Arch bridge damage identification method based on the curvature of the influence line of the thrust of a parabolic variable cross-section arch

Abstract

The present invention discloses an arch bridge damage identification method based on the curvature of the influence line of the arch thrust with a parabolic variable cross-section, belonging to the technical field of arch bridge damage detection. Based on the force method and Ritter's formula, through the setting of the variable cross-section and the fitting of the arch axis shape, a new index - the thrust influence line difference curvature TILDC for reflecting the damage degree of the two-hinged arch structure is proposed. Through the TILDC index, the damage location of the parabolic variable cross-section two-hinged arch structure can be realized, which can be used as the design basis for the strength of the arch seat foundation under the action of moving loads and the theoretical reference for the rapid detection application of arch bridges, verifying the feasibility of using the influence line of the thrust as a damage identification method.

Description

Technical Field

[0001] The present invention relates to the technical field of arch bridge damage detection, and particularly to an arch bridge damage identification method based on the curvature of the thrust influence line of a parabolic variable cross-section arch. Background Art

[0002] Two-hinged arches have no moment effect at the arch feet, are less affected by concrete creep, shrinkage, and temperature, and have lower requirements for the foundation structure. Therefore, two-hinged arches are widely used in actual arch bridge projects. Since the main arch ring and arch ribs are the main load-bearing components of the arch bridge, once damage occurs, it will cause a significant decrease in the bearing capacity of the arch bridge and even dangerous situations such as collapse. Therefore, it is of great practical significance to quickly locate the damage of the two-hinged arch structure and evaluate the degree of damage. In reality, when conducting bridge damage detection, a large number of sensors need to be arranged, which costs a lot of manpower and material resources. The above problems need to be solved urgently. For this reason, an arch bridge damage identification method based on the curvature of the thrust influence line of a parabolic variable cross-section arch is proposed. Summary of the Invention

[0003] The technical problem to be solved by the present invention is: how to solve the problem that a large number of sensors need to be arranged and a lot of manpower and material resources are spent during the damage detection of the existing two-hinged arch, and an arch bridge damage identification method based on the curvature of the thrust influence line of a parabolic variable cross-section arch is provided.

[0004] The present invention solves the above technical problems through the following technical solutions. The present invention includes the following steps:

[0005] S1: Establish the basic system of the two-hinged arch in the damaged state, simulate the damage by reducing the local elastic modulus of the structure and thus reducing the stiffness. The flexural stiffness after damage is E'I, and the integral expression of the thrust influence line at the arch feet of the two-hinged arch structure in the damaged state is derived;

[0006] S2: Subtract the thrust influence line at the arch feet of the two-hinged arch structure in the damaged state from the thrust influence line at the arch feet of the two-hinged arch structure in the undamaged state, and take the second derivative of the difference of the thrust influence lines for analysis. Determine whether the structure is damaged through the curvature of the difference of the thrust influence lines, and then locate the damage position.

[0007] Furthermore, in the step S1, the integral expression of the thrust influence line at the arch feet of the two-hinged arch structure in the damaged state is as follows:

[0008] F′ H =-Δ′ 1p / δ′ 11

[0009] where, Δ′ 1p is the load displacement in the damaged state, and δ′ 11 is the self-displacement in the damaged state.

[0010] Furthermore, the self-displacement calculation formula under the damaged state is as follows:

[0011]

[0012] Where: E is the elastic modulus, I is the moment of inertia, f is the rise, E' is the elastic modulus at the time of damage, [b - ε, b + ε] is the damaged section, x is the horizontal distance from any section on the arch structure to the crown of the arch.

[0013] Furthermore, the load-displacement calculation formula under the damaged state is as follows:

[0014] When -L ≤ x p <b - ε,

[0015] When b - ε ≤ x p ≤ b + ε,

[0016] When b + ε < x p ≤ L,

[0017] Where, Δ1' p is the load-displacement under the damaged state, x p is the horizontal distance from the concentrated load to the crown of the arch.

[0018] Furthermore, in the step S2, the expression of the influence line of the thrust at the arch feet of the two-hinged arch structure in the undamaged state is as follows:

[0019] F H =-Δ 1p / δ 11

[0020] Where, Δ 1p is the load-displacement in the undamaged state, δ 11 is the self-displacement in the undamaged state.

[0021] Furthermore, the calculation formula of the load-displacement in the undamaged state is as follows:

[0022]

[0023] Furthermore, the calculation formula of the self-displacement in the undamaged state is as follows:

[0024]

[0025] Furthermore, in the step S2, the parabolic expression of the two-hinged arch is as follows:

[0026] y=(4fx 2 ) / (2L)2

[0027] Among them, L is the length of half of the span of the two-hinged arch, x is the horizontal distance from any section on the arch structure to the crown of the arch, and y is the corresponding ordinate of x.

[0028] Furthermore, in the step S2, when the moving load is in the non-damaged section, the difference curvature of the thrust influence line is zero. When the moving load is in the damaged area, the difference curvature of the thrust influence line is non-zero and undergoes a mutation, thereby identifying the damage location.

[0029] Furthermore, in the step S3, for the damage degree of the two-hinged arch structure, in actual engineering, first, the damage location is determined through the difference curvature index of the thrust influence line. After both the measuring points and the damage points are determined, the amplitude curve of the difference curvature of the thrust influence line under different damage conditions is simulated by finite element software, and the relationship formula between the damage degree and the amplitude is fitted to invert the damage degree to achieve the quantification of the damage.

[0030] The present invention has the following advantages compared with the prior art: This method for identifying the damage of an arch bridge based on the curvature of the thrust influence line of a parabolic variable-section arch, based on the force method and Ritter's formula, through the setting of the variable section and the fitting of the arch axis shape, proposes a new index - Thrust Influence Lines Difference Curvature (TILDC) for reflecting the damage degree of the two-hinged arch structure. Through the TILDC index, the damage location of the parabolic variable-section two-hinged arch structure can be realized, which can be used as the design basis for the strength of the arch seat foundation under the action of moving loads and the theoretical reference for the rapid detection application of arch bridges, verifying the feasibility of using the thrust influence line as a damage identification method. Description of the Drawings

[0031] Figure 1 is a schematic flow chart of the method for identifying the damage of an arch bridge based on the curvature of the thrust influence line of a parabolic variable-section arch in the first embodiment of the present invention;

[0032] Figure 2 is a schematic diagram of the basic system of the two-hinged arch in the damaged state in the first embodiment of the present invention;

[0033] Figure 3a is a three-dimensional diagram comparing the different damage degrees of damaged elements with non-damaged elements in the single-point damage state in the second embodiment of the present invention;

[0034] Figure 3b is Figure 3a the top view of

[0035] Figure 4a is a three-dimensional diagram comparing the different damage degrees of damaged elements with non-damaged elements in the multi-point damage state in the second embodiment of the present invention;

[0036] Figure 4b is Figure 4a the top view of Specific embodiments

[0037] The following is a detailed description of the embodiments of the present invention. These embodiments are implemented on the premise of the technical solution of the present invention, and detailed implementation manners and specific operation processes are given. However, the protection scope of the present invention is not limited to the following embodiments.

[0038] Embodiment 1

[0039] As Figure 1 shown, this embodiment provides a technical solution: an arch bridge damage identification method based on the curvature of the influence line of the arch thrust with a parabolic variable cross-section, including the following steps:

[0040] (1): Establish the basic system of a two-hinged arch in the damaged state (see Figure 2 ), simulate the damage by reducing the local elastic modulus of the structure and thus reducing the stiffness. The flexural stiffness after damage is E'I, and the integral expression of the influence line of the (horizontal) thrust at the arch foot of the two-hinged arch structure in the damaged state is derived;

[0041] In this step, the integral expression of the influence line of the (horizontal) thrust at the arch foot of the two-hinged arch structure in the damaged state is as follows:

[0042] F' H =-Δ' 1p / δ' 11

[0043] where Δ' 1p is the load displacement in the damaged state, and δ' 11 is the self-displacement in the damaged state.

[0044] The load displacement Δ' 1p : The displacement along the X1 direction generated by the load P; the self-displacement δ' 11 : The displacement along the X1 direction generated by the unit force X1 = 1, often called the flexibility coefficient.

[0045] In this step, the calculation formula for the self-displacement in the damaged state is as follows:

[0046]

[0047] where E is the elastic modulus, I is the moment of inertia, f is the rise, E' is the elastic modulus at the time of damage, [b - ε, b + ε] is the damaged section, x is the horizontal distance from any section on the arch structure to the crown of the arch.

[0048] In this step, the calculation formula for the load displacement in the damaged state is as follows:

[0049] When -L ≤ x p <b - ε,

[0050] When b - ε ≤ x p ≤ b + ε,

[0051] When b + ε < x p ≤ L,

[0052] Wherein, Δ1′ p is the load displacement in the damaged state, and x p is the horizontal distance from the concentrated load to the crown of the arch.

[0053] (2): Subtract the influence line of the thrust at the arch foot of the two - hinged arch structure in the damaged state from the influence line of the thrust at the arch foot of the two - hinged arch structure in the undamaged state, and take the second - derivative of the difference of the influence lines of the thrust for analysis. When the moving load is in the undamaged area of the structure, the curvature of the difference of the influence lines of the thrust is zero. Determine whether the structure is damaged by the curvature of the difference of the influence lines of the thrust, and then locate the damage position.

[0054] In this step, the expression of the influence line of the thrust at the arch foot of the two - hinged arch structure in the undamaged state is as follows:

[0055] F H = - Δ 1p / δ 11

[0056] Wherein, Δ 1p is the load displacement in the undamaged state, and δ 11 is the self - displacement in the undamaged state.

[0057] In this step, the calculation formula of the load displacement in the undamaged state is as follows:

[0058]

[0059] In this step, the calculation formula of the self - displacement in the undamaged state is as follows:

[0060]

[0061] In this step, the parabolic expression of the two - hinged arch is as follows:

[0062] y=(4fx 2 ) / (2L) 2

[0063] Wherein, L is the length of the half - span of the two - hinged arch, x is the horizontal distance from any section on the arch structure to the crown of the arch, and y is the corresponding ordinate of x.

[0064] This expression is for the case when the arch axis curve is a parabola and is used for the line type setting of the two-hinged arch structure.

[0065] The theoretical analysis is as follows:

[0066] According to the force method principle, it can be known that:

[0067]

[0068]

[0069] Among them: M1 is the bending moment generated by the unit force in any section of the basic structure, and M p is the bending moment generated by the load in the basic structure.

[0070] The bending moment generated by the unit force:

[0071] M1 = -(f - y)

[0072] Among them, y is the expression of the arch axis curve.

[0073] Let the moving load F p be the unit 1. Under its action, the bending moment M at any point of the arch section p is:

[0074]

[0075] Among them, L is the half-span length of the arch structure (a two-hinged arch in this embodiment).

[0076] Table 1 Self-displacements and load-displacements of a variable-section parabolic two-hinged arch during damage

[0077]

[0078]

[0079] Table 2 Difference table of thrust influence lines

[0080]

[0081] According to the curvature formula In the formula, F H The first derivative of the thrust is approximately 0, that is:

[0082] 1 / ρ(x) = |F″ H |

[0083] The curvature of the difference in arch springing thrust before and after damage:

[0084] 1 / ρ(x) = |(F H - F′ H )″|

[0085] By deriving the difference curvature of the influence line of the arch springing thrust before and after damage, it can be known that when the moving load is in the non-damaged section of -L ≤ x p <b - ε, b + ε < x p ≤ L and when it is in the damaged section of b - ε ≤ x p ≤ b + ε, the difference in the expressions is reflected in the coefficients. When the moving load is in the damaged area, (F H - F′ H )″ is not 0 and there is a mutation. Therefore, the difference in the expression coefficients can be used as the basis for damage location. According to the second-order expression, the degree of section stiffness reduction can be deduced, so as to realize the analysis of the damage degree.

[0086] Example 2

[0087] In this example, a two-hinged arch model is used to verify the method in Example 1. The span l of the two-hinged arch is 50.934 m, the height of the mid-span section is 1 m, the width of the arch rib section is 1 m, the arch axis coefficient m is taken as 1.988, the arch thickness variation coefficient n is taken as 0.4, k is taken as 1.31, and the rise-span ratio f is 1 / 4. Let the 1# node of the arch springing be the measuring point. The identification effect of TILDC is verified by simulating single-point damage and multi-point damage. The degree of damage is simulated by the reduction degree of the elastic modulus of the section element material. The obtained data is processed by moving average to eliminate waves. The specific working conditions are shown in Table 3. The data processing results are shown in Figure 3a 、 3b 、 Figure 4a 、 4b .

[0088] Table 3 Damage working conditions in this example

[0089]

[0090] In Figure 3a 、 3b 、4a, 4b, it can be seen from the position where the TILDC amplitude appears that there is a relatively obvious mutation near the damaged element. This method has a good identification effect on both single-point damage and multi-point damage; as the degree of damage increases, the mutation peak also continues to rise, and thus the degree of damage can be quantitatively evaluated.

[0091] To better study the relationship between the degree of damage and the TILDC amplitude and accurately quantify the degree of damage, the TILDC amplitudes of 8 degrees of damage without moving average processing in the single-point damage working condition are selected, as shown in Table 4, and the corresponding damage inversion formula is fitted to invert the damage TILDC amplitude of a certain damaged element under different degrees of damage.

[0092] Table 4 TILDC values under different degrees of damage

[0093]

[0094] Inversion formula: TILDC(x) = 0.131827x 5 - 0.305734x 4 + 0.27661x 3 - 0.118643x 2 + 2.46949e -2 x - 1.85e -3

[0095] Analyze the goodness of fit and determine the coefficient R 2 = 0.9999, close to 1, indicating a good fitting correlation. Through finite element simulation, it is known that the TILDC amplitude is 0.000341 when the damage degree is 45%. By comparing this data with the numerical result 0.000339 obtained from the inversion formula, it is found that the amplitude deviation is about 0.59%. Considering the error in curve fitting, the result can be considered reasonable. Therefore, through the inversion formula of this working condition, the accurate quantification of the damage degree of a two-hinged arch with a variable cross-section of 1 / 4 rise-span ratio can be achieved.

[0096] In this embodiment, through formula derivation and model examples, it is proved that the newly proposed damage identification index TILDC can locate the damage of a variable cross-section two-hinged arch structure, and has good identification effects on single-point damage and multi-point damage; the magnitude of the sudden change peak of the TILDC curve is proportional to the damage degree, that is, an accurate calculation theoretical method for the damage degree of a two-hinged arch structure under specific working conditions is obtained.

[0097] In summary, the arch bridge damage identification method based on the curvature of the influence line of the arch thrust of a parabolic variable cross-section arch in the above embodiment, based on the force method and Ritter formula, through the setting of the variable cross-section and the fitting of the arch axis shape, proposes a new index - the difference curvature of the influence line of the thrust TILDC for reflecting the damage degree of a two-hinged arch structure. Through the TILDC index, the damage location of a parabolic variable cross-section two-hinged arch structure can be realized, which can be used as the design basis for the strength of the arch seat foundation under moving loads and the theoretical reference for the rapid detection of arch bridges, and proves the feasibility of using the influence line of the thrust as a damage identification method.

[0098] Although the embodiments of the present invention have been shown and described above, it can be understood that the above embodiments are exemplary and should not be construed as limiting the present invention. Those of ordinary skill in the art can make changes, modifications, substitutions, and variations to the above embodiments within the scope of the present invention.

Claims

1. An arch bridge damage identification method based on the curvature of the influence line of the arch thrust with a parabolic variable cross-section, characterized in that, It includes the following steps: S1: Establish the basic system of the two-hinged arch under the damaged state. Simulate the damage by reducing the local elastic modulus of the structure to reduce the stiffness. The flexural stiffness after damage is E′I, and the integral expression of the influence line of the thrust at the arch foot of the two-hinged arch structure under the damaged state is derived; In the step S1, the integral expression of the influence line of the thrust at the arch foot of the two-hinged arch structure under the damaged state is as follows: F′ H = -Δ′ 1p / δ′ 11 where, Δ′ 1p is the load displacement under the damaged state, and δ′ 11 is the self-displacement under the damaged state; The calculation formula for the self-displacement under the damaged state is as follows: Where: E is the elastic modulus, I is the moment of inertia, f is the rise, E′ is the elastic modulus at damage, [b - ε, b + ε] is the damaged section, x is the horizontal distance from any section on the arch structure to the crown of the arch, and y is the corresponding ordinate of x; The calculation formula for the load-displacement under the damaged state is as follows: where, Δ′ 1p is the displacement under the damaged state, and x p is the horizontal distance from the concentrated load to the crown of the arch; S2: Subtract the influence line of the thrust at the arch foot of the two-hinged arch structure under the damaged state from the influence line of the thrust at the arch foot of the two-hinged arch structure under the undamaged state, and take the second derivative of the difference of the influence lines of the thrust for analysis. Judge whether the structural damage occurs through the curvature of the difference of the influence lines of the thrust, and then locate the damage position.

2. The arch bridge damage identification method based on the curvature of the influence line of the arch thrust with a parabolic variable cross-section according to claim 1, wherein: In the step S2, the expression of the influence line of the thrust at the arch foot of the two-hinged arch structure under the undamaged state is as follows: F H = -Δ 1p / δ 11 Among them, Δ 1p is the load displacement in the lossless state, and δ 11 is the self-displacement in the lossless state.

3. The arch bridge damage identification method based on the curvature of the influence line of the arch thrust with a parabolic variable cross-section according to claim 2, wherein: The calculation formula for the load-displacement under the undamaged state is as follows: Wherein, L is the length of the half-span of the two-hinged arch.

4. The arch bridge damage identification method based on the curvature of the influence line of the arch thrust with a parabolic variable cross-section according to claim 3, characterized in that: The calculation formula for the self-displacement under the undamaged state is as follows:

5. The arch bridge damage identification method based on the curvature of the influence line of the arch thrust with a parabolic variable cross-section according to claim 4, characterized in that: In the step S2, the parabolic expression of the two-hinged arch is as follows: y = (4fx 2 ) / (2L) 2 Wherein, L is the length of the half-span of the two-hinged arch, x is the horizontal distance from any section on the arch structure to the crown of the arch, and y is the ordinate corresponding to x.

6. The arch bridge damage identification method based on the curvature of the influence line of the arch thrust with a parabolic variable cross-section according to claim 1, characterized in that: In the step S2, when the moving load is located in the undamaged section, the curvature of the difference of the influence lines of the thrust is zero. When the moving load is located in the damaged area, the curvature of the difference of the influence lines of the thrust is not zero and there is a mutation, thereby identifying the damage position.

7. The arch bridge damage identification method based on the curvature of the influence line of the arch thrust with a parabolic variable cross-section according to claim 6, wherein: In the step S3, for the damage degree of the two-hinged arch structure, in actual engineering, first locate the damage through the curvature index of the difference of the influence lines of the thrust. After both the measuring points and the damage points are determined, simulate the curvature amplitude curve of the difference of the influence lines of the thrust under different damage conditions by finite element software, and fit the relationship formula between the damage degree and the amplitude to invert the damage degree to realize the quantification of the damage.

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