch and is expressed as follows: μ ( x ): 。 2. The method for calculating the reasonable arch axis of a through arch bridge according to claim 1 is characterized in that: The boundary conditions for establishing the governing differential equations include: Based on the coordinate system used in the computational model, the boundary conditions are obtained as follows: 。 3. The method for calculating the reasonable arch axis of a through arch bridge according to claim 1 or 2, characterized in that: Obtaining the simplified second-order linear ordinary differential equation form of the governing differential equation and obtaining the analytical solution of the second-order linear ordinary differential equation includes: Establishing a dead load concentration μ ( x ) linear function; based on special coordinate points, the expression of the arch axis coefficient, the unknown parameters of the linear function and the concentration of the dead load are obtained. μ ( x )’s linear expression; Based on the concentration of dead load μ ( x ) to obtain the simplified form of the second-order linear ordinary differential equation governing the differential equation; Based on the arch axis coefficient, the solution of the second-order linear ordinary differential equation is obtained; Accordingly, each solution of the second-order linear ordinary differential equation corresponds to a reasonable arch axis equation.

4. The method for calculating the reasonable arch axis of a through arch bridge according to claim 3 is characterized in that: Obtain the expression of the arch axis coefficient, the undetermined parameters of the linear function and the concentration of the dead load μ ( x ) include: Establishing a dead load concentration μ ( x ) is a linear function: ； Where, a 、 b is a parameter to be determined; Based on the static equilibrium condition, when x =0, y =0, dy / dx =0, substitute (0,0) into the constant load concentration μ ( x ) expression (10), we get μ ( 0 ) expression: ; Substitute the constant load concentration μ ( x ) of the linear function (12), and obtain the parameters a The expression: ； Based on the rectangular coordinate system, obtain the arch foot B The coordinates of the point are ( L / 2 , f ), will arch the foot B Points and Vaults O The ratio of the dead load concentration is defined as the arch axis coefficient m , we can get the expression of arch axis coefficient: ； Based on the concentration of dead load μ ( x ), we can obtain μ ( 0 ) =a , μ ( L / 2 ) =a+bf ;Will μ ( 0 ) =a , μ ( L / 2 ) =a+bf Substituting into the expression (14) of the arch axis coefficient, we obtain the parameter b The expression: ； Parameter-based a Expressions and parameters b Substituting equations (13) and (15) into equation (12), we can obtain the constant load concentration μ ( x ) is the linear expression: 。 5. The method for calculating the reasonable arch axis of a through arch bridge according to claim 4 is characterized in that: Based on the concentration of dead load μ ( x ), the simplified form of the second-order linear ordinary differential equation of the governing differential equation is obtained as follows: Based on formula (16), substitute formula (16) into the governing differential equation of the reasonable arch axis to obtain the second-order linear ordinary differential form of the governing differential equation: ； Where, K is a constant, and the constant K The expression is as follows: 。 6. The method for calculating the reasonable arch axis of a through arch bridge according to claim 5 is characterized in that: Based on the arch axis coefficient, the solutions of the second-order linear ordinary differential equations include: Based on constant K The expression of the second-order linear ordinary differential equation depends on the arch axis coefficient m The relationship with the constant 1, accordingly, is based on m>1、m=1 and m<1 , the second-order linear ordinary differential equation has three solutions; when m=1 hour, K=0 , combined with the boundary conditions, the solution of the second-order linear ordinary differential equation is a parabola: ； when m>1 hour, K>0 ,make k 2 = K , and obtain the following formula: ； Combined with the boundary conditions, when x =0, y =0, dy / dx =0, the solution of the second-order linear ordinary differential equation is: ； Where cosh represents the hyperbolic cosine function; The boundary conditions x=L / 2,y=f Substituting into formula (21), we obtain the following formula: ； Where, arch represents the inverse hyperbolic cosine function; From formula (21), we can get y The first-order derivative of is: Where sinh represents the hyperbolic sine function; when m<1 hour, K<0 ,make , and obtain the following formula: ； Based on formula (24), the solution of the second-order linear ordinary differential equation is obtained as: ； The boundary conditions x=L / 2,y=f Substituting into formula (25), we obtain the following formula: 。 7. The method for calculating the reasonable arch axis of a through arch bridge according to claim 6, characterized in that: Based on the explicit approximate calculation formula of the arch axis coefficient, the arch axis coefficient is obtained including: Based on the dead load division working condition, the analytical solution of the reasonable arch axis corresponding to the working condition is obtained by solving the control differential equation; Based on the function fitting method, the approximate value of the corresponding 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.

8. The method for calculating the reasonable arch axis of a through arch bridge according to claim 7 is characterized in that: The analytical solutions to the reasonable arch axis corresponding to the working condition obtained by solving the governing differential equations include: Based on the constant load concentration expression (10) ,set up: ； The entire dead load is divided into two simple working conditions. Working condition 1 is to consider only the dead load of the main arch. p The second working condition is to consider the dead load of the bridge system. q and boom dead load γ The role of; The analytical solution of the reasonable arch axis of the corresponding 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 is obtained from formula (28): dy / dx , and substitute it into the approximate calculation formula of the arch axis coefficient to obtain the main arch Functional: ； In formula (28) and formula (29), H is a parameter to be determined; Similarly, the reasonable arch axis equation of working condition 2 is obtained: 。 9. The method for calculating the reasonable arch axis of a through arch bridge according to claim 8, characterized in that: The analytical solutions for obtaining the reasonable arch axis corresponding to the working condition include: Based on boundary conditions x=L / 2,y=f , the working condition 1 is obtained by function fitting method η(L / 2 ) β 1: ； Based on boundary conditions x=L / 2,y=f , the second working condition is obtained by function fitting method η(L / 2 ) β 2: ； The constant load concentration of working condition 1 and working condition 2 p and q+γf As weight, take β 1 and β 2 and take the weighted average value as the reasonable arch axis under all dead loads η(L / 2) Approximate value of : ； Explicit approximate calculation formulas for obtaining the arch axis coefficient include: Based on formula (14) and formula (10), we can get: Substituting equation (33) into equation (34), we can obtain the arch axis coefficient m The explicit approximate calculation formula of is: 。

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