Geometric nonlinear structure response calculation method for large-span arch bridge

Abstract

The invention discloses a large-span arch bridge geometric nonlinear structure response calculation method, and relates to the technical field of bridge engineering structure analysis. Comprising the steps that a reasonable arch axis is set, and a second-order deflection nonlinear equilibrium equation of the large-span arch bridge is established; discretizing a reasonable arch axis into a polynomial capable of being solved singly, and deriving a nonlinear deflection expression and a nonlinear bending moment expression of the large-span arch bridge based on a second-order deflection nonlinear equilibrium equation; solving unknown parameters in the nonlinear deflection expression and the nonlinear bending moment expression on the basis of the principle that the elastic compression amount and the compression strain are equal along the integral of the arc length of the arch axis, and obtaining the nonlinear deflection and the nonlinear bending moment of the large-span arch bridge; verifying the calculation method by adopting a numerical simulation method considering geometric nonlinearity; and analyzing a change rule and a spatial distribution rule of the non-linear structure response of the large-span arch bridge along with a loading process through numerical simulation. According to the method, a reliable basis is provided for structural safety reserve evaluation and optimization design.

Description

Technical Field

[0001] This invention relates to the field of bridge engineering structural analysis technology, and more specifically to a method for calculating the geometric nonlinear structural response of a long-span arch bridge. Background Technology

[0002] Long-span arch bridges are widely used in major transportation infrastructure construction due to their strong span capacity, excellent load-bearing performance, and beautiful appearance. As the span continues to increase, the axial compressive stress of the arch ribs increases significantly, and the geometric nonlinear effects of the structure become more prominent, directly affecting the structural load-bearing performance and safety reserve.

[0003] Existing nonlinear analysis methods for long-span arch bridges mainly include numerical simulation and analytical methods. Numerical simulation methods (such as the finite element method) can handle complex structural forms, but the calculation process is cumbersome and time-consuming, and it is difficult to intuitively reveal the intrinsic laws of nonlinear response. Traditional analytical methods are mostly based on simplifying assumptions, ignoring the discrete characteristics of the arch axis and the nonlinear changes of redundant horizontal forces, resulting in insufficient calculation accuracy and difficulty in meeting the requirements of refined design for long-span arch bridges. In addition, existing methods lack systematic research on the distribution law of nonlinear bending moments and the applicable range of bending moment amplification factors, and cannot provide accurate nonlinear effect assessment basis for structural design.

[0004] Therefore, proposing a method for calculating the geometric nonlinear structural response of long-span arch bridges to address the difficulties in existing technologies is a problem that urgently needs to be solved by those skilled in the art. Summary of the Invention

[0005] In view of this, the present invention provides a method for calculating the geometric nonlinear structural response of long-span arch bridges to solve the problems existing in the prior art.

[0006] To achieve the above objectives, the present invention provides the following technical solution: A method for calculating the geometric nonlinear structural response of a long-span arch bridge includes the following steps: S1. Set a reasonable arch axis and establish the second-order nonlinear equilibrium equation for the deflection of a long-span arch bridge. S2. Discretize the reasonable arch axis into a polynomial that can be solved individually, and derive the nonlinear deflection expression and nonlinear bending moment expression for long-span arch bridges based on the second-order nonlinear equilibrium equation of deflection. S3. Based on the principle that the integral of elastic compression and compressive strain along the arc length of the arch axis is equal, the unknown parameters in the nonlinear deflection expression and nonlinear bending moment expression are solved to obtain the nonlinear deflection and nonlinear bending moment of the long-span arch bridge. S4. Verify the calculation methods from S1 to S3 using a numerical simulation method that considers geometric nonlinearity. S5. Numerical simulation analysis was conducted to study the variation and spatial distribution of the nonlinear structural response of a long-span arch bridge as it loads.

[0007] Optionally, the second-order flexural nonlinear equilibrium equation in S1 is established based on the control differential equation of the arch structure, and the total bending moment of the structure that takes into account the second-order effect includes the linear elastic bending moment of the structure.

[0008] Optionally, the arch structure corresponding to the reasonable arch axis in S1 is a statically indeterminate hingeless arch, and the additional bending moment is the bending moment generated by the superposition of the additional bending moment of the three-hinged arch and the redundant force, which includes the redundant horizontal force and bending moment.

[0009] Optionally, the nonlinear deflection and nonlinear bending moment expressions in S2 consist of cosine terms and higher-order polynomials, and are functions of the structural loading process and response space distribution.

[0010] Optionally, when the reasonable arch axis is a parabola, the nonlinear displacement expression consists of a cosine curve and a parabola, and the nonlinear bending moment expression is a cosine curve.

[0011] Optionally, in the process of solving the unknown parameters in S3, it is necessary to combine the variation characteristics of the geometric stiffness of the arch rib and consider the nonlinear variation of the redundant horizontal force. The redundant horizontal force includes the redundant horizontal force solved according to the linear elastic theory and the additional redundant force after considering the nonlinear effect.

[0012] Optionally, the numerical simulation in S4 adopts the Newton-Raphson iterative method. The simulation verification indicators include the total deflection and total bending moment of the long-span arch bridge. The calculation error of the total deflection is less than 4.4%, and the calculation error of the total bending moment is less than 5.9%.

[0013] Optionally, the nonlinear structural response in S5 includes: nonlinear bending moment, the spatial distribution of which is M-shaped, passing through two zero-moment points within the semi-arch range, and exhibiting two positive bending moment peaks and one negative bending moment peak.

[0014] Optionally, the load in the loading process of S5 is set to 1.75 times the dead load of the arch rib, and is divided into 10 levels for loading analysis.

[0015] Optionally, it also includes correcting the crown moment amplification factor for variable-height arches, and constructing a correction formula based on the crown moment amplification factor for variable-width arches, combined with the antisymmetric first-order stability factor of the arch rib, the axial compressive stress under dead load, and the span.

[0016] As can be seen from the above technical solution, compared with the prior art, the present invention discloses a method for calculating the geometric nonlinear structural response of a long-span arch bridge, the beneficial effects of which are: 1) This invention establishes a discretized polynomial analytical model, and combines it with the principle of elastic compression integral to achieve accurate solution of nonlinear response, with total deflection error <4.4% and total bending moment error <5.9%, balancing accuracy and efficiency; 2) The system reveals the M-shaped distribution of nonlinear bending moment, the movement law of inflection point, and the suppression effect of arch rib stiffness, providing theoretical support for structural stress analysis; 3) A correction formula for the bending moment amplification factor of variable-height arches was proposed, its applicable range was clarified, and the problem of nonlinear effect evaluation of arch bridges with different cross-sectional forms was solved; 4) It can meet the dual requirements of refined design and rapid design for long-span arch bridges, and provide a reliable basis for structural safety reserve assessment and optimization design. Attached Figure Description

[0017] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the provided drawings without creative effort.

[0018] Figure 1 A flowchart illustrating a method for calculating the geometric nonlinear structural response of a long-span arch bridge, provided by this invention. Figure 2a A schematic diagram of the geometric nonlinear deformation of the hingeless arch provided by the present invention; Figure 2b The schematic diagram of the equivalent internal forces of the hingeless arch provided by the present invention is as follows: Figure 3a The 400m arch rib provided for this invention ( Comparison chart of total deflection calculation results; Figure 3b The 400m arch rib provided for this invention ( Comparison chart of total deflection calculation results; Figure 3c The 600m arch rib provided for this invention ( Comparison chart of total deflection calculation results; Figure 3d The 600m arch rib provided for this invention ( Comparison chart of total deflection calculation results; Figure 3e The 800m arch rib provided for this invention ( Comparison chart of total deflection calculation results; Figure 3f The 1000m arch rib provided for this invention ( Comparison chart of total deflection calculation results; Figure 4a The 400m arch rib provided for this invention ( Comparison chart of total bending moment calculation results; Figure 4b The 400m arch rib provided for this invention ( Comparison chart of total bending moment calculation results; Figure 4c The 600m arch rib provided for this invention ( Comparison chart of total bending moment calculation results; Figure 4d The 600m arch rib provided for this invention ( Comparison chart of total bending moment calculation results; Figure 4e The 800m arch rib provided for this invention ( Comparison chart of total bending moment calculation results; Figure 4f The 1000m arch rib provided for this invention ( Comparison chart of total bending moment calculation results; Figure 5a The 600m arch rib provided for this invention ( The total bending moment distribution and variation diagram at half span; Figure 5b The 600m arch rib provided for this invention ( The total bending moment distribution and variation diagram at half span; Figure 6a The graph showing the variation of the total bending moment of the key section of the 600m arch rib as a function of loading history, provided by this invention; Figure 6b The present invention provides a graph showing the variation of the total bending moment at a key section of an 800m arch rib as a function of the loading process. Figure 6c The graph showing the variation of the total bending moment of the key section of the 1000m arch rib with the loading process provided by this invention; Figure 7a The 400m arch rib provided for this invention ( Comparison chart of nonlinear bending moment calculation results for ( ) time period; Figure 7b The 400m arch rib provided for this invention ( Comparison chart of nonlinear bending moment calculation results for ( ) time period; Figure 7c The 600m arch rib provided for this invention ( Comparison chart of nonlinear bending moment calculation results for ( ) time period; Figure 7d The 600m arch rib provided for this invention ( Comparison chart of nonlinear bending moment calculation results for ( ) time period; Figure 7eThe 800m arch rib provided for this invention ( Comparison chart of nonlinear bending moment calculation results for ( ) time period; Figure 7f The 1000m arch rib provided for this invention ( Comparison chart of nonlinear bending moment calculation results for ( ) time period; Figure 8 A comparison chart of the differences in the bending moment amplification coefficients for variable-height and variable-width arches provided for this invention; Figure 9a The diagram shows the distribution and variation of the moment amplification factor for an arch rib with a constant load linear elastic stability coefficient of 2, provided by this invention. Figure 9b The diagram shows the distribution and variation of the moment amplification factor for an arch rib with a constant load linear elastic stability coefficient of 4, provided by this invention. Figure 9c The diagram shows the distribution and variation of the moment amplification factor for an arch rib with a dead load linear elastic stability coefficient of 6, provided by this invention. Figure 10a The curve of the bending moment increase factor of the arch crown section when L:400m and σ:8MPa is provided for the present invention; Figure 10b The curve of the bending moment increase factor of the arch crown section when L: 600m and σ: 8MPa is provided for the present invention; Figure 10c The curve of the bending moment increase factor of the arch crown section when L: 800m and σ: 8MPa is provided for the present invention; Figure 10d The curve of the bending moment increase factor of the arch crown section when L:400m and σ:12MPa is provided for the present invention; Figure 10e A curve showing the increase in bending moment at the crown section when L: 600m and σ: 12MPa is provided for this invention. Figure 10f The curve of the bending moment increase factor of the arch crown section when L: 800m and σ: 12MPa is provided for the present invention; Figure 10g The curve of the bending moment amplification factor at the arch crown section when L: 400m and σ: 16MPa is provided for this invention; Figure 10h The curve of the bending moment amplification factor at the arch crown section when L: 600m and σ: 16MPa is provided for this invention; Figure 10i The curve of the bending moment increase factor of the arch crown section when L: 800m and σ: 16MPa is provided for the present invention. Detailed Implementation

[0019] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0020] See Figure 1 As shown, this invention discloses a method for calculating the geometric nonlinear structural response of a long-span arch bridge, comprising the following steps: S1. Set a reasonable arch axis and establish the second-order nonlinear equilibrium equation for the deflection of a long-span arch bridge. S2. Discretize the reasonable arch axis into a polynomial that can be solved individually, and derive the nonlinear deflection expression and nonlinear bending moment expression for long-span arch bridges based on the second-order nonlinear equilibrium equation of deflection. S3. Based on the principle that the integral of elastic compression and compressive strain along the arc length of the arch axis is equal, the unknown parameters in the nonlinear deflection expression and nonlinear bending moment expression are solved to obtain the nonlinear deflection and nonlinear bending moment of the long-span arch bridge. S4. Verify the calculation methods from S1 to S3 using a numerical simulation method that considers geometric nonlinearity. S5. Numerical simulation analysis was conducted to study the variation and spatial distribution of the nonlinear structural response of a long-span arch bridge as it loads.

[0021] Furthermore, the second-order flexural nonlinear equilibrium equation in S1 is established based on the control differential equation of the arch structure, and the total bending moment of the structure that takes into account the second-order effect includes the linear elastic bending moment of the structure.

[0022] Furthermore, the arch structure corresponding to the reasonable arch axis in S1 is a statically indeterminate hingeless arch, and the additional bending moment is the bending moment generated by the superposition of the additional bending moment of the three-hinged arch and the redundant force, which includes the redundant horizontal force and bending moment.

[0023] Furthermore, the nonlinear deflection and nonlinear bending moment expressions in S2 are composed of cosine terms and higher-order polynomials, and are functions of the structural loading process and response space distribution.

[0024] Furthermore, when the reasonable arch axis is a parabola, the nonlinear displacement expression consists of a cosine curve and a parabola, and the nonlinear bending moment expression is a cosine curve.

[0025] Specifically, arch bridges deform under load, causing the pressure line to deviate from the arch axis. According to mechanics, this deviation generates an additional bending moment within the arch. For a statically determinate three-hinged arch, the additional bending moment of the arch rib is expressed as the product of the horizontal reaction force and the deflection. However, for statically indeterminate hingeless arches with redundant constraints, their additional bending moment cannot be simply calculated according to... Instead of using the additional bending moment of the three-hinged arch, the additional bending moment of the hingeless arch should be represented by the bending moment generated by the superposition of the additional bending moment and the redundant force. Figure 2a , Figure 2b As shown. Due to the symmetry of the arch structure, there is a redundant horizontal force. With bending moment Then the additional bending moment of the hingeless arch is expressed as equation (1).

[0026] (1)

[0027] Based on the governing differential equation of the hingeless arch, the second-order deflection differential equation is given by equation (2), where... The total structural bending moment, taking into account second-order effects, is expressed as equation (3). In equation (3), For the structural line elastic bending moment .

[0028] (2) (3) in, The optimal arch axis for the arch structure. The elastic center of the arch, Let be the second derivative of the arch deflection equation. Let be the axial compressive deformation function of the arch. The first derivative of the reasonable arch axis of the arch structure. The elastic modulus of the arch structure material. The equation for the variation of the moment of inertia of the arch section along the arch span direction is given. For the linear elastic bending moment of the arch structure.

[0029] For an arch bridge with constant height and varying width and uniform stress, the compressive strain is constant, and we have: , The equilibrium equation is equation (4). Taking the second derivative of equation (4), we obtain equation (5).

[0030] (4) (5) in, The moment of inertia of the arch crown section is [value missing]. Let be the second derivative of the reasonable arch axis of the arch structure. The elastic horizontal excess force of the arch. The elastic center of the arch, The redundant fixed-end moment of the arch. It is the fourth derivative of the arch deflection equation.

[0031] make , , Equation (5) simplifies to Equation (6), at which point the redundant horizontal force of the arch structure becomes ,in This refers to the redundant horizontal force obtained by solving according to linear elasticity theory. This is the additional redundant force considering nonlinear effects. In equation (6) This represents the vertical displacement caused solely by bending deformation. Furthermore, equation (6) satisfies the boundary conditions in equation (7).

[0032] (6) (7) in, Let be the fourth derivative of the arch bending deflection. Let be the second derivative of the arch bending deflection. The nonlinear horizontal redundant force coefficient of the arch. The loading process characterization coefficients are used.

[0033] Expanding the arch axis, we obtain the second derivative. Substituting into equation (6), and according to the boundary condition equation (7), the bending deflection is obtained. With total deflection The polynomial analytical expressions (8) and (9) are given.

[0034] (8) (9) in, The coefficient characterizing the arch axis, also known as the arch crown curvature, It is half the span of the arch. , Calculate coefficients for the process. These are the coordinate values ​​of the arch along its span.

[0035] Substituting equation (9) into equation (1), we obtain the analytical expression (10) for the nonlinear bending moment of the arch. From equations (9) and (10), it can be seen that the nonlinear deflection and the nonlinear deflection-bending moment of the arch are related to... and The function, where Indicates the loading process of the structure. This represents the spatial distribution of the response. Meanwhile, the arch's nonlinear deflection and nonlinear deflection moment are composed of cosine terms and higher-order polynomials.

[0036] (10)

[0037] For parabolic arch bridges, there are At this point, equations (9) and (10) are simplified to equations (11) and (12). From equations (11) and (12), it can be seen that the nonlinear displacement of the parabolic arch bridge is composed of a cosine curve and a parabola, and the nonlinear bending moment is a cosine curve.

[0038] (11) (12) This led to the establishment of analytical expressions for the nonlinear deflection and nonlinear bending moment of long-span arch bridges.

[0039] Furthermore, in the process of solving the unknown parameters in S3, it is necessary to combine the variation characteristics of the geometric stiffness of the arch rib and consider the nonlinear variation of the redundant horizontal force. The redundant horizontal force includes the redundant horizontal force solved according to the linear elastic theory and the additional redundant force after considering the nonlinear effect.

[0040] Specifically, the analytical expressions for the nonlinear deflection and bending moment of the arch (9) and (10) still contain... The coefficients are yet to be solved. During loading, the geometric stiffness of the arch rib changes continuously, and the redundant horizontal force is related to the arch rib stiffness. Therefore, the redundant horizontal force will exhibit nonlinear changes, as calculated from the elastic redundant horizontal force. Change to Based on the principle that elastic compression and compressive strain are equal along the arc length integral, we obtain equation (13).

[0041] (13)

[0042] in, Let be the distribution function of the cross-sectional area of ​​the arch rib along the span direction. This represents the axial compressive strain value of the arch.

[0043] Equation (13) involves integral operations. To simplify the formula, we will... , , , Substituting into equation (9), we get The expressions (14) and (15) are given.

[0044] (14) (15) in, For the bending deflection of the arch, , The coefficients of the quadratic and quartic terms obtained by Taylor expansion of the arch axis, ~ , ~ These are all simplified formulas, and the calculation coefficients used are for ease of calculation; Combine equation (15) with Substituting into equation (13), the left side of equation (13) integrates to equation (13), and the right side integrates to equation (14). Therefore, The coefficients can be solved according to equation (18).

[0045] (16) (17) (18) Substituting equation (18) into (9) and (10) yields the nonlinear deflection and nonlinear bending moment of the arch. For a parabola, Substituting into equation (18) yields equation (19), which can be further simplified to obtain equation (20).

[0046] (19) (20) For equal stress arch bridges Substituting into (15) yields the coefficient. Expression (21).

[0047] (twenty one)

[0048] Will Substituting equation (21) into equation (18) yields equation (22).

[0049] (twenty two)

[0050] Will Substituting back into equations (9) and (10), the nonlinear deflection and nonlinear bending moment of the arch structure can be obtained, thus completing the nonlinear response solution of a long-span arch bridge under a reasonable arch axis.

[0051] Furthermore, the numerical simulation in S4 adopts the Newton-Raphson iterative method. The simulation verification indicators include the total deflection and total bending moment of the long-span arch bridge. The calculation error of the total deflection is less than 4.4%, and the calculation error of the total bending moment is less than 5.9%.

[0052] Furthermore, the nonlinear structural response in S5 includes: nonlinear bending moment, the spatial distribution of which is M-shaped, passing through two zero-moment points within the semi-arch range, and exhibiting two positive bending moment peaks and one negative bending moment peak.

[0053] Furthermore, the load in the loading process of S5 is set to 1.75 times the dead load of the arch rib, and is divided into 10 levels for loading analysis.

[0054] Furthermore, it also includes a correction to the crown moment amplification factor for variable-height arches, and a correction formula is constructed based on the crown moment amplification factor for variable-width arches, combined with the antisymmetric first-order stability factor of the arch rib, the axial compressive stress under constant load, and the span.

[0055] Specifically, in one particular embodiment, (1) Comparison of total deflection To verify the accuracy of the calculation results obtained by this method, the results were compared with those obtained from the geometric nonlinear analysis of the finite element method. The calculated values ​​are shown as scatter points in the figure, while the simulated values ​​are shown as curves. The parameters compared include spans of 400m, 600m, 800m, and 1000m; axial compressive stresses of 14MPa and 16MPa; and linear elastic stability coefficients of 2 and 7.

[0056] Considering that current design codes use a safety factor of 1.75 for the ultimate bearing capacity design of long-span arch bridges, meaning the ultimate bearing capacity of the structure must be greater than 1.75 times the design load, subsequent nonlinear analyses will use a total calculated load of 1.75 times the arch rib dead load. With a total calculated load of 1.75 times the arch rib dead load, the Newton-Raphson iterative method is used in the finite element method for geometric nonlinear analysis, with each analysis step using a load of 0.1 times the total calculated load.

[0057] Depend on Figures 3a-3f It can be seen that the maximum calculation error is located at the 400m arch rib ( At the mid-span position, the calculated value by this method is -919.5mm, while the simulated value is -951.7mm. The calculated result by this method is 32.2mm smaller than the simulated result, with an error percentage of 4.4%.

[0058] (2) Comparison of total bending moments

[0059] To verify the accuracy of the total bending moment calculation results of this method, the finite element method and the calculation results of this method are plotted in Figure 4.

[0060] Depend on Figures 4a-4f It can be seen that the maximum calculation error is located at the 400m arch rib ( At the mid-span position, the calculated value by this method is 782.7 kN×m, while the simulated value is 831.8 kN×m. The calculated result by this method is 49.1 kN×m smaller than the simulated result, with an error percentage of 5.9%.

[0061] The calculation error for total deflection using this method is less than 4.4%, and the calculation error for total bending moment is less than 5.9%. The calculation error for total bending moment decreases as the stability coefficient of the arch rib increases. The first-order stability coefficient of the arch rib is usually designed to be greater than 2, and the calculation error for total bending moment using this method will be further reduced in practical applications. Therefore, the calculation method of this invention can achieve accurate solutions for the deflection and bending moment of long-span arch bridges.

[0062] (1) Total bending moment

[0063] In the aforementioned derivation, the total bending moment of a long-span arch bridge is expressed as... and The function, where This indicates the loading process of the structure, when the elastic modulus E of the material is constant. Horizontal reaction force of arch ribs under load and the moment of inertia of the arch section Related, among which The impact can be analyzed by increasing the calculated load, while It is linearly related to the linear elastic stability coefficient, and can be analyzed by setting different linear elastic stability coefficients. The effect on nonlinear bending moment.

[0064] Considering that current design codes use a safety factor of 1.75 for the ultimate bearing capacity design of long-span arch bridges, the total load is calculated as 1.75 times the dead load of the arch ribs. To analyze the horizontal reaction forces of the arch ribs... The effect on the total bending moment was calculated by dividing the total load into 10 levels. This was done to analyze the moment of inertia of the arch crown section. Impact on total bending moment Figure 5a , Figure 5b The critical loads for linear elastic instability of the medium structure are designed to be 2 times and 7 times the dead load, respectively.

[0065] Depend on Figure 5a It can be seen that when the load on the arch rib is close to the critical load for instability, the nonlinear effect of the arch rib bending moment is significant. Specifically, the positive bending moment at the arch crown increases nonlinearly with increasing loading force, while the negative bending moment at the arch foot first increases and then decreases. The peak value of the negative bending moment and the inflection point of the arch rib bending moment also shift from the arch foot position to the arch crown position. When the loading force reaches the calculated total load, the peak value of the negative bending moment moves to around L / 8, the inflection point of the arch rib bending moment moves from L / 5 to 2L / 7, and the maximum value of the positive bending moment is located at the arch crown section. Therefore, when the linear elastic instability critical load of the structure is twice that of the load, under the action of 1.75 times the symmetrical dead load, the arch crown section is the control section of the bending moment.

[0066] The total bending moment of the arch structure consists of a cosine term and a polynomial. As the loading force increases, the horizontal reaction force of the arch rib increases, which in turn leads to a cosine period. As the period decreases, the inflection point of the arch rib bending moment shifts towards the arch crown. When the period is less than l, i.e. At this time, the negative bending moment at the arch foot decreases. (From...) Figure 5b It can be seen that when the load on the arch rib is less than the critical load for instability, the nonlinear effect of the arch rib bending moment is not obvious, and the bending moment of the arch rib across the entire span is approximately linearly increasing. When the loading force reaches the total load, the inflection point of the arch rib bending moment only moves L / 30 towards the top of the arch.

[0067] To further clarify the bending moment variation law of the key sections of the main arch under the reasonable arch axis, a parametric analysis was carried out with the span, axial compressive stress and first-order antisymmetric stability coefficient as variables.

[0068] Figures 6a-6c The calculation results are shown for arch rib spans of 600m, 800m, and 1000m, respectively. As the arch rib stiffness increases, the nonlinear effect of the bending moment is suppressed. When the first-order antisymmetric instability critical load of the arch rib reaches 4 times the dead load, the bending moment at the arch crown increases by approximately 29.5% under 1.75 times the dead load, while the bending moment at the arch foot decreases by approximately 30%. The stress level of the arch rib and the span have little effect on the nonlinear effect of the bending moment; the nonlinear effect of the arch rib bending moment is mainly controlled by the arch rib stiffness.

[0069] (2) Nonlinear incremental bending moment

[0070] To independently analyze the nonlinear incremental effect of the arch structure, the nonlinear effect is isolated from the overall effect and solved separately. The nonlinear bending moment of the arch structure can be expressed as Equation (23), where Equation (23) .

[0071] (twenty three)

[0072] Boundary conditions The formula can be obtained. As can be seen from the formula, the half-span integral of the arch rib bending moment is equal to 0. This equation is based on the variable-width arch. When the relative rotation angle between the arch crown and the arch foot is 0, it also indicates that a large-span arch bridge with a reasonable arch axis is constructed, and the stiffness of its arch ribs follows the principle of... When distributed, the sum of its positive bending moments equals the sum of its negative bending moments. This conclusion applies not only to nonlinear bending moments but also to linear bending moments.

[0073] (twenty four) (25) According to the definition of a center of elasticity, we have At the same time, nonlinear deflection is introduced. Expression (26) can be solved to obtain The parsing expression (27).

[0074] (26) (27) Depend on Figures 7a-7f It can be seen that the trend of the results obtained by this method is consistent with that of the finite element method, and the numerical values ​​are close. The maximum calculation error is located at the 400m arch rib ( At the mid-span position, the calculated value by this method is 39.8 kN×m, while the simulated value is 45.2 kN×m. The calculated result by this method is 5.4 kN×m smaller than the simulated result, with an error percentage of 11.9%.

[0075] The nonlinear bending moment curves of the arch all exhibit an M-shape. Within the semi-arch area, the nonlinear bending moment passes through two zero-moment points, exhibiting two positive bending moment peaks at the arch crown and arch foot, and one negative bending moment peak. When the elastic stability coefficient of the arch rib is 2, the two zero-moment points are located near L / 13 and L / 3, respectively, and the negative bending moment peak is located near 3L / 16. When the elastic stability coefficient of the arch rib is 7, the two zero-moment points are located near L / 20 and L / 3, respectively, and the negative bending moment peak is located near L / 6.

[0076] (3) Moment amplification factor

[0077] The moment amplification factor is the ratio of the total bending moment to the linear bending moment. The linear bending moment under a reasonable arch axis is... The nonlinear bending moment is Then the bending moment increase factor It can be expressed as equation (28).

[0078] (28)

[0079] Through the analysis of nonlinear bending moment, the arch crown section is the control section of bending moment under symmetrical dead load. The variation law of the arch crown bending moment amplification coefficient is further discussed. Substituting into equation (28) The analytical expression for the crown moment amplification factor is obtained (29).

[0080] (29)

[0081] The aforementioned is an analytical method for the nonlinear response of an arch bridge with constant width and varying height. For an arch bridge with constant width and varying height, the equilibrium equation becomes equation (30). For about The function is a second-order differential equation with variable coefficients, and an analytical solution cannot be obtained directly.

[0082] (30)

[0083] Therefore, to further analyze the moment amplification factor of the equal-width variable-height arch, nonlinear analysis was conducted using 288 established simulation models. The moment amplification factors of the variable-width and variable-height arches were compared to analyze the differences between them. The differences in the moment amplification factor at the crown of the 288 variable-width and variable-height arch ribs were statistically analyzed. Figure 8 , Figure 8The axial compressive stress ranges of the central arch rib under constant load are 8–12 MPa and 14–16 MPa, respectively. The error data in the figure is expressed as (bending moment amplification factor of variable height arch - bending moment amplification factor of variable width arch) / bending moment amplification factor of variable width arch.

[0084] Depend on Figure 8 It can be seen that, under the same parameters, the increase factor of the crown moment of the variable-height arch is greater than that of the variable-width arch, and the error is less than 9%. The error decreases with the increase of axial compressive stress and arch rib stiffness under dead load, and increases with the increase of span. The error is approximately inversely proportional to the square of the axial compressive stress and the stability coefficient, and approximately directly proportional to the span. Therefore, the corrected calculation formula (31) for the increase factor of the crown moment of the variable-height arch can be obtained.

[0085] (31)

[0086] In equation (31), , These represent the coefficients for increasing the bending moment at the crown of arches with varying heights and widths, respectively. , , These represent the antisymmetric first-order stability coefficient of the arch rib, the axial compressive stress under dead load (unit: MPa), and the span (unit: m), respectively. As a correction factor, based on the calculation results according to the present invention, it is statistically obtained according to equation (32). The mean is 0.0071.

[0087] (32)

[0088] (4) Applicability analysis of the moment amplification factor

[0089] To further analyze the spatial distribution and variation of the moment amplification factor of the arch rib with the loading process, this method was used to calculate the moment amplification factor of a 400m arch rib. The total load was calculated to be 1.75 times the dead load, and the linear elastic stability coefficients of the arch rib under dead load were set to 2, 4, and 6, respectively. The calculation results were plotted on [date missing]. Figures 9a-9c As shown.

[0090] Depend on Figures 10a-10i It can be seen that during the entire loading process, the moment amplification factor at the arch foot is <1, while the moment amplification factor at the arch crown is >1. As the loading force increases, the moment amplification factor at the arch foot decreases while that at the arch crown increases. Consistent with the nonlinear effect of bending moment, the moment amplification factor of the arch rib is also suppressed by stiffness. When the stability coefficient of the arch rib is set to 2, 4, and 6, the moment amplification factor at the arch foot ranges from (0.2 to 1), (0.7 to 1), and (0.8 to 1), while the moment amplification factor at the arch crown ranges from (1 to 2), (1 to 1.3), and (1 to 1.2).

[0091] To further analyze the variation law of the moment amplification factor of key sections, the moment amplification factor of the arch crown section was calculated using this method, and parametric analysis was carried out with the span, axial compressive stress and first-order antisymmetric stability coefficient as variables.

[0092] The moment amplification factor of the arch crown section increases with the increase of the loading force and the axial compressive stress of the arch rib under dead load, and decreases with the increase of the span and the stiffness of the arch rib. When the stability coefficient of the arch rib is set to 2 and the axial compressive stress of the arch rib under dead load is 16MPa, the moment amplification factors of the arch crown section of the 400m, 600m, and 800m arch bridges are 2.3, 2.05, and 1.9, respectively.

[0093] The above analysis shows that the moment amplification factor is a function of the horizontal reaction force of the arch rib. and location The coefficient of change. Looking at the distribution of the moment amplification coefficient across the entire span, during the entire loading process, the moment amplification coefficient increases progressively from the arch foot to L / 16 and is always <1, approximating 1 near L / 16. The moment amplification coefficient in the range from L / 16 to L / 3 initially increases non-linearly from 1, and at the arch's elastic center... (Near L / 4), the linear bending moment is 0, and the bending moment amplification factor increases to positive infinity. Within the range of L / 4 to L / 3, the bending moment amplification factor increases from negative infinity to 1. From L / 3 to the arch crown, the bending moment amplification factor increases progressively and is always greater than 1. At this point, if internal force adjustments are made based on a reasonable arch axis to reduce the linear bending moment during the completed bridge stage under dead load, then a very large bending moment amplification factor will appear on all arch ribs.

[0094] Therefore, the moment amplification factor varies significantly across different sections of the arch rib span, and it does not increase across all sections; some regions have a value less than 1, while others exhibit extremely high values. Furthermore, the moment amplification factor changes differently across different sections with increasing loading force, decreasing at the arch foot and increasing at the arch crown. In this case, the moment amplification factor is no longer suitable for characterizing the nonlinear effect of the bending moment.

[0095] When performing rapid design, considering the symmetrical dead load, the moment amplification factor increases progressively from the arch foot to L / 16 and is always <1. Although the moment amplification factor is large in the range from L / 16 to L / 3, the linear elastic moment is small and does not control the design, so it can be conservatively considered as 1, i.e., the moment amplification factor is not considered. However, the moment amplification factor increases progressively from L / 3 to the arch crown and is always >1. In this case, the nonlinear effect of the moment can be approximated by the moment amplification factor.

[0096] The various embodiments in this specification are described in a progressive manner, with each embodiment focusing on the differences from other embodiments. The same or similar parts between the various embodiments can be referred to each other.

[0097] The above description of the disclosed embodiments enables those skilled in the art to make or use the invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the invention. Therefore, the invention is not to be limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A method for calculating the geometric nonlinear structural response of a long-span arch bridge, characterized in that, Includes the following steps: S1. Set a reasonable arch axis and establish the second-order nonlinear equilibrium equation for the deflection of a long-span arch bridge. S2. Discretize the reasonable arch axis into a polynomial that can be solved individually, and derive the nonlinear deflection expression and nonlinear bending moment expression for long-span arch bridges based on the second-order nonlinear equilibrium equation of deflection. S3. Based on the principle that the integral of elastic compression and compressive strain along the arc length of the arch axis is equal, the unknown parameters in the nonlinear deflection expression and nonlinear bending moment expression are solved to obtain the nonlinear deflection and nonlinear bending moment of the long-span arch bridge. S4. Verify the calculation methods from S1 to S3 using a numerical simulation method that considers geometric nonlinearity. S5. Numerical simulation analysis was conducted to study the variation and spatial distribution of the nonlinear structural response of a long-span arch bridge as it loads.

2. The method for calculating the geometric nonlinear structural response of a long-span arch bridge according to claim 1, characterized in that, The second-order flexural nonlinear equilibrium equation in S1 is established based on the control differential equation of the arch structure. The total bending moment of the structure that takes into account the second-order effect includes the linear elastic bending moment of the structure.

3. The method for calculating the geometric nonlinear structural response of a long-span arch bridge according to claim 1, characterized in that, The arch structure corresponding to the reasonable arch axis in S1 is a statically indeterminate hingeless arch. The additional bending moment is the bending moment generated by the superposition of the additional bending moment of the three-hinged arch and the redundant force. The redundant force includes the redundant horizontal force and bending moment.

4. The method for calculating the geometric nonlinear structural response of a long-span arch bridge according to claim 1, characterized in that, The nonlinear deflection and nonlinear bending moment expressions in S2 consist of cosine terms and higher-order polynomials, and are functions of the structural loading process and response space distribution.

5. The method for calculating the geometric nonlinear structural response of a long-span arch bridge according to claim 4, characterized in that, When the reasonable arch axis is a parabola, the nonlinear displacement expression consists of a cosine curve and a parabola, and the nonlinear bending moment expression is a cosine curve.

6. The method for calculating the geometric nonlinear structural response of a long-span arch bridge according to claim 1, characterized in that, In the process of solving the unknown parameters in S3, it is necessary to combine the variation characteristics of the geometric stiffness of the arch rib and consider the nonlinear variation of the redundant horizontal force. The redundant horizontal force includes the redundant horizontal force solved according to the linear elastic theory and the additional redundant force after considering the nonlinear effect.

7. The method for calculating the geometric nonlinear structural response of a long-span arch bridge according to claim 1, characterized in that, The numerical simulation in S4 uses the Newton-Raphson iterative method. The simulation verification indicators include the total deflection and total bending moment of the long-span arch bridge. The calculation error of the total deflection is less than 4.4%, and the calculation error of the total bending moment is less than 5.9%.

8. The method for calculating the geometric nonlinear structural response of a long-span arch bridge according to claim 1, characterized in that, The nonlinear structural response in S5 includes: nonlinear bending moment, the spatial distribution of which is M-shaped, passing through two zero-moment points within the semi-arch range, and exhibiting two positive bending moment peaks and one negative bending moment peak.

9. The method for calculating the geometric nonlinear structural response of a long-span arch bridge according to claim 1, characterized in that, In S5, the load in the loading process is set to 1.75 times the dead load of the arch rib, and is divided into 10 levels for loading analysis.

10. The method for calculating the geometric nonlinear structural response of a long-span arch bridge according to claim 1, characterized in that, It also includes a correction to the crown moment amplification factor for variable-height arches, and a correction formula based on the crown moment amplification factor for variable-width arches, combined with the antisymmetric first-order stability factor of the arch rib, the axial compressive stress under dead load, and the span.

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