Fast algorithm for mechanical properties of cross section of box girder

Abstract

The invention discloses a fast algorithm for mechanical properties of a box beam section. The fast algorithm comprises the following steps: establishing a parameterized calculation model of the box beam section; obtaining an area analytical expression of the cross section of the box girder; obtaining a torsional rigidity analytical expression; obtaining a centroid coordinate analytical expression; obtaining an effective shear area analytical expression; obtaining an inertia moment analytical expression; obtaining an analytical expression of the distance from the neutral axis to the edge fiber; obtaining a shear coefficient analytical expression along the local coordinate system direction of the unit; obtaining an analytical expression of inner and outer perimeters; obtaining a plane coordinate analytical expression of the four angular points; and substituting the obtained characteristic values into expressions of a stiffness matrix, a mass matrix and a damping matrix of the corresponding spatial beam units to obtain unit characteristic matrix values. According to the calculation method, the accurate mapping relation between the parameter space and the mechanical property space can be established, the limitation of a traditional method in the aspects of calculation dimensionality and efficiency is broken through, and an efficient and reliable section property calculation tool is provided.

Description

Technical Field

[0001] The invention belongs to the technical field of bridge engineering in the transportation industry, and particularly relates to a fast algorithm for mechanical properties of a box beam section. Background Art

[0002] In the finite element analysis of bridge engineering, the precise characterization of the mechanical properties of box girder sections plays a decisive role in the accuracy of structural simulation. These properties encompass a multi-dimensional parameter system, including cross-sectional area, torsional stiffness, centroid coordinates, effective shear area, and moment of inertia. Their numerical quality directly affects the accuracy of the unit stiffness matrix constructed from material properties and cross-sectional characteristics. With the increasing demand for refined modeling in large-scale and long-span bridge engineering, the solution of the massive mechanical property parameters of box girder sections has become a key bottleneck restricting computational efficiency. A fast algorithm that combines computational accuracy and efficiency is urgently needed to overcome the bottleneck of the traditional method in balancing computational completeness and timeliness.

[0003] Existing methods for calculating cross-sectional properties mainly include three technical routes: block superposition method, integral method, and numerical simulation based on mesh subdivision. Research shows that the traditional block superposition method and integral method have the defect of incomplete calculation dimensions, which makes it difficult to meet the requirements of simultaneous solution of multivariate mechanical parameters; although the numerical simulation method based on mesh subdivision can realize full characteristic calculation, its complex modeling process and high timing cost seriously restrict the efficiency of engineering application. It is worth noting that the box beams in railway and highway bridges generally present the characteristics of standardized geometric configuration and gradual change of cross-sectional dimensions. This regular distribution provides a theoretical basis for parametric modeling. By establishing a parameter-driven cross-sectional characteristic analytical model and constructing an explicit mathematical relationship between mechanical parameters and geometric variables, it is theoretically possible to achieve rapid batch solution of all characteristic indicators.

[0004] This paper addresses the technical bottlenecks in the calculation of box girder cross-sectional properties by innovatively proposing a new algorithm that integrates analytical methods with parametric modeling. This method systematically constructs a cross-sectional geometric parameter system and establishes a precise mapping relationship between parameter space and mechanical property space, thus overcoming the limitations of traditional methods in terms of computational dimensionality and efficiency. The algorithm is designed to balance engineering applicability and program compatibility, and its modular architecture can be easily integrated into finite element pre-processing systems, providing an efficient and reliable cross-sectional property calculation tool for the digital design of bridge engineering. Summary of the Invention

[0005] The purpose of this invention is to provide a fast algorithm for calculating the mechanical properties of box girder sections. This algorithm can accurately characterize and rapidly calculate the mechanical properties of box girder sections in transportation, such as railway, highway, and municipal bridge projects. This patented calculation method establishes a precise mapping between parameter space and mechanical property space, overcoming the limitations of traditional methods in terms of computational dimensionality and efficiency.

[0006] The present invention provides a fast algorithm for calculating the mechanical properties of a box beam section, comprising the following steps:

[0007] Step A, establishing a parameterized calculation model of the box beam section;

[0008] Step B: Based on the parametric calculation model established in step A, derive an analytical expression for the area of ​​the box beam cross section;

[0009] Step C, based on the parameterized calculation model established in step A, deriving an analytical expression for the torsional stiffness of the box beam section;

[0010] Step D, based on the parametric calculation model established in step A, deriving an analytical expression for the centroid coordinates of the box beam section;

[0011] Step E: Based on the parameterized calculation model established in step A, derive an analytical expression for the effective shear area of ​​the box beam section;

[0012] Step F, based on the parametric calculation model established in step A, deriving an analytical expression for the moment of inertia of the box beam section;

[0013] Step G, based on the parameterized calculation model established in step A, deriving an analytical expression for the distance from the neutral axis to the edge fiber of the box beam section;

[0014] Step H: Based on the parameterized calculation model established in step A, derive an analytical expression for the shear coefficient of the box beam section along the unit local coordinate system direction;

[0015] Step I: Based on the parametric calculation model established in step A, derive analytical expressions for the inner and outer perimeters of the box beam section;

[0016] Step J: Based on the parametric calculation model established in step A, derive analytical expressions for the plane coordinates of the four corner points of the box beam section;

[0017] In step K, the characteristic values ​​calculated based on the analytical expressions of the calculation parameters of the box beam section established in steps B-J are substituted into the expressions of the stiffness matrix, mass matrix, and damping matrix of the corresponding spatial beam unit to obtain a complete set of unit characteristic matrix values ​​applied to the finite element internal force and displacement calculation of the bridge.

[0018] Furthermore, step A establishes a parameterized calculation model for the box beam section. The specific process is as follows:

[0019] Firstly, according to the structural characteristics of the box beam, a digital model of the box beam section that can be represented parametrically is abstracted.

[0020] Secondly, the parameters of the dimensions at different positions of the box beam section are calibrated;

[0021] Thirdly, in order to ensure the geometric compatibility of the parameters of the box beam section, the geometric constraints of the parameters are established. The corresponding constraint equations are as follows:

[0022]

[0023] Where, t f1 , t f2 is the thickness parameter of the top and bottom plates of the box beam section, H is the height parameter of the box beam section, t w is the side wall thickness of the box beam section, C is the distance between the centers of the side walls on both sides of the box beam section, and B is the distance between the outer edges of the side walls on both sides of the box beam section.

[0024] Furthermore, step B derives an analytical expression for the area of ​​the box beam section based on the parametric calculation model established in step A. The specific expression is as follows:

[0025] S=B(tf1+tf2)+2t w (H-tf1-tf2)(2)

[0026] Where S is the area of ​​the box beam cross section.

[0027] Furthermore, in step C, based on the parametric calculation model established in step A, an analytical expression for the torsional stiffness of the box beam section is derived. The specific expression is as follows:

[0028]

[0029] Where, I xx is the torsional stiffness of the box beam section.

[0030] Furthermore, in step D, based on the parametric calculation model established in step A, an analytical expression for the centroid coordinates of the box beam section is derived. The specific expression is as follows:

[0031]

[0032] Where Y c 、Z c is the centroid of the box beam section.

[0033] Furthermore, in step E, based on the parametric calculation model established in step A, an analytical expression for the effective shear area of ​​the box beam section is derived. The specific expression is as follows:

[0034]

[0035] Where A sy 、A sz is the effective shear area of ​​the box beam in two directions.

[0036] Furthermore, step F derives an analytical expression for the moment of inertia of the box beam section based on the parametric calculation model established in step A. The specific expression is as follows:

[0037]

[0038]

[0039] Where, I yy , I zz Represents the section bending inertia moment of the box beam section in two directions. These two variables are used to calculate the bending stiffness of the section under the action of bending moment and are relative to the neutral axis of the section. y1 , I y2 , I y3 To calculate I yy The intermediate variable represents the bending inertia moment around the y-axis at different parts of the box girder section, I z1 , I z2 , I z3 To calculate I zz The intermediate variable represents the bending inertia moment around the z-axis at different parts of the box girder section.

[0040] Furthermore, step G is based on the parametric calculation model established in step A to derive an analytical expression for the distance from the neutral axis to the edge fiber of the box beam section. The specific expression is as follows:

[0041]

[0042] Where C yp is the distance from the neutral axis of the element section to the edge fiber along the +y axis of the element local coordinate system, C ym is the distance from the neutral axis of the element section to the edge fiber along the -y axis of the element local coordinate system, C zp is the distance from the neutral axis of the element section to the edge fiber along the +z axis of the element local coordinate system, C zm It is the distance from the neutral axis of the element section to the edge fiber along the -z axis of the element local coordinate system.

[0043] Furthermore, in step H, based on the parameterized calculation model established in step A, an analytical expression for the shear coefficient of the box beam section along the unit local coordinate system is derived. The specific expression is as follows:

[0044]

[0045] Where Q yb is the shear coefficient along the z-axis of the local coordinate system of the element, Q zbis the shear coefficient along the y-axis of the element's local coordinate system.

[0046] Furthermore, in step I, based on the parametric calculation model established in step A, an analytical expression for the inner and outer perimeters of the box beam section is derived. The specific expression is as follows:

[0047]

[0048] Where peri1 and peri2 are the perimeters of the outer and inner contours of the box beam section.

[0049] The beneficial effects of the present invention are as follows:

[0050] This invention enables precise characterization and rapid calculation of the mechanical properties of box girder cross-sections in transportation, such as railway, highway, and municipal bridge projects. This patented calculation method establishes a precise mapping between parameter space and mechanical property space, overcoming the limitations of traditional methods in terms of computational dimensionality and efficiency.

[0051] The algorithm of the present invention takes into account both engineering applicability and program compatibility. Its modular architecture can be conveniently integrated into the finite element pre-processing system, providing an efficient and reliable cross-sectional property calculation tool for the digital design of bridge engineering. BRIEF DESCRIPTION OF THE DRAWINGS

[0052] Figure 1 Schematic diagram of the calculation process of the present invention;

[0053] Figure 2 is a schematic diagram of a box beam cross section in the present invention;

[0054] Figure 3 Schematic diagram of the parameterized calculation model of the box beam section in the present invention;

[0055] Figure 4 It is a schematic diagram of the cross-sectional dimensions of the actual engineering box girder of the present invention. DETAILED DESCRIPTION

[0056] Hereinafter, the present invention will be described in detail with reference to the accompanying drawings and embodiments:

[0057] like Figures 1 to 4 As shown in Figure 1, a fast algorithm for calculating the mechanical properties of a box beam section includes the following steps:

[0058] Step A, establish a parameterized calculation model of the box beam section; the box beam section is as follows Figure 2 As shown, its calculation model is as follows Figure 3 shown.

[0059] Step B: Based on the parametric calculation model established in step A, derive an analytical expression for the area of ​​the box beam cross section;

[0060] Step C, based on the parameterized calculation model established in step A, deriving an analytical expression for the torsional stiffness of the box beam section;

[0061] Step D, based on the parametric calculation model established in step A, deriving an analytical expression for the centroid coordinates of the box beam section;

[0062] Step E: Based on the parameterized calculation model established in step A, derive an analytical expression for the effective shear area of ​​the box beam section;

[0063] Step F, based on the parametric calculation model established in step A, deriving an analytical expression for the moment of inertia of the box beam section;

[0064] Step G, based on the parameterized calculation model established in step A, deriving an analytical expression for the distance from the neutral axis to the edge fiber of the box beam section;

[0065] Step H: Based on the parameterized calculation model established in step A, derive an analytical expression for the shear coefficient of the box beam section along the unit local coordinate system direction;

[0066] Step I: Based on the parametric calculation model established in step A, derive analytical expressions for the inner and outer perimeters of the box beam section;

[0067] Step J: Based on the parametric calculation model established in step A, derive analytical expressions for the plane coordinates of the four corner points of the box beam section;

[0068] In step K, the characteristic values ​​calculated based on the analytical expressions of the calculation parameters of the box beam section established in steps B-J are substituted into the expressions of the stiffness matrix, mass matrix, and damping matrix of the corresponding spatial beam unit to obtain a complete set of unit characteristic matrix values ​​applied to the finite element internal force and displacement calculation of the bridge.

[0069] Step A: Establish a parameterized calculation model for the box beam section. The specific process is as follows:

[0070] Firstly, according to the structural characteristics of the box beam, a digital model of the box beam section that can be represented parametrically is abstracted.

[0071] Secondly, the parameters of the dimensions at different positions of the box beam section are calibrated;

[0072] Thirdly, in order to ensure the geometric compatibility of the parameters of the box beam section, the geometric constraints of the parameters are established. The corresponding constraint equations are as follows:

[0073]

[0074] Where, t f1 , t f2is the thickness parameter of the top and bottom plates of the box beam section, H is the height parameter of the box beam section, t w is the side wall thickness of the box beam section, C is the distance between the centers of the side walls on both sides of the box beam section, and B is the distance between the outer edges of the side walls on both sides of the box beam section.

[0075] In step B, based on the parametric calculation model established in step A, the analytical expression for the area of ​​the box beam section is derived. The specific expression is as follows:

[0076] S=B(tf1+tf2)+2t w (H-tf1-tf2)(2)

[0077] Where S is the area of ​​the box beam cross section.

[0078] In step C, based on the parametric calculation model established in step A, the analytical expression for the torsional stiffness of the box beam section is derived. The specific expression is as follows:

[0079]

[0080] Where, I xx is the torsional stiffness of the box beam section.

[0081] In step D, based on the parametric calculation model established in step A, the analytical expression of the centroid coordinates of the box beam section is derived. The specific expression is as follows:

[0082]

[0083] Where Y c 、Z c is the centroid of the box beam section.

[0084] In step E, based on the parametric calculation model established in step A, an analytical expression for the effective shear area of ​​the box beam section is derived. The specific expression is as follows:

[0085]

[0086] Where A sy 、A sz is the effective shear area of ​​the box beam in two directions.

[0087] In step F, based on the parametric calculation model established in step A, the analytical expression for the moment of inertia of the box beam section is derived. The specific expression is as follows:

[0088]

[0089]

[0090] Where, I yy , Izz Represents the section bending inertia moment of the box beam section in two directions. These two variables are used to calculate the bending stiffness of the section under the action of bending moment and are relative to the neutral axis of the section. y1 , I y2 , I y3 To calculate I yy The intermediate variable represents the bending inertia moment around the y-axis at different parts of the box girder section, I z1 , I z2 , I z3 To calculate I zz The intermediate variable represents the bending inertia moment around the z-axis at different parts of the box girder section.

[0091] In step G, based on the parametric calculation model established in step A, an analytical expression for the distance from the neutral axis to the edge fiber of the box beam section is derived. The specific expression is as follows:

[0092]

[0093] Where C yp is the distance from the neutral axis of the element section to the edge fiber along the +y axis of the element local coordinate system, C ym is the distance from the neutral axis of the element section to the edge fiber along the -y axis of the element local coordinate system, C zp is the distance from the neutral axis of the element section to the edge fiber along the +z axis of the element local coordinate system, C zm It is the distance from the neutral axis of the element section to the edge fiber along the -z axis of the element local coordinate system.

[0094] In step H, based on the parameterized calculation model established in step A, the analytical expression of the shear coefficient of the box beam section along the unit local coordinate system is derived. The specific expression is as follows:

[0095]

[0096] Where Q yb is the shear coefficient along the z-axis of the local coordinate system of the element, Q zb is the shear coefficient along the y-axis of the element's local coordinate system.

[0097] In step I, based on the parametric calculation model established in step A, the analytical expressions for the inner and outer perimeters of the box beam section are derived. The specific expressions are as follows:

[0098]

[0099] Where peri1 and peri2 are the perimeters of the outer and inner contours of the box beam section.

[0100] Specifically, step J is based on the parametric calculation model established in step A to derive the analytical expressions of the plane coordinates of the four corner points of the box beam section. The specific expressions are as follows:

[0101]

[0102] Where y1 and z1 are the y and z coordinates of the upper left edge point of the cross section, y2 and z2 are the y and z coordinates of the upper right edge point of the cross section, y3 and z3 are the y and z coordinates of the lower right edge point of the cross section, and y4 and z4 are the y and z coordinates of the lower left edge point of the cross section.

[0103] In step K, the characteristic values ​​calculated based on the analytical expressions of the calculation parameters of the box beam section established in steps B-J are substituted into the expressions of the stiffness matrix, mass matrix, and damping matrix of the corresponding spatial beam unit to obtain a complete set of unit characteristic matrix values ​​that can be applied to the finite element internal force and displacement calculation of the bridge.

[0104] The present invention will be further described in detail below with reference to the accompanying drawings and embodiments:

[0105] Example 1

[0106] like Figure 4 As shown, the single-box, single-cell box girder has a beam height of 3.8m at the end supports, along the straight sections of the side spans, and at the midspan, and 6.8m at the center support. The beam height follows a circular curve with a radius of 360.493m. The inner width of the deck ballast retaining wall is 4.50m. The box girder has a top width of 7.00m and a bottom width of 5.0m, widening to 6.2m within 5.2m of the center support. The box girder utilizes straight webs.

[0107] The top plate is 36cm thick; the bottom plate is 40 to 90cm thick, changing along a circular curve to the root of the center support beam, where it thickens to 144.6cm. The web plate thickness varies from 40 to 60cm and 60 to 80cm, changing along a broken line. The bridge has seven cross beams, located at the center support, end support, and mid-span. A 2.0m thick cross beam is installed at the center support, a 1.5m thick end beam is installed at the side support, and a 0.8m thick cross beam is installed at the mid-span. Holes are provided in the cross beams for inspection personnel to pass through. The letter R in the figure represents the structural chamfer radius at different locations on the box girder section.

[0108] The algorithm of the present invention was compiled into an executable program to perform calculations for this engineering case, and was verified by back-to-back comparison calculations with the commercial finite element software MIDAS. The error values ​​between the calculation results of the present invention and the calculation results of MIDAS were calculated and listed in the following table (the symbols listed in the table have the same meanings as listed above):

[0109] Table 1 Accuracy test results of fast algorithm for mechanical properties of box beam section

[0110]

[0111]

[0112] As can be seen from Table 1, the calculation results of the present invention are very close to those of the commercial software MIDAS, with an error within 2%, indicating that the calculation results of the patented method are very accurate for the calculation of the mechanical properties of the box beam section, meeting the engineering calculation requirements.

[0113] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the above embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the above embodiments, or replace some or all of the technical features therein with equivalents. However, these modifications or replacements do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A fast algorithm for calculating the mechanical properties of a box beam section, characterized by: The following steps are involved: Step A, establishing a parameterized calculation model of the box beam section; Step B: Based on the parametric calculation model established in step A, derive an analytical expression for the area of ​​the box beam cross section; Step C, based on the parameterized calculation model established in step A, deriving an analytical expression for the torsional stiffness of the box beam section; Step D, based on the parametric calculation model established in step A, deriving an analytical expression for the centroid coordinates of the box beam section; Step E: Based on the parameterized calculation model established in step A, derive an analytical expression for the effective shear area of ​​the box beam section; Step F, based on the parametric calculation model established in step A, deriving an analytical expression for the moment of inertia of the box beam section; Step G, based on the parameterized calculation model established in step A, deriving an analytical expression for the distance from the neutral axis to the edge fiber of the box beam section; Step H: Based on the parameterized calculation model established in step A, derive an analytical expression for the shear coefficient of the box beam section along the unit local coordinate system direction; Step I: Based on the parametric calculation model established in step A, derive analytical expressions for the inner and outer perimeters of the box beam section; Step J: Based on the parametric calculation model established in step A, derive analytical expressions for the plane coordinates of the four corner points of the box beam section; In step K, the characteristic values ​​calculated based on the analytical expressions of the calculation parameters of the box beam section established in steps B-J are substituted into the expressions of the stiffness matrix, mass matrix, and damping matrix of the corresponding spatial beam unit to obtain a complete set of unit characteristic matrix values ​​applied to the finite element internal force and displacement calculation of the bridge.

2. The fast algorithm for calculating the mechanical properties of a box beam section according to claim 1, characterized in that: Step A: Establish a parameterized calculation model for the box beam section. The specific process is as follows: Firstly, according to the structural characteristics of the box beam, a digital model of the box beam section that can be represented parametrically is abstracted. Secondly, the parameters of the dimensions at different positions of the box beam section are calibrated; Thirdly, in order to ensure the geometric compatibility of the parameters of the box beam section, the geometric constraints of the parameters are established. The corresponding constraint equations are as follows: Where, t f1 , t f2 is the thickness parameter of the top and bottom plates of the box beam section, H is the height parameter of the box beam section, t w is the side wall thickness of the box beam section, C is the distance between the centers of the side walls on both sides of the box beam section, and B is the distance between the outer edges of the side walls on both sides of the box beam section.

3. The fast algorithm for calculating the mechanical properties of a box beam section according to claim 1, characterized in that: In step B, based on the parametric calculation model established in step A, the analytical expression for the area of ​​the box beam section is derived. The specific expression is as follows: S=B(tf1+tf2)+2t w (H-tf1-tf2)(2) Where S is the area of ​​the box beam cross section.

4. The fast algorithm for calculating the mechanical properties of a box beam section according to claim 1, characterized in that: In step C, based on the parametric calculation model established in step A, the analytical expression for the torsional stiffness of the box beam section is derived. The specific expression is as follows: Where, I xx is the torsional stiffness of the box beam section.

5. The fast algorithm for calculating the mechanical properties of a box beam section according to claim 1 is characterized in that: In step D, based on the parametric calculation model established in step A, the analytical expression of the centroid coordinates of the box beam section is derived. The specific expression is as follows: Where Y c 、Z c is the centroid of the box beam section.

6. The fast algorithm for calculating the mechanical properties of a box beam section according to claim 1, characterized in that: In step E, based on the parametric calculation model established in step A, an analytical expression for the effective shear area of ​​the box beam section is derived. The specific expression is as follows: Where A sy 、A sz is the effective shear area of ​​the box beam in two directions.

7. The fast algorithm for calculating the mechanical properties of a box beam section according to claim 1, characterized in that: In step F, based on the parametric calculation model established in step A, the analytical expression for the moment of inertia of the box beam section is derived. The specific expression is as follows: Where, I yy , I zz Represents the section bending inertia moment of the box beam section in two directions. These two variables are used to calculate the bending stiffness of the section under the action of bending moment and are relative to the neutral axis of the section. y1 , I y2 , I y3 To calculate I yy The intermediate variable represents the bending inertia moment around the y-axis at different parts of the box girder section, I z1 , I z2 , I z3 To calculate I zz The intermediate variable represents the bending inertia moment around the z-axis at different parts of the box girder section.

8. The fast algorithm for calculating the mechanical properties of a box beam section according to claim 1, characterized in that: In step G, based on the parametric calculation model established in step A, an analytical expression for the distance from the neutral axis to the edge fiber of the box beam section is derived. The specific expression is as follows: Where C yp is the distance from the neutral axis of the element section to the edge fiber along the +y axis of the element local coordinate system, C ym is the distance from the neutral axis of the element section to the edge fiber along the -y axis of the element local coordinate system, C zp is the distance from the neutral axis of the element section to the edge fiber along the +z axis of the element local coordinate system, C zm It is the distance from the neutral axis of the element section to the edge fiber along the -z axis of the element local coordinate system.

9. The fast algorithm for calculating the mechanical properties of a box beam section according to claim 1, characterized in that: In step H, based on the parameterized calculation model established in step A, the analytical expression of the shear coefficient of the box beam section along the unit local coordinate system is derived. The specific expression is as follows: Where Q yb is the shear coefficient along the z-axis of the local coordinate system of the element, Q zb is the shear coefficient along the y-axis of the element's local coordinate system.

10. The fast algorithm for calculating the mechanical properties of a box beam section according to claim 1, characterized in that: In step I, based on the parametric calculation model established in step A, the analytical expressions for the inner and outer perimeters of the box beam section are derived. The specific expressions are as follows: Where peri1 and peri2 are the perimeters of the outer and inner contours of the box beam section.

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