Shear lag estimation method for box girder with stiffening structure

Abstract

The application provides a shear lag estimation method for a box girder with a reinforcing structure, and comprises the following steps: S10: establishing a structural model for the box girder based on an analogy bar method, distributing the area of the reinforcing structure to simulation bars and simulation plates, and outputting the equivalent area of the simulation bars and the equivalent thickness of the simulation plates; S20: establishing stress balance equations and deformation coordination equations for the simulation bars, and outputting differential equations of shear flow; S30: outputting the shear flow of the simulation bars and the shear flow of the reinforcing structure according to the differential equations of the shear flow; and S40: outputting the stress of the simulation bars. The shear lag estimation method for the box girder with the reinforcing structure has the beneficial effects of wide application range and being capable of being used for guiding engineering design, and solves the defects of the traditional analogy bar method that cannot process the reinforcing structure.

Description

Technical Field

[0001] This invention relates to the field of bridge construction technology, and in particular to a method for estimating shear lag in box girders with reinforced structures. Background Technology

[0002] Shear lag refers to the phenomenon of uneven distribution of normal stress in thin-walled box girders. In engineering applications, shear lag can lead to instability or failure of the box girder, resulting in defects and even accidents. Modern bridge designs increasingly use wider cross-sections, which increases the shear lag effect. Therefore, research on the shear lag effect of thin-walled box girders has received increasing attention in recent years.

[0003] Shear lag effect leads to instability in the mechanical properties of thin-walled box girders and may cause their mechanical behavior to differ from the theoretical analysis of beam elements. Therefore, a reasonable analysis of the shear lag effect is crucial for structural safety. To address this, Risner studied the shear lag effect of a rectangular box girder without side cantilever plates based on minimum potential energy and used a quadratic parabola to simulate the displacement function of shear lag warping. Besides using a quadratic parabola to simulate shear warping displacement, cubic parabolas and cosine functions are also used to analyze the shear lag effect. To ensure structural safety, many practical engineering projects also use the shear lag coefficient and effective width to describe the shear lag effect.

[0004] Shear lag effect has been studied for decades, and related theories have developed to some extent, but these have been limited to simple cross-sectional forms, such as single-cell, double-cell, or T-shaped cross-sections. However, with the increasing number of box girder bridges, their cross-sections have become increasingly complex. During construction, numerous reinforcing ribs are typically added to the top and bottom slabs to improve mechanical properties, but this also complicates the shear lag effect. Therefore, understanding the mechanical characteristics of these structures is necessary to improve the design of reinforced steel box girders.

[0005] The analysis of shear lag effects generally employs the energy variational method and the analogy bar method, but these methods lack applicability to the analysis of shear lag effects in box girders with reinforced structures. The energy variational method requires the assumption of a reasonable shear warping displacement function, which is difficult to obtain for box girder bridges with reinforced structures; traditional quadratic, cubic, and cosine functions are not applicable. Unlike the energy variational method, which requires the assumption of a warping displacement function, the analogy bar method focuses on establishing equilibrium differential equations by considering the mechanical properties of the structure; however, existing analogy bar methods cannot analyze reinforced structures. Summary of the Invention

[0006] In view of the above, this application proposes a shear lag estimation method for box girders with reinforced structures based on the traditional analog rod method, in order to solve the above problems.

[0007] This application provides a method for estimating the shear lag of a box girder with a reinforced structure, including:

[0008] S10: Based on the analog rod method, a structural model of the box girder is established, the area of ​​the strengthening structure is allocated to the simulated rod and the simulated plate, and the equivalent area of ​​the simulated rod and the equivalent thickness of the simulated plate are output.

[0009] S20: Establish the force balance equation and deformation compatibility equation for the simulated rod, and output the differential equation of shear flow;

[0010] S30: Based on the differential equation of shear flow, output the shear flow of the simulated rod and the shear flow of the reinforced structure;

[0011] S40: Outputs the stress of the analog rod.

[0012] In at least one embodiment, step S10 includes:

[0013] Half of the reinforced structural area is allocated to the simulated rod and the simulated plate respectively.

[0014] In at least one embodiment, the reinforcing structure is a stalk axilla.

[0015] In at least one embodiment, step S10 includes:

[0016] The moment of inertia I of the simulated rod axle on the top plate is calculated using equation (1). top :

[0017]

[0018] The additional area ΔA1 of each simulated rod on the top plate is calculated using equation (2):

[0019]

[0020] Based on the fact that the bending stresses of the top plate and the bottom plate are equal, the equivalent area A of the top plate is calculated. eu The equivalent area A of the base plate eb :

[0021] A eu =2λ1t w H+β1at1+r1a4h (3)

[0022] A eb =2λ2t w H+β2bt2 (4)

[0023] Calculate the equivalent thickness t of the top plate eu and the equivalent thickness t of the base plate eb :

[0024]

[0025] t eb =β2t2 (6)

[0026] Where a4 and h are the width and height of the axle, respectively; a and b are the lengths of the top and bottom plates, respectively; h1 and h2 represent the distances from the centroid of the cross section to the centerline of the top and bottom plates, respectively; H is the distance from the centerline of the top plate to the centerline of the bottom plate; and t1, t2, and t w λ1 and λ2 are the thicknesses of the top plate, bottom plate, and web plate, respectively; λ1 and λ2 are the areas of the upper and lower flanges of the web plate, respectively; β1 and β2 are the heights of the top plate and bottom plate, respectively; and r1 and r2 are the equivalent coefficients of the upper and lower flanges, respectively.

[0027] In at least one embodiment, step S10 includes:

[0028] The structural model is divided into 18 simulated rods, with 13 rods in the top plate and 5 rods in the bottom plate. i The equivalent area of ​​the i-th simulated rod is calculated using the following formula:

[0029]

[0030]

[0031]

[0032]

[0033]

[0034] A 15 =A 16 =A 17 =a r3 t eb (12)

[0035] Among them, a r1 a is the spacing between every two adjacent simulated rods within the left and right cantilever arms of the top plate. r2 a is the spacing between every two adjacent simulated rods in the middle of the top plate. r3 This represents the spacing between every two adjacent simulated rods on the base plate.

[0036] In at least one embodiment, step S20 includes:

[0037] The force equilibrium equations are established by selecting simulation rods 1 to 7 and 14 to 16:

[0038]

[0039]

[0040]

[0041]

[0042]

[0043]

[0044]

[0045]

[0046]

[0047]

[0048] Where, q e The shear flow of the e-th simulation plate is distributed evenly to the 4th and 6th simulation rods, and considered as two stiffening ribs. The shear flows of the two stiffening ribs are respectively represented by q. r3 With q r4 It means that q El Let F represent the shear flow of the l-th web. i This represents the axial force of the i-th simulated rod;

[0049] Calculate the shear deformation γ of the e-th simulated plate. e :

[0050]

[0051] Among them, u i and u i+1 Let be the longitudinal displacements of the i-th and (i+1)-th simulated rods, respectively; Δx be the differential element of the simulated plate length; Di be the differential element of the simulated plate width; and ε be the differential element of the simulated plate width. i and ε i+1 The strains of the i-th and (i+1)-th simulated rods are σ, respectively. i and σ i+1 These are the stresses of the i-th and (i+1)-th simulated rods, respectively, and E represents the Young's modulus of the material.

[0052] Based on the relationship between the shear flow and the shear strain of the simulated plate, the following equation is constructed:

[0053] q e =γ e t eu G (24)

[0054] Where G is the shear modulus of the material;

[0055] Substituting equation (24) into equation (23), the following equation is output:

[0056]

[0057] The following equation is output from equation (27):

[0058]

[0059] Substituting the force equilibrium equations into equation (28), the following differential equations are output:

[0060]

[0061]

[0062]

[0063]

[0064]

[0065]

[0066]

[0067]

[0068] in,

[0069] In at least one embodiment, step S30 includes:

[0070] Determine the boundary conditions of the differential equation:

[0071]

[0072] Where L is the length of the box girder.

[0073] In at least one embodiment, step S30 includes:

[0074] The structural model is divided into n segments. Each segment is assumed to have m+p unknown shear flows, where m is the number of shear flows on the top plate and p is the number of shear flows on the bottom plate. A cubic spline function is constructed to simulate the shear flows in each segment. For the top plate, the following equation is output:

[0075] q i ={S i1 ,S i2 ,S i3 ,…,S in}(i=1,2,3,…,m) (36)

[0076] S ij =a ij0 +a ij1 x 1 +a ij2 x 2 +a ij3 x 3 (i=1,2,…,m; j=1,2,…,n)(37)

[0077] Among them, S ij For the shear flow of the j-th segment and the i-th simulated rod, the value of x ranges from x. j-1 ≤x≤x j (j=1,2,…,n; x0=0; x n =L), a ij0 a ij1 a ij2 a ij3 All are the fitting coefficients of the i-th simulated rod in the j-th segment; Substituting equation (39) into equations (29) to (36):

[0078]

[0079] Considering the deformation, the compatibility equation is satisfied in the j-th segment, and the following equation is constructed:

[0080]

[0081] Based on equations (40) and (41), output the coefficient C in the spline function. ij Substituting into equation (38), the axial force of the simulated rod is output.

[0082] In at least one embodiment, step S40 includes:

[0083] Based on the shear lag effect of the reinforced structure, the stress of the simulated rod is output:

[0084]

[0085]

[0086]

[0087]

[0088]

[0089]

[0090]

[0091]

[0092]

[0093]

[0094] In at least one embodiment, the reinforcing structure is a rectangular rib.

[0095] Compared with existing technologies, the shear lag estimation method for reinforced box girders in this application solves the problem that the traditional analog rod method cannot take into account the reinforced structure by incorporating the reinforced structure into the simulated rod and simulated plate. It has the beneficial effect of wide applicability and can be used to guide engineering design. Attached Figure Description

[0096] Figure 1 This is a flowchart of the steps for estimating the shear lag of a box girder with a reinforced structure in one embodiment of the present invention;

[0097] Figure 2 yes Figure 1 The diagram shows a structural schematic of a box girder with a truncated haunch in the method described above;

[0098] Figure 3 yes Figure 2 A schematic diagram of the cross-section of a box girder with a spur-shaped haunch is shown.

[0099] Figure 4 yes Figure 3 A schematic diagram of the structural model of the box girder with a spur-shaped haunch shown;

[0100] Figure 5 yes Figure 4 A schematic diagram of a partial structural model of a box girder with a spur-shaped haunch shown;

[0101] Figure 6 yes Figure 2 The diagram shows a simulated plate shear deformation in a box girder with a stalk and a haunch.

[0102] Figure 7 is Figure 2 The diagram shows a simply supported box girder with a stalk and a haunch under uniformly distributed load and concentrated load.

[0103] Figure 8 is Figure 2 The diagram shows a structural schematic of a box girder with a stalk and a haunch in a bridge structure.

[0104] Figure 9 yes Figure 2 The box girder with a stalk and a haunch shown is Figure 1 A schematic diagram comparing the calculation results of the method with the finite element analysis results;

[0105] Figure 10 yes Figure 1 The diagram shows a structural schematic of a rectangular ribbed box girder in the method described.

[0106] Figure 11 is Figure 10 A schematic diagram of the cross-section of a box girder with rectangular ribs is shown.

[0107] Figure 12 is Figure 10 The rectangular ribbed box girder shown is Figure 1 A schematic diagram comparing the calculation results of the method with the results of finite element analysis. Detailed Implementation

[0108] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention. All other embodiments obtained by those skilled in the art based on the embodiments of this invention without inventive effort are within the scope of protection of this invention.

[0109] In the description of this invention, it should be understood that the terms "center," "longitudinal," "lateral," "length," "width," "thickness," "upper," "lower," "front," "rear," "left," "right," "vertical," "horizontal," "top," "bottom," "inner," "outer," "clockwise," and "counterclockwise," etc., indicating orientations or positional relationships based on the orientations or positional relationships shown in the accompanying drawings, are only for the convenience of describing the invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation, and therefore should not be construed as a limitation of the invention. Furthermore, the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of indicated technical features. Thus, features defined with "first" and "second" may explicitly or implicitly include one or more of the stated features. In the description of this invention, "a plurality of" means two or more, unless otherwise explicitly specified.

[0110] In the description of this invention, it should be noted that, unless otherwise explicitly specified and limited, the terms "installation," "connection," and "linking" should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral connection; they can refer to a mechanical connection, an electrical connection, or a connection that allows for communication; they can refer to a direct connection or an indirect connection through an intermediate medium; they can refer to the internal communication of two components or the interaction between two components. Those skilled in the art can understand the specific meaning of the above terms in this invention according to the specific circumstances.

[0111] In this invention, unless otherwise explicitly specified and limited, "above" or "below" the second feature can include direct contact between the first and second features, or contact between the first and second features through another feature between them. Furthermore, "above," "over," and "on top" of the second feature includes the first feature being directly above or diagonally above the second feature, or simply indicates that the first feature is at a higher horizontal level than the second feature. "Below," "below," and "under" the second feature includes the first feature being directly above or diagonally above the second feature, or simply indicates that the first feature is at a lower horizontal level than the second feature.

[0112] The following disclosure provides many different embodiments or examples for implementing various structures of the invention. To simplify the disclosure, specific examples of components and arrangements are described below. These are merely examples and are not intended to limit the invention. Furthermore, reference numerals and / or letters may be repeated in different examples; such repetition is for simplification and clarity and does not in itself indicate a relationship between the various embodiments and / or arrangements discussed. In addition, examples of various specific processes and materials are provided in this invention, but those skilled in the art will recognize the application of other processes and / or the use of other materials.

[0113] Please see Figure 1 A method for estimating shear lag in a reinforced box girder, comprising:

[0114] S10: Based on the analog rod method, a structural model of the box girder is established, the area of ​​the strengthening structure is allocated to the simulated rod and the simulated plate, and the equivalent area of ​​the simulated rod and the equivalent thickness of the simulated plate are output.

[0115] S20: Establish the force balance equation and deformation compatibility equation for the simulated rod, and output the differential equation of shear flow;

[0116] S30: Based on the differential equation of shear flow, output the shear flow of the simulated rod and the shear flow of the reinforced structure;

[0117] S40: Outputs the stress of the analog rod.

[0118] Please refer to [link / reference needed] for further explanation. Figure 2 and Figure 9 The box girder structure includes a top plate, a bottom plate, a web, and reinforcing structures. The web consists of two plates, left and right, used to connect the top and bottom plates. The top plate is wider than the bottom plate. The reinforcing structures may be haunches located at the connection between the web and the top plate, or rectangular ribs facing each other on the top or bottom plate.

[0119] Traditional analog rod methods only produce structural models that simulate rods and plates, failing to address other structural variations. The inventors argue that the analog rod method assumes that the vertical shear force at any cross-section of the box girder is entirely borne by the web and uniformly distributed across it. However, the actual load on a bridge primarily resides near the web, thus simplifying the load distribution. The inventors chose to allocate a portion of the reinforced structure's area to the simulated plate and another portion to the simulated rod, thereby transforming the reinforced box girder into a standard box girder structure, thus bringing it within the scope of the analog rod method. The area of ​​the reinforced structure refers to the cross-sectional area along the horizontal direction.

[0120] In one embodiment, step S10 includes:

[0121] Half of the reinforced structural area is allocated to the simulated rod and the simulated plate respectively.

[0122] Through analysis and experimentation, the inventors concluded that allocating half of the reinforcing structure area to the simulated rod and the simulated plate yields results that are closer to the actual results, making it the preferred implementation method.

[0123] Please see Figure 2 As shown in Figure 7, in one embodiment, the reinforcing structure is a stalk axilla.

[0124] In one embodiment, step S10 includes:

[0125] The moment of inertia I of the simulated rod axle on the top plate is calculated using equation (1). top :

[0126]

[0127] The additional area ΔA1 of each simulated rod on the top plate is calculated using equation (2):

[0128]

[0129] Based on the fact that the bending stresses of the top plate and the bottom plate are equal, the equivalent area A of the top plate is calculated. eu The equivalent area A of the base plate eb :

[0130] A eu =2λ1t w H+β1at1+r1a4h (3)

[0131] A eb =2λ2t w H+β2bt2 (4)

[0132] Calculate the equivalent thickness t of the top plate eu and the equivalent thickness t of the base plate eb:

[0133]

[0134] t eb =β2t2 (6)

[0135] Where a1 and a3 are the lengths of the left and right cantilever arms of the top plate, a2 is the length of the middle part of the top plate, a4 and h are the width and height of the haunch, a and b are the lengths of the top plate and the bottom plate, h1 and h2 represent the distances from the centroid of the cross section to the center line of the top plate and the center line of the bottom plate, respectively, H is the distance from the center line of the top plate to the center line of the bottom plate, and t1, t2 and t w λ1 and λ2 are the thicknesses of the top plate, bottom plate, and web plate, respectively; λ1 and λ2 are the areas of the upper and lower flanges of the web plate, respectively; β1 and β2 are the heights of the top plate and bottom plate, respectively; and r1 and r2 are the equivalent coefficients of the upper and lower flanges, respectively.

[0136] Please continue reading. Figure 4 In one embodiment, step S10 includes:

[0137] The structural model is divided into 18 simulated rods, with 13 rods in the top plate and 5 rods in the bottom plate. i The equivalent area of ​​the i-th simulated rod is calculated using the following formula:

[0138]

[0139]

[0140]

[0141]

[0142]

[0143] A 15 =A 16 =A 17 =a r3 t eb (12)

[0144] Among them, a r1 a is the spacing between every two adjacent simulated rods within the left and right cantilever arms of the top plate. r2 a is the spacing between every two adjacent simulated rods in the middle of the top plate. r3 This represents the spacing between every two adjacent simulated rods on the base plate.

[0145] It's important to explain that the more simulated bars the structural model is divided into, the greater the subsequent computational load, but the higher the accuracy. Since the final calculation yields the stress of each simulated bar, plotting the stresses of each simulated bar sequentially facilitates observation and analysis of the shear lag effect. Therefore, the more simulated bars there are, the smaller the spacing between the results in the graph, and the higher the accuracy. This example uses 18 simulated bars for calculation.

[0146] Please continue reading. Figure 5 In one embodiment, step S20 includes:

[0147] The force equilibrium equations are established by selecting simulation rods 1 to 7 and 14 to 16:

[0148]

[0149]

[0150]

[0151]

[0152]

[0153]

[0154]

[0155]

[0156]

[0157]

[0158] Where, q e The shear flow of the e-th simulation plate is distributed evenly to the 4th and 6th simulation rods, and considered as two stiffening ribs. The shear flows of the two stiffening ribs are respectively represented by q. r3 With q r4 It means that q El Let F represent the shear flow of the l-th web. i This represents the axial force of the i-th simulated rod.

[0159] Since the box girder structure is axisymmetric, only half of the structure needs to be analyzed to obtain the stress situation of the entire box girder structure. This embodiment divides the structure into 18 simulated rods; therefore, half of them are selected, i.e., the following... Figure 4 and Figure 5 The simulation rods 1-7 and 14-16 shown are used for calculation.

[0160] Please continue reading. Figure 6 The diagram shows the stress and deformation of the simulated plate. Step S20 may further include: calculating the shear deformation γ of the e-th simulated plate. e :

[0161]

[0162] Among them, u i and u i+1 Let be the longitudinal displacements of the i-th and (i+1)-th simulated rods, respectively; Δx be the differential element of the simulated plate length; Di be the differential element of the simulated plate width; and ε be the differential element of the simulated plate width. i and ε i+1 The strains of the i-th and (i+1)-th simulated rods are σ, respectively. i and σ i+1 These are the stresses of the i-th and (i+1)-th simulated rods, respectively, and E represents the Young's modulus of the material.

[0163] Based on the relationship between the shear flow and the shear strain of the simulated plate, the following equation is constructed:

[0164] q e =γ e t eu G (24)

[0165] Where G is the shear modulus of the material;

[0166] Substituting equation (24) into equation (23), the following equation is output:

[0167]

[0168] The following equation is output from equation (27):

[0169]

[0170] Substituting the force equilibrium equations into equation (28), the following differential equations are output:

[0171]

[0172]

[0173]

[0174]

[0175]

[0176]

[0177]

[0178]

[0179] in,

[0180] Please continue referring to Figure 7. In one embodiment, step S30 includes:

[0181] The boundary conditions for the differential equation are determined as follows: regardless of whether the structural model is subjected to a uniformly distributed load as shown in Figure 7(a) or a concentrated load as shown in Figure 7(b), the boundary conditions are the same:

[0182]

[0183] Where L is the length of the box girder.

[0184] In one embodiment, step S30 includes:

[0185] The structural model is divided into n segments. Each segment is assumed to have m+p unknown shear flows, where m is the number of shear flows on the top plate and p is the number of shear flows on the bottom plate. A cubic spline function is constructed to simulate the shear flows in each segment. For the top plate, the following equation is output:

[0186] q i ={S i1 ,S i2 ,S i3 ,…,S in}(i=1,2,3,…,m) (36)

[0187] S ij =a ij0 +a ij1 x 1 +a ij2 x 2 +a ij3 x 3 (i=1,2,…,m; j=1,2,…,n)(37)

[0188] Among them, S ij For the shear flow of the j-th segment and the i-th simulated rod, the value of x ranges from x. j-1 ≤x≤x j (j=1,2,…,n; x0=0; x n =L), a ij0 a ij1 a ij2 a ij3 All are fitting coefficients for the i-th simulated rod in the j-th segment. Equations (36) and (37) have a total of 4nm unknowns. Substituting equation (39) into equations (29) to (36):

[0189]

[0190] Equation (38) contains m(n+1) equations, and the aforementioned boundary condition equations number 2m. Therefore, the total number of equations reaches 4nm. Combining the aforementioned total of 4nm unknowns, the solution can be obtained through 4nm equations. Due to the large number of formulas, software tools such as MATLAB can be used for solving the problem.

[0191] Considering the deformation, the compatibility equation is satisfied in the j-th segment, and the following equation is constructed:

[0192]

[0193] Based on equations (38) and (39), output the coefficient C in the spline function. ij Substituting into equation (36), the axial force of the simulated rod is output.

[0194] In one embodiment, step S40 includes:

[0195] Based on the shear lag effect of the reinforced structure, the area of ​​the reinforced structure is converted into an additional area and applied to part of the simulated rod, and the stress of the simulated rod is output:

[0196]

[0197]

[0198]

[0199]

[0200]

[0201]

[0202]

[0203]

[0204]

[0205]

[0206] By calculating the stress in each simulated member, the stress distribution of the box girder can be obtained, facilitating the analysis of shear lag effects. Then, dividing the axial force of each simulated member by the average axial force of all simulated members yields the shear lag coefficient for each member. Furthermore, the shear lag coefficient can be visualized more intuitively through graphs or other means, making it easier for relevant personnel to understand.

[0207] In one embodiment, the reinforcing structure is a rectangular rib.

[0208] It should be explained that the shear lag estimation method for reinforced box girders in this application can be set in computer software and run automatically, which has the advantage of high computational efficiency, enabling relevant personnel to quickly and intuitively understand and analyze the shear lag effect after inputting parameters.

[0209] The method proposed in this application is verified through the following examples.

[0210] This embodiment proposes a model with the following parameters:

[0211] Line loads were selected in the numerical simulation. Considering that the actual load on the bridge is mainly located near the web, the simplification of the load conditions is reasonable. The geometric and load parameters of the structure are shown in Figure 8. This model is a simply supported structure with a total length of 2m and a uniformly distributed load of 100kN / m. The Young's modulus E and shear modulus G of the material are 206GPa and 79GPa, respectively. The cross-sectional height is 0.2m; the cantilever length is 0.25m; the thickness of the top and bottom plates is 8mm; the web thickness is 6mm; the length and height of the haunches are 0.2m and 8mm, respectively; and the lengths of the top and bottom plates are 1m and 0.4m, respectively.

[0212] Under the boundary conditions and loads shown in Figure 8, the shear lag of this box girder bridge was studied using ANSYS software, and the structural calculation model was simulated using Shell181 shell elements. To compare the results of the finite element analysis with those of the improved method of this invention, the mid-span cross-section was analyzed.

[0213] To evaluate the accuracy of the improved method of this invention, we compared its results with those of the finite element method. Figure 9 The improved method of this invention and the results of finite element analysis are shown. It can be seen that the results generated by the improved method of this invention are similar to those of finite element analysis. Table 1 shows a detailed comparison between the results. Analysis of the data in the table shows that both the absolute and relative errors are small, with a maximum absolute error of 5.78 MPa and a maximum relative error of 7.29%. This embodiment illustrates that the improved method of this invention has practical value and can be used for actual analysis of the shear lag effect of thin-walled box girders with spurs.

[0214] Table 1 Comparison between the improved method of the present invention and the results of the finite element method.

[0215]

[0216] When the reinforcing structure is a rectangular rib, the same approach can be used: the area of ​​the rectangular rib is evenly distributed across the simulated rod and the simulated plate, and then the analog rod method is used for calculation. The following examples will verify this.

[0217] This embodiment proposes a model with the following parameters: Please refer to Figure 11. This model is a simply supported structure with a total length of 14.4m and a uniformly distributed load of 216kN / m. The Young's modulus E and shear modulus G of the material are 206GPa and 79GPa, respectively. The cross-sectional height is 1.2m, the cantilever length is 1.2m, the thickness of the top and bottom plates is 24mm, the web thickness is 24mm, the length and thickness of the rectangular ribs are 0.1m and 16mm, respectively, the spacing between the rectangular ribs is 0.3m, and the lengths of the top and bottom plates are 3.6m and 1.2m, respectively.

[0218] Under the same boundary conditions and loads, a finite element model with 48,384 quadrilateral elements and a unit size of 5 mm was established as a control. The method proposed in this application was verified by comparing the results of the finite element analysis with those of the method proposed in this invention.

[0219] Referring to Figure 12 and Table 2, it can be seen that the results obtained by the method proposed in this invention are similar to those of the finite element method, with a relative error range of 3% to 6.5%. Table 2 details the comparison of the specific results of the two methods. The data in Table 2 show that both the absolute and relative errors are small, with a maximum absolute error of 3.1 MPa and a maximum relative error of 6.3%. This embodiment demonstrates that the improved method of this invention has practical value and can be used for the actual analysis of the shear lag effect of thin-walled box girders with rectangular ribs.

[0220] Table 2 Comparison of results from the present invention and finite element analysis.

[0221]

[0222]

[0223] The shear lag estimation method for reinforced box girders in this application overcomes the limitation of traditional analog rod methods, which cannot obtain accurate numerical values ​​of the shear lag effect in reinforced box girder bridges, by equivalently treating the reinforced structure. Furthermore, comparison shows that the results obtained in this application are only slightly different from those calculated by finite element methods, verifying the accuracy of the proposed method. The method proposed in this invention aims to provide an efficient and intuitive way to understand the shear lag effect of reinforced thin-walled box girder bridges in bridge design.

[0224] It will be apparent to those skilled in the art that the present invention is not limited to the details of the exemplary embodiments described above, and that the invention can be implemented in other specific forms without departing from the spirit or essential characteristics of the invention. Therefore, the embodiments should be considered in all respects as exemplary and non-limiting, and the scope of the invention is defined by the appended claims rather than the foregoing description. Thus, it is intended that all variations falling within the meaning and scope of equivalents of the claims be incorporated into the present invention.

[0225] The above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit it. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can be made to the technical solutions of the present invention without departing from the spirit and scope of the technical solutions of the present invention.

Claims

1. A method for estimating shear lag in a box girder with a reinforced structure, characterized in that, include: S10: Based on the analog rod method, a structural model of the box girder is established, the area of ​​the strengthening structure is allocated to the simulated rod and the simulated plate, and the equivalent area of ​​the simulated rod and the equivalent thickness of the simulated plate are output. S20: Establish the force balance equation and deformation compatibility equation for the simulated rod, and output the differential equation of shear flow; S30: Based on the differential equation of shear flow, output the shear flow of the simulated rod and the shear flow of the reinforced structure; S40: Stress of the output analog rod Step S10 includes: The moment of inertia of the simulated rod axle on the top plate is calculated using equation (1). : （1） The additional area of ​​each simulated rod on the top plate is calculated using equation (2). : （2） Based on the fact that the bending stresses of the top plate and the bottom plate are equal, the equivalent area of ​​the top plate is calculated. Equivalent area of ​​base plate : （3） （4） Calculate the equivalent thickness of the top plate and the equivalent thickness of the base plate : （5） （6） in, a 4 、h These are the width and height of the axilla, respectively. a and b These are the lengths of the top plate and the bottom plate, respectively. h 1 and h 2 represents the distance from the centroid of the cross section to the centerline of the top plate and the centerline of the bottom plate, respectively. H It is the distance between the center line of the top slab and the center line of the bottom slab. t 1 ,t 2 and t w These refer to the thicknesses of the top plate, bottom plate, and web plate, respectively. , These are the areas of the upper and lower flanges of the web, respectively. , These are the heights of the top and bottom plates, respectively. , These are the equivalent coefficients for the upper and lower flanges, respectively; Step S10 includes: The structural model is divided into 18 simulated rods, with 13 rods in the top plate and 5 rods in the bottom plate. i For the first i The equivalent area of ​​the simulated rod is calculated using the following formula: （7） （8） （9） （10） （11） （12） in, This represents the spacing between every two adjacent simulated rods within the left and right cantilever arms of the top plate. The spacing between every two adjacent simulated rods in the middle of the top plate. This represents the spacing between every two adjacent simulated rods on the base plate.

2. The method for estimating shear lag in a reinforced box girder as described in claim 1, characterized in that, Step S10 includes: Half of the reinforced structural area is allocated to the simulated rod and the simulated plate respectively.

3. The method for estimating shear lag in a reinforced box girder as described in claim 2, characterized in that, The reinforcing structure is a stalk axilla.

4. The method for estimating shear lag in a reinforced box girder as described in claim 2, characterized in that, The reinforcing structure is a rectangular rib.

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