Method for analyzing cross section bearing capacity of large-span arch bridge under dead load leading

Abstract

The invention discloses a method for analyzing the section bearing capacity of a large-span arch bridge under dead load leading, and relates to the technical field of bridge engineering.The method comprises the steps that firstly, the failure mode of the large-span arch bridge under a positive and antisymmetric deformation mode is analyzed, and the eccentric limit when the section of the large-span arch bridge is damaged under the high axial force level is judged; secondly, large-span arch bridge in-plane bearing capacity damage experiment research is carried out, and weak links and a control section of an arch structure are defined; and finally, further analyzing and determining influence factors of the in-plane bearing capacity of the large-span arch bridge based on a refined simulation model verified by a test. Through integration of theoretical derivation, test verification and analogue simulation, a stress mechanism and a failure law of the large-span arch bridge in positive and antisymmetric deformation modes are systematically analyzed, the section bearing capacity is accurately calculated, and the design is optimized; according to the method, the problem of a traditional equivalent beam column method overestimation bending moment nonlinear effect is solved, accurate analysis of the cross section bearing capacity of the large-span arch bridge is achieved, and technical support is provided for efficient design of the large-span arch bridge.

Description

Technical Field

[0001] This invention relates to the field of bridge engineering technology, and more specifically to a method for analyzing the cross-sectional bearing capacity of long-span arch bridges under constant load. Background Technology

[0002] Currently, with the continuous increase in bridge spans, the ultimate bearing capacity problem of long-span arch bridges is becoming increasingly prominent. Current bridge codes use a simplified "equivalent beam-column" method to calculate the bearing capacity of reinforced concrete arch rib sections. However, this method does not consider the actual positive and negative symmetric deformation coupling stress mode of arch bridges, leading to an overestimation of the bending moment amplification factor. This severely underestimates the true bearing capacity of long-span arch bridges, resulting in design redundancy and hindering the further development of arch bridge spans.

[0003] Existing analytical methods often treat the stress on arch bridges as equivalent to that on a compression-bending beam, neglecting the independent characteristics and coupling effects of the symmetrical deformation dominated by symmetrical dead load and the antisymmetrical deformation dominated by asymmetrical live load. This fails to accurately reflect the section failure mechanism of long-span arch bridges under high axial force levels.

[0004] Therefore, there is an urgent need for a cross-sectional bearing capacity analysis method based on the actual deformation mode to improve the accuracy of bearing capacity assessment and the rationality of design for long-span arch bridges. Summary of the Invention

[0005] In view of this, the present invention provides a method for analyzing the cross-sectional bearing capacity of a long-span arch bridge under the dominance of dead load, in order to solve the problems existing in the background art.

[0006] To achieve the above objectives, the present invention adopts the following technical solution: A method for analyzing the cross-sectional bearing capacity of a long-span arch bridge under dead load dominance includes: S1. Obtain the basic parameters of the long-span arch bridge, including the arch bridge span, rise-to-span ratio, material mechanical properties, cross-sectional dimensions, and dead and live load distribution characteristics; S2. Based on the aforementioned basic parameters, and in conjunction with the plane section assumption, Hooke's law, and finite displacement theory, the governing differential equations of the arch structure under the positive symmetric deformation mode and the antisymmetric deformation mode are derived respectively. S3. Based on the aforementioned control differential equation, decompose the symmetrical dead load and asymmetrical live load borne by the arch bridge, analyze the stress characteristics dominated by axial compression under the positive symmetrical deformation mode and the stress characteristics dominated by bending compression under the antisymmetrical deformation mode, and clarify the cross-sectional failure eccentricity limit and failure mode under the coupled action of the two deformation modes. S4. Based on the principle of stiffness equivalence, design a scaled-down model of a long-span arch bridge, build an unconstrained adaptive loading system and a lateral limiting device, conduct an ultimate bearing capacity loading test on the scaled-down model, and obtain data on displacement, strain and bending moment amplification factor throughout the loading process. S5. Based on experimental data, establish a dual nonlinear simulation model that considers both geometric nonlinearity and material nonlinearity, and calibrate the material constitutive parameters and boundary conditions of the simulation model using experimental data; S6. Using the calibrated simulation model, simulate the stress distribution and plastic hinge formation process of the cross section under the coupling action of positive and negative symmetric deformation, and calculate the ultimate bearing capacity of the cross section; S7. Analyze the influence of live load ratio, structural stiffness, reinforcement method, initial bending moment and initial defects on the section bearing capacity, output the section bearing capacity analysis results and optimize the arch bridge structure design.

[0007] Optionally, the mechanical properties of the materials include the compressive strength and modulus of elasticity of concrete, and the yield strength and modulus of elasticity of steel; the dead load includes the self-weight of the arch ring and the load of the building on the arch, and the live load includes the moving load of highway or railway, and the distribution characteristics of the live load are quantified by the ratio of the live load to the dead load across the entire span.

[0008] Optionally, in S2, the governing differential equation corresponding to the positive symmetric deformation mode is:

[0009] The governing differential equation for the antisymmetric deformation mode is:

[0010] Wherein, for any infinitesimal element mn of the arch along the arch axis, the horizontal displacement of point m is: The vertical displacement is Then the horizontal displacement of point n is The vertical displacement is M is the bending moment of the arch section. Let be the axial force at any point in the arch. The elastic modulus of the arch material. , Let be the bending stiffness and cross-sectional area at any point in the arch. , .

[0011] Optionally, in S3, the failure mode under the symmetrical deformation mode is the five-hinged mechanism failure caused by the crushing of the top plate of the arch section, and the failure mode under the antisymmetrical deformation mode is the four-hinged mechanism failure caused by the crushing of the bottom plate of the arch foot section on the loading side. The eccentricity limit of the section failure is determined by the ratio of axial compressive stress to bending stress. When the axial compressive stress is greater than the bending stress, it is a small eccentric compression failure; when the axial compressive stress is less than the bending stress, it is a large eccentric compression failure. Optionally, in S4, the design of the scaled model satisfies the stiffness equivalence principle, and the similarity ratio is determined according to the dimensional relationship of length, area, volume, and stress. The scale ratio is 1:30 to 1:50. The unconstrained adaptive loading system adopts a lever-pulley structure to realize the synchronous application of dead load and half-span live load. The lateral limiting device restricts the out-of-plane deformation of the model to ensure that the in-plane force dominates. Optionally, in S5, the dual nonlinear simulation model uses C3D8R solid elements to simulate concrete and C4R plate elements to simulate steel. The concrete uses a plastic damage constitutive model, and the steel uses an ideal elastoplastic constitutive model. The calibration process involves comparing the load-displacement curves and strain distribution data from simulation and experiment, adjusting the constitutive parameters until the error is less than 5%. Optionally, in S6, the calculation of the ultimate bearing capacity of the section includes: obtaining the maximum compressive stress of the section based on the simulation model; when the maximum compressive stress reaches the design strength of the concrete, the corresponding load is the section strength bearing capacity; when the structure forms a plastic hinge mechanism, the corresponding load is the overall ultimate bearing capacity of the structure, and the smaller of the two values ​​is taken as the final section bearing capacity. Optionally, in S7, the specific method for optimizing the structural design is: reducing the initial bending moment by adjusting the shape of the arch axis, optimizing the section stiffness to homogenize the stress throughout the arch, configuring reinforcement on both sides of the top and bottom plates to improve structural stiffness, and controlling the initial defect amplitude within ±50mm. Optionally, the specific construction process of the dual nonlinear simulation model is as follows: (1) Input the basic information of the model in MATLAB, including the linear shape, the size of the arch cross section and the form of cross section variation; (2) Input the number of longitudinal discrete points N in the model, and calculate the axis inclination angle of the discrete points based on the arch axis of the model; (3) Set the grid size on the cross section, calculate the two-dimensional coordinates of the grid nodes at each discrete point based on the cross section size information, and generate a two-dimensional grid plane group for the arch. (4) Rotate and translate the two-dimensional grid plane group of the arch top in sequence according to the coordinates of the discrete points of the arch axis and the inclination angle, and place each two-dimensional grid plane in the correct position at the discrete point to obtain the three-dimensional node coordinate information of the model; (5) Connect the corresponding nodes of the two two-dimensional mesh planes to generate solid elements, thereby obtaining the element information of the model; if there are one-dimensional line elements on the two-dimensional mesh plane, connect the corresponding nodes of the line elements to generate plate elements. (6) Generate an initialization inp text in Abaqus, and write the calculated model node coordinates and element information into the inp text using MATLAB to complete the establishment of the dual nonlinear simulation model; (7) After the model inp text is created, it can be directly imported into Abaqus to generate a visualization model; or MATLAB can be used to call the Abaqus / Standard solver for calculation and analysis.

[0012] As can be seen from the above technical solution, compared with the prior art, the present invention discloses a method for analyzing the cross-sectional bearing capacity of long-span arch bridges under both positive and negative deformation modes. By integrating theoretical derivation, experimental verification and simulation, the method systematically analyzes the stress mechanism and failure law of long-span arch bridges under both positive and negative deformation modes, accurately calculates the cross-sectional bearing capacity and optimizes the design. The present invention solves the problem of overestimating the nonlinear effect of bending moment in the traditional equivalent beam-column method, realizes the accurate analysis of the cross-sectional bearing capacity of long-span arch bridges, and provides technical support for the efficient design of long-span arch bridges. Attached Figure Description

[0013] Figure 1 This is a schematic diagram of the method flow provided by the present invention; Figure 2 This is a schematic diagram of the arch provided by the present invention in a rectangular coordinate system; Figure 3 This is a schematic diagram of the stress state of a long-span arch bridge provided by the present invention; Figure 4 This invention provides a schematic diagram of the load-displacement curve under multipath failure of a structure. Figure 5 This is a schematic diagram of the antisymmetric failure mode of the arch structure provided by the present invention; Figure 6 This is a schematic diagram of the symmetrical deformation of the arch structure provided by the present invention; Figure 7 This is a schematic diagram of the positive symmetric failure mode of the arch structure provided by the present invention; Figure 8 This is a schematic diagram of the failure mode of the arch structure section provided by the present invention; Figure 9 A schematic diagram of the geometric dimensions of the scaled model provided by the present invention; Figure 10 This is a schematic diagram comparing the calculation results of the model provided by this invention with those of the actual bridge. Figure 11 Elevation layout of the unrestrained loading system for scaled-down testing of a large-span arch bridge provided by the present invention; Figure 12 Three-dimensional diagram of the lateral limiting device for scaled-down test of a large-span arch bridge provided by the present invention; Figure 13A schematic diagram illustrating the key process of establishing the finite element model provided by this invention; Figure 14 A comparison chart of measured and simulated vertical displacement values ​​of a steel-concrete composite truss arch bridge provided by this invention; Figure 15 The diagram shows the calculation results of the model failure mode provided by this invention; Figure 16 This is a schematic diagram of the failure state of a long-span arch bridge under constant load provided by the present invention. Figure 17 The key section load-principal compressive stress curve provided for this invention; Figure 18 The calculation results of the ultimate bearing capacity and failure mode of the arch rib under different stiffnesses provided by this invention are shown in the figure. Figure 19 The load-displacement curve diagram of the key section provided for this invention; Figure 20 The load-displacement curve diagram of the key section provided for this invention; Detailed Implementation

[0014] This invention discloses a method for analyzing the cross-sectional bearing capacity of a long-span arch bridge under the dominance of dead load, such as... Figure 1 As shown, it includes: S1. Obtain the basic parameters of the long-span arch bridge, including the arch bridge span, rise-to-span ratio, material mechanical properties, cross-sectional dimensions, and dead and live load distribution characteristics; S2. Based on the aforementioned basic parameters, and in conjunction with the plane section assumption, Hooke's law, and finite displacement theory, the governing differential equations of the arch structure under the positive symmetric deformation mode and the antisymmetric deformation mode are derived respectively. S3. Based on the aforementioned control differential equation, decompose the symmetrical dead load and asymmetrical live load borne by the arch bridge, analyze the stress characteristics dominated by axial compression under the positive symmetrical deformation mode and the stress characteristics dominated by bending compression under the antisymmetrical deformation mode, and clarify the cross-sectional failure eccentricity limit and failure mode under the coupled action of the two deformation modes. S4. Based on the principle of stiffness equivalence, design a scaled-down model of a long-span arch bridge, build an unconstrained adaptive loading system and a lateral limiting device, conduct an ultimate bearing capacity loading test on the scaled-down model, and obtain data on displacement, strain and bending moment amplification factor throughout the loading process. S5. Based on experimental data, establish a dual nonlinear simulation model that considers both geometric nonlinearity and material nonlinearity, and calibrate the material constitutive parameters and boundary conditions of the simulation model using experimental data; S6. Using the calibrated simulation model, simulate the stress distribution and plastic hinge formation process of the cross section under the coupling action of positive and negative symmetric deformation, and calculate the ultimate bearing capacity of the cross section; S7. Analyze the influence of live load ratio, structural stiffness, reinforcement method, initial bending moment and initial defects on the section bearing capacity, output the section bearing capacity analysis results and optimize the arch bridge structure design.

[0015] In one specific embodiment, the material mechanical properties include the compressive strength and elastic modulus of concrete, and the yield strength and elastic modulus of steel; the dead load includes the self-weight of the arch ring and the load of the building on the arch; the live load includes the moving load of highway or railway, and the live load distribution characteristics are quantified by the ratio of the live load to the dead load across the entire span.

[0016] In a specific embodiment, the derivation of the governing differential equations of the arch also requires the following basic assumptions: Adhering to the plane section assumption, the angle between the normal direction and the tangent direction of the arch section remains unchanged before and after deformation; it obeys Hooke's law; and it follows the finite displacement theory, where small strain results in large displacement. When the arch undergoes large deformation, the cross-sectional shape and area remain unchanged.

[0017] like Figure 2 As shown, for any infinitesimal element mn of an arch along its axis, the horizontal displacement of point m is... The vertical displacement is Then the horizontal displacement of point n is The vertical displacement is Displacement increment and Due to the corner and length expansion cause, This represents axial strain. The displacement increment of the arch structure can be obtained from elasticity theory. and The expressions are (1) and (2).

[0018] (1) (2) From formulas (1) and (2), we can obtain: (3) (4) Differentiating equation (4) yields: (5) According to Hooke's Law, the normal stress at any point on the arch cross section is... and normal strain The following relationship exists: (6) In the formula: —Elastic modulus of the arch material; —Normal stress in the arch section; — Normal strain of the arch section.

[0019] Depend on Given the bending moment M, shear force Q, and axial force N at the composite section, and assuming that the axial force N is positive when the arch section is under compression and the bending moment is positive when the lower edge of the arch section is under tension, the following relationship can be obtained: (7) From the formulas of mechanics of materials: (8) Combination Then (8) can be transformed into the following formula: (9) Substituting equations (7) and (9) into equation (5) yields: (10) From the derivative relationship Equation (10) can be transformed into equation (11).

[0020] (11)

[0021] In the formula, Let be the bending moment at any cross section in the arch; The axial force at any point in the arch; , Let be the bending stiffness and cross-sectional area at any point in the arch; The shear force at any point in the arch; This indicates the vertical displacement of the arch axis. This represents the change in the angle between the tangent to the arch axis and the horizontal line.

[0022] By combining equations (3) and (4), the horizontal displacement can be obtained. and vertical displacement The relationship between them is (12). Equation (12) can be transformed into equation (13).

[0023] (12) (13) Equations (11) and (13) are the governing differential equations of the deformation and internal forces of the arch. If the influence of shear force on displacement is ignored, then equations (11) and (13) are transformed into equations (14) and (15), respectively.

[0024] (14) (15) Neglecting the influence of axial force, equations (11) and (13) are transformed into equations (16) and (17), respectively. Equations (16) and (17) are the basic equations of elasticity theory.

[0025] (16) (17) Analysis of the force mode of an arch Arch bridges in actual operation must withstand both basically symmetrical dead loads. It also needs to withstand asymmetric live loads with antisymmetric components. For its function, see Figure 3 The concentrated forces transmitted from the hangers and columns to the arch ring can be replaced by distributed forces based on the equivalent membrane tensor. In this case, the structural response trends produced by the concentrated and distributed forces are consistent, with only numerical differences. Therefore, the load borne by the arch rib can be approximately decomposed into symmetrically distributed loads. With antisymmetric distributed load Symmetrical load The axial force of the arch bridge structure under load is relatively large, and its axial compression effect cannot be ignored. Structural calculations should be performed according to equations (14) and (15). At this time, the dead load bending moment of the structure consists of two parts: the bending moment M1 caused by the offset of the designed arch axis from the pressure line and the bending moment M2 caused by axial compression. Since the design of the arch axis of the long-span arch bridge takes into account a reasonable arch axis theory, the value of M1 is relatively small. At this time, the structure undergoes symmetrical deformation. Furthermore, the deformation is relatively small, mainly due to axial compression deformation. With bending deformation Together they form, see Figure 3 (b).

[0026] Symmetrical structures under antisymmetric loads and antisymmetric structures (including initial imperfections) under symmetrical loads This will generate a large bending moment in the arch ribs, resulting in significant bending deformation. Since the horizontal force of the arch ribs is symmetrical, the antisymmetric deformation state will not cause symmetrical horizontal forces. At this time, the axial force of the arch ribs is relatively small, and the structure can be calculated according to equations (16) and (17). In this case, the arch bridge is mainly subjected to bending deformation, and the structure is an antisymmetric deformation. And the deformation is relatively large.

[0027] Therefore, arch bridges in actual operation exhibit an asymmetric deformation mode, encompassing both symmetric deformation under symmetric dead loads and antisymmetric deformation under asymmetric live loads.

[0028] In one specific embodiment, arch failure mode analysis is also included; As the stress on an arch bridge increases, the geometry of the arch changes, leading to an increase in the eccentricity of the axial force relative to the arch axis. The resulting additional internal forces further exacerbate the deformation of the arch structure. This synergistic effect of interaction significantly reduces the stiffness of the components, manifesting as a marked softening of the arch's bending stiffness caused by the axial force. This phenomenon is called the second-order effect, and calculations of arch structures incorporating the second-order effect constitute a geometrical nonlinear analysis of the arch bridge. The second-order effect becomes increasingly significant as the percentage of the axial force relative to the Euler critical axial force increases. When the axial force increases to the Euler critical axial force, the arch structure completely loses its bending stiffness and becomes unstable. If material damage and softening are not considered at this point, it is a linear elastic stability analysis; if material damage and softening are considered, it is an elastoplastic stability analysis. The corresponding load at this point is the ultimate load of the structure.

[0029] Although many finite element analysis software programs now offer dual nonlinear analysis capabilities and can determine ultimate bearing capacity, for arch structures, elasto-plastic instability failure and strength failure often intertwine, and the failure mode is related not only to the structural system and material properties but also to the load. For example... Figure 4 As shown, the same structure may have many extreme points or branch points under different loading methods, and the extreme point A corresponding to the minimum ultimate load PA is the desired result.

[0030] Therefore, when solving for the bearing capacity of an arch structure, it is necessary to determine that the ultimate limit state of the arch structure is the ultimate limit state corresponding to the minimum instability limit load, to ensure that no lower ultimate load has occurred before this state. Given this, conducting ultimate bearing capacity and failure mode analysis on long-span arch bridges first requires identifying the most unfavorable arrangement of variable loads. However, different arch structures have different stress states, and live loads and dead loads are coupled, so the corresponding most unfavorable live load arrangement is uncertain. Therefore, it is impossible to determine the failure mode of the structure solely through live load analysis. However, in an arch structure, the coupling of symmetrical reaction forces and symmetrical deformation exacerbates the symmetrical effect but does not affect the antisymmetrical structural response. Conversely, the coupling of symmetrical reaction forces and antisymmetrical deformation exacerbates the antisymmetrical effect but does not affect the symmetrical structural response. Therefore, the symmetrical and antisymmetrical deformations of an arch are independent of each other. Thus, the arch deformation can be independently decomposed into symmetrical deformation and antisymmetrical deformation, and the final failure mode of the arch is the linear superposition of these two deformation modes.

[0031] Antisymmetric failure under asymmetric live load

[0032] The failure of a reinforced concrete arch under concentrated or asymmetrically distributed forces involves a predominantly compressive-bending structure. The continuously increasing bending moment causes tensile cracking of the concrete and yielding of the reinforcing steel at several key sections, drastically weakening the section stiffness and generating significant rotational deformation. This forms plastic hinges, and eventually, as the number of plastic hinges exceeds the statically indeterminate number, a mechanism is formed, leading to the structure's loss of load-bearing capacity and failure. Figure 5 As shown, under these conditions, the load-displacement curve exhibits a significant nonlinear change. The rate of development of the plastic region of the arch towards the axis is greater than the rate of development towards the section height, and the deformation of the structure continues to develop within a relatively long plastic region.

[0033] Under antisymmetric loads, arch bridges mainly bear bending moments and have large bending deformations, but relatively small axial forces. In this case, the structure should be calculated according to equation (16). Taking the second derivative of equation (16) yields the change in the cofactor of the arch rib stiffness. This satisfies the distributed force balance equation (17) for the arch structure. In equation (18), For the linear elastic deformation of an arch under antisymmetric load, it can be determined by... The quadruple integral (18) and antisymmetric boundary conditions get.

[0034] (16) (17) (18) At this point, the arch ribs primarily bear bending moments, resulting in relatively low axial forces and uneven stress distribution. The most unfavorable section reaches its ultimate stress state, while other areas experience lower stress. Material in the ultimate stress state cannot continue to distribute the load, and the load is transferred to materials with lower stress. This internal force / stress redistribution process is lengthy, and the arch ribs exhibit significant ductility, leading to large eccentric failure. At this stage, the concrete side of the most unfavorable section of the arch first experiences tensile cracking, causing the neutral axis to shift and resulting in crushing of the concrete on the other side. After section failure, the structure still possesses a certain residual bearing capacity; that is, the instability failure load that ultimately causes the structure to completely lose its bearing capacity is greater than the hinge load. At section failure, since the concrete's tensile strength is limited, the placement of reinforcing steel at this point can better bear the tensile force, delaying the tensile cracking process and allowing more material to enter a higher stress level, resulting in higher material utilization and thus improving the ultimate bearing capacity.

[0035] Therefore, under antisymmetric loads, the arch structure exhibits an antisymmetric failure mode. After the most unfavorable section reaches strength failure, the arch structure can still continue to bear loads. The structure will have a relatively long plastic region, at which point the strength bearing capacity of the section is less than the ultimate bearing capacity of the entire arch rib.

[0036] Symmetrical failure under symmetrical constant load

[0037] When an arch bridge is subjected to axial loads, the arch axis shortens. To maintain the continuity of the arch crown, the arch undergoes bending deformation to "close" the crown, thus inducing bending moment internal forces within the arch. This bending deformation process is a forced displacement, such as... Figure 6 As shown.

[0038] Under symmetrical loads, the arch bridge mainly bears axial force, and the structure should be calculated according to equation (14). Double integration of equation (16) yields the displacement expression (17) for the arch structure. In equation (18), For the linear elastic structural deformation of an arch under symmetrical loads, it can be determined by... The double integral (19) and symmetric boundary conditions get.

[0039] (19)

[0040] Under symmetrical dead load, the entire arch rib experiences a relatively uniform axial compressive stress distribution. The bending stress at the arch crown increases non-linearly while the bending stress at the arch foot decreases non-linearly. Therefore, the arch crown is the control section for the structure's ultimate bearing capacity. Under symmetrical dead load, the arch rib primarily bears axial force, resulting in a smaller bending moment and a more uniform stress distribution. At the same load level, most of the material reaches its ultimate stress state. Due to structural deformation and material damage, internal force / stress redistribution occurs, preventing further transmission. The entire arch rib fails under a very small load increment, exhibiting brittle, small-eccentric failure. At this point, one side of the arch crushes first. Since concrete can withstand significant pressure, reinforcing steel in this area cannot bear the enormous pressure released by the crushing concrete. The steel can only share the pressure with the concrete before it crushes, thus increasing the load on the crushed concrete. However, the steel is under compression, and due to the mismatch between the elastic modulus and ultimate strength of the steel and concrete, its strength cannot be fully utilized. When the concrete on one side of the arch is crushed, the neutral axis of the arch crown shifts downward, causing a significant change in the internal forces of the arch structure in an instant. The enormous pressure released by the crushed concrete is transferred to the less stressed web. Simultaneously, the stiffness of the arch crown section rapidly decreases, altering the boundary conditions of the arch crown, causing it to rotate clockwise on the left and counterclockwise on the right, resulting in significant downward deflection at the arch crown and upward deflection at the quarter points. This deformation generates substantial positive bending moments at the arch crown and arch feet, and negative bending moments at the quarter points, forming plastic hinges in these five regions. This causes the structure to transform into a mechanical system and lose its load-bearing capacity. Figure 7 As shown.

[0041] If, for the sake of economic conservation, the bearing capacity of each section of the structure is designed to be relatively close, and if one section reaches its ultimate bearing capacity, the overload capacity of other sections is not stronger than that section. After one section fails, the other sections will also fail quickly, leading to the failure of the entire structure. In this case, it can be considered that the bearing capacity of the section is basically close to the bearing capacity of the entire structure.

[0042] At this point, due to the significant displacement caused by the structural system transformation, the reinforcement on the tension side of the five zones is insufficient to suppress concrete cracking, and the reinforcement cannot effectively enhance the ultimate bearing capacity of the structure. After section failure, the load-displacement curve exhibits a clear nonlinear change, with structural deformation concentrated in a very short plastic region. At this point, the remaining bearing capacity of the structure is low; that is, the structure will rapidly lose its bearing capacity after reaching the hinge load of a single section, leading to instability failure. Therefore, under symmetrical dead load, the arch structure exhibits a positively symmetrical failure mode. After the arch crown section reaches strength failure, the arch structure will develop into overall instability failure under a very small load increment. The strength bearing capacity of the arch crown section can represent the ultimate bearing capacity of the entire arch rib.

[0043] Coupled failure state of long-span arch bridge

[0044] Arch bridges in actual operation simultaneously bear both essentially symmetrical dead loads and asymmetrical live loads. In long-span arch bridges, because the dead load accounts for a very large proportion of the total load, the geometric nonlinearity of the structure is not obvious in the early stages of loading, at which point the structure mainly bears compressive stress while the bending moment is relatively small. As the loading force increases, since the antisymmetric deformation growth rate is greater than that of the orthosymmetric deformation, when the loading force reaches the first-order stability factor, the antisymmetric deformation component reaches its maximum value, and the structure basically exhibits antisymmetric deformation. However, since the first-order stability factor is generally greater than 4 in actual structural design, while the load safety factor for bearing capacity is 1.75, the structure then exhibits a coupling of two deformation modes.

[0045] In linear analysis, the stress in the cross-section of the arch rib can be decomposed into axial compressive stress. With bending stress The initial design will definitely involve small eccentric compression, meaning the axial compressive stress is greater than the bending stress. Nonlinear effects will be factored into the ultimate bearing capacity design. Since the increase in axial compressive stress is insensitive to geometric nonlinearity, it increases linearly with the continuous application of load. However, bending stress is more sensitive to geometric nonlinearity; it increases nonlinearly with the continuous application of load, and the degree of nonlinearity is related to the structural stiffness.

[0046] If only geometric nonlinearity is considered, the bending stress will continue to increase, reaching its maximum value at the Euler critical force. However, if material nonlinearity is further considered, the continuous increase in bending stress is halted when the sum of the bending stress and axial compressive stress reaches the material's design strength, at which point the section reaches small eccentric compression failure. Figure 8 (a) If the bending stress has increased to a level greater than the axial compressive stress before the sum of the bending stress and axial compressive stress reaches the material's design strength, and the reinforcing steel on the compression side has yielded, the cross-section exhibits large eccentric tensile failure, such as Figure 8 (b) When the sum of bending stress and axial compressive stress increases to the material's design strength, the bending stress is exactly equal to the axial compressive stress, such as... Figure 8 (c)

[0047] (20)

[0048] In equation (20), For the design of the nonlinear stability coefficient, the specification limit is 1.75. The design axial compressive stress is such that its value increases linearly with the increase of the applied force. The bending stress at which the arch bridge fails varies non-linearly with increasing loading force. From equation (20), the relationship between the design axial compressive stress and the material strength can be obtained as equation (21). Further substituting... The specification limit is 1.75. At this point, to satisfy the condition that the axial compressive stress at section failure is less than or equal to the bending stress, Then, equation (22) needs to be satisfied.

[0049] (twenty one) (twenty two) Therefore, the failure mode of the arch rib section is the result of the competition between axial compressive stress and bending stress. The axial compressive stress is mainly provided by the dead load, while the bending stress is mainly provided by the live load, and the nonlinear growth effect of the bending stress is suppressed by the stiffness of the arch rib. For long-span arch bridges, the axial compressive stress of the arch rib accounts for a relatively high proportion of the material's design strength, which is sufficient to resist the bending stress caused by live load disturbance. However, the space left for bending stress is insufficient to offset the axial compressive stress. Therefore, in the instant before the final failure, the entire cross-section is under compression, and the arch rib section will inevitably exhibit small eccentric compression failure.

[0050] In one specific embodiment, a scaled-down model design based on stiffness equivalence is also included; Taking into account factors such as size effect, test space, and feasibility of model fabrication, and considering that this experiment mainly examines the in-plane stress state of a long-span arch bridge, the model adopts a single-rib arrangement and a scale of 1:40. Based on stress equivalence, the similarity ratio of relevant physical parameters between the actual bridge and the model is shown in Table 1.

[0051] Table 1. Similarity ratio of physical parameters between the actual bridge and the model

[0052] Figure 9 This is a layout diagram for the scaled-down model. The scaled-down model has a main span of 15m and a rise of 3.125m. Directly scaling down to the scale results in a minimum theoretical concrete slab thickness of only 11.25mm, making fabrication impossible. Considering the basic structural requirements for the thickness of the reinforced concrete cover, and to avoid local distortion in the model, the composite section of the actual bridge is designed as an equivalent reinforced concrete section based on its compressive and in-plane bending stiffness. The steel pipes are equivalent to reinforcing bars based on their reinforcement ratio. While the similarity of out-of-plane stiffness is not considered, it is necessary to ensure that the out-of-plane stiffness of the arch rib is not too small, so that it can guarantee the out-of-plane stability of the arch rib under lateral constraints. At this point, the in-plane internal force response of the model and the actual bridge conforms to a similar ratio. Considering that this experiment needs to simulate the actual stress state of a large-span arch bridge when it reaches its bearing capacity, the model needs to simulate the stress state of the actual bridge structure. The axial compressive stress calculation formula is used... The axial compressive stress of an arch structure is controlled by the axial force and the cross-sectional area, while the structural compressive stiffness is provided by the cross-sectional area. Therefore, when the cross-sectional compressive stiffness meets the similarity ratio, the axial compressive stress of the model and the actual bridge automatically approximates each other. This is based on the bending stress calculation formula. The bending stress of the arch structure is controlled by the bending moment, the bending stiffness of the section, and the height of the section. When the bending stiffness of the section is similar, the bending moment of the arch rib automatically meets the similarity ratio. In order to ensure that the bending stress of the model is close to that of the actual bridge, the height of the model section must also meet the similarity ratio of the actual bridge.

[0053] After stiffness equivalence, the model's cross-section width is 140mm, and the cross-section height varies from 180mm at the arch crown to 300mm at the arch foot. The thickness of the top and bottom slabs is 40mm, and the web thickness is 30mm. The model uses single-layer reinforcement, with Φ5 ribbed HRB400 steel bars as the main reinforcement, spaced at 40mm intervals according to the reinforcement ratio equivalence and structural requirements. A 100mm long variable cross-section section is set at the arch foot for local reinforcement, increasing the thickness of the top and bottom slabs to 60mm and the web thickness to 40mm. The top and bottom slabs use double-layer reinforcement. Simultaneously, to avoid local bearing failure of the thin-walled top slab, local bearing pads and a 60mm thick diaphragm are set at the loading point to share the load.

[0054] Figure 10 To compare the calculation results of the 15m model with the adjusted dimensions with those of the 600m actual bridge, the structural response of the 600m actual bridge was scaled down according to the similarity ratio for ease of comparison. Figure 10 (a) to (f) compare the calculated results of the arch rib deflection under dead load, axial force under dead load, bending moment due to self-weight, bending moment under dead load, axial compressive stress under dead load, and bending stress under dead load, respectively. Figure 10It can be seen that the structural response of the model after size adjustment, except for the constant load axial force, is close to that of the prototype. The maximum deflection difference between the model and the actual bridge is only 0.4 mm, accounting for 4.5% of the total deflection of the model. Therefore, it is reasonable and feasible to make local adjustments to the model size by adopting the design principle of stiffness similarity in this experiment.

[0055] Depend on Figure 10 (b) It can be seen that the ratio of the axial force in the model to the axial force in the actual bridge is in the range of 1.3 to 1.4, while... Figure 10 (c) It can be seen that the axial compressive stress of the model matches the actual bridge well, with the maximum deviation located at the arch foot. The calculated axial compressive stress of the model is 12 MPa, while the calculated axial compressive stress of the actual bridge is 10 MPa, with an error of 2 MPa. The ratio of the model section stiffness to the actual bridge section stiffness is mostly within the range of 1.2 to 1.3. Since the structural compressive stiffness is provided by the cross-sectional area, the ratio of the model cross-sectional area to the actual bridge cross-sectional area is also within the range of 1.2 to 1.3. Although the axial force of the model is larger than that of the actual bridge, the area of ​​the model is also increased proportionally compared to that of the actual bridge. Therefore, the difference in axial compressive stress between the model and the actual bridge is not significant.

[0056] Figure 10 (c) and (d) show the calculation results of the arch rib bending moment, from Figure 10 As shown in (c) and (d), due to the adoption of the stiffness equivalence scaling design method, the bending moment and internal force values ​​of the model and the actual bridge are in good agreement. Since the model's self-weight cannot be simulated using distributed forces during the actual loading process, and can only be replaced by concentrated forces, the effect of concentrated force loads on the internal force concentration of the arch ribs was further analyzed, see... Figure 10 The blue curves in (c) and (d) show that, after applying multi-point concentrated force, a peak bending moment appears at the location of the concentrated force. The overall bending moment curve of the arch rib exhibits a wavy pattern, but the overall trend and the location of the peak bending moment remain unchanged. However, due to... Figure 10 (f) shows that since the arch rib bending stiffness, arch rib bending moment and section height of the model and the actual bridge are not much different, the bending stress distribution of the arch rib is highly consistent. The maximum deviation of the bending stress is located at the arch foot. The calculated value of the bending stress of the model is 0.2 MPa, while the calculated value of the bending stress of the actual bridge is 1.6 MPa, a difference of 1.4 MPa.

[0057] Unconstrained adaptive loading system

[0058] To avoid the constraint effect of the loading system on the arch structure, a lever-pulley loading device was designed for load application. Utilizing the tension consistency and self-sliding characteristics of the steel wire ropes in the pulley system, unconstrained free deformation of the model and synchronous load application were achieved during loading. The load amplification characteristic of the lever was utilized to achieve counterweight loading of large-tonnage dead loads on the model. Simultaneously, considering the inconsistent required load values ​​at each loading point, an adjustable amplification lever device was designed to achieve the application of non-uniform loads at different loading points of the model, such as… Figure 11 As shown.

[0059] Lateral limiting device

[0060] The experiment mainly investigated the in-plane stress state of a long-span arch bridge. The model adopted a single-rib arrangement. To ensure the lateral stability of the arch rib, two lateral restraint devices were arranged longitudinally along the arch rib, such as... Figure 12 As shown. This device can limit the lateral deformation of the single-rib model, preventing lateral instability during the arch rib loading process, and can also serve as a safety protection measure against collapse. Calculations show that without the lateral restraint device, the first-order instability mode of the model under constant load is lateral bending, with a critical load factor of 0.75. With the lateral restraint device included, the first-order instability mode of the model is vertical bending, with a critical load factor of 4.25.

[0061] Analysis of Experimental Results

[0062] The experimental results show that, due to the large axial stress, within a reasonable stiffness range, the bending tensile stress caused by live load disturbance in long-span arch bridges is insufficient to offset the compressive stress generated by the axial force. The final failure mode of the cross-section is small eccentric failure, with relatively low stress in the reinforcing steel at failure. When one section fails, the entire arch rib fails under a very small load increment, and the entire arch rib loses its bearing capacity. However, further research is needed on the nonlinearity of small eccentric failure under the combined action of dead and live loads, the effect of arch rib stiffness on suppressing geometric nonlinear effects, the contribution of reinforcing steel to bearing capacity, the initial bending moment state, and the potential influence of initial geometric imperfections on bearing capacity.

[0063] In one specific embodiment, the method also includes establishing a dual nonlinear simulation model; The conventional modeling process of generating mesh models from geometric models is complex and time-consuming, and the quality of the final mesh is difficult to guarantee. Since all information in an Abaqus model is stored in an inp file, this inp file can be compiled and written using MATLAB. Therefore, this paper proposes a dual nonlinear finite element model establishment method using MATLAB for secondary development of Abaqus, such as... Figure 13 As shown, the specific process is as follows: (1) Input the basic information of the model in MATLAB, including the linear shape, the size of the arch cross section and the form of cross section variation; (2) Input the number of longitudinal discrete points N in the model, and calculate the axis inclination angle of the discrete points based on the arch axis of the model; (3) Set the grid size on the cross section, calculate the two-dimensional coordinates of the grid nodes at each discrete point based on the cross section size information, and generate a two-dimensional grid plane group for the arch. (4) Rotate and translate the two-dimensional grid plane group of the arch top in sequence according to the coordinates of the discrete points of the arch axis and the inclination angle, and place each two-dimensional grid plane in the correct position at the discrete point to obtain the three-dimensional node coordinate information of the model; (5) Connect the corresponding nodes of the two two-dimensional mesh planes to generate solid elements, thereby obtaining the element information of the model. If there are one-dimensional line elements on the two-dimensional mesh plane, the corresponding nodes of the line elements can be connected to generate plate elements.

[0064] (6) Generate an initialization text file in Abaqus. This text file stores information such as nonlinear analysis steps, materials, loads, and boundary conditions. Write the calculated model node coordinates and element information into the text file using MATLAB to complete the establishment of the dual nonlinear simulation model.

[0065] (7) After the model inp text is established, it can be directly imported into Abaqus to generate a visual model, or MATLAB can be used to call the Abaqus / Standard solver for calculation and analysis. In a specific embodiment, the experimental verification of the simulation model in S6 is as follows: Figure 14 This paper compares the measured and simulated values ​​of the load-vertical displacement curves at three key sections (L / 4, L / 2, and 3L / 4) of a steel-concrete composite truss arch bridge. Figure 14 It can be seen that the measured value of the vertical displacement of the model is in good agreement with the simulation value. The calculated value of the bearing capacity of the arch is 132.2kN, and the measured value is 134kN. The calculated value is 1.3% smaller than the measured value. The finite element simulation method used can effectively simulate the bearing capacity and deformation of the arch structure.

[0066] Analysis of the failure state of arch structures

[0067] Antisymmetric destruction of arch

[0068] To explore the failure state of a long-span arch bridge under asymmetric loading, simulation calculations were performed on a model arch under single-point loading at four quarter points. The calculation results of the failure mode of the model arch are as follows: Figure 15 As shown.

[0069] Depend on Figure 15It can be seen that when the model arch failed, the upper and lower chord steel pipes yielded in four regions: the two arch feet and the two quarter points. The stress in the web members near these regions also exhibited a high stress state. Simultaneously, the C80 concrete inside the pipes also showed extensive damage. Specifically, at 3L / 4 and the left arch foot, the damage state of the concrete inside the pipes was the same: the upper chord pipe showed compressive damage while the lower chord pipe showed tensile damage. Conversely, at L / 4 and the right arch foot, the upper chord of the concrete inside the pipes was under tension while the lower chord was under compression. Consistent with the experimental results, the entire arch rib formed four plastic hinges at the two arch feet and the two quarter points upon failure, resulting in the structure losing its load-bearing capacity.

[0070] Symmetrical failure state of an arch

[0071] To explore the failure state of long-span arch bridges under high axial force, simulation calculations were conducted on model arches with spans of 600m, 300m, and 15m. The calculation results show that different spans do not change the failure state of the arch. Figure 16 This shows the failure state of a long-span arch bridge under dead load. Figure 16 (a) and (b) respectively illustrate the stress and damage distribution during structural failure. Figure 16 It is evident that when the model arch fails, five regions—the arch crown, the two quarter-point sections on both sides, and the two arch feet—experience high stress. Specifically, the top plate of the arch crown and arch feet sections is under compression, while the bottom plate is under tension. Conversely, the top plate of the two quarter-point sections is under tension, while the bottom plate is under compression. At this point, the arch crown and arch feet sections bear positive bending moments, while the two quarter-point sections bear negative bending moments. Simultaneously, the five key sections also exhibit significant compressive damage, with the top plate of the arch crown and arch feet sections and the bottom plate of the quarter-point sections showing the most pronounced compressive damage. Upon failure, the entire arch rib forms five plastic hinges at the arch crown, the two arch feet, and the two quarter-point sections, resulting in the structure losing its load-bearing capacity.

[0072] Figure 17 The load-principal compressive stress curve at the 8th point of the 600m arch rib is given by... Figure 17 It can be seen that in the elastic stage, the maximum compressive stress in the bottom and top plates occurs at the arch foot and arch crown sections, respectively, and the stress growth rate remains essentially constant at this stage. In the elastoplastic stage, the compressive stress growth rate in the bottom plate of the arch foot section gradually decreases, and at failure, the compressive stress exhibits negative growth, with the peak compressive stress in the bottom plate shifting to L / 8 and 2L / 8. The compressive stress growth rate in the top plate of the arch crown section gradually increases, and at failure, the top plate of the arch crown section collapses first, forming a plastic hinge in the arch crown section. After the top plate of the arch crown section reaches the material design strength, the load essentially does not increase while the arch ribs continue to deform, at which point the structure has lost its load-bearing capacity. Simultaneously, due to… Figure 17 It can be seen that before the structure fails, the stress at each measuring point increases linearly, while at the moment of failure, the stress at each measuring point increases sharply. The material between the stress point and the critical failure point can be approximated as a linear elastic state.

[0073] In a specific embodiment, in S7, the specific method of optimizing the structural design is as follows: the initial bending moment is reduced by adjusting the shape of the arch axis, the cross-sectional stiffness is optimized to make the stress of the entire arch uniform, the top and bottom plates are reinforced with steel bars on both sides to improve the structural stiffness, and the initial defect amplitude is controlled within ±50mm.

[0074] Specifically, the factors affecting the ultimate bearing capacity are analyzed as follows: To further explore the influence of structural parameters such as live load ratio, structural stiffness, reinforcement ratio, initial bending moment, and initial defects on the in-plane bearing capacity of long-span arch bridges, a validated finite element model was used for parameter analysis.

[0075] Live load percentage

[0076] Because asymmetrical live loads generate antisymmetric deformation components in arch structures, and the nonlinear growth rate of antisymmetric deformation is relatively fast, potentially causing a rapid decrease in the bearing capacity of arch bridges, it is necessary to investigate the impact of live loads on the in-plane bearing capacity of long-span arch bridges. Therefore, a model with a span of 400m, an arch rib dead load stress level of 15MPa, a reinforcement ratio of 0%, a defect amplitude of 0, and a dead load linear elastic stability coefficient of 4 was designed for dual nonlinear simulation calculations. The live loads in the model were arranged for lanes 0 to 4, which translate to line loads of 0kN×m, 10.8kN×m, 32.4kN×m, and 43.2kN×m, respectively, corresponding to live load percentages of 0%, 3.3%, 6.6%, 9.9%, and 13.2%.

[0077] Figure 18 The figure shows the calculation results of the ultimate bearing capacity and failure mode of the arch rib under different live load ratios under the combined action of dead load and half-span live load. c η1 and η2 represent the live load ratio and nonlinear stability coefficient of the structure, respectively. The live load ratio is expressed as the ratio of the vertical resultant force of the live load to the vertical resultant force of the dead load. Figure 18 It can be seen that under different live load percentages, the arch rib exhibits an asymmetric failure mode, with four regions—the two quarter points and the two arch feet—showing high damage levels. The structural failure is caused by the crushing of the bottom plate of the arch foot section on the live load side. The nonlinear stability coefficient of the arch rib decreases with increasing live load percentage, but the rate of decrease gradually slows down. The nonlinear stability coefficients for live load percentages of 0%, 3.3%, 6.6%, 9.9%, and 13.2% are 2.53, 2.04, 1.74, 1.53, and 1.35, respectively. Compared to the live load percentage of 0%, the nonlinear stability coefficient of the arch rib decreases by 19.4%, 31.2%, 39.5%, and 46.6% respectively when the live load percentage is from 3.3% to 13.2%.

[0078] Structural stiffness

[0079] When the stiffness is low, the deflection curves of each section show a significant change in slope before entering the yield plateau, reflecting the continuous deterioration of the arch rib stiffness and a longer failure process. As the stiffness of the arch rib increases, the deflection curves of each section become closer to straight lines, at which point the nonlinear effect of the arch rib is suppressed. When the stiffness reaches a constant load and the Euler stability coefficient is 3, the deflection curves of each section are all sloping straight lines before entering the yield plateau, and then enter the yield plateau at a sudden angle. At this point, the structure enters a longer yielding plastic deformation stage under a very small load increment, and the brittle failure characteristics of the structure are more obvious.

[0080] To further analyze the nonlinear variation of bending moment, a moment amplification factor is used to characterize the nonlinear effect of bending moment. The moment amplification factor is expressed as the ratio of the total structural bending moment considering nonlinear effects to the linear elastic bending moment. Since the structure is within the linear elastic stress range when the load factor is 1, the nonlinear effect is not significant; therefore, the bending moment with a load factor of 1 is used as the linear elastic bending moment for analysis. Under different stiffnesses, the moment amplification factor of each section at the arch crown decreases with increasing arch rib stiffness. Upon reaching the yield plateau, the moment amplification factor of all sections except L / 4 is less than 3. The L / 4 section, located near the elastic center of the arch rib, has a smaller linear elastic bending moment, resulting in a larger moment amplification factor.

[0081] As the stiffness of the arch rib increases, the displacement at each section decreases non-linearly. When the loading coefficient is 1, compared to when the linear elastic stability coefficient of the arch rib is 4, the deflection of the arch rib crown section decreases by 3.5%, 5.6%, 7.0%, 8.0%, and 8.8% respectively when the stability coefficient is 5-9. When the stability coefficient increases by 1, the deflection at the crown section decreases by 3.5%, 2.2%, 1.5%, 1.1%, and 0.9%. When the stability coefficient is 6, the deflection decreases by only 2.2%, and further increasing the arch rib stiffness at this point has little effect on the arch rib deformation.

[0082] Under the combined action of dead load and half-span live load, when the stiffness is low, the plastic development before structural failure is prolonged. When the stiffness is high, the structure undergoes large deformation under a small load increment, and the brittle failure characteristics of the structure are obvious.

[0083] Under different stiffnesses, the moment amplification factor of each section of the arch crown decreases as the stiffness of the arch rib increases. When reaching the yield plateau, the moment amplification factor of each section of the arch rib is less than 2.

[0084] Reinforcement ratio

[0085] To further analyze the contribution of steel reinforcement to the in-deck bearing capacity of long-span arch bridges, a dual nonlinear simulation calculation was performed on a model with a span of 400m, an arch rib dead load stress level of 15MPa, a live load ratio of 3.3% (equivalent to one lane load), reinforcement ratios of 0%, 0.1%, 0.5%, 1%, and 2%, a defect amplitude of 0, and a dead load linear elastic stability coefficient of 4.

[0086] Under the same reinforcement ratio, the arch rib exhibits an asymmetric failure mode, with four regions—the two quarter points and the two arch feet—showing high levels of damage. The structural failure is caused by the crushing of the bottom plate of the arch foot section on the live load side. The nonlinear stability coefficient of the arch rib increases essentially linearly with the increase of the reinforcement ratio. The nonlinear stability coefficients for reinforcement ratios of 0%, 0.1%, 0.5%, 1%, and 2% are 1.97, 2.01, 2.03, 2.12, and 2.23, respectively. Compared to a plain concrete arch, the nonlinear stability coefficient of the arch rib increases by 2.0%, 3.0%, 7.6%, and 13.2% respectively for reinforcement ratios of 0.1%, 0.5%, 1%, and 2%. With the increase of the reinforcement ratio, the ultimate bearing capacity of the arch rib increases essentially linearly.

[0087] When the load is relatively small, the load-vertical displacement curves of each section of the arch rib basically coincide, and the initial linear elastic stiffness of the structure is not significantly different. As the loading force increases, the concrete continuously deteriorates and its stiffness decreases, while the proportion of stiffness provided by the steel reinforcement increases. At this point, the model with a higher reinforcement ratio has greater stiffness.

[0088] The load-displacement curve of the arch section exhibits obvious brittleness, indicating that the structure fails under compression. In this case, the contribution of the reinforcing steel to the ultimate bearing capacity is manifested in its cooperative resistance to compression with the concrete before section failure, reducing the compressive stress in the concrete. After section failure, the reinforcing steel cannot withstand the enormous bending moment caused by the crushing and unloading of the concrete and will yield under a very short load increment; the tensile effect of the reinforcing steel is completely ineffective. Therefore, long-span arch bridges only require structural reinforcement.

[0089] To further analyze the influence of the distribution of reinforcing bars on the bearing capacity, based on the model with a total cross-section reinforcement ratio of 1%, finite element models with only top slab reinforcement, only bottom slab reinforcement, and top slab + bottom slab reinforcement were established to further analyze the influence of top slab reinforcement, bottom slab reinforcement, and web reinforcement on the bearing capacity.

[0090] The nonlinear stability coefficients of the arch rib are 1.97, 2.04, 2.04, 2.15, and 2.12 for unreinforced, top-slab-only, bottom-slab-only, top-slab + bottom-slab reinforcement, and full-section reinforcement, respectively. Compared to plain concrete, the arch rib bearing capacity increases by 3.6% with single-sided reinforcement, while double-sided reinforcement effectively improves the flexural stiffness of the arch rib, increasing the bearing capacity by 9.1%. Web reinforcement cannot increase the arch rib stiffness, and its weight increases the dead load on the arch rib, thus causing a 1.4% decrease in the arch rib bearing capacity.

[0091] initial bending moment

[0092] The aforementioned analysis shows that the failure of large-span arch bridges under the combined action of dead and live loads is caused by the crushing of the concrete at the arch foot. Furthermore, once the arch foot reaches its cross-sectional bearing capacity, the arch rib will fail under a very small load increment, at which point the ultimate bearing capacity of the arch rib is approximately equal to the cross-sectional bearing capacity of the arch foot. Therefore, without changing the axial compressive stress of the arch rib, the bending stress at the arch foot becomes the direct factor affecting the ultimate bearing capacity of the arch rib. Since the bending stress at the arch foot is generated by axial compression under dead load, the offset of the dead load pressure line from the arch axis, and asymmetrically distributed live load, the influence of the live load proportion on the ultimate bearing capacity of the arch rib has been discussed previously. To further analyze the contribution of the initial bending moment to the in-plane bearing capacity of large-span arch bridges, and considering that the initial bending moment under a reasonable arch axis is generated by axial compression, while the overall temperature load effect is consistent with axial compression, an overall temperature rise and fall is used to change the initial bending moment. This is achieved using a predefined temperature field in the ABAQUS simulation model. A dual nonlinear simulation was performed on a model with a span of 400m, an arch rib dead load stress level of 15MPa, a live load ratio of 3.3% (equivalent to the load of one lane), a reinforcement ratio of 1%, and dead load bending stresses increased by 50%, increased by 20%, 0%, decreased by 20%, and decreased by 50% respectively, with corresponding overall temperatures of -40℃, -20℃, 0℃, 20℃, and 40℃, a defect amplitude of 0, and a dead load linear elastic stability coefficient of 4. Figure 19 The calculation results of the load-vertical displacement curves of the arch rib 3L / 4 and the arch crown section under different initial bending moment states are derived from... Figure 19 It can be seen that when the bending stress under dead load changes by ±50%, the nonlinear stability coefficient changes between 2.48 and 2.72. When the bending stress under dead load increases by 50%, the ultimate bearing capacity of the arch rib under dead load decreases by 8.8%. The smaller the initial bending stress, the higher the bearing capacity. Optimizing the initial bending stress can significantly improve the bearing capacity of the arch bridge.

[0093] Initial defects

[0094] The "Standards for Quality Inspection and Evaluation of Highway Engineering, Volume 1: Civil Engineering" (JTG F801—2017) stipulates that for large-span arch bridges that cannot be cast in situ using arch frames, when the arch span is greater than 60m, the elevation deviation of the arch ring should be less than L / 3000 and not greater than 50mm. When the span is greater than 150m, 50mm is the limit for the elevation deviation of the arch ring. Therefore, to explore the influence of initial geometric defects on the in-plane bearing capacity of the arch bridge, and considering that the probability of first-order eigenvalue buckling is relatively high and close to the most unfavorable defect distribution, a model with a span of 400m, an arch rib dead load stress level of 15MPa, a live load ratio of 3.3% (equivalent to one lane load), a reinforcement ratio of 1%, a defect mode of first-order eigenvalue buckling, defect amplitudes of -50, -20, 0, 20, and 50mm, and a dead load linear elastic stability coefficient of 4 were designed for dual nonlinear simulation calculations.

[0095] Figure 20 The calculation results of the load-vertical displacement curves of the arch rib 3L / 4 and the arch crown section under different defect amplitudes are derived from... Figure 20 It can be seen that the deformation curves of the arch section under different defect amplitudes basically coincide, the nonlinear stability coefficient is between 2.02 and 2.04, the relative change in ultimate bearing capacity is less than 1%, and the initial defects within the specification limit have little impact on the ultimate bearing capacity of the arch bridge.

Claims

1. A method for analyzing the cross-sectional bearing capacity of a long-span arch bridge under the dominance of dead load, characterized in that, include: S1. Obtain the basic parameters of the long-span arch bridge, including the arch bridge span, rise-to-span ratio, material mechanical properties, cross-sectional dimensions, and dead and live load distribution characteristics; S2. Based on the aforementioned basic parameters, and in conjunction with the plane section assumption, Hooke's law, and finite displacement theory, the governing differential equations of the arch structure under the positive symmetric deformation mode and the antisymmetric deformation mode are derived respectively. S3. Based on the aforementioned control differential equation, decompose the symmetrical dead load and asymmetrical live load borne by the arch bridge, analyze the stress characteristics dominated by axial compression under the positive symmetrical deformation mode and the stress characteristics dominated by bending compression under the antisymmetrical deformation mode, and clarify the cross-sectional failure eccentricity limit and failure mode under the coupled action of the two deformation modes. S4. Based on the principle of stiffness equivalence, design a scaled-down model of a long-span arch bridge, build an unconstrained adaptive loading system and a lateral limiting device, conduct an ultimate bearing capacity loading test on the scaled-down model, and obtain data on displacement, strain and bending moment amplification factor throughout the loading process. S5. Based on experimental data, establish a dual nonlinear simulation model that considers both geometric nonlinearity and material nonlinearity, and calibrate the material constitutive parameters and boundary conditions of the simulation model using experimental data; S6. Using the calibrated simulation model, simulate the stress distribution and plastic hinge formation process of the cross section under the coupling action of positive and negative symmetric deformation, and calculate the ultimate bearing capacity of the cross section; S7. Analyze the influence of live load ratio, structural stiffness, reinforcement method, initial bending moment and initial defects on the section bearing capacity, output the section bearing capacity analysis results and optimize the arch bridge structure design.

2. The method for analyzing the cross-sectional bearing capacity of a long-span arch bridge under the dominant dead load as described in claim 1, characterized in that, The mechanical properties of the materials include the compressive strength and modulus of elasticity of concrete, and the yield strength and modulus of elasticity of steel; the dead load includes the self-weight of the arch ring and the load of the building on the arch; the live load includes the moving load of highway or railway, and the distribution characteristics of the live load are quantified by the ratio of the live load to the dead load across the entire span.

3. The method for analyzing the cross-sectional bearing capacity of a long-span arch bridge under constant load as described in claim 1, characterized in that, In S2, the governing differential equation corresponding to the orthosymmetric deformation mode is: The governing differential equation for the antisymmetric deformation mode is: Wherein, for any infinitesimal element mn of the arch along the arch axis, the horizontal displacement of point m is: The vertical displacement is Then the horizontal displacement of point n is The vertical displacement is M is the bending moment of the arch section. Let be the axial force at any point in the arch. The elastic modulus of the arch material. , Let be the bending stiffness and cross-sectional area at any point in the arch. , .

4. The method for analyzing the bearing capacity of a long-span arch bridge section under constant load as described in claim 1, characterized in that, In S3, the failure mode under the positive symmetric deformation mode is the failure of the five-hinged mechanism caused by the crushing of the top plate of the arch section, and the failure mode under the anti-symmetric deformation mode is the failure of the four-hinged mechanism caused by the crushing of the bottom plate of the arch foot section on the loading side. The eccentricity limit of section failure is determined by the ratio of axial compressive stress to bending stress. When the axial compressive stress is greater than the bending stress, it is small eccentricity compressive failure, and when the axial compressive stress is less than the bending stress, it is large eccentricity compressive failure.

5. The method for analyzing the bearing capacity of a long-span arch bridge section under the dominant dead load according to claim 1, characterized in that, In S4, the design of the scaled model satisfies the stiffness equivalence principle, and the similarity ratio is determined according to the dimensional relationship of length, area, volume, and stress. The scale ratio is 1:30 to 1:

50. The unconstrained adaptive loading system adopts a lever-pulley block structure to realize the synchronous application of dead load and half-span live load. The lateral limiting device restricts the out-of-plane deformation of the model to ensure that the in-plane force dominates.

6. The method for analyzing the cross-sectional bearing capacity of a long-span arch bridge under the dominant dead load according to claim 1, characterized in that, In S5, the dual nonlinear simulation model uses C3D8R solid elements to simulate concrete and C4R plate elements to simulate steel. The concrete adopts a plastic damage constitutive model, and the steel adopts an ideal elastic-plastic constitutive model. The calibration process adjusts the constitutive parameters until the error is less than 5% by comparing the load-displacement curves and strain distribution data of the simulation and the experiment.

7. The method for analyzing the cross-sectional bearing capacity of a long-span arch bridge under the dominant dead load according to claim 1, characterized in that, In S6, the calculation of the ultimate bearing capacity of the section includes: obtaining the maximum compressive stress of the section based on the simulation model; when the maximum compressive stress reaches the design strength of the concrete, the corresponding load is the section strength bearing capacity; when the structure forms a plastic hinge mechanism, the corresponding load is the overall ultimate bearing capacity of the structure, and the smaller of the two is taken as the final section bearing capacity.

8. The method for analyzing the cross-sectional bearing capacity of a long-span arch bridge under the dominant dead load according to claim 1, characterized in that, In S7, the specific method of optimizing the structural design is as follows: the initial bending moment is reduced by adjusting the shape of the arch axis, the cross-sectional stiffness is optimized to make the stress of the entire arch uniform, the top and bottom plates are reinforced with steel bars on both sides to improve the structural stiffness, and the initial defect amplitude is controlled within ±50mm.

9. The method for analyzing the bearing capacity of a long-span arch bridge section under the dominant dead load according to claim 6, characterized in that, The specific construction process of the dual nonlinear simulation model is as follows: (1) Input the basic information of the model in MATLAB, including the linear shape, the size of the arch cross section and the form of cross section variation; (2) Input the number of longitudinal discrete points N in the model, and calculate the axis inclination angle of the discrete points based on the arch axis of the model; (3) Set the grid size on the cross section, calculate the two-dimensional coordinates of the grid nodes at each discrete point based on the cross section size information, and generate a two-dimensional grid plane group for the arch. (4) Rotate and translate the two-dimensional grid plane group of the arch top in sequence according to the coordinates of the discrete points of the arch axis and the inclination angle, and place each two-dimensional grid plane in the correct position at the discrete point to obtain the three-dimensional node coordinate information of the model; (5) Connect the corresponding nodes of the two two-dimensional mesh planes to generate solid elements, thereby obtaining the element information of the model; if there are one-dimensional line elements on the two-dimensional mesh plane, connect the corresponding nodes of the line elements to generate plate elements. (6) Generate an initialization inp text in Abaqus, and write the calculated model node coordinates and unit information into the inp text using MATLAB to complete the establishment of the dual nonlinear simulation model; (7) After the model inp text is created, it can be directly imported into Abaqus to generate a visualization model; or MATLAB can be used to call the Abaqus / Standard solver for calculation and analysis.

Images