Method for determining section of cable for reinforcing arch foot of deck arch bridge

Abstract

The present application relates to the technical field of deck arch bridge reinforcement, and particularly discloses a method for determining the section of a cable for reinforcing the arch foot of a deck arch bridge, which comprises the following steps: (1) establishing a stress model of an original arch structure and a stress model of a reinforced structure after setting a cable, respectively; (2) obtaining the most unfavorable load position of the arch foot negative bending moment of the stress model of the original arch structure and the stress model of the reinforced structure, and applying the same load; (3) deriving the arch foot bending moment expression of the stress model of the original arch structure and the stress model of the reinforced structure; (4) fitting the expression y(t) of the reduction amplitude y of the arch foot negative bending moment with respect to the stiffness ratio t; and (5) establishing a relationship between the reduction amplitude of the arch foot section bending resistance and the reduction amplitude of the arch foot negative bending moment after setting the cable. The method for determining the section of the cable for reinforcing the arch foot of the deck arch bridge can reduce the waste of cable materials and save the construction cost while ensuring good reinforcement effect.

Description

Technical Field

[0001] The present invention relates to the technical field of top-decker arch bridge reinforcement, and in particular to a method for determining the cross-section of a cable for reinforcing an arch foot of a top-decker arch bridge. Background Art

[0002] Arch bridges are a distinctive form of bridge construction in my country. They are widely used in Chinese bridge construction due to their material-saving, low-cost, simple construction, beautiful design, and smooth curves. However, with increasing traffic volume, increasing service life, and the aging of some materials, these damaged arch bridges are no longer able to meet normal operational requirements. Because the main arch rings of arch bridges are compression-bending members, excessive bending moments often lead to increased cracking in the arch rings. In particular, cracks in the arch foot reduce cross-sectional resistance, leading to a decrease in structural bearing capacity. Demolishing and rebuilding a large number of these old and dangerous bridges would consume enormous financial and material resources, even requiring traffic disruption and prolonging the construction period. However, reinforcing and renovating old bridges would only cost 10% to 30% of the cost of building a new bridge. This demonstrates that reinforcing and renovating a large number of dangerous bridges not only meets the needs of modern transportation but also has significant economic and social benefits.

[0003] At present, the methods commonly used to reinforce arch bridges include increasing the cross-section of the main arch ring, adjusting the constant load of the building on the arch, changing the structural system, pasting steel plates and fiber composite materials, and prestressed reinforcement. For existing reinforcement methods, researchers mainly start from the aspect of improving the resistance of the components themselves, but there is relatively little research on changing the structural system. In addition, although the existing reinforcement methods have a certain reinforcement effect, it is inevitable that after being put into operation, they face problems such as the separation of new and old materials after the bridge reinforcement and the excessive bending moment of the main arch ring caused by the external constant load, resulting in the main arch ring of the bridge to be reinforced. The main arch ring has excessive bending moment and cracks and other defects cannot be improved. Therefore, the reinforcement method proposed in the present invention can effectively solve the technical difficulties of cracks and other defects caused by excessive negative bending moment of the arch foot of the main arch ring without increasing the constant load, thereby achieving the purpose of arch bridge reinforcement.

[0004] like Figure 1 and Figure 2As shown in the figure, slots are cut in the bridge deck above the arch ribs on both sides of the main arch ring, and two cables are embedded in the slots, located on the same axis. One end of each cable is connected to the arch crown, and the other end is connected to the abutments at each end. Without increasing the dead load of the arch bridge, the installation of the cables changes the original arch structure, thereby changing the original structure's force transmission path and redistributing the internal forces of the arch ribs. At the same time, the axial force of the cables and the concentrated load generate opposing moments at the arch foot, reducing the absolute value of the negative bending moment at the arch foot. The cables provide additional force to the arch ribs, effectively improving cracks and other defects caused by excessive negative bending moment at the arch foot of the main arch ring, thereby achieving the purpose of strengthening the arch bridge. The most unfavorable load positions are mainly the following two working conditions: one is that the concentrated load acts on the arch crown, the arch crown does not produce horizontal displacement, and the cable has no axial force; the other is that the concentrated load acts on other arch ribs except the arch crown, and the arch crown produces horizontal displacement. Since the arch ribs and the cables are hingedly constrained at the arch crown, according to the principle of action and reaction, the horizontal displacement of the arch crown causes the cables to produce axial force. From the force analysis, it can be seen that the concentrated load and the axial force of the cable produce opposite moments on the arch foot, which makes the absolute value of the bending moment of the arch foot smaller.

[0005] Therefore, the cable arrangement utilizes the synergy between the arch ribs and the cables, which can help improve the arch bridge's arch foot and other defects such as cracks, thereby achieving the desired effect. However, the selection of the cable cross-section used in this reinforcement method is a topic worthy of further research. Summary of the Invention

[0006] The present invention aims to solve at least one of the technical problems mentioned above and proposes a method for determining the cross-section of cables for reinforcing the arch foot of a top-supported arch bridge. By determining a reasonable range of the tensile stiffness of the cables, the waste of cable materials can be reduced while ensuring a good reinforcement effect, thereby achieving the purpose of saving construction costs.

[0007] To achieve the above-mentioned object, the present invention adopts a technical solution: a method for determining the cross-section of a cable for reinforcing an arch foot of a deck arch bridge, comprising the following steps:

[0008] (1) Establish the stress model of the original arch structure and the stress model of the reinforced structure after installing cables;

[0009] (2) Obtain the most unfavorable load position of the negative bending moment at the arch foot of the two arch bridge load models, and apply the same concentrated load at the most unfavorable load position of the two arch bridge load models respectively;

[0010] (3) Using the basic principle of force method of structural mechanics, the arch foot bending moment expression of the original arch structure stress model and the arch foot bending moment expression of the reinforced structure stress model are derived;

[0011] (4) Set the stiffness ratio t = (EA) 索 / (EA)拱肋 The negative bending moment ratio of the arch foot before and after reinforcement is used as the characterization of the bending moment change, and the expression of the reduction amplitude y of the negative bending moment of the arch foot with respect to the stiffness ratio t is fitted: y = -1.7734t 2 +1.3823t+0.0352;

[0012] (5) Analyze the relationship between the bending bearing capacity M1 before the arch foot section is cracked, the bending bearing capacity M2 after cracking, and the stiffness ratio t, and establish the relationship: The reasonable range of stiffness ratio t that meets the bearing capacity requirements is obtained, and the corresponding cable cross-sectional area is determined according to the material properties of the cable.

[0013] Preferably, the influence line curves of the arch foot bending moments of the original arch structure and the reinforced structure are simulated by the moving load module of the Midas Civil software to determine the most unfavorable load position and uniformly distributed load loading range of the arch foot negative bending moment of the arch bridge stress model.

[0014] Preferably, at the most unfavorable load position x of the original arch structure stress model F Apply load F at the position, and use the superposition principle to superimpose the internal forces of the positive symmetric load and the antisymmetric load respectively to satisfy the typical equation of the force method: Where: X1, X2, X3 are the bending moment, axial force, and shear force at the arch section respectively; the coefficient δ ij The unknown force When acting alone, the i Displacement in the direction; free term Δ iF The load F is the load along X caused by the i The displacement in the direction of the arch is derived; the expression of the arch foot bending moment of the original arch structure stress model is derived: Where: F is the concentrated load, x F is the horizontal coordinate of the concentrated load, l, f, y s The arch span, arch rise and elastic center ordinate are respectively.

[0015] Preferably, the most unfavorable load position x of the negative bending moment of the arch foot of the stress model of the reinforced structure after the cables are set is F Apply load F at the position, and use the superposition principle to superimpose the internal forces of the positive symmetric load and the antisymmetric load respectively to satisfy the typical equation of the force method: Where: X1, X2, X3, X4 are the bending moment, axial force, shear force and cable axial force at the arch section respectively; the coefficient δ ij The unknown force When acting alone, the i Displacement in the direction; free term Δ iF The load along X caused by the concentrated load F acting alonei The displacement in the direction of the reinforcement structure is derived as follows: Where: F is the concentrated load, x F is the horizontal coordinate of the concentrated load, l, f, y s The arch span, arch rise and elastic center ordinate are respectively.

[0016] Preferably, finite element models of the original arch structure and the reinforced structure are established respectively, and the same load is applied to the most unfavorable load position of the negative bending moment of the arch foot of the two finite element models and the two force models. The output results of the arch foot bending moment of the two finite element models are compared with the calculation results of the arch foot bending moment calculated in the corresponding two force models, and the difference between the output results of the finite element model and the calculation results of the force model is calculated.

[0017] The beneficial effect is that, compared with the prior art, the cable cross-section determination method for the arch foot reinforcement of a deck arch bridge of the present invention constructs the force models of the original arch structure and the reinforced structure respectively, and deduces them in combination with the basic principle of the force method of structural mechanics, thereby obtaining the bending moment equations of the force models of the original arch structure and the reinforced structure at the arch foot respectively. This method can quickly calculate the bending moment magnitudes of the unreinforced and reinforced arch bridges at the arch foot. At the same time, by introducing the parameter t=(EA) 索 / (EA) 拱肋 , analyze the influence of the tensile stiffness EA of the cable on the negative bending moment of the arch foot of the arch bridge, determine the reasonable range of the tensile stiffness of the cable at the arch foot that meets the bearing capacity requirements, and determine the corresponding cable cross-sectional area based on the material properties of the cable, so as to reduce the waste of cable materials and achieve the purpose of saving construction costs while ensuring a good reinforcement effect. BRIEF DESCRIPTION OF THE DRAWINGS

[0018] The specific embodiments of the present invention are further described in detail below with reference to the drawings, wherein:

[0019] Figure 1 This is a schematic diagram of the unreinforced structure of the arch bridge;

[0020] Figure 2 This is a structural diagram of the arch bridge after reinforcement;

[0021] Figure 3 This is a schematic diagram of the stress analysis of the original arch structure;

[0022] Figure 4 Schematic diagram of stress analysis of the reinforced structure;

[0023] Figure 5 This is a schematic diagram of the finite element model of the original arch structure;

[0024] Figure 6 Schematic diagram of the finite element model of the reinforced structure;

[0025] Figure 7 The curve of the percentage reduction of negative bending moment at the arch foot changes with the stiffness ratio t. DETAILED DESCRIPTION

[0026] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the figures in the embodiments of the present invention. Obviously, the embodiments described are only part of the embodiments of the present invention, not all of the embodiments. All other embodiments obtained by ordinary technicians in this field based on the embodiments of the present invention without making any creative efforts shall fall within the scope of protection of the present invention.

[0027] It should be noted that when a component is referred to as being "fixed to" another component, it may be directly on the other component or there may be a central component. When a component is considered to be "connected" to another component, it may be directly connected to the other component or there may be a central component at the same time. When a component is considered to be "set on" another component, it may be directly set on the other component or there may be a central component at the same time. When a component is referred to as being "set in the middle", it does not only mean being set in the middle, but also being set at both ends within the range defined by the middle. The terms "vertical", "horizontal", "left", "right" and similar expressions used herein are for illustrative purposes only.

[0028] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by those skilled in the art to which this invention pertains. The terms used in this specification of the present invention are for the purpose of describing specific embodiments only and are not intended to limit the present invention. The term "and / or" as used herein includes any and all combinations of one or more of the associated listed items.

[0029] like Figures 1 to 6 As shown in FIG, the method for determining the cross-section of the cables for reinforcing the arch foot of a deck arch bridge includes the following steps:

[0030] (1) Establish the stress model of the original arch structure and the stress model of the reinforced structure after installing cables;

[0031] (2) Obtain the most unfavorable load position of the negative bending moment at the arch foot of the two arch bridge load models, and apply the same concentrated load at the most unfavorable load position of the two arch bridge load models;

[0032] (3) Using the basic principle of force method of structural mechanics, the arch foot bending moment expression of the original arch structure stress model and the arch foot bending moment expression of the reinforced structure stress model are derived;

[0033] (4) Set the stiffness ratio t = (EA) 索 / (EA) 拱肋The negative bending moment ratio of the arch foot before and after reinforcement is used as the characterization of the bending moment change, and the expression of the reduction of the negative bending moment of the arch foot with respect to the stiffness ratio t is fitted: y = -1.7734t 2 +1.3823t+0.0352;

[0034] (5) Analyze the relationship between the bending bearing capacity M1 before the arch foot section is cracked, the bending bearing capacity M2 after cracking, and the stiffness ratio t, and establish the relationship: The reasonable range of the stiffness ratio t that meets the bearing capacity requirements is obtained, and the cross-sectional area of ​​the cable can be determined according to the material properties of the cable.

[0035] Preferably, the influence line curves of the arch foot bending moments of the original arch structure and the reinforced structure are simulated by the moving load module of the Midas Civil software to determine the most unfavorable load position and uniformly distributed load loading range of the arch foot negative bending moment of the arch bridge stress model.

[0036] Preferably, at the most unfavorable load position x of the original arch structure stress model F Apply load F at Figure 3 , the superposition principle is used to superimpose the internal forces of the positive symmetric load and the antisymmetric load respectively, satisfying the typical equation of the force method: Where: X1, X2, X3 are the bending moment, axial force, and shear force at the arch section respectively; the coefficient δ ij The unknown force When acting alone, the i Displacement in the direction; free term Δ iF The load F is the load along X caused by the i The displacement in the direction of the arch is derived; the expression of the arch foot bending moment of the original arch structure stress model is derived: Where: F is the concentrated load, x F is the horizontal coordinate of the concentrated load, l, f, y s The arch span, arch rise and elastic center ordinate are respectively.

[0037] Preferably, the most unfavorable load position x of the negative bending moment of the arch foot of the reinforced structure stress model after the cable reinforcement is set F Apply load F at Figure 4 , the internal forces of the positive symmetric load and the antisymmetric load are superimposed using the superposition principle to satisfy the typical equation of the force method: Where: X1, X2, X3, X4 are the bending moment, axial force, shear force and cable axial force at the arch section respectively; the coefficient δ ij The unknown force When acting alone, the i Displacement in the direction; free term Δ iFThe load along X caused by the concentrated load F acting alone i The displacement in the direction of the reinforcement structure is derived as follows: Where: F is the concentrated load, x F is the horizontal coordinate of the concentrated load, l, f, y s The arch span, arch rise and elastic center ordinate are respectively.

[0038] Preferably, finite element models of the original arch structure and the reinforced structure are established respectively, the same load is applied at the most unfavorable load position of the two finite element models and the two force models, the output results of the arch foot bending moment of the two finite element models are compared with the calculation results of the arch foot bending moment calculated in the corresponding two force models, and the difference between the output results of the finite element model and the calculation results of the force model is calculated.

[0039] The method for determining the cross-section of the cable for reinforcing the arch foot of a deck arch bridge of the present invention constructs the stress models of the original arch structure and the reinforced structure respectively, and deduces them in combination with the basic principle of the force method of structural mechanics to obtain the bending moment expressions of the stress models of the original arch structure and the reinforced structure at the arch foot respectively. The bending moment magnitudes of the stress models of the unreinforced and reinforced arch bridges at the arch foot can be quickly calculated. At the same time, by introducing the parameter t=(EA) 索 / (EA) 拱肋 , analyze the influence of the tensile stiffness EA of the cable on the reduction of the negative bending moment at the arch foot of the bridge arch, determine the reasonable range of the tensile stiffness of the cable at the arch foot that meets the bearing capacity requirements, and determine the corresponding cable cross-sectional area based on the material properties of the cable, so as to reduce the waste of cable materials and achieve the purpose of saving construction costs while ensuring a good reinforcement effect.

[0040] Specifically, a simple system top-supported arch bridge is a multi-times hyperstatic spatial structure, and its main arch ring is in the form of a bare arch as the main load-bearing component. For an actual arch bridge, the arch structure (such as the belly arch, arch filler and vertical wall) and the main arch ring work together to resist the load. Its prominent characteristics are: (1) the main arch ring will deform under the action of external loads, but the deflection of the main arch ring will be reduced due to the constraint between the arch structure and the main arch ring; (2) the elastic displacement of the main arch ring will affect the internal force of the arch structure, and the internal force of the arch structure constrains the displacement of the main arch ring. After considering the joint action of the arch structure, the bending moment of the main arch ring will be reduced to a certain extent. According to the stress characteristics of the arch bridge, when the joint action of the arch structure and the main arch ring is considered, the deformation of the main arch ring can be reduced to a certain extent and the structural stiffness of the entire bridge can be improved. The simplified mechanical model is also based on the fact that the impact of superstructures on bearing capacity is generally not considered in arch bridge design. The gravity and live loads of the superstructure are evenly distributed across each main arch ring unit, meaning that each arch unit experiences an equal load. This consideration, while ignoring the beneficial effects of superstructures on the structure, also provides a certain degree of safety margin for arch bridges.

[0041] Based on the above analysis, the stress model of the main arch ring was simplified as follows: one of the arch ribs of the main arch ring was used as the calculation unit, and the structure above the arch was considered as a local load-transmitting member that transfers the load to the main arch ring. The vertical load was simply transferred to the arch rib, that is, the arch bridge structure was simplified to a hingeless arch structure. In order to analyze the reduction in the maximum negative bending moment at the arch rib foot before and after reinforcement, the most unfavorable load position of the original arch structure and the reinforced structure was first determined. The moving load module of the Midas Civil software was used to simulate the influence line curves of the arch foot bending moment of the original arch structure and the reinforced structure to determine the most unfavorable load position and uniformly distributed load loading range of the arch foot negative bending moment.

[0042] Based on the simplified stress model of the original arch structure, the most unfavorable load position x of the negative bending moment at the arch foot is F Apply load F, and the force analysis is as follows Figure 3 As shown. Using the basic principle of the force method of structural mechanics, that is, cutting at the cross-section of the arch, the unknown forces of bending moment X1, axial force X2, and shear force X3 are used to replace them. Since the original arch structure is under asymmetric load, the superposition principle can be used to superimpose the internal forces of the positive symmetric load (c) and the antisymmetric load (d), and the typical equation of the force method is satisfied:

[0043]

[0044] Where: X1, X2, X3 are the bending moment, axial force, and shear force at the arch section respectively; the coefficient δ ij The unknown force When acting alone, the i Displacement in the direction; free term Δ iFThe load along X caused by the concentrated load F acting alone i Assume that the equivalent radius of the arch bridge is R and the central angle of the half arch is And satisfy certain geometric relationships:

[0045]

[0046]

[0047] In addition, in order to simplify the calculation, polar coordinates are used, namely:

[0048] Expressions for bending moment, shear force and axial force under basic structure:

[0049]

[0050] The internal force equation of the right half arch structure under the action of asymmetric concentrated load is:

[0051]

[0052] The coefficient δ can be obtained by using the graph multiplication method and the elastic center method ij and the free term Δ iF , and then substitute it back into formula (1) to find the unknown forces X1, X2, and X3:

[0053]

[0054] Therefore, the expression of the arch foot bending moment of the original arch structure can be obtained by using the superposition principle:

[0055]

[0056] Where: F is the concentrated load, x F is the horizontal coordinate of the concentrated load, l, f, y s The arch span, arch rise and elastic center ordinate are respectively.

[0057] On the basis of the original arch structure, cables are added to connect the arch ribs so that the two can bear the force together. One end of the cable is connected to the abutment, and the other end is hinged to the arch top. Figure 4As shown. Perform stress analysis on the reinforced structure and determine the most unfavorable load position. The most unfavorable load position is mainly the following two working conditions: one is that the concentrated load acts on the arch crown, the arch crown does not produce horizontal displacement, and the cable has no axial force; the other is that the concentrated load acts on the arch ribs other than the arch crown, and the arch crown produces horizontal displacement. Since the arch ribs and the cables are hinged constraints at the arch crown, according to the principle of action and reaction, the horizontal displacement of the arch crown causes the cables to produce axial force. From the stress analysis, it can be seen that the concentrated load and the axial force of the cable produce opposite moments on the arch foot, which makes the absolute value of the bending moment of the arch foot smaller. The derivation process of the mechanical formula is as follows:

[0058] Using the basic principle of the force method of structural mechanics, that is, cutting at the cross-section of the arch, the four reaction forces of bending moment X1, axial force X2, shear force X3, and cable reaction force X4 are used to replace them. For asymmetric load structures, the internal forces can be superimposed by the internal forces under positive symmetric load (c) and antisymmetric load (d), and satisfy the typical equation of the force method:

[0059]

[0060] Where: X1, X2, X3, X4 are the bending moment, axial force, shear force and cable axial force at the arch section respectively; the coefficient δ ij The unknown force X j When acting alone, the i Displacement in the direction; free term Δ iF The load along X caused by the concentrated load F acting alone i Direction displacement.

[0061] Expressions for bending moment, shear force and axial force under basic structure:

[0062]

[0063] The internal force equation of the right half arch structure under concentrated load is the same as formula (5). i , I i 、A i The meaning of , i = q, i = g are the elastic modulus, moment of inertia and cross-sectional area of ​​cable BD and side arch rib AC respectively. Those without subscripts are side arch rib AC.

[0064] The coefficients and free terms of the basic structure under positive symmetrical load (c) are:

[0065]

[0066]

[0067] δ 12 =δ 21 =0 (10.3)

[0068]

[0069]

[0070] Under the antisymmetric load (d), the axial force of the cable on the arch rib side is 0, and the axial force of the cable on the arch rib side is X4. Due to the symmetry of the structure, the two sides of the arch are subjected to The additional force of the basic structure is:

[0071]

[0072]

[0073]

[0074]

[0075]

[0076] When the cross-sectional characteristics of the arch rib and the cable are determined, the tensile stiffness EA and bending stiffness EI of the arch rib and the tensile stiffness EI of the cable in the above formula can be determined. q A q (Parameter t=E q A q / EA), using the coefficient δ obtained above ij and the free term Δ iP Substituting back into equation (8), we can solve for the unknown forces X1, X2, X3, and X4.

[0077] Therefore, the superposition principle can be used to obtain the bending moment expression of the arch foot of the reinforced structure:

[0078]

[0079] Where: F is the concentrated load, x F is the horizontal coordinate of the concentrated load, l, f, y s The arch span, arch rise and elastic center ordinate are respectively.

[0080] In a specific engineering application example, for example, an arch bridge with a span of l = 30 m and a rise of f = 4.5 m, the arch axis is a parabola and the arch ribs are concrete rectangular sections, EI = 3.24 × 10 8 (N·m 2 ), EA=1.08×10 10(N), with a column spacing of 2.5m. The main arch ring and the piers are consolidated at both ends, and the longitudinal beams of the bridge deck are simply supported. Due to long service life and increasing traffic volume, the negative bending moment at the arch foot of the main arch ring of the bridge is excessive, resulting in numerous cracks on the arch back at the arch foot. The continuous extension and development of these cracks exacerbates the carbonization of the concrete and the corrosion of the steel bars. Therefore, a reinforcement method is needed to effectively reduce the negative bending moment at the arch foot of the main arch ring. By slotting the bridge deck above the arch ribs along the longitudinal side of the main arch ring on both sides of the bridge, one end of the cable is hinged to the arch crown, with the cable on the same horizontal line, and the other end of the cable is hinged to the abutment. The installation of the cable changes the original structural system of the arch bridge, thereby changing the force transmission path of the simple arch structure main arch ring. Without increasing the dead load, it can effectively reduce the negative bending moment at the arch foot, thereby achieving the purpose of strengthening the arch bridge.

[0081] In order to verify the rationality of the above mechanical derivation formula, two finite element models were established using Midas / Civil software. Model 1 (original arch structure) is as follows: Figure 5 As shown: the arch rib is a beam unit with two ends consolidated; Model 2 (reinforced structure) is as follows Figure 6 As shown, the arch ribs are constructed using beam elements with fixed ends, while the cables are constructed using truss elements subjected only to tension and not directly bearing vertical loads. In both models, the arch structures are hinged, and only load transfer is considered. The loading method utilizes the most unfavorable negative bending moment at the arch foot, with a load of 100 kN. The cross-sectional parameters of the cables and arch ribs are shown in Table 1.

[0082] Table 1 Section parameters

[0083]

[0084] According to the finite element calculation results, the key cross-sectional internal forces of the original arch structure and the reinforced structure can be obtained:

[0085] 1) The arch foot bending moment of the original arch structure is -198.7 kN·m.

[0086] 2) The bending moment at the arch foot of the right half arch of the reinforced structure is -166.9 kN·m; the internal force of the cable of the right half arch is F 右 =58.7kN.

[0087] (2) Derivation and calculation of mechanical formulas

[0088] 1) Original arch structure

[0089] Combining the parameters in the example, a concentrated load of 100 kN is applied to the most unfavorable load position of the negative bending moment at the arch foot of the original arch structure. The calculation results are as follows:

[0090] Arch rib equivalent radius

[0091] Elasticity Center Central angle of a semi-arch

[0092] The equivalent radius R, the vertical coordinate y of the elastic center s 、The central angle of the semi-arch And the concentrated load F is substituted into the above formula to obtain the coefficients δ ij and the free term Δ iP , combined with formula (1), we can obtain: X1 = 43.4 kN·m; X2 = 48.7 kN; X3 = 7.4 kN.

[0093] The arch foot bending moment of the original arch structure can be obtained:

[0094]

[0095] 2) Strengthen the structure

[0096] The stiffness ratio t is set to 0.1, and a concentrated load of 100 kN is applied to the most unfavorable load position of the negative bending moment at the arch foot of the reinforced structure. The calculation results are as follows:

[0097] The tensile stiffness E of the cable q A q =t·EA=0.1×1.08×10 10 =1.08×10 9 (N)

[0098] The equivalent radius R, elastic center y s 、The central angle of the semi-arch Load F and cable tensile stiffness E q A q Substitute the above formula to find the coefficients δ ij and the free term Δ iP , combined with formula (8), we can obtain: X1 = 43.4 kN·m; X2 = 48.7 kN; X3 = 1.3 kN; X4 = 57.4 kN.

[0099] The bending moment of the arch foot of the right half arch of the reinforced structure can be obtained:

[0100]

[0101] Cable axial force: F 索 =X4=57.4kN.

[0102] 2. Comparative analysis of finite element and mechanical calculation results

[0103] The following comparison is made based on the output results of the finite element model and the calculated values ​​derived from the mechanical formula, and the two are mutually verified. The internal force values ​​of the reinforced structure are taken from the right half arch structure. The comparative analysis results are shown in Table 2 below.

[0104] Table 2 Comparative analysis of the calculated values ​​using the finite element model and mechanical formulas

[0105]

[0106] It can be concluded from Table 2 that the internal forces of the key sections calculated by the finite element model and the mechanical formula are basically consistent, and the errors are within 3%, which proves the rationality of the mechanical formula derivation and the finite element model.

[0107] Furthermore, the influence of the tensile stiffness EA of the cable on the negative bending moment of the arch foot can be analyzed. Based on the above calculation model, under the premise of keeping the arch rib cross section unchanged, the stiffness ratio t is expanded and the stiffness ratio t = (EA) is established. 索 / (EA) 拱肋 The loading method derived from the finite element model and mechanical formula is to load the most unfavorable load position of the negative bending moment of the arch foot with a load size of 100kN. The results are shown in Table 3 and Figure 7 shown.

[0108] Table 3 Comparison of arch foot bending moment results before and after reinforcement (f / L=0.15)

[0109]

[0110] The finite element calculation results in Table 3 were subjected to nonlinear regression fitting to obtain the expression for the stiffness ratio t: y = -1.7734t 2 +1.3823t+0.0352, variance R 2 =0.9978, where y is the reduction of the arch foot bending moment.

[0111] Assuming that the bending bearing capacity of the arch foot section is M1 when there is no cracking, the arch foot section will crack as the traffic volume and service life increase and the vehicle load is too heavy. At this time, the bending bearing capacity of the arch foot section is M2. The effect produced when the arch foot section is under the most unfavorable load is M3. When M1≥M3≥M2, that is, due to cracking and other defects at the arch foot of the arch bridge, the bending bearing capacity requirements are not met and reinforcement methods are urgently needed. When cables are used for reinforcement, there is

[0112]

[0113] Where, Indicates the reduction in the bending bearing capacity of the arch foot section.

[0114] when When t=0.050≤t≤0.729, the bearing capacity requirement can be met when the stiffness ratio t=0.05. When the cable adopts epoxy steel strand, the cable cross-sectional area A=2769.2mm 2 , 10 bundles of steel strands with a nominal diameter of 21.6mm (1×7) are required.

[0115] Therefore, the following rules can be drawn from the above comparative analysis:

[0116] 1) The mechanical formula derivation calculation formula is basically consistent with the bending moment variation curve of the finite element model, and the error is within 3%. Therefore, the rationality and feasibility of both the mechanical formula derivation and the finite element model can be proved;

[0117] 2) When the side arch rib cross section is constant, the greater the stiffness ratio t, the greater the reduction in the negative bending moment at the arch foot, indicating that within a certain range, the larger the cross-sectional area of ​​the cable, the better the reinforcement effect;

[0118] 3) From the expression obtained by nonlinear fitting, it can be seen that its first-order derivative y'=-3.5468t+1.3823 is a descending straight line, that is, the trend of the arch foot bending moment change curve is a process that gradually tends to be gentle as the stiffness ratio t increases;

[0119] 4) For the above engineering example, when using the present invention for reinforcement, setting the tensile stiffness of the cables to 0.05 times that of the side arch ribs can reduce the maximum negative bending moment of the arch foot by 10%. In this case, 10 bundles of steel strands with a nominal diameter of 21.6 mm (1×7) are required. This can save construction costs while achieving good reinforcement effects, providing a certain reference value for actual engineering applications.

[0120] The above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit the same. Any modifications or equivalent substitutions that do not depart from the spirit and scope of the present invention should be included in the scope of the technical solutions of the present invention.

Claims

1. A method for determining the cross-section of cables for reinforcing the arch foot of a deck arch bridge, characterized in that: The following steps are involved: (1) Establish the stress model of the original arch structure and the stress model of the reinforced structure after installing cables; (2) Obtain the most unfavorable load position of the negative bending moment of the arch foot of the two arch bridge load models, and apply the same concentrated load at the most unfavorable load position of the two arch bridge load models respectively; (3) Using the force method principle of structural mechanics, the arch foot bending moment expression of the original arch structure stress model and the arch foot bending moment expression of the reinforced structure stress model are derived; (4) Set the stiffness ratio t = (EA) 索 / (EA) 拱肋 The negative bending moment ratio of the arch foot before and after reinforcement is used as the characterization of the bending moment change, and the expression of the reduction amplitude y of the negative bending moment of the arch foot with respect to the stiffness ratio t is fitted. 2 +1.3823t+0.0352; (5) Analyze the relationship between the bending bearing capacity M1 before the arch foot section is cracked, the bending bearing capacity M2 after cracking, and the stiffness ratio t, and establish the relationship: Find the range of stiffness ratio t that meets the bearing capacity requirements, and determine the corresponding cable cross-sectional area based on the material properties of the selected cable.

2. The method for determining the cross-section of cables for reinforcing the arch foot of a deck arch bridge according to claim 1, characterized in that: The moving load module of Midas Civil software was used to simulate the influence line curves of the arch foot bending moment of the original arch structure and the reinforced structure to determine the most unfavorable load position and uniformly distributed load loading range of the arch foot negative bending moment of the arch bridge stress model.

3. The method for determining the cross-section of cables for reinforcing the arch foot of a deck arch bridge according to claim 1, characterized in that: At the most unfavorable load position x of the original arch structure stress model F A concentrated load F is applied at the position, and the internal forces of the positive symmetric load and the antisymmetric load are superimposed using the superposition principle to satisfy the typical equation of the force method: Where: X1, X2, X3 are the bending moment, axial force, and shear force at the arch section respectively; the coefficient δ ij The unknown force When acting alone, the i Displacement in the direction; free term Δ iF The load F is the load along X caused by the i The displacement in the direction is used to derive the expression of the arch foot bending moment of the original arch structure stress model: Where: F is the concentrated load, x F is the horizontal coordinate of the concentrated load, l, f, y s are the arch span, arch rise and elastic center ordinate respectively.

4. The method for determining the cross-section of cables for reinforcing the arch foot of a deck arch bridge according to claim 1, characterized in that: At the most unfavorable load position x of the reinforced structure stress model F A concentrated load F is applied at the position, and the internal forces of the positive symmetric load and the antisymmetric load are superimposed using the superposition principle to satisfy the typical equation of the force method: Where: X1, X2, X3, X4 are the bending moment, axial force, shear force and cable axial force at the arch section respectively; the coefficient δ ij The unknown force When acting alone, the i Displacement in the direction; free term Δ iF The load F is the load along X caused by the i The displacement in the direction is used to derive the expression of the arch foot bending moment of the reinforced structure stress model: Where: F is the concentrated load, x F is the horizontal coordinate of the concentrated load, l, f, y s are the arch span, arch rise and elastic center ordinate respectively.

5. The method for determining the cross-section of cables for reinforcing the arch foot of a deck arch bridge according to claim 1, characterized in that: Finite element models of the original arch structure and the reinforced structure were established respectively, and the same concentrated load was applied to the most unfavorable load position of the negative bending moment of the arch foot of the two finite element models and the two force models. The output results of the arch foot bending moment of the two finite element models were compared with the calculation results of the arch foot bending moment obtained in the corresponding two force models, and the difference between the output results of the finite element model and the calculation results of the force model was calculated.

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