Simplified calculation method for transverse stress of concrete box girder

Abstract

The invention relates to the technical field of prestressed concrete bridges, in particular to a simplified calculation method for transverse stress of a concrete box girder. The method comprises the following steps: S1, confirming a box girder structure needing to be analyzed; in the step S1, the box girder structure needing to be analyzed is confirmed, specifically, whether the box girder structure needing to be analyzed is a single-box single-chamber section or not is confirmed; whether the box girder structure needing to be analyzed is a single-box multi-chamber section or not is determined; and S2, carrying out transverse force analysis on the box girder confirmed in the step S1. According to the simplified calculation method for the transverse stress of the concrete box girder provided by the invention, two simple and feasible transverse stress analysis methods for the concrete box girder are provided by improving the method, so that the calculation precision of the transverse bending moment and the axial force is met, meanwhile, the calculation precision can be provided for bridge design reference, and the method is simple, visual, high in precision and wide in application range; the method is not only suitable for a single-box uniform-section simply supported beam, but also suitable for a single-box multi-chamber variable-section continuous beam.

Description

Technical Field

[0001] The invention relates to the technical field of prestressed concrete bridges, and in particular to a simplified calculation method for the transverse stress of a concrete box girder. Background Art

[0002] The most commonly used section for long-span prestressed concrete bridges is the box girder. In order to reduce the number of webs, the box girder chamber and cantilever width should be increased as much as possible in the project. Among the concrete bridges built in my country, the maximum width of the box girder has reached more than 30 meters, and the maximum cantilever length exceeds 8 meters. For example, the Jialing River Bridge has a deck width of 41 meters and a maximum cantilever length of 8 meters.

[0003] As the width increases, the spatial behavior of the box girder becomes more prominent. The box section of the prestressed concrete bridge often has longitudinal cracks related to the transverse force. Therefore, the transverse force analysis method of the box girder, as a key technology, cannot be ignored in the design of the box girder. The transverse force may become a controlling factor in the design of the box girder. Most of the theories of the spatial analysis of the box girder can be traced back to more than 50 years ago. The classical method of the transverse analysis of the box girder has complicated formulas and is not widely used in the engineering community. The commonly used transverse support frame method has poor accuracy and has the following disadvantages:

[0004] 1. Guo Jinqiong [1] An elastic support frame method was proposed, in which springs were set at the frame boundaries to make the cross-section deformation consistent with the deformation in the overall structure. The disadvantages are as follows: the lateral bending moment calculated according to the lateral frame in my country's "Railway Bridge Design Code" and "Highway Bridge Code" is relatively accurate, but the lateral axial force accuracy is poor. The calculated bottom plate and web axial forces may be opposite to the actual axial forces, which is unsafe in engineering applications.

[0005] 2. American Manual of Precast Segmental Box Girder [2] The TYL transverse force analysis is introduced, and the transverse frame axial force under symmetrical load of rectangular single box section is given; and the transverse frame internal force under anti-symmetric load is given based on the relationship between the shear force difference between the front and back of the beam and the cross-section distortion internal force. Guo Jinqiong extended it to the trapezoidal single box section with cantilever [3] Wang Chenguang, Wang Yangsheng and others considered more detailed equilibrium conditions and improved the accuracy of horizontal frame analysis. [4]-[5] The above methods are collectively referred to as TYL methods. The disadvantages are as follows: Although the TYL method is a manual calculation method, the process is extremely cumbersome and is only applicable to single-box single-chamber cross-sections.

[0006] 3. The American AASHTO specification and the Canadian highway bridge design specification CHBDC recommend four methods for analyzing box girder bridges, including the beam grid method, the folded plate method, the finite strip method and the finite element method. [6]-

[10] .

[0007] 1. CHBDC stipulates that the beam grid method can be used for single-cell and double-cell sections.

[0008] 2. The folded plate method is to split the box girder into rectangular plates along the longitudinal intersection of the plate, take the longitudinal shear force and plate bending moment at the intersection as redundant forces, expand the external load and redundant forces into triangular series, and solve the displacement coordination equations of each series at the intersection. The folded plate method assumes that the box girder is a straight beam simply supported beam, does not consider the middle diaphragm, and has rigid diaphragms at both ends of the bridge span.

[0009] 3. The finite strip method can be regarded as between the folded plate method and the finite element method. It is divided into a series of strip units in the longitudinal direction. The shape function of this unit in the longitudinal direction is a trigonometric series, and the shape function in the transverse direction is a polynomial.

[0010] The disadvantages are as follows:

[0011] CHBDC stipulates that grillage analysis is not recommended for sections with more than two chambers.

[0012] The folded plate method and the finite strip method have less calculation amount than the finite element method, but lack a universal and user-friendly folded plate method program, and the folded plate method and the finite strip method are only applicable to simply supported beams with uniform cross-sections. The simplified calculation method for the transverse stress of concrete box beams proposed in the present invention has a wider application range, and is not only applicable to simply supported beams with uniform cross-sections, but also to simply supported beams with variable cross-sections and continuous beams with variable cross-sections.

[0013] The finite element method has the best adaptability to structural details and boundary conditions, but the workload of modeling, calculation and post-processing is relatively large. The simplified plate-shell element analysis method of the present invention can effectively simplify the modeling and post-processing of the plate-shell finite element analysis of the transverse force of the box girder, and greatly improve the calculation efficiency while meeting the requirements of the transverse internal force accuracy.

[0014] Therefore, it is urgent to design a simple and easy simplified calculation method for the lateral stress of concrete box girders to meet the engineering accuracy, and at the same time provide a reference for bridge design to provide another technical solution to the above technical problems. Summary of the invention

[0015] Based on this, it is necessary to provide a simplified calculation method for the transverse stress of a concrete box girder in order to solve the technical problems raised in the above background technology.

[0016] In order to solve the above technical problems, the present invention adopts the following technical solutions:

[0017] The simplified calculation method of the transverse stress of concrete box beam is as follows:

[0018] S1: Confirm the box girder structure to be analyzed;

[0019] S2: Perform lateral force analysis on the box girder confirmed in step S1.

[0020] As a preferred implementation of the simplified calculation method for the transverse stress of a concrete box girder provided by the present invention, in the step S1, the box girder structure to be analyzed is confirmed, specifically:

[0021] Confirm whether the box girder structure to be analyzed is a single-box single-chamber section.

[0022] As a preferred implementation of the simplified calculation method for the transverse stress of a concrete box girder provided by the present invention, in the step S1, the box girder structure to be analyzed is confirmed, specifically:

[0023] Confirm whether the box girder structure to be analyzed is a single-box multi-chamber section.

[0024] As a preferred implementation of the simplified calculation method for the transverse stress of a concrete box girder provided by the present invention, when it is confirmed that the box girder structure to be analyzed is a single-box single-chamber section, the calculation method is as follows:

[0025] Calculation of straight web box girders;

[0026] Calculation of inclined web box girders;

[0027] Calculation of mid-span section with variable cross-section;

[0028] Calculate the variable cross-section 1 / 4 section.

[0029] As a preferred implementation of the simplified calculation method for the transverse stress of a concrete box girder provided by the present invention, the calculation of the straight web box girder is performed in the following steps:

[0030] The top plate calculation expression is as follows:

[0031]

[0032] Among them, y t is the distance from the center of the top plate, t t is the top plate thickness, N yy represents the axial force per unit length along the y direction, q t (x) is the uniformly distributed lateral load on the top plate;

[0033] The base plate calculation expression is as follows:

[0034]

[0035] Among them, y b is the distance from the center of the bottom plate, t b is the bottom plate thickness, N yy represents the axial force per unit length along the y direction, q b (x)q b (x) is the uniformly distributed lateral load on the bottom plate;

[0036] In the straight web box girder, the top transverse axial force is zero, the bottom transverse axial force is an external load, and the middle is approximately a straight line.

[0037] As a preferred implementation of the simplified calculation method for the transverse stress of a concrete box girder provided by the present invention, the calculation of the inclined web box girder is performed in the following steps:

[0038] The top plate calculation expression is as follows:

[0039]

[0040] Among them, y t is the distance from the center of the top plate, t t is the top plate thickness, N yy represents the axial force per unit length along the y direction, q t (x) is the uniformly distributed lateral load on the top plate;

[0041] The base plate calculation expression is as follows:

[0042]

[0043] Among them, y b is the distance from the center of the bottom plate, t b is the bottom plate thickness, N yy represents the axial force per unit length along the y direction, q b (x)q b (x) is the uniformly distributed lateral load on the bottom plate;

[0044] In the inclined web box girder, the top plate and the bottom plate superimpose the horizontal component of the web axial force, and the vertical force of the web axial force is balanced with the external load.

[0045] As a preferred implementation of the simplified calculation method for the transverse stress of a concrete box girder provided by the present invention, the calculation steps for the mid-span section of the variable cross-section are as follows:

[0046] If the transverse axial forces of the top plate, bottom plate and web of the variable cross-section box girder in the mid-span section are consistent with those of the simply supported box girder with equal cross-section, the transverse internal forces of the mid-span section of the variable cross-section box girder are analyzed by the top plate calculation expression and the bottom plate calculation expression;

[0047] The top plate calculation expression is as follows:

[0048]

[0049] Among them, y t is the distance from the center of the top plate, t t is the top plate thickness, N yy represents the axial force per unit length along the y direction, q t(x) is the uniformly distributed lateral load on the top plate;

[0050] The base plate calculation expression is as follows:

[0051]

[0052] Among them, y b is the distance from the center of the bottom plate, t b is the bottom plate thickness, N yy represents the axial force per unit length along the y direction, q b (x)q b (x) is the uniformly distributed lateral load on the bottom plate;

[0053] If the bottom curve of the variable-section box girder is a quadratic parabola, the expression is as follows:

[0054]

[0055] As a preferred implementation of the simplified calculation method for the transverse stress of a concrete box girder provided by the present invention, the calculation steps for the variable cross-section 1 / 4 section are as follows:

[0056] Calculate the correction coefficients β1 and β2 using the following formula:

[0057]

[0058]

[0059] The transverse axial force of the variable cross-section can be obtained by multiplying the transverse axial force of the top plate and the bottom plate at the equal cross-section by the correction coefficients β1 and β2, and the expression is as follows:

[0060]

[0061]

[0062] As a preferred implementation of the simplified calculation method for the transverse stress of a concrete box girder provided by the present invention, it is confirmed that the box girder structure to be analyzed is a single-box multi-chamber section, and the analysis method is as follows:

[0063] Establish a straight beam model with uniform cross-section;

[0064] The cross section has nodal loads applied at the corners;

[0065] Rigid diaphragms are set at the boundaries;

[0066] A line-supported simply-supported boundary is set at the lower edge of the rigid diaphragm;

[0067] Extract the transverse axial force and bending moment of a beam in the mid-span of the box girder as the analysis results;

[0068] The lateral internal force of the box girder is obtained by superimposing the analysis results of the plate-shell element model with the lateral frame model.

[0069] It can be seen without a doubt that the above-mentioned technical solution of the present application can definitely solve the technical problem to be solved by the present application.

[0070] At the same time, through the above technical solutions, the present invention has at least the following beneficial effects:

[0071] The simplified calculation method for the transverse stress of a concrete box girder provided by the present invention provides two simple and easy concrete box girder transverse force analysis methods by improving the method, so as to meet the calculation accuracy of the transverse bending moment and the axial force, and can be used as a reference for bridge design, thereby being simple and intuitive, having high accuracy and a wide range of applications; it is not only applicable to a single-box chamber simply supported beam with equal cross-section, but also to a single-box multi-chamber variable cross-section continuous beam;

[0072] The simplified analysis method of transverse internal force is simple, intuitive and highly accurate. It provides a method for drawing transverse axial force diagrams under symmetrical loads on a single-chamber cross section. The transverse axial force of the top and bottom plates is analogous to the bending moment diagram of the beam, and the transverse axial force of the web plate is obtained according to the node equilibrium condition. It is consistent with the results of the finite element method, and the error of the uniform cross-section simply supported beam is controlled within 5%. The transverse axial force obtained by the traditional transverse frame analysis method has a large error, and the transverse force design of the bottom plate may be unsafe.

[0073] The simplified shell unit finite element method has a wide range of applications. A simplified finite element method for analyzing the transverse force of box girders using finite element analysis of simply supported plates and shells with equal cross-sections is given. Numerical examples show that this method is applicable to single-box multi-chamber continuous beams with equal cross-sections under symmetrical and asymmetrical loads, and can be extended to continuous beams with variable cross-sections. The mid-span section of a box girder with variable cross-sections is cut off to perform equivalent simply supported plate and shell analysis. The transverse internal force of the box girder obtained is close to the original structure and can be used for the design and calculation of bridge sections. Most of the traditional transverse frame analysis methods are only applicable to single-box single-chamber sections, and have a narrow range of applications. BRIEF DESCRIPTION OF THE DRAWINGS

[0074] Figure 1 It is a schematic cross-sectional view of a simply supported beam model of uniform cross-section according to the present invention;

[0075] Figure 2 It is a schematic diagram of the stress state and symbol convention of the box girder microelement of the present invention;

[0076] Figure 3 It is a schematic diagram of the force state of the microelement in the x direction of the present invention;

[0077] Figure 4 It is a schematic diagram of the force state of the microelement in the y direction of the present invention;

[0078] Figure 5 It is a schematic diagram of the bending moment of the top plate under the uniformly distributed transverse load of the present invention;

[0079] Figure 6 It is a schematic diagram of the top plate corner point balance of the present invention;

[0080] Figure 7 It is a schematic diagram of the bottom plate corner point balance of the present invention;

[0081] Figure 8 It is a schematic diagram of the cross-sectional dimensions / m of a single-box single-chamber inclined web box girder of the present invention;

[0082] Fig. 9 It is a schematic diagram of the transverse axial force distribution of the mid-span section of a single-chamber simply supported beam under symmetrical load of the present invention;

[0083] Fig.10 It is a schematic diagram of the cross-sectional dimensions / m of a single-box single-chamber straight web box girder of the present invention;

[0084] Fig.11 It is a schematic diagram of the space model of the variable cross-section continuous beam shell unit of the present invention;

[0085] Fig.12 It is a schematic diagram of the transverse axial force distribution of the mid-span section of the single-chamber variable-section continuous beam under symmetrical load of the present invention;

[0086] Fig.13 It is a schematic diagram of the transverse axial force distribution of 1 / 4 section of a single-chamber variable-section continuous beam under symmetrical load of the present invention;

[0087] Fig.14 It is a schematic diagram of the mid-span cross section / m of a single-box double-chamber uniform cross-section continuous beam of the present invention;

[0088] Fig.15 It is a schematic diagram of the transverse internal force distribution of the mid-span section of the single-box double-chamber uniform-section continuous beam under deadweight load of the present invention;

[0089] Fig.16 It is a schematic diagram of the transverse internal force distribution of the mid-span section of the single-box double-chamber uniform cross-section continuous beam under asymmetric load of the present invention;

[0090] Fig.17 It is a schematic diagram of a 1 / 4 cross section / m of a single-box double-chamber variable-section continuous beam of the present invention;

[0091] Fig.18 It is a schematic diagram of the transverse internal force distribution of the mid-span section of the single-box double-chamber variable-section continuous beam under deadweight load of the present invention;

[0092] Fig.19 It is a schematic diagram of the transverse internal force distribution of the mid-span section of the single-box double-chamber variable-section continuous beam under asymmetric load of the present invention;

[0093] Fig. 20It is a schematic diagram of the transverse internal force distribution of 1 / 4 section of the single-box double-chamber variable-section continuous beam under deadweight load of the present invention;

[0094] Fig.21 It is a schematic diagram of the lateral internal force distribution of 1 / 4 cross section of the single-box double-chamber variable-section continuous beam under asymmetric load of the present invention. DETAILED DESCRIPTION

[0095] In order to make the purpose, technical solution and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention.

[0096] In order to enable those skilled in the art to better understand the solutions of the present invention, the technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the accompanying drawings.

[0097] It should be noted that, in the absence of conflict, the embodiments of the present invention and the features and technical solutions in the embodiments may be combined with each other.

[0098] It should be noted that similar reference numerals and letters denote similar items in the following drawings, and therefore, once an item is defined in one drawing, further definition and explanation thereof is not required in subsequent drawings.

[0099] Reference Figure 1-Figure 21 , the simplified calculation method of the transverse stress of concrete box beam is as follows:

[0100] S1: Confirm the box girder structure to be analyzed;

[0101] Confirm that the box girder structure to be analyzed is a single-box multi-cell section or a single-box single-cell section;

[0102] Then confirm whether the analysis needs to be done is a single box multi-chamber section or a single box single-chamber section;

[0103] S2: Perform lateral force analysis on the box girder confirmed in step S1;

[0104] Method 1: In step S2, the lateral force analysis needs to be performed on the single-box single-chamber section, and then the single-box single-chamber mid-span and 1 / 4 section under symmetrical load, including uniform cross-section simply supported beams and variable cross-section continuous beams; a simple method for drawing the lateral axial force diagram is given; and the rigid support frame method is used to analyze the bending moment;

[0105] like Figure 1 The simply supported box girder shown has a web spacing of B and a cantilever width of b. t , the section height is H. The distance from the neutral axis to the center point of the top plate and the bottom plate is y t ,y bThe thickness of the top plate, bottom plate and web plate are t t ,t b and t w The internal forces and sign conventions of the box beam slab unit are as follows: Figure 2 As shown. Where N xx Represents the axial force per unit length along the x direction, N yy Represents the axial force per unit length along the y direction; N xy It represents the shear force of the cross section, and the positive value is the rotation of the infinitesimal element around the positive z direction.

[0106] Figure 3 The equilibrium equation of the microelement in the x direction is:

[0107]

[0108] Among them, dx and dy represent the derivatives of x and y on both sides of the equation, respectively. and Both represent N xx Find the partial derivative with respect to x, and Both represent N yx Taking the partial derivative with respect to y, the shear force N on the cross section yx The axial force N in the longitudinal direction xx The increment is expressed as N xx It represents the axial force per unit length in the x direction.

[0109]

[0110] Among them, y t is the distance from the center of the top plate, t t is the top plate thickness, N yy represents the axial force per unit length along the y direction, q t (x) is the uniformly distributed lateral load on the top plate.

[0111] Figure 4 The equilibrium equation of the microelement in the y direction is:

[0112]

[0113] Among them, dx and dy represent the derivatives of x and y on both sides of the equation, respectively. and Both represent N xx Find the partial derivative with respect to x, and Both represent N yx Taking the partial derivative with respect to y, the shear force N on the cross section yx The axial force N in the longitudinal direction xx The increment is expressed as N xx It represents the axial force per unit length in the x direction.

[0114] And by N xy =-N yx , then the right side of formula (3) is

[0115]

[0116] Then the axial force in the transverse direction is N yy The increment can be determined by the shear force N on the cross section. yx Represented. Taking partial derivatives of both sides of equation (4) with respect to the y coordinate and substituting them into equation (2), we can get

[0117]

[0118] For a beam of uniform cross-section, the moment of inertia I of the section along the longitudinal direction and the vertical coordinate y of the top plate t are all constants, then we have

[0119]

[0120] Formula (6) calculates the second-order derivative of the x coordinate and then substitutes it into formula (5) to obtain:

[0121]

[0122] From the differential relationship between bending moment and distributed load concentration q(x) For the top plate, equation (7) can be obtained:

[0123]

[0124] For the bottom plate, the same logic can be obtained:

[0125]

[0126] In formula (8), the lateral axial force N yy With the distributed load q t (x) is the second-order derivative relationship about the coordinate y, which is similar to the relationship between bending moment and distributed load, except that q is transversely t (x) is applied to the top plate, and the resulting bending moment is the lateral axial force of the top plate. Similarly, the top plate distributed load q b (x) is applied to the bottom plate, and the lateral axial force of the bottom plate also takes the value of the bending moment;

[0127] For straight web box girder, the top plate is based on the cantilever simply supported beam system, and the top plate is subjected to uniformly distributed transverse load q t (x), the obtained "bending moment" value is the lateral axial force of the box girder top plate, such as Figure 5 Similarly, the bottom plate is subjected to a uniformly distributed lateral load q according to the simply supported beam system. b (x).

[0128] When the symmetrical load acts on the corner points of the bottom plate, the node balance shows that the lateral axial force at the top of the web is zero, the lateral axial force at the bottom is the external load, and the middle is approximately a straight line.

[0129] For inclined web box girders, the top and bottom plates are superimposed with the horizontal component of the web axial force, and the vertical force of the web axial force is balanced with the external load, such as Figure 6 , Figure 7 shown.

[0130] For variable cross-section box girders, the longitudinal section inertia moment I(x) and the top plate vertical coordinate y t (x) is a function of x, then

[0131]

[0132] Formula (10) calculates the second-order derivative of the x coordinate and then substitutes it into formula (5) to obtain:

[0133]

[0134] According to the differential relationship between bending moment, shear force and distributed load concentration q(x), V(x), Then formula (11) can be written as:

[0135]

[0136] For the mid-span section, the lateral axial forces of the top, bottom and web of the variable-section box girder are the same as those of the uniform-section simply supported box girder. Therefore, the calculation formula (8) and (9) of the uniform-section box girder can be used to analyze the lateral internal forces of the mid-span section of the variable-section box girder. If the bottom curve of the variable-section box girder is a quadratic parabola, y in formula (12) t (x) The first-order derivative is zero at the mid-span section. At this time, the quadratic term can be ignored, and equation (12) can be simplified to

[0137]

[0138] For the 1 / 4 section, let V be equal to the self-weight G of the box beam from the 1 / 4 section position to the middle span. The positive and negative values ​​of V are determined by the slope of the tangent at the 1 / 4 position in the beam element bending moment diagram. Let x = 0.25L, Δx = 0.1m, and we can get the y t (x), y b (x), I(x), and the correction coefficients are obtained from the relationship between the first-order derivative and the limit, which are defined as β1 and β2, and the expressions are as follows:

[0139]

[0140]

[0141] Therefore, the lateral axial force of the variable cross-section can be calculated by multiplying the lateral axial force of the top and bottom plates at the equal cross-section by the correction coefficients of equations (16) and (17).

[0142]

[0143]

[0144] The above-mentioned method of lateral force analysis for a single-box single-chamber section is applied and verified as follows:

[0145] ① Calculate the balance of straight web box girder using formula (8) (9) and the top and bottom plate corner points.

[0146] ② Calculate the inclined web box girder using formula (8) (9) and Figure 6 Figure 7 The nodes shown are balanced.

[0147] ③ Calculate the mid-span section of the variable cross-section using formulas (8) (9) and Figure 6 Figure 7 The nodes shown are balanced.

[0148] ④ To calculate the variable cross-section 1 / 4 section, first calculate the correction coefficients β1 and β2, then use formulas (16) (17) and Figure 6 Figure 7 The nodes shown are balanced.

[0149] The shell unit spatial model and beam unit transverse frame model were established for the simply supported beam with uniform cross-section and the continuous beam with variable cross-section, and the calculation results were compared with those of method one.

[0150] (1) Simply supported beam with uniform cross section

[0151] Use literature

[10] Calculate the example in Figure 8 The span of the concrete simply supported beam bridge shown is 40 m, the concrete strength is C60, the elastic modulus is E=3.6 GPa, and the Poisson's ratio μ=0.2.

[0152] The bridge analysis software Midas Civil was used to establish the beam unit transverse frame model, and a uniformly distributed load of q = 10.9 kN / m2 was applied to the top plate. The concrete box beam shell unit space model was established, with the longitudinal unit size of 0.5 m and the transverse unit size of 0.1 m, totaling 15732 units. The corner load applied by the plate shell unit analysis is the support reaction of the transverse frame model with an opposite sign, uniformly distributed along the longitudinal direction, and the load value is F = -73 kN / m.

[0153] It is known that under the action of symmetrical load, the longitudinal distributed load value q(x) = 2F, and the uniformly distributed transverse load q on the top and bottom plates can be obtained from formula (8): t (x), q b(x), the top plate and bottom plate are calculated as cantilever simply supported beam and simply supported beam system respectively. The bottom plate bending moment is added with the horizontal component of the web axial force. The vertical force of the web axial force is balanced with the external load F, as shown in Figure 7 shown.

[0154] The beam unit frame model and shell unit space model are used to calculate the lateral axial force of the top plate, bottom plate and inclined web of the mid-span section, and compared with the theoretical calculation results, such as Fig. 9 As shown, the origin of the top plate horizontal coordinate is the cantilever end position, the origin of the bottom plate horizontal coordinate is the bottom plate corner position, and the origin of the web plate horizontal coordinate is the top plate corner position.

[0155] Table 1 lists the lateral axial forces at the top plate corners, the middle of the top plate, the bottom plate corners and the middle of the bottom plate under symmetrical loads for comparison.

[0156] Table 1 Comparison of transverse axial forces in the mid-span section of single-box single-chamber simply supported beams under symmetrical loads

[0157]

[0158] The distribution law of the transverse axial force is the same as that of the calculated shell unit results. When the box girder bottom plate is subjected to symmetrical load, the middle part of the top plate is under compression and the corner points are under tension. The entire cross-section of the bottom plate is under tension, and the maximum transverse tension in the lower middle part per unit width is 68.7kN / m. The transverse tension at the upper end of the web is close to zero, and the maximum transverse tension at the lower end is 91.2kN / m, and the middle section approaches a straight line.

[0159] The results show that the lateral internal force calculated by the box beam lateral axial force proposed by the present invention is consistent with the Midas shell element result. The maximum error of 4.4% occurs at the top plate corner point, and the bottom plate and web plate have high accuracy. The lateral axial force can be used to calculate the lateral internal force of a single-box single-chamber simply supported beam of equal cross-section under symmetrical load.

[0160] Compared with the shell unit results, the lateral axial force errors obtained from the traditional lateral frame analysis are relatively large, and the top plate, bottom plate, and web plate all have opposite signs. The design based on the lateral support frame method may be unsafe.

[0161] (2) Variable cross-section continuous beam

[0162] Fig.10 The concrete continuous beam bridge shown has a mid-span span of 60m and a side span of 36m. The height of the box beam changes along the span in a quadratic curve. The beam height of the support section is 3.32m, and the beam height of the mid-span section is 2.21m. The top plate, bottom plate, and web plate have equal thickness in the longitudinal direction, and the cross-sectional dimensions at the mid-span are in meters. The beam width is 7.4m, and the span length is 60m in the uniform cross-section simply supported beam model. The concrete material is the same as that in 2.1, and the support reaction force under the self-weight of the frame model with opposite signs is applied to the corner points of the bottom plate of the shell unit model.

[0163] The variable cross-section continuous beam model has a total of 23424 units, such as Fig.11 As shown, the transverse axial force and bending moment of a beam piece in the mid-span section and 1 / 4 section of the box girder are extracted as analysis results.

[0164] The uniform cross-section formulas (8) and (9) are used to calculate the mid-span cross-section and the 1 / 4 cross-section. The self-weight G of the box girder from the 1 / 4 cross-section to the mid-span is 1795 kN. t ,y b , I, and the correction coefficients β1 = -0.40 and β2 = -0.48 are approximately taken from equations (16) and (17) to obtain the corrected lateral axial forces of the top and bottom plates.

[0165] Compare the lateral axial forces of the top and bottom plates and the inclined web of the shell unit space model, such as Fig.12 , Fig.13 As shown, the origin of the horizontal axis in the figure is the corner point of the top and bottom plates.

[0166] Tables 2 and 3 list the lateral axial forces at the top plate corners, the top plate middle, the bottom plate corners and the bottom plate middle under symmetrical loads.

[0167] Table 2 Comparison of transverse internal forces in the mid-span section of single-box single-chamber variable-section continuous beams under symmetrical loads

[0168]

[0169] Table 3 Comparison of transverse internal forces of 1 / 4 section of single-box single-chamber variable-section continuous beam under symmetrical load

[0170]

[0171] The distribution law of transverse axial force in the calculation and shell unit results is the same. For the mid-span section, the positions subject to transverse tension are mainly in the middle of the bottom plate and the lower end of the web. The maximum relative error is 6.2%, and the transverse tension at the lower end of the web is close to the corner load.

[0172] For the 1 / 4 section, the lateral axial force in the middle of the bottom plate calculated by the variable section formula is 7.91 kN / m, which is closer to the shell element result with a relative error of 3.5%. The lateral axial force at the corner point of the shell element method is close to zero, which is consistent with the node equilibrium law.

[0173] The results show that the transverse internal force of the box beam calculated by the transverse axial force is in good agreement with the Midas shell element results. The most unfavorable sections under symmetrical loads are the web and bottom plates, and the calculation accuracy of the bottom plate and the web of the mid-span section is relatively high. The transverse axial force can be used to calculate the transverse internal force of the mid-span section of a single-box single-chamber variable-section beam under symmetrical loads, and is also applicable to the 1 / 4 section top plate and bottom plate.

[0174] For variable cross-section continuous beams, the traditional transverse support frame method results in the transverse axial force having opposite signs at the top plate end and the web, and the accuracy of the bottom plate axial force is poor.

[0175] Method 2: In step S2, the single-box multi-chamber cross-section that needs lateral force analysis is adopted, and the uniform cross-section simply supported plate-shell finite element method is used; the simplified plate-shell finite element method is used to analyze the modeling and post-processing of the lateral force of the box girder; the plate-shell finite element is used to reproduce the results of the folded plate method or the finite strip method;

[0176] The key points of spatial plate and shell element modeling and post-processing are as follows:

[0177] (1) Establish a straight beam model with uniform cross-section. For example, establish a transverse frame wireframe model in MIDAS software, and generate a spatial plate and shell mesh by longitudinal scanning.

[0178] (2) Nodal loads are applied only at the corners of the section.

[0179] (3) Rigid diaphragms are set at the boundary. The in-plane thickness of the plate is set to be more than 10 times the thickness of the box girder plate, and the out-of-plane plate thickness is set to be less than 1 / 1000 of the thickness of the outer box girder plate, so as to be close to the boundary diaphragm assumption in the finite strip method.

[0180] (4) A line-supported simply supported boundary is set at the lower edge of the rigid diaphragm.

[0181] (5) Extract the transverse axial force and bending moment of a beam segment in the mid-span of the box girder as the analysis results.

[0182] The corner loads applied by the plate-shell element analysis are the support reactions of the transverse frame model with opposite signs. The analysis results of the plate-shell element model are superimposed on the transverse frame model to obtain the transverse internal force of the box girder.

[0183] The application and inspection of the lateral force analysis of single-box multi-chamber sections are as follows:

[0184] (1) Single-box double-chamber uniform cross-section continuous beam

[0185] ① Establish shell unit space model and beam unit transverse frame model

[0186] like Fig.14 The concrete continuous beam bridge shown has a span of 48+60+48m and a beam height of 2.21m. The beam width is 1.7+4+4+1.7m, and the concrete material is the same as the model in Section 3. A single-box double-chamber continuous beam shell unit space model and a beam unit transverse frame model are established. Self-weight and uniform load on the right top plate are applied. The concrete bulk density γ = 25kN / m3, and the uniform load q = 10kN / m2. The measuring points ① to ⑥ are the corners of the top plate and the middle web plate, the middle of the top plate, the corners of the top plate and the side web plate, the corners of the bottom plate and the middle web plate, the middle of the bottom plate, and the corners of the bottom plate and the side web plate.

[0187] ②Establish a simplified shell element model

[0188] A uniform cross-section simply supported beam plate-shell model is established, with a span of 60m, the mid-span of a continuous beam. The bottom plate corner load applied by the plate-shell unit analysis is the support reaction of the opposite transverse frame model. A rigid diaphragm is set at the boundary, with an in-plane thickness of 2.5m and an out-of-plane thickness of 0.0001m. A line-supported simply supported boundary is set at the lower edge of the rigid diaphragm to constrain the vertical displacement of the node. A node in the middle of the lower edge constrains the longitudinal and lateral displacements.

[0189] ③Extraction results

[0190] The lateral axial force and bending moment of the beam in the middle of the continuous beam under self-weight load and asymmetric load are extracted as the spatial shell element analysis results. The analysis results of the simply supported beam-plate shell element model are superimposed on the transverse frame model to obtain the transverse internal force of the box beam. The comparison of the transverse internal force results of the box beam after shell element spatial analysis, frame and superposition is shown in the figure. Fig.15 , Fig.16 As shown in the figure, the origin of the horizontal axis is the corner point of the top and bottom plates. The lateral internal forces of the top plate corner point, the middle of the top plate, the bottom plate corner point and the middle of the bottom plate of the right box room under the two loads are shown in Tables 4 and 5.

[0191] Table 4 Comparison of transverse internal forces in the mid-span section of single-box double-chamber continuous beams under self-weight load

[0192]

[0193] Table 5 Comparison of transverse internal forces in the mid-span section of single-box double-chamber continuous beams under asymmetric loads

[0194]

[0195]

[0196] ④Results analysis

[0197] Under symmetrical load, the lateral internal force distribution law of the space shell element and simplified shell element models of the uniform cross-section continuous beam is the same, and the relative error between the top plate and the middle of the bottom plate is small, within 5%. The most unfavorable position appears in the middle of the bottom plate, with the maximum lateral axial force of 29.07kN / m and the maximum lateral bending moment of 3.45kN·m / m.

[0198] Under asymmetric loads, the transverse axial force and bending moment of the frame analysis method showed opposite signs under asymmetric loads, and the results of the continuous beam shell element and the simplified shell element were in good agreement. The most unfavorable position appeared at the corner of the middle web plate and the top and bottom plates, with the maximum transverse axial force of 13.22 kN / m and the maximum transverse bending moment of 10.00 kN·m / m.

[0199] In summary, the simplified plate-shell element analysis method can extend the single-box single-chamber cross-section to the uniform cross-section continuous beam under multiple chambers. It is applicable to both symmetrical and asymmetrical loads with high accuracy. The traditional rigid support frame method has poor accuracy in analyzing the lateral axial force and can only be used to calculate the lateral bending moment under symmetrical loads.

[0200] (2) Single-box double-chamber variable-section continuous beam

[0201] The plate-shell element model is simplified to a simply supported box beam with uniform cross-section. The present invention extends the simplified model to a common continuous beam with variable cross-section in engineering to test its applicability.

[0202] ① Establish shell unit space model and beam unit transverse frame model

[0203] Fig.17 The span of the concrete continuous beam bridge shown is 48+60+48m. The height of the box beam changes along the span in a quadratic curve. The height of the support section is 3.32m, and the height of the mid-span section is 2.21m. The top plate, bottom plate, and web plate have the same thickness along the longitudinal direction. The concrete material, load, and mid-span cross section are the same as those in Section (1). A space model of a variable-section single-box double-chamber continuous beam shell unit is established, and symmetric loads and asymmetric loads are applied respectively. A transverse frame model is established for the mid-span section, and the applied load is consistent with the space model.

[0204] ②Establish a simplified shell element model

[0205] The span of the mid-span simply supported beam-plate-shell model is 60m, which is the mid-span of the continuous beam, and the corner load is applied. A rigid diaphragm is set at the support point, and a line support boundary is set at the lower edge of the rigid diaphragm to constrain the vertical displacement of the node. A node in the middle of the lower edge constrains the longitudinal and lateral displacements.

[0206] ③Extraction results

[0207] The results of the lateral internal force of the simply supported beam-plate shell element model and the lateral frame model are compared. The results of the lateral internal force of the frame analysis, shell element space analysis and box beam after superposition are compared. Figure 18-19 As shown in the figure, the origin of the horizontal coordinate is the corner point of the top and bottom plates. The lateral internal forces of the top and bottom plates of the right box room under two loads at the mid-span section are shown in Tables 6 and 7.

[0208] ④Results analysis

[0209] A transverse frame model is established for the mid-span section and the 1 / 4 section, and the applied load is consistent with the spatial model. The span of the mid-span and 1 / 4 section simply supported beam plate shell model is taken as the mid-span span of the continuous beam, 60m, and the corner load is applied. The transverse internal force results of the simply supported beam plate shell element model and the transverse frame model are superimposed. The transverse internal force results of the frame analysis, shell element spatial analysis and superimposed box beam are compared. Figure 18 to Figure 21 As shown in the figure, the origin of the horizontal coordinate is the corner point of the top and bottom plates. The lateral internal forces of the top and bottom plates of the right box room under two loads at the mid-span section are shown in Tables 6 and 7.

[0210] Table 6 Comparison of transverse internal forces in the mid-span of single-box double-chamber variable-section continuous beams under self-weight load

[0211]

[0212] Table 7 Comparison of transverse internal forces in the mid-span section of single-box double-chamber variable-section continuous beams under asymmetric loads

[0213]

[0214]

[0215] Under the symmetrical load of the mid-span section, the distribution law of the lateral internal force of the variable cross-section continuous beam space shell element and the simplified plate-shell element model is basically the same. The relative error of the lateral axial force in the middle of the top and bottom plates is small, within 5%. The most unfavorable position occurs in the middle of the bottom plate; under asymmetric loads, the results of the two models are in good agreement, and the most unfavorable position occurs at the corner point between the middle web plate and the top and bottom plates. The maximum lateral axial force is 13.22kN / m, and the maximum lateral bending moment is 10.00kN·m / m. Under the two loads, the distribution law of the simplified model of the 1 / 4 section is similar to the spatial shell element result, but the internal force value given by the simplified method is too large, which can be used as a conservative estimate of the lateral internal force.

[0216] In summary, it is feasible to simplify the calculation of the lateral internal force of the variable cross-section continuous beam multi-box chamber into a uniform cross-section simply supported beam model by the simplified plate-shell element analysis method. It is applicable to the top and bottom plates of the mid-span section under symmetrical and asymmetrical loads, but not to the 1 / 4 section. The lateral internal force of the simplified analysis model of the 1 / 4 section under both loads is greater than the continuous beam shell element result, which is conservative. For variable cross-section continuous beams, the frame analysis method is only applicable to the calculation of the lateral bending moment under symmetrical loads.

[0217] References:

[0218] [1] Zheng Zhen, Guo Jinqiong. Computer method for calculating transverse internal forces of box beam bridges[J]. Journal of Fuzhou University (Natural Science Edition), 1995(01):60-66.

[0219] ZHENG Zhen,GUO Jinqiong.A Computer Method of Calculating theTransversal Internal Force in Box Girder Bridge[J].

[0220] Journal of Fuzhou University:Natural Science,1995,23(1):60-66.(inChinese)

[0221] [2]Precast Segmental Box Girder Bridge Manual,Published by Post-tensioning Institute,Prestressed Concrete Institute,1978.

[0222] [3] Zheng Zhen, Guo Jinqiong. Analysis of lateral internal forces of trapezoidal box girder bridges[J]. Journal of Fuzhou University (Natural Science Edition), 1989(02):43-50.

[0223] ZHENG Zhen, GUO Jinqiong.Analysis of Transverse Internal Forces in BoxGirder of Trapezoid Section[J].Journal of Fuzhou University(Natural SciencesEdtion),1989,17(2):43-50.(in Chinese)

[0224] [4] Wang Chenguang, Zhang Yuanhai, Wang Yangsheng. Improved analysis method of transverse internal forces of thin-walled box girders with inclined webs[J]. Journal of the China Railway Society, 2021, 43(09): 127-132.

[0225] WANG Chenguang, ZHANG Yuanhai, WANG Yangsheng.An Improved Method forAnalyzing Transverse Internal Forces of Thin-walled Box Girders with InclinedWebs[J].Journal of the China Railway Society,2021,43(09):127-132.(in Chinese)

[0226] [5] Wang Yangsheng, Zhang Yuanhai. Analysis of transverse internal forces of thin-walled box girder with inclined web and study on the influence of its parameters[J]. Journal of the China Railway Society, 2017, 39(03): 104-111.

[0227] WANG Yangsheng,ZHANG Yuanhai.Research on Transverse Internal Forceand Its Parameter Influence of Thin-walled Box Girders with Inclined Webs[J].Journal of the China Railway Society,2017,39(3):104-111.(in Chinese)

[0228] [6]Brengle K W,Speedway C.American Association of State Highway andTransportation Officials[J].Washington,1993.

[0229] [7]International C.Canadian Highway Bridge Design Code[J].Design ofHighway Bridges,2000.

[0230] [8]Sennah K M,Kennedy J B.Literature Review in Analysis of Box-GirderBridges[J].Journal of Bridge Engineering,2002(1 / 3):7.

[0231] [9]Shushkewich K W.MEMBRANE FORCES ACTING ON A BOX GIRDER BRIDGE[J].Journal of Structural Engineering,1986,112(8):1900-1907.

[0232]

[10] Shushkewich,Kenneth W.Approximate Analysis of Concrete Box GirderBridges[J].Journal of Structural Engineering,1988,114(7):1644-1657.

[0233] The preferred embodiments of the present invention disclosed above are only used to help explain the present invention. The preferred embodiments do not describe all the details in detail, nor do they limit the present invention to the specific implementation methods described. Obviously, many modifications and changes can be made according to the content of this specification. This specification selects and specifically describes these embodiments in order to better explain the principles and practical applications of the present invention, so that those skilled in the art can understand and use the present invention well. The present invention is limited only by the claims and their full scope and equivalents.

Claims

1. A simplified calculation method for the transverse stress of a concrete box girder is characterized by: Here are the steps: S1: Confirm the box girder structure to be analyzed; S2: Perform lateral force analysis on the box girder confirmed in step S1.

2. The simplified calculation method for transverse stress of concrete box beam according to claim 1 is characterized in that: In the step S1, the box girder structure to be analyzed is confirmed, specifically: Confirm whether the box girder structure to be analyzed is a single-box single-chamber section.

3. The simplified calculation method for transverse stress of concrete box beam according to claim 1 is characterized in that: In the step S1, the box girder structure to be analyzed is confirmed, specifically: Confirm whether the box girder structure to be analyzed is a single-box multi-chamber section.

4. The simplified calculation method for transverse stress of concrete box beam according to claim 2 is characterized in that: When it is confirmed that the box girder structure to be analyzed is a single-box single-chamber section, the calculation method is as follows: Calculation of straight web box girders; Calculation of inclined web box girders; Calculation of mid-span section with variable cross-section; Calculate the variable cross-section 1 / 4 section.

5. The simplified calculation method for transverse stress of concrete box beam according to claim 4 is characterized in that: The calculation steps for the straight web box girder are as follows: The top plate calculation expression is as follows: Among them, y t is the distance from the center of the top plate, t t is the top plate thickness, N yy represents the axial force per unit length in the y direction, q t (x) is the uniformly distributed lateral load on the top plate; The base plate calculation expression is as follows: Among them, y b is the distance from the center of the bottom plate, t b is the bottom plate thickness, N yy represents the axial force per unit length along the y direction, q b (x)q b (x) is the uniformly distributed lateral load on the bottom plate; In the straight web box girder, the top transverse axial force is zero, the bottom transverse axial force is an external load, and the middle is approximately a straight line.

6. The simplified calculation method for transverse stress of concrete box beam according to claim 4 is characterized in that: The calculation steps for the inclined web box girder are as follows: The top plate calculation expression is as follows: Among them, y t is the distance from the center of the top plate, t t is the top plate thickness, N yy represents the axial force per unit length in the y direction, q t (x) is the uniformly distributed lateral load on the top plate; The base plate calculation expression is as follows: Among them, y b is the distance from the center of the bottom plate, t b is the bottom plate thickness, N yy represents the axial force per unit length in the y direction, q b (x)q b (x) is the uniformly distributed lateral load on the bottom plate; In the inclined web box girder, the top plate and the bottom plate superimpose the horizontal component of the web axial force, and the vertical force of the web axial force is balanced with the external load.

7. The simplified calculation method for transverse stress of concrete box beam according to claim 4 is characterized in that: The steps for calculating the mid-span section of the variable section are as follows: If the transverse axial forces of the top plate, bottom plate and web of the variable cross-section box girder in the mid-span section are consistent with those of the simply supported box girder with equal cross-section, the transverse internal forces of the mid-span section of the variable cross-section box girder are analyzed by the top plate calculation expression and the bottom plate calculation expression; The top plate calculation expression is as follows: Among them, y t is the distance from the center of the top plate, t t is the top plate thickness, N yy represents the axial force per unit length in the y direction, q t (x) is the uniformly distributed lateral load on the top plate; The base plate calculation expression is as follows: Among them, y b is the distance from the center of the bottom plate, t b is the bottom plate thickness, N yy represents the axial force per unit length in the y direction, q b (x)q b (x) is the uniformly distributed lateral load on the bottom plate; If the bottom curve of the variable-section box girder is a quadratic parabola, the expression is as follows:

8. The simplified calculation method for transverse stress of concrete box beam according to claim 4 is characterized in that: The steps for calculating the variable cross-section 1 / 4 cross-section are as follows: Calculate the correction coefficients β1 and β2 using the following formula: The transverse axial force of the variable cross-section can be obtained by multiplying the transverse axial force of the top plate and the bottom plate at the equal cross-section by the correction coefficients β1 and β2, and the expression is as follows:

9. The simplified calculation method for transverse stress of concrete box beam according to claim 3 is characterized in that: Confirm that the box girder structure to be analyzed is a single-box multi-chamber section. The analysis method is as follows: Establish a straight beam model with uniform cross-section; The cross section has nodal loads applied at the corners; Rigid diaphragms are set at the boundaries; A line-supported simply-supported boundary is set at the lower edge of the rigid diaphragm; Extract the transverse axial force and bending moment of a beam in the mid-span of the box girder as the analysis results; The lateral internal force of the box girder is obtained by superimposing the analysis results of the plate-shell element model with the lateral frame model.

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