Method for judging anti-overturning stability of single-column-pier curve beam bridge

Abstract

The invention relates to a method for judging the anti-overturning stability of a single-column-pier curve beam bridge, and the method comprises the following steps: S1, calculating the anti-overturning stability effect of a bridge under the dead load action according to the basic principle of structural mechanics on the basis of the mechanical analysis of the displacement of a simply supported statically indeterminate curve beam and the reaction force of a support; s2, on the basis of mechanical analysis of simply supported statically indeterminate curve beam displacement and support reaction force, according to the basic principle of structural mechanics, the overturning instability effect of the bridge under the live load effect is calculated; s3, calculating the anti-overturning stability coefficient of the bridge according to the calculation results of the dead load anti-overturning stability effect and the live load overturning instability effect; and S4, based on the calculation result of the anti-overturning stability coefficient of the single-column-pier curve beam bridge, judging the anti-overturning stability of the bridge according to a standard threshold value. According to the method, the efficiency of calculating and judging the anti-overturning stability of the single-column-pier curve beam bridge can be improved, the popularization of a bridge health monitoring technology is facilitated, and the method has the advantages of simplicity, high efficiency and rapidness.

Description

Technical Field

[0001] This invention relates to bridge health monitoring technology, specifically a method for determining the overturning stability of a single-column pier curved beam bridge. Background Technology

[0002] The superstructure of a single-column pier curved beam bridge typically uses an integral box girder, while the piers in the substructure usually use single-column piers. Its structure is lightweight and aesthetically pleasing, which can meet the needs of the overall road alignment design. In addition, the substructure occupies a small area, which can increase the usable space under the bridge. Therefore, it is widely used in urban bridges and highway ramp bridges.

[0003] Single-column pier curved beam bridges typically feature double supports on the abutments or side pier tops, and single supports on the middle pier tops. This makes it difficult to provide effective torsional restraint across the cross-section. When vehicles travel along the convex side of the curved beam bridge, the side piers are prone to experiencing compression on the convex support and detachment of the concave support, meaning the concave support is a potential failure point, thus posing a risk of lateral overturning to the entire bridge. In recent years, numerous lateral overturning accidents have occurred in single-column pier curved beam bridges, causing significant direct and indirect economic losses and various adverse effects. Therefore, there is an urgent need for in-depth analysis and calculation of the overturning stability of single-column curved beam bridges to facilitate the implementation and promotion of bridge health monitoring technologies.

[0004] The classic method for calculating the bending-torsional coupling of curved beams combines the structural equilibrium equations, physical equations, and geometric equations to construct the displacement-load equations of the curved beam. Solving these equations allows for the determination of the forces and displacements within the curved beam. However, applying this method to analyze the overturning stability of a single-column pier curved beam bridge requires solving a system of high-order differential equations, which is extremely difficult and tedious. Therefore, the classic method for calculating the bending-torsional coupling of curved beams has limited practical application in engineering and hinders the promotion of bridge health monitoring technology. Summary of the Invention

[0005] The purpose of this invention is to provide a method for determining the overturning stability of a single-column pier curved beam bridge. This method is based on the mechanical analysis of displacement and support reactions of a simply supported, statically indeterminate curved beam. According to the basic principles of structural mechanics, it calculates the overturning stability effect under dead load and the overturning instability effect under live load on a single-column pier curved beam bridge. Then, it calculates the bridge's overturning stability coefficient and determines the bridge's overturning stability based on the result of the overturning stability coefficient. Compared with classical bending-torsional coupling calculation methods and conventional finite element methods, this invention improves the efficiency of calculating and determining the overturning stability of single-column pier curved beam bridges, facilitates the promotion of bridge health monitoring technology, and has the advantages of simplicity, efficiency, and speed.

[0006] To achieve the above objectives, the present invention provides the following technical solution: A method for determining the overturning stability of a single-column pier curved beam bridge includes the following steps: S1: Based on the mechanical analysis of displacement and support reaction of simply supported statically indeterminate curved beams, the anti-overturning stability effect of single-column pier curved beam bridges under dead load is calculated according to the basic principles of structural mechanics. S2: Based on the mechanical analysis of displacement and support reaction of simply supported statically indeterminate curved beams, the overturning instability effect of single-column pier curved beam bridge under live load is calculated according to the basic principles of structural mechanics. S3: Based on the calculation results of the dead load anti-overturning stability effect described in S1 and the live load overturning instability effect described in S2, the anti-overturning stability coefficient of the single-column pier curved beam bridge is calculated. S4: Based on the calculation results of the overturning stability coefficient of the single-column pier curved beam bridge in S3, the overturning stability of the bridge is judged according to the threshold of the specification.

[0007] In step S1, based on the fundamental principles of structural mechanics, the calculation of the overturning stability effect of a single-column pier curved beam bridge under dead load is performed. The curved beam of the superstructure is represented by its planar axis, on which a uniformly distributed dead load as required by the specifications is applied. q 1. Establishing a single-column pier curved beam bridge under uniformly distributed dead load q The basic model M1 of the statically indeterminate structure under load is based on the fundamental principle of the statically indeterminate structural force method in structural mechanics, with a simply supported statically indeterminate curved beam as the basic structure U. Using the basic structure U as a reference, redundant constraints of the basic model M1 are removed, and the corresponding constraint reactions are applied. X 1. X 2. Replace the model with a basic model M1 and use the force method to solve for the corresponding basic system Q1. Based on the basic principles of structural mechanics, the basic system Q1 is equivalent to the basic model M1. X 1. X 2 represents the fundamental unknowns for solving the basic model M1 using the force method. Based on the displacement analysis results of simply supported statically indeterminate curved beams in structural mechanics, the relevant compliance coefficients and free terms for solving the basic model M1 using the force method are calculated. The fundamental equations for solving the basic model M1 using the force method are established according to the deformation compatibility principle, and the fundamental unknowns are obtained by solving these equations. X 1. X 2. Based on the analysis results of the support reactions of simply supported statically indeterminate curved beams in structural mechanics, the basic structure U under dead load is calculated respectively. q 1. Redundant constraints X 1. X 2. Support reactions of potentially failing supports A2 and D2 under load; the superposition principle is used to calculate the basic model M1 under dead load. qThe support reactions of potentially failing supports A2 and D2 under load 1; based on the pure torsional analysis theory of curved beam bridges, the anti-overturning stability effect of a single-column pier curved beam bridge under dead load is calculated.

[0008] As a further improvement to the calculation method for the overturning stability of a single-column curved beam bridge, in step S1, based on the basic principles of the statically indeterminate structural force method in structural mechanics and the displacement analysis results of simply supported statically indeterminate curved beams in structural mechanics, the relevant compliance coefficients of the basic model M1, solved by the force method, are calculated. d 11 , d 22 , d 12 and d 21 They are respectively: in, d 11 To simulate unit load Acting on the basic structure U X Position 1 X The displacement generated at position 1 d 22 To simulate unit load Acting on the basic structure U X At position 2 X The displacement generated at position 2 d 12 To simulate unit load Acting on the basic structure U X At position 2 X The displacement generated at position 1 d 21 To simulate unit load Acting on the basic structure U X Position 1 X The displacement generated at position 2; E The elastic modulus of the curved beam material. I Let be the moment of inertia of the curved beam cross section. k This is the ratio of the bending stiffness to the torsional stiffness of the curved beam's cross-section. k=EI / GI d , G The shear modulus of the curved beam material. I d Let be the torsional moment of inertia of the cross section of the curved beam. R Let be the radius of curvature of the curved beam. f 0 is the central radian angle of the curved beam. f 1 represents the radian angle between the left end support line A1A2 of the bridge and the left middle pier support B. f 2 represents the radian angle between the center of the left middle pier support B and the right middle pier support C of the bridge. f 3 represents the arc angle between the right middle pier support C and the right end support line D1D2 of the bridge.

[0009] As a further improvement to the calculation method for the overturning stability of a single-column curved beam bridge, in step S1, based on the basic principles of the statically indeterminate structural force method in structural mechanics and the displacement analysis results of simply supported statically indeterminate curved beams in structural mechanics, the relevant free terms of the basic model M1 are calculated using the force method. D 1p , D 2p They are respectively: in, D 1p For uniformly distributed constant load q 1. Acting on the basic structure U X The displacement generated at position 1 D 2p For uniformly distributed constant load q 1. Acting on the basic structure U X The displacement generated at position 2.

[0010] As a further improvement to the calculation method for the overturning stability of a single-column curved beam bridge, in step S1, the basic equations for solving the basic model M1 using the force method based on the deformation compatibility principle are as follows: As a further improvement to the calculation method for the overturning stability of a single-column curved beam bridge, in step S1, the basic equations of the force method for the basic model M1 are solved to obtain the basic unknowns of the force method solution for the basic model M1. X 1. X 2 are respectively: As a further improvement to the calculation method for the overturning stability of a single-column curved beam bridge, in step S1, the basic model M1 is calculated based on the superposition principle under uniformly distributed dead load. q Support reaction force at the potential failure location A2 under action 1 F A21 Support reaction force at the potential failure support D2 location F D21 They are respectively: in, , , Uniformly distributed dead loads q 1. Redundant constraints X 1. Redundant constraints X The support reaction force of the basic structure U under the action of 2, which indicates potential failure of the support A2. , , Uniformly distributed dead loads q 1. Redundant constraints X 1. Redundant constraints X The support reaction force at support D2 of the basic structure U under potential failure under load 2. Calculated based on the analysis results of support reactions of simply supported statically indeterminate curved beams in structural mechanics. , , and , , They are respectively: in, L 1 represents the distance between supports A1 and A2 on the left support line A1A2. L 2 represents the distance between supports D1 and D2 on the right-end support line D1D2.

[0011] As a further improvement to the calculation method for the overturning stability of a single-column curved beam bridge, in step S1, the overturning stability effect of a single-column pier curved beam bridge under dead load is calculated based on the pure torsional analysis theory of curved beam bridges. G 1 is: In step S2, the overturning instability effect of a single-column pier curved beam bridge under live load is calculated based on the basic principles of structural mechanics. The curved beam of the superstructure is represented by its planar axis, and the live loads required by the specifications, including concentrated forces, act on it. P 2. Concentrated torque T 2. Uniformly distributed force q 2. Uniformly distributed torque t2. Establish a basic statically indeterminate structural model M2 of the single-column pier curved beam bridge under live load. Based on the fundamental principles of the statically indeterminate structural force method in structural mechanics, a simply supported statically indeterminate curved beam is used as the basic structure U. Using the basic structure U as a reference, the redundant constraints of the basic model M2 are removed, and the corresponding redundant constraint reactions are used. X 3. X 4. Replace the model with the basic model M2 and use the force method to solve for the corresponding basic system Q2. Based on the basic principles of structural mechanics, the basic system Q2 is equivalent to the basic model M2. X 3. X 4 represents the fundamental unknowns for solving the basic model using the M2 force method. Based on the displacement analysis results of simply supported statically indeterminate curved beams in structural mechanics, the relevant compliance coefficients and free terms for solving the basic model using the M2 force method are calculated. The fundamental equations for solving the basic model using the M2 force method are established based on the deformation compatibility principle, and the fundamental unknowns are obtained by solving these equations. X 3. X 4. Based on the analysis results of the support reactions of simply supported statically indeterminate curved beams in structural mechanics, the basic structure U is calculated under various live loads and redundant constraint forces. X 3. X 4. The support reaction forces of potentially failed supports A2 and D2 under live load are calculated using the superposition principle. The overturning instability effect of a single-column pier curved beam bridge under live load is calculated based on the pure torsion analysis theory of curved beam bridges.

[0012] As a further improvement to the calculation method for the overturning stability of a single-column curved beam bridge, in step S2, based on the basic principles of the statically indeterminate structural force method in structural mechanics and the displacement analysis results of simply supported statically indeterminate curved beams in structural mechanics, the relevant compliance coefficients of the basic model solved by the M2 force method are calculated. d 33 , d 44 , d 34 and d 43 They are respectively: in, d 33 To simulate unit load Acting on the basic structure U X At position 3 X The displacement generated at position 3. d 44 To simulate unit load Acting on the basic structure UX At position 4 X The displacement generated at position 4 d 34 To simulate unit load Acting on the basic structure U X At position 4 X The displacement generated at position 3 d 43 To simulate unit load Acting on the basic structure U X At position 3 X The displacement generated at position 4.

[0013] As a further improvement to the calculation method for the overturning stability of a single-column curved beam bridge, in step S2, based on the basic principles of the statically indeterminate structural force method and the superposition principle in structural mechanics, the relevant free terms of the basic model M2 force method are calculated. D 3p , D 4p They are respectively: in, D 3p For live loads acting on the basic structure U X The displacement generated at position 3. D 4p For live loads acting on the basic structure U X The displacement generated at position 4 , , , Concentrated live load force P 2. Concentrated torque T 2. Uniformly distributed force q 2. Uniformly distributed torque t Under the action of 2, the basic structure U is subject to redundant constraint forces. X The displacement generated at position 3. , , , Concentrated live load force P 2. Concentrated torque T 2. Uniformly distributed force q 2. Uniformly distributed torque t Under the action of 2, the basic structure U is subject to redundant constraint forces. X The displacement at position 4 is calculated based on the displacement analysis results of a simply supported statically indeterminate curved beam in structural mechanics. , , , and , , , They are respectively: As a further improvement to the calculation method for the overturning stability of a single-column curved beam bridge, in step S2, the basic equations for solving the basic model using the M2 force method based on the deformation compatibility principle are as follows: As a further improvement to the calculation method for the overturning stability of a single-column curved beam bridge, in step S2, the basic equations of the force method for the basic model M2 are solved to obtain the basic unknowns of the force method solution for the basic model M2. X 3. X 4 are respectively: As a further improvement to the calculation method for the overturning stability of a single-column curved beam bridge, in step S2, the support reaction force of the potentially failing support A2 of the basic model M2 under live load is calculated based on the superposition principle. F A22 Support reaction force of potentially failed support D2 F D22 They are respectively: in, , , , , , Concentrated live load force P 2. Concentrated torque T 2. Uniformly distributed force q 2. Uniformly distributed torque t 2. Redundant constraints X 3 and X The support reaction force of the basic structure U under the action of potential failure support A2. , , , , , Concentrated live load force P 2. Concentrated torque T 2. Uniformly distributed force q 2. Uniformly distributed torque t 2. Redundant constraints X 3 and X The support reaction force of the basic structure U at potential failure support D2 under load 4. Calculated based on the analysis results of the support reaction force of a simply supported statically indeterminate curved beam in structural mechanics. , , , , , and , , , , , They are respectively: As a further improvement to the calculation method for the overturning stability of a single-column curved girder bridge, in step S2, the overturning instability effect of a single-column pier curved girder bridge under live load is calculated based on the pure torsional analysis theory of curved girder bridges. G 2 is: As a further improvement to the calculation method for the overturning stability of a single-column curved girder bridge, in step S3, based on the pure torsional analysis theory of curved girder bridges and according to the calculation results of the overturning stability effect under dead load and the overturning instability effect under live load, the overturning stability coefficient of the single-column pier curved girder bridge is calculated. G 1 / G 2 is: As a further improvement to the calculation method for the overturning stability of a single-column curved beam bridge, in step S4, based on the calculation results of the overturning stability coefficient of the single-column pier curved beam bridge, and according to the requirements of the current standard, when the overturning stability coefficient of the single-column pier curved beam bridge... G 1 / G When 2 ≥ 2.5, it can be determined that the bridge will not overturn or become unstable; when G 1 / G When 2 < 2.5, it can be determined that the bridge will overturn and become unstable. Attached Figure Description

[0014] Figure 1 This is a flowchart of a method for determining the overturning stability of a single-column pier curved beam bridge.

[0015] Figure 2 M1 is the basic model of a statically indeterminate structure of a single-column pier curved beam bridge under dead load.

[0016] Figure 3 The basic structure U is solved by the force method for the basic model M1 and the basic model M2.

[0017] Figure 4 The force method is used to solve for the basic system Q1 corresponding to the basic model M1.

[0018] Figure 5 This is the force decomposition diagram of the basic system Q1.

[0019] Figure 6 M2 is the basic model of a statically indeterminate structure of a single-column pier curved beam bridge under live load.

[0020] Figure 7 The basic system Q2 is solved using the force method of the basic model M2.

[0021] Figure 8 This is the force decomposition diagram of the basic system Q2. Detailed Implementation

[0022] The following is a unified explanation of the parameters appearing in this manual: q 1. Uniformly distributed dead loads acting on single-column pier curved beam bridges according to specifications; M1, a single-column pier curved beam bridge under uniformly distributed dead load q Basic model of a statically indeterminate structure under the action of 1; U. Basic structure of simply supported statically indeterminate curved beams solved by the force method for single-column pier curved beam bridges; Q1. The basic system for solving the force method for dead load on a single-column pier curved beam bridge; X 1. In the basic system Q1, remove the redundant constraint of support B and replace the corresponding constraint reaction force; X 2. In the basic system Q1, remove the redundant constraints at support C and replace them with the corresponding constraint reactions; d 11 False unit load Acting on the basic structure U X Position 1 X The displacement generated at position 1; d 22 False unit load Acting on the basic structure U X At position 2 X The displacement generated at position 2; d 12 False unit load Acting on the basic structure U X At position 2 X The displacement generated at position 1; d 21 False unit load Acting on the basic structure U X Position 1 X The displacement generated at position 2; D 1p Uniformly distributed constant load q 1. Acting on the basic structure U X The displacement generated at position 1; D 2p Uniformly distributed constant load q 1. Acting on the basic structure U X The displacement generated at position 2; F A21 Basic model M1 under uniformly distributed constant load q Support reaction force of potentially failing support A2 under action 1; F D21 Basic model M1 under uniformly distributed constant load q Support reaction force of potentially failing support D2 under action 1; Uniformly distributed constant load q Support reaction force of support A2 of basic structure U under action of 1, which is at potential failure. Redundant constraints X Support reaction force of support A2 of basic structure U under action of 1, which is at potential failure. Redundant constraintsX Support reaction force of support A2 of basic structure U under potential failure under action 2; Uniformly distributed constant load q Support reaction force of support D2 of basic structure U under action of 1, which may fail; Redundant constraints X Support reaction force of support D2 of basic structure U under action of 1, which may fail; Redundant constraints X Support reaction force of support D2 of basic structure U under potential failure under action 2; G 1. Overturning stability effect of a single-column pier curved beam bridge under dead load; L 1. The distance between supports A1 and A2 on the left support line A1A2; L 2. The distance between supports D1 and D2 on the right end support line D1D2; P 2. Concentrated live load forces acting according to specifications on single-column pier curved beam bridges; T 2. Concentrated torque of live load acting according to specifications on a single-column pier curved beam bridge; q 2. Uniformly distributed live load force acting according to specifications on a single-column pier curved beam bridge; t 2. Uniformly distributed torque of live load acting on a single-column pier curved beam bridge according to specifications; M2. Basic model of a statically indeterminate structure of a single-column pier curved beam bridge under live load; Q2. The basic system for solving the force method for live load on a single-column pier curved beam bridge; X 3. In the basic system Q2, remove the redundant constraint at support B and replace the corresponding constraint reaction force; X 4. In the basic system Q2, remove the redundant constraints at support C and replace the corresponding constraint reactions; d 33 False unit load Acting on the basic structure U X At position 3 X The displacement generated at position 3; d 44 False unit load Acting on the basic structure U X At position 4 XThe displacement generated at position 4; d 34 False unit load Acting on the basic structure U X At position 4 X The displacement generated at position 3; d 43 False unit load Acting on the basic structure U X At position 3 X The displacement generated at position 4.

[0023] D 3p Live loads acting on the basic structure U X The displacement generated at position 3; D 4p Live loads acting on the basic structure U X The displacement generated at position 4; Live load concentration force P Under the action of 2, the basic structure U is subject to redundant constraint forces. X The displacement generated at position 3; Concentrated torque of live load T Under the action of 2, the basic structure U is subject to redundant constraint forces. X The displacement generated at position 3; Uniformly distributed live load q Under the action of 2, the basic structure U is subject to redundant constraint forces. X The displacement generated at position 3; Live load uniformly distributed torque t Under the action of 2, the basic structure U is subject to redundant constraint forces. X The displacement generated at position 3; Live load concentration force P Under the action of 2, the basic structure U is subject to redundant constraint forces. X The displacement generated at position 4; Concentrated torque of live load T Under the action of 2, the basic structure U is subject to redundant constraint forces. X The displacement generated at position 4; Uniformly distributed live load q Under the action of 2, the basic structure U is subject to redundant constraint forces. X The displacement generated at position 4; Live load uniformly distributed torque t Under the action of 2, the basic structure U is subject to redundant constraint forces. X The displacement generated at position 4; F A22 The support reaction force of the potential failure support A2 of the basic model M2 under live load; F D22 The support reaction force of the potential failure support D2 of the basic model M2 under live load; Live load concentration force P Support reaction force of support A2 of basic structure U under potential failure under action 2; Concentrated torque of live load T Support reaction force of support A2 of basic structure U under potential failure under action 2; Uniformly distributed live load q Support reaction force of support A2 of basic structure U under potential failure under action 2; Live load uniformly distributed torque t Support reaction force of support A2 of basic structure U under potential failure under action 2; Redundant constraints X Support reaction force of support A2 of basic structure U under potential failure under action 3; Redundant constraints X Support reaction force of support A2 of basic structure U under potential failure under action 4; Live load concentration force P Support reaction force of support D2 of basic structure U under potential failure under action 2; Concentrated torque of live load T Support reaction force of support D2 of basic structure U under potential failure under action 2; Uniformly distributed live load q Support reaction force of support D2 of basic structure U under potential failure under action 2; Live load uniformly distributed torque t Support reaction force of support D2 of basic structure U under potential failure under action 2; Redundant constraints X Support reaction force of support D2 of basic structure U under potential failure under action 3; Redundant constraints XSupport reaction force of support D2 of basic structure U under potential failure under action 3; G 2. Overturning instability effect of a single-column pier curved beam bridge under live load; G 1 / G 2. Overturning stability coefficient of a single-column pier curved beam bridge; E The elastic modulus of the curved beam material; I The moment of inertia of the cross section of the curved beam; k The ratio of bending stiffness to torsional stiffness of the cross section of a curved beam; G The shear modulus of the curved beam material; I d Torsional moment of inertia of the cross section of the curved beam; Curve ABCD, curved beam; Hinge point A1, outer support at the left end of the curved beam; Hinge point A2, inner support at the left end of the curved beam; Hinge point D1, right end outer support of the curved beam; Hinge point D2, inner support at the right end of the curved beam; Hinge point B, left middle pier support; Hinge point C, right middle pier support; O, the center of the curved beam; R The radius of curvature of a curved beam; f 0. The radian angle of the center of the curved beam; f 1. The arc angle between the left end support line A1A2 of the bridge and the left middle pier support B; f 2. The arc angle between the center of the left middle pier support B and the right middle pier support C of the bridge; f 3. The arc angle between the right middle pier support C and the right end support line D1D2 of the bridge.

[0024] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention.

[0025] Please see Figure 1 : This invention discloses a method for determining the overturning stability of a single-column pier curved beam bridge, comprising the following steps: S1: Based on the mechanical analysis of displacement and support reaction of simply supported statically indeterminate curved beams, the anti-overturning stability effect of single-column pier curved beam bridges under dead load is calculated according to the basic principles of structural mechanics. S2: Based on the mechanical analysis of displacement and support reaction of simply supported statically indeterminate curved beams, the overturning instability effect of single-column pier curved beam bridge under live load is calculated according to the basic principles of structural mechanics. S3: Based on the calculation results of the overturning stability effect under S1 dead load and the overturning instability effect under S2 live load, the overturning stability coefficient of the single-column pier curved beam bridge is calculated. S4: Based on the calculation results of the overturning stability coefficient of the single-column pier curved beam bridge, the overturning stability of the bridge is judged according to the threshold of the specification.

[0026] The algorithm principle is explained as follows: In S1, please refer to Figure 2 , Figure 3 , Figure 4 , Figure 5 : In the diagram, curve ABCD and a curved beam are shown. Hinge point A1, outer support at the left end of the curved beam; Hinge point A2, inner support at the left end of the curved beam; Hinge point D1, right end outer support of the curved beam; Hinge point D2, inner support at the right end of the curved beam; Hinge point B, left middle pier support; Hinge point C, right middle pier support; L 1. The distance between supports A1 and A2 on the left support line A1A2; L 2. The distance between supports D1 and D2 on the right end support line D1D2; O, the center of the curved beam; R The radius of curvature of a curved beam; f 0. The radian angle of the center of the curved beam; f 1. The arc angle between the left end support line A1A2 of the bridge and the left middle pier support B; f 2. The arc angle between the center of the left middle pier support B and the right middle pier support C of the bridge; f 3. The arc angle between the center of the right middle pier support C and the right end support line D1D2 of the bridge; q 1. Uniformly distributed dead loads acting on single-column pier curved beam bridges according to specifications; X1. In the basic system Q1, remove the redundant constraint of support B and replace the corresponding constraint reaction force; X 2. In the basic system Q1, remove the redundant constraints at support C and replace them with the corresponding constraint reactions; S1.1 Establish the basic model M1 and basic structure U of the single-column pier curved beam bridge under dead load: Figure 2 As shown, based on the fundamental principles of structural mechanics, a basic statically indeterminate structural model M1 of a single-column pier curved beam bridge under dead load is established. The plane axis of the curved beam represents the main superstructure of the curved beam bridge, on which a uniformly distributed dead load as required by the code is applied. q 1.

[0027] Figure 3 As shown, based on the fundamental principle of the statically indeterminate structural force method in structural mechanics, a basic model M1 is established for the basic structure U to be solved by the force method, which is a simply supported statically indeterminate curved beam.

[0028] S1.2 Establish the basic model M1, solve the corresponding basic system Q1 using the force method, and determine the basic unknowns for the force method solution. X 1. X 2: Figure 4 As shown, taking the basic structure U as a reference, the redundant constraints of the basic model M1 are removed using the corresponding constraint reaction forces. X 1. X 2. Replace the model with a basic model M1 and use the force method to solve for the corresponding basic system Q1; based on the basic principles of structural mechanics, the basic system Q1 is equivalent to the basic model M1. X 1. X 2 represents the basic unknowns that can be solved using the force method for the basic model M1.

[0029] S1.3 Decomposes the forces acting on the basic system Q1: Figure 5 As shown, to simplify calculations, the forces on the basic system Q1 are decomposed into three parts based on the fundamental principles of structural mechanics, with each part corresponding to the case of a single force or load acting alone.

[0030] S1.4 Calculation of the compliance coefficient of the basic model M1 obtained by force method d 11 , d 22 , d 12 and d 21 and free items D 1p , D 2p : S1.4.1 Based on the fundamental principles of the statically indeterminate structural force method in structural mechanics and the displacement analysis of simply supported statically indeterminate curved beams in structural mechanics, the relevant compliance coefficients of the basic model M1 are calculated using the force method. d 11 , d 22 , d 12 and d 21 They are respectively: (1) (2) (3) in, d 11 To simulate unit load Acting on the basic structure U X Position 1 X The displacement generated at position 1 d 22 To simulate unit load Acting on the basic structure U X At position 2 X The displacement generated at position 2 d 12 To simulate unit load Acting on the basic structure U X At position 2 X The displacement generated at position 1 d 21 To simulate unit load Acting on the basic structure U X Position 1 X The displacement generated at position 2; E The elastic modulus of the curved beam material. I Let be the moment of inertia of the curved beam cross section. k This is the ratio of the bending stiffness to the torsional stiffness of the curved beam's cross-section. k=EI / GI d , G The shear modulus of the curved beam material. I d Let be the torsional moment of inertia of the cross section of the curved beam. R Let be the radius of curvature of the curved beam. f 0 is the central radian angle of the curved beam. f 1 represents the radian angle between the left end support line A1A2 of the bridge and the left middle pier support B. f 2 represents the radian angle between the center of the left middle pier support B and the right middle pier support C of the bridge. f 3 represents the arc angle between the right middle pier support C and the right end support line D1D2 of the bridge.

[0031] S1.4.2 Based on the fundamental principles of the statically indeterminate structural force method in structural mechanics and the displacement analysis of simply supported statically indeterminate curved beams in structural mechanics, the relevant free terms of the basic model M1 force method are calculated. D 1p , D 2p They are respectively: (4) (5) in, D 1p For uniformly distributed constant load q 1. Acting on the basic structure U X The displacement generated at position 1 D 2p For uniformly distributed constant load q 1. Acting on the basic structure U X The displacement generated at position 2.

[0032] S1.5 Based on the principle of deformation compatibility, the basic equations for solving the M1 force method of the basic model are as follows: (6) Solving the fundamental equations of the force method for the basic model M1 yields the basic unknowns for solving the force method for the basic model M1. X 1. X 2 are respectively: (7) (8) S1.6 Calculate the support reactions of potentially failing supports under the dead load U on the basic structure: Based on the analysis of support reactions of simply supported statically indeterminate curved beams in structural mechanics, the calculation determines , , and , , They are respectively: (11) (12) (13) (14) (15) (16) in, L 1 represents the distance between supports A1 and A2 on the left support line A1A2. L 2 represents the distance between supports D1 and D2 on the right-end support line D1D2.

[0033] S1.7 Calculation of the basic model M1 under various dead loads and X 1. X Support reaction force of potentially failed support under 2 action: Figure 5 As shown, the basic model M1 is calculated based on the superposition principle under a uniformly distributed constant load. q Support reaction force at the potential failure location A2 under action 1 F A21 Support reaction force at the potential failure support D2 location F D21 They are respectively: (9) (10) in, , , Uniformly distributed dead loads q 1. Redundant constraints X 1. Redundant constraints X The support reaction force of the basic structure U under the action of 2, which indicates potential failure of the support A2. , , Uniformly distributed dead loads q 1. Redundant constraints X 1. Redundant constraints X Support reaction force of support D2 of basic structure U under potential failure under action 2.

[0034] S1.8 Based on pure torsional analysis of curved beam bridges, calculate the overturning stability effect of a single-column pier curved beam bridge under dead load. G 1 is: (17) In S2, please refer to Figure 6 , Figure 7 , Figure 8 : In the diagram, curve ABCD and a curved beam are shown. Hinge point A1, outer support at the left end of the curved beam; Hinge point A2, inner support at the left end of the curved beam; Hinge point D1, right end outer support of the curved beam; Hinge point D2, inner support at the right end of the curved beam; Hinge point B, left middle pier support; Hinge point C, right middle pier support; O, the center of the curved beam; R The radius of curvature of a curved beam; f 0. The radian angle of the center of the curved beam; f 1. The arc angle between the left end support line A1A2 of the bridge and the left middle pier support B; f 2. The arc angle between the center of the left middle pier support B and the right middle pier support C of the bridge; f 3. The arc angle between the center of the right middle pier support C and the right end support line D1D2 of the bridge; P 2. Concentrated live load forces acting according to specifications on single-column pier curved beam bridges; T 2. Concentrated torque of live load acting according to specifications on a single-column pier curved beam bridge; q 2. Uniformly distributed live load force acting according to specifications on a single-column pier curved beam bridge; t 2. Uniformly distributed torque of live load acting on a single-column pier curved beam bridge according to specifications; X 3. In the basic system Q2, remove the redundant constraint at support B and replace the corresponding constraint reaction force; X 4. In the basic system Q2, remove the redundant constraints at support C and replace them with the corresponding constraint reaction forces.

[0035] S2.1 Establish the basic model M2 and basic structure U of a single-column pier curved beam bridge under live load: Figure 6 As shown, based on the fundamental principles of structural mechanics, a basic statically indeterminate structural model M2 of a single-column pier curved beam bridge under live load is established. The plane axis of the curved beam represents the main superstructure of the curved beam bridge, on which the live loads required by the specifications, including concentrated forces, are applied. P 2. Concentrated torque T 2. Uniformly distributed forceq 2. Uniformly distributed torque t 2. Please refer to Figure 3 Based on the fundamental principle of the statically indeterminate structural force method in structural mechanics, a basic model is established for the basic structure U to be solved by the M2 force method, which is a simply supported statically indeterminate curved beam.

[0036] S2.2 Establish the basic model M2, solve the corresponding basic system Q2 using the force method, and determine the basic unknowns for the force method solution. X 3. X 4: Figure 7 As shown, based on the fundamental principle of the statically indeterminate structural force method in structural mechanics, taking the basic structure U as a reference, the redundant constraints of the basic model M2 are removed and the corresponding redundant constraint reactions are used. X 3. X 4. Replace the model with the basic model M2 and use the force method to solve for the corresponding basic system Q2; based on the basic principles of structural mechanics, the basic system Q2 is equivalent to the basic model M2. X 3. X 4 represents the basic unknowns for solving the basic model using the M2 force method.

[0037] S2.3 Decomposes the forces acting on the basic system Q2: Figure 8 As shown, to simplify calculations, the forces on the basic system Q2 are decomposed into six parts based on the fundamental principles of structural mechanics, with each part corresponding to the case of a single force or load acting alone.

[0038] S2.4 Calculation of the compliance coefficients of the basic model obtained by the M2 force method d 33 , d 44 , d 34 and d 43 and free items D 3p , D 4p : S2.4.1 Based on the fundamental principles of the statically indeterminate structural force method in structural mechanics and the displacement analysis results of simply supported statically indeterminate curved beams in structural mechanics, the relevant compliance coefficients of the basic model solved by the M2 force method are calculated. d 33 , d 44 , d 34 and d 43 They are respectively: (18) (19) (20) in, d 33 To simulate unit load Acting on the basic structure U X At position 3 X The displacement generated at position 3. d 44 To simulate unit load Acting on the basic structure U X At position 4 X The displacement generated at position 4 d 34 To simulate unit load Acting on the basic structure U X At position 4 X The displacement generated at position 3 d 43 To simulate unit load Acting on the basic structure U X At position 3 X The displacement generated at position 4.

[0039] S2.4.2 Figure 8 As shown, based on the fundamental principles of the statically indeterminate structural force method and the superposition principle in structural mechanics, the relevant free terms of the basic model M2 force method are calculated. D 3p , D 4p They are respectively: (twenty one) (twenty two) in, D 3p For live loads acting on the basic structure U X The displacement generated at position 3. D 4p For live loads acting on the basic structure U X The displacement generated at position 4 , , , Concentrated live load force P 2. Concentrated torque T 2. Uniformly distributed force q 2. Uniformly distributed torque t Under the action of 2, the basic structure U is subject to redundant constraint forces. X The displacement generated at position 3. , , , Concentrated live load force P 2. Concentrated torque T 2. Uniformly distributed force q 2. Uniformly distributed torque t Under the action of 2, the basic structure U is subject to redundant constraint forces. X The displacement at position 4 is calculated based on the displacement analysis results of a simply supported statically indeterminate curved beam in structural mechanics. , , , and , , , They are respectively: (twenty three) (twenty four) (25) (26) (27) (28) (29) (30) In step S2, the basic equations for solving the basic model M2 force method based on the deformation compatibility principle are as follows: (31) S2.5 solves the basic equations of the force method for the basic model M2, obtaining the basic unknowns for solving the force method of the basic model M2. X 3. X 4 are respectively: (32) (33) S2.6 Calculate the support reactions of the potentially failing supports under the live load U on the basic structure: Based on the analysis of support reactions of simply supported statically indeterminate curved beams in structural mechanics, calculation , , , , , and , , , , , They are respectively: (36) (37) (38) (39) (40) (41) (42) (43) (44) (45) (46) (47) S2.7 Calculation of the basic model M2 under various live loads and X 3. X Support reaction force of potentially failed support under 4 loads: Figure 8 As shown, the support reaction force of the basic model M2 under live load is calculated based on the superposition principle. F A22 Support reaction force of potentially failed support D2 F D22 They are respectively: (34) (35) in, , , , , , Concentrated live load force P 2. Concentrated torque T 2. Uniformly distributed force q 2. Uniformly distributed torque t 2. Redundant constraints X 3 and X The support reaction force of the basic structure U under the action of potential failure support A2. , , , , , Concentrated live load force P 2. Concentrated torque T 2. Uniformly distributed force q 2. Uniformly distributed torquet 2. Redundant constraints X 3 and X Support reaction force of support D2 of basic structure U under potential failure under action 4.

[0040] S2.8 Calculation of Overturning Instability Effect of a Single-Column Pier Curved Girder Bridge under Live Load Based on Pure Torsional Analysis Theory of Curved Girder Bridges S2.8 G 2 is: (48) In step S3, based on the pure torsional analysis theory of curved beam bridges, and according to the calculation results of the overturning stability effect under dead load and the overturning instability effect under live load, the overturning stability coefficient of the single-column pier curved beam bridge is calculated. G 1 / G 2 is: (49) In step S4, based on the calculation results of the overturning stability coefficient of the single-column pier curved beam bridge, and according to the threshold requirements of the current specifications, when the overturning stability coefficient of the single-column pier curved beam bridge... G 1 / G When 2 ≥ 2.5, it can be determined that the bridge will not overturn or become unstable; when G 1 / G When 2 < 2.5, it can be determined that the bridge will overturn and become unstable.

[0041] Calculation example: The basic design parameters of a single-column pier curved beam bridge are shown in the table below: The calculation process and results of the bridge dead load stability effect using the single-column pier curved beam bridge anti-overturning stability discrimination method provided by this invention are shown in the table below: The calculation process and results of the live load instability effect of the bridge using the overturning stability discrimination method for single-column pier curved beam bridges provided by this invention are shown in the table below: Using the overturning stability discrimination method for single-column pier curved beam bridges provided by this invention, the calculation result of the bridge's overturning stability coefficient is as follows: G 1 / G Since 2 = 2.69 > 2.5, it can be determined that the single-column pier curved beam bridge in the embodiment will not experience overturning instability.

[0042] As can be seen, the method for determining the overturning stability of a single-column pier curved beam bridge in this invention can easily and quickly calculate the overturning stability coefficient of the bridge after compiling a calculation table, and determine the overturning stability of the bridge. Compared with the classic curved beam bending-torsional coupling calculation method and the conventional finite element method, it has the advantages of simple calculation, speed and efficiency.

[0043] Therefore, the widespread application of the method of this invention can improve the efficiency of calculating and judging the overturning stability of single-column pier curved beam bridges, facilitate the promotion of bridge health monitoring technology, and has the advantages of being simple, efficient and fast.

Claims

1. A method for determining the overturning stability of a single-column pier curved beam bridge, characterized in that: Includes the following steps: S1: Based on the mechanical analysis of displacement and support reaction of simply supported statically indeterminate curved beams, the anti-overturning stability effect of single-column pier curved beam bridges under dead load is calculated according to the basic principles of structural mechanics. S2: Based on the mechanical analysis of displacement and support reaction of simply supported statically indeterminate curved beams, the overturning instability effect of single-column pier curved beam bridge under live load is calculated according to the basic principles of structural mechanics. S3: Based on the calculation results of the dead load anti-overturning stability effect described in S1 and the live load overturning instability effect described in S2, the anti-overturning stability coefficient of the single-column pier curved beam bridge is calculated. S4: Based on the calculation results of the overturning stability coefficient of the single-column pier curved beam bridge in S3, the overturning stability of the bridge is judged according to the threshold of the specification.

2. The method for determining the overturning stability of a single-column pier curved beam bridge according to claim 1, characterized in that: In step S1, based on the fundamental principles of structural mechanics, the calculation of the overturning stability effect of a single-column pier curved beam bridge under dead load is performed. The curved beam of the superstructure is represented by its planar axis, on which a uniformly distributed dead load as required by the specifications is applied. q 1. Establishing a single-column pier curved beam bridge under uniformly distributed dead load q The basic model M1 of the statically indeterminate structure under the action of 1 is based on the basic principle of the statically indeterminate structural force method in structural mechanics, with a simply supported statically indeterminate curved beam as the basic structure U. Using the basic structure U as a reference, remove the redundant constraints of the basic model M1 using the corresponding constraint reaction forces. X 1. X 2. Replace the model with a basic model M1 and use the force method to solve for the corresponding basic system Q1. Based on the basic principles of structural mechanics, the basic system Q1 is equivalent to the basic model M1. X 1. X 2 refers to the basic unknowns that can be solved using the force method for the basic model M1; Based on the displacement analysis results of simply supported statically indeterminate curved beams in structural mechanics, the relevant compliance coefficients and relevant free terms of the basic model M1 force method are calculated. Based on the principle of deformation compatibility, the basic equations of the basic model M1 are established and solved using the force method to obtain the basic unknowns. X 1. X 2; Calculate the basic structure U under dead load respectively q 1. Redundant constraints X 1. X Support reactions of potentially failing supports A2 and D2 under action 2; The basic model M1 under dead load is calculated using the superposition principle. q Support reactions of potentially failing supports A2 and D2 under action 1; Based on the pure torsional analysis theory of curved beam bridges, the overturning stability effect of a single-column pier curved beam bridge under dead load is calculated.

3. The method for determining the overturning stability of a single-column pier curved beam bridge according to claim 2, characterized in that: In step S1, based on the fundamental principles of the force method for solving statically indeterminate structures in structural mechanics and the displacement analysis results of simply supported statically indeterminate curved beams in structural mechanics, the relevant compliance coefficients of the basic model M1 are calculated using the force method. δ 11 , δ 22 , δ 12 and δ 21 ; in: δ 11 To simulate unit load Acting on the basic structure U X Position 1 X The displacement generated at position 1 δ 22 To simulate unit load Acting on the basic structure U X At position 2 X The displacement generated at position 2 δ 12 To simulate unit load Acting on the basic structure U X At position 2 X The displacement generated at position 1 δ 21 To simulate unit load Acting on the basic structure U X Position 1 X The displacement generated at position 2; Based on the fundamental principles of the statically indeterminate structural force method in structural mechanics and the displacement analysis results of simply supported statically indeterminate curved beams in structural mechanics, the relevant free terms of the basic model M1 are calculated using the force method. Δ 1p , Δ 2p , Δ 1p For uniformly distributed constant load q 1. Acting on the basic structure U X The displacement generated at position 1 Δ 2p For uniformly distributed constant load q 1. Acting on the basic structure U X The displacement generated at position 2; The basic equations for solving the basic model M1 using the force method based on the principle of deformation compatibility are as follows: Solving the fundamental equations of the force method for the basic model M1 yields the basic unknowns for solving the force method for the basic model M1. X 1. X 2 are respectively: 。 4. The method for determining the overturning stability of a single-column pier curved beam bridge according to claim 2, characterized in that: In step S1, the basic model M1 is calculated based on the superposition principle under uniformly distributed constant load. q Support reaction force of potentially failing support A2 under action 1 F A21 Support reaction force of potentially failed support D2 F D21 for: in , , Uniformly distributed dead loads q 1. Redundant constraints X 1. Redundant constraints X The support reaction force of the basic structure U under the action of 2, which indicates potential failure of the support A2. , , Uniformly distributed dead loads q 1. Redundant constraints X 1. Redundant constraints X The support reaction force of the basic structure U at the potential failure support D2 under the action of 2. , , and , , The results were calculated based on the analysis results of the support reactions of simply supported statically indeterminate curved beams in structural mechanics.

5. The method for determining the overturning stability of a single-column pier curved beam bridge according to claim 2, characterized in that: In step S1, the overturning stability effect of a single-column pier curved beam bridge under dead load is calculated based on the pure torsional analysis theory of curved beam bridges. G 1 is: in, L 1 represents the distance between supports A1 and A2 on the left support line A1A2. L 2 represents the distance between supports D1 and D2 on the right-end support line D1D2.

6. The method for determining the overturning stability of a single-column pier curved beam bridge according to claim 1, characterized in that: In step S2, the overturning instability effect of a single-column pier curved beam bridge under live load is calculated based on the basic principles of structural mechanics. The curved beam of the superstructure is represented by its planar axis, and the live loads required by the specifications, including concentrated forces, act on it. P 2. Concentrated torque T 2. Uniformly distributed force q 2. Uniformly distributed torque t 2. Establish a statically indeterminate basic model M2 of a single-column pier curved beam bridge under live load. Based on the basic principle of solving statically indeterminate structural forces in structural mechanics, a simply supported statically indeterminate curved beam is used as the basic structure U. Using the basic structure U as a reference, remove the redundant constraints of the basic model M2 using the corresponding redundant constraint reaction forces. X 3. X 4. Replace the model with the basic model M2 and use the force method to solve for the corresponding basic system Q2. Based on the basic principles of structural mechanics, the basic system Q2 is equivalent to the basic model M2. X 3. X 4 represents the basic unknowns for solving the basic model using the M2 force method; Based on the displacement analysis results of simply supported statically indeterminate curved beams in structural mechanics, the relevant compliance coefficients and relevant free terms of the basic model are calculated by the M2 force method. Based on the principle of deformation compatibility, the basic equations of the basic model are established and solved using the M2 force method to obtain the basic unknowns. X 3. X 4; Based on the analysis of support reactions of simply supported statically indeterminate curved beams in structural mechanics, the basic structure U is calculated under various live loads and redundant constraint forces. X 3. X Support reactions of potentially failing supports A2 and D2 under the action of 4; The support reactions of the basic model M2 under live load were calculated using the superposition principle. Based on the pure torsional analysis theory of curved beam bridges, the overturning instability effect of a single-column pier curved beam bridge under live load is calculated.

7. The method for determining the overturning stability of a single-column pier curved beam bridge according to claim 6, characterized in that: Based on the fundamental principles of the statically indeterminate structural force method in structural mechanics and the displacement analysis results of simply supported statically indeterminate curved beams in structural mechanics, the relevant compliance coefficients of the basic model solved by the M2 force method are calculated. δ 33 , δ 44 , δ 34 and δ 43 ; in, δ 33 To simulate unit load Acting on the basic structure U X At position 3 X The displacement generated at position 3. δ 44 To simulate unit load Acting on the basic structure U X At position 4 X The displacement generated at position 4 δ 34 To simulate unit load Acting on the basic structure U X At position 4 X The displacement generated at position 3 δ 43 To simulate unit load Acting on the basic structure U X At position 3 X The displacement generated at position 4; Based on the fundamental principles of the statically indeterminate structural force method and the superposition principle in structural mechanics, the relevant free terms of the basic model solved by the M2 force method are calculated. Δ 3p , Δ 4p for: in, Δ 3p For live loads acting on the basic structure U X The displacement generated at position 3. Δ 4p For live loads acting on the basic structure U X The displacement generated at position 4 , , , Concentrated live load force P 2. Concentrated torque T 2. Uniformly distributed force q 2. Uniformly distributed torque t Under the action of 2, the basic structure U is subject to redundant constraint forces. X The displacement generated at position 3. , , , Concentrated live load force P 2. Concentrated torque T 2. Uniformly distributed force q 2. Uniformly distributed torque t Under the action of 2, the basic structure U is subject to redundant constraint forces. X The displacement generated at position 4 , , , and , , , The displacement analysis results of simply supported statically indeterminate curved beams in structural mechanics were used to determine the displacement. The basic equations for solving the basic model based on the deformation compatibility principle using the M2 force method are as follows: Solving the fundamental equations of the force method for the basic model M2 yields the basic unknowns for solving the force method for the basic model M2. X 3. X 4 are respectively: 。 8. The method for determining the overturning stability of a single-column pier curved beam bridge according to claim 6, characterized in that: The support reaction force of the basic model M2 under live load is calculated based on the superposition principle. F A22 Support reaction force of potentially failed support D2 F D22 They are respectively: in, , , , , , Concentrated live load force P 2. Concentrated torque T 2. Uniformly distributed force q 2. Uniformly distributed torque t 2. Redundant constraints X 3 and X The support reaction force of the basic structure U under the action of potential failure support A2. , , , , , Concentrated live load force P 2. Concentrated torque T 2. Uniformly distributed force q 2. Uniformly distributed torque t 2. Redundant constraints X 3 and X The support reaction force of the basic structure U at the potential failure support D2 under the action of 4. , , , , , and , , , , , The results were calculated and determined based on the analysis results of the support reactions of simply supported statically indeterminate curved beams in structural mechanics. Calculation of overturning instability effect of single-column pier curved beam bridge under live load based on pure torsional analysis theory of curved beam bridge G 2 is: 。 9. The method for determining the overturning stability of a single-column pier curved beam bridge according to claim 1, characterized in that: In step S3, the overturning stability coefficient of the single-column pier curved beam bridge is calculated based on the pure torsional analysis theory of curved beam bridges and the calculation results of the dead load overturning stability effect and the live load overturning instability effect. G 1 / G 2 is: 。 10. The method for determining the overturning stability of a single-column pier curved beam bridge according to claim 1, characterized in that: In step S4, when determining the overturning stability of the bridge according to the threshold specified in the standard, based on the calculation results of the overturning stability coefficient of the single-column pier curved beam bridge, and according to the requirements of the standard, when the overturning stability coefficient of the single-column pier curved beam bridge... G 1 / G When 2 ≥ 2.5, it can be determined that the bridge will not overturn or become unstable; when G 1 / G When 2 < 2.5, it can be determined that the bridge will overturn and become unstable.

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