Eccentric pressure correction method considering torsion constraint rigidity of end diaphragm

Abstract

The invention discloses an eccentric pressure correction method considering torsion constraint rigidity of an end diaphragm. The method comprises the following steps: (1) obtaining related parameters of a beam bridge according to engineering data; (2) calculating the torsional constraint rigidity of the end diaphragm to the main beam; (3) calculating a load transverse distribution coefficient, a single-web section main beam and a double-main-beam bridge at a fulcrum through a lever principle method, calculating by adopting a single-beam model, and calculating by adopting a double-beam model for a double-web three-main-beam and more-piece main-beam bridge; (4) calculating a load transverse distribution coefficient of any position in the longitudinal bridge direction of the main beam through a correction eccentric pressure method considering beam end torsion constraint; and (5) considering the change of the transverse distribution of the load along the longitudinal bridge direction for the concentrated load, taking the transverse distribution coefficient of the load at the midspan position for the uniformly distributed load, and then calculating the load effect. According to the method, the elastic support of the rigidity of the end diaphragm on the torsional deformation of the main beam is considered, and a novel high-precision method is provided for calculating the load transverse distribution of the box-shaped section beam bridge.

Description

Technical Field

[0001] This invention belongs to the field of engineering technology and relates to a method for calculating the internal forces of bridge engineering structures. Specifically, it relates to a modified eccentric pressure method that considers the torsional constraint stiffness of the end crossbeam. Background Technology

[0002] The calculation methods for the lateral load distribution coefficient differ internationally. Internationally, this involves analyzing bridge parameters to derive influencing parameters and then using regression to arrive at a calculation formula. Domestically, methods such as the lever principle method, eccentric pressure method, modified eccentric pressure method, rigid (hinged) beam (plate) method, and analogous orthotropic plate method are generally used to calculate the lateral distribution influence lines of each main beam, thereby obtaining the lateral distribution coefficient. The rigid beam method is more accurate than the formula in the American AASHTO standard and has a wider range of applications.

[0003] Research on methods for calculating the lateral load distribution coefficient began early. For example, Li Guohao and Shi Dong published a monograph in 1987 titled "Calculation of Lateral Load Distribution in Highway Bridges," which provided numerous tables based on the characteristics of the methods to facilitate calculations using various lateral distribution methods. Hu Zhaozi published a monograph in 1996 titled "Simplified Analysis of Bridge Span Structures—Lateral Load Distribution," which detailed the theory of lateral distribution, especially the modified eccentric pressure method. Early literature mostly used T-beam bridges for experiments. With the application of small box girder, hollow slab, and composite beam bridges (such as channel steel box composite beams and narrow steel box composite beams), some studies have found that the theoretical lateral load distribution coefficient calculated by existing methods is about 7% smaller than the experimental values. The biggest difference between box girder and T-beam sections is that box girder has a larger torsional stiffness. It is possible that existing theories overestimate the influence of the torsional stiffness of box girders. Therefore, for box girder bridges, how to improve the calculation method of the lateral load distribution coefficient and improve the calculation accuracy has important engineering research value. Summary of the Invention

[0004] To address the issue of the theoretically calculated value of the lateral load distribution coefficient for box girder bridges being too small, this invention proposes a modified eccentric pressure method that considers the torsional constraint stiffness of the end transverse diaphragms.

[0005] The modified eccentric pressure method considering the torsional constraint stiffness of the end diaphragm beam described in this invention comprises the following steps:

[0006] (1) Based on the engineering data, obtain the relevant parameters of the simply supported beam bridge: bridge span L, cross-sectional layout, bending moment of inertia and torsional moment of inertia of each main beam section, cross-sectional dimensions of the end diaphragm, material elastic modulus E, shear modulus G, Poisson's ratio ν;

[0007] (2) Calculate the torsional stiffness of the end diaphragm beam to the main beam;

[0008] (3) Calculate the lateral load distribution coefficient at the fulcrum using the lever principle method:

[0009] a) For T-shaped and I-shaped single-web section main beams, a single-beam model is used for calculation;

[0010] b) For small box girders, hollow slab type double web section main beams, and double main beam bridges, a single beam model is used for calculation; for multi-main beam bridges with three or more main beams, a double beam model is used for calculation.

[0011] The lateral load distribution factor calculated using the single beam model is denoted as m. 01 The lateral load distribution coefficient calculated using the double-beam model is denoted as m. 02 ;

[0012] (4) The load distribution coefficient at any position in the longitudinal direction of the main beam is calculated by the modified eccentric pressure method considering the torsional constraint at the beam end.

[0013] (5) For concentrated loads, consider the variation of the lateral distribution of the load along the longitudinal direction of the bridge. For uniformly distributed loads, take the lateral distribution coefficient of the load at the mid-span position. Reduce the load by the lateral distribution coefficient of the load and then calculate the load effect.

[0014] In step (2), according to the lever principle, the end diaphragm beam is simplified to a simply supported beam. For the main beam on the adjacent side, its torsional elastic support stiffness k is:

[0015] ;

[0016] Among them: EI h For the bending stiffness of the end diaphragm, l h Let G be the span of the end diaphragm, G be the shear modulus of the material, and A be the length of the end diaphragm. sz κ is the effective shear area of ​​the end diaphragm, and k is the shear coefficient of the beam section. R The elastic support stiffness of the support;

[0017] For the edge main beam, it is connected to only one end transverse diaphragm, and the torsional restraint stiffness at the beam end is taken as k. For the middle main beam, it is connected to two end transverse diaphragms, and the torsional restraint stiffness at the beam end is taken as 2k, that is:

[0018] ;

[0019] Where: i is the main beam number, n is the number of main beams, and k is the main beam number. i The torsional constraint stiffness at the end of main beam i is given. Main beams 1 and n are side main beams, and the other numbered main beams are middle main beams.

[0020] In step (4), the steps for calculating the transverse load distribution coefficient at any position in the longitudinal direction of the main girder using the modified eccentric pressure method considering torsional constraints at the beam ends are as follows:

[0021] a) Calculate the influence line η of the lateral distribution of the load. i

[0022] ;

[0023] ;

[0024] Where: η ie This is the influence line value of the lateral load distribution of the main girder i when an eccentric load is applied at position e in the transverse direction of the bridge. e represents the transverse position of the load application, with a negative sign to the left of the centroid of the section and a positive sign to the right. i Let be the moment of inertia of the section of main beam i, n be the number of main beams, and a be the moment of inertia of the section of main beam i. i Let be the distance from the i-th main beam to the centroid of the cross-section. The main beam to the left of the centroid is considered negative, and the main beam to the right is considered positive. E is the elastic modulus of the material, L is the span of the main beam, x is the position of the eccentric load acting in the longitudinal direction of the bridge, with the left end support of the beam as the origin and the right end as the positive direction. k i Let GJ be the torsional restraint stiffness at the end of main beam i. i Let i be the torsional stiffness of the main beam.

[0025] pass The influence line values ​​of the lateral load distribution at multiple locations along the transverse direction of the main girder i were obtained, such as... , Connecting these lines with a straight line yields the influence line η of the lateral load distribution of the main beam i. i ;

[0026] ;

[0027] ;

[0028] ;

[0029] Where: y represents the transverse bridge position, the origin of the coordinate axis is the centroid of the cross section, and negative signs are taken to the left of the centroid of the cross section and positive signs are taken to the right of the centroid.

[0030] b) Calculate the lateral load distribution factor

[0031] The influence line η of the lateral load distribution of the main beam i is obtained. i Then, the vehicle load is distributed according to the most unfavorable load position, and the lateral load distribution coefficient of the vehicle load is calculated:

[0032] ;

[0033] Where: m iqki Let ξ be the lateral load distribution coefficient of the vehicle load on main beam i, ξ be the lateral lane distribution coefficient of the vehicle load (i.e., the lateral multi-lane reduction coefficient), N be the number of vehicles loaded, and η be the lateral load distribution coefficient of the vehicle load on main beam i. ijThe value of the lateral load distribution influence line of main beam i when the vehicle tires act on position j in the transverse direction of the bridge.

[0034] For the lateral load distribution coefficient m of the crowd load on main beam i... irki The calculation is as follows:

[0035] ;

[0036] Where: η ir The influence line value of the lateral distribution of the load on the main beam i when the crowd load acts on the center point of the resultant force of the sidewalk;

[0037] In step (5), for concentrated loads, when it is a double main girder bridge, the value is directly taken as... For multi-main-girder bridges with three or more main girders, the load lateral distribution along the longitudinal direction varies as follows:

[0038] a) Calculate the transition zone length x t

[0039] The length of the transition zone is calculated using the following formula:

[0040] ;

[0041] Where: L is the span of the main girder, and x is the longitudinal position of the bridge. Let x be the lateral load distribution factor of the vehicle load at position x. , These are the lateral load distribution coefficients for vehicle loads at the fulcrum and mid-span positions, respectively.

[0042] b) Regarding the transition region length x t Adjust the lateral load distribution coefficient value within the range.

[0043] ;

[0044] in: The lateral load distribution factor of the vehicle load after adjustment in the transition zone is m. 02 The lateral load distribution coefficient is calculated for the double-beam model.

[0045] Specifically, in step (1), the bending moment of inertia and torsional moment of inertia of the main beam section are calculated by finite element software such as midascivil, bridge doctor, and ansys, or by AutoCAD software, or by theoretical mechanics methods.

[0046] Specifically, in step (1), the shear modulus G is calculated by the following formula:

[0047] .

[0048] Specifically, in step (1), if the beam bridge is a continuous beam bridge or a continuous rigid frame bridge, the equivalent simply supported beam method is used to correct the bending stiffness and torsional stiffness of the main beam section.

[0049] Specifically, in step (2), the beam section shear coefficient κ is calculated using the following formula:

[0050] ;

[0051] Among them: A h Let A be the cross-sectional area of ​​the end diaphragm. sz This represents the effective shear area of ​​the end crossbeam.

[0052] Specifically, in step (2), when the span-to-depth ratio of the end transverse diaphragm is greater than 10, take... That is, the effect of shear deformation is not considered.

[0053] Specifically, in step (2), for the steel support, take... That is, without considering the deformation of the support, for plate rubber bearings and pot rubber bearings, Calculate using the following formula:

[0054] ;

[0055] Where: E e Let A be the compressive elastic modulus of the rubber bearing. e h is the area of ​​the rubber bearing. e This refers to the total thickness of the rubber layer in the rubber bearing.

[0056] Specifically, step (2) is for a conventional beam bridge where each main girder has one support. If other support arrangements are used, the torsional stiffness k at the beam end of the main girder is derived based on mechanical principles. i .

[0057] Specifically, in step (3), the single beam model refers to the bridge deck being broken at the centroid line of the main beam section, and the double beam model refers to the bridge deck being broken at the junction of the two webs of the box-section main beam and the bridge deck. The bridge deck is calculated as a multi-span simply supported beam.

[0058] Specifically, in step (5), the calculation method for the load effect is as follows:

[0059] The general formula for calculating the effect of vehicle load on main beam i is:

[0060] ;

[0061] Wherein: S q The effect of vehicle load is represented by μ, where μ is the impact coefficient of the vehicle load, and P is the effect of vehicle load. k This represents the concentrated load value in the vehicle load. q represents the coordinate value of the influence line of the longitudinal bridge action effect. k Ω represents the uniformly distributed load value in the vehicle load, x represents the area of ​​the influence line of the effect, and x represents the longitudinal bridge position.

[0062] The general formula for calculating the effect of crowd load on main beam i is:

[0063] ;

[0064] Wherein: S r For the effect of crowd load, q r Ω represents the population load value, and Ω represents the area of ​​the influence line of the effect.

[0065] The beneficial effects of this invention are as follows: The method of this invention considers the elastic support of the end diaphragm stiffness on the torsional deformation of the main beam, rather than the rigid support, so that the main beam at the support point can generate a limited torsional angle, increasing the correction coefficient value of the modified eccentric pressure method, making the calculated value of the load lateral distribution coefficient slightly larger than that of the ordinary modified eccentric pressure method, which is more consistent with the actual situation of the bridge. Through three narrow-span steel box girder bridge examples, the effect of the method of this invention is verified, providing a new high-precision method for calculating the load lateral distribution of box girder bridges. Attached Figure Description

[0066] Figure 1 This is a flowchart of the load lateral distribution calculation process of the present invention.

[0067] Figure 2 This is a schematic diagram of the deformation of a simply supported beam under torque with torsional constraint at both ends according to the present invention.

[0068] Figure 3 This is a schematic diagram of bridge deformation under the eccentric pressure method load of the present invention.

[0069] Figure 4 This is a schematic diagram of the decomposition of the unit eccentric load force of the present invention.

[0070] Figure 5 This is a schematic diagram of deformation under unit center load of the present invention.

[0071] Figure 6 This is a schematic diagram of the deformation caused by the unit eccentric torque of the present invention.

[0072] Figure 7 This is the bending moment diagram of the simply supported beam under the action of unit end bending moment according to the present invention.

[0073] Figure 8 This is the shear force diagram of the simply supported beam under the action of unit end bending moment according to the present invention.

[0074] Figure 9 This is a deformation diagram of the simply supported beam end under bending moment in the elastic support of the present invention.

[0075] Figure 10 This refers to the torsional constraint stiffness at the ends of each main beam in this invention.

[0076] Figure 11 This is a cross-sectional view of the three main beams in Embodiment 1 of the present invention (unit: cm).

[0077] Figure 12 This is the finite element model of the three main beam bridge in Embodiment 1 of the present invention.

[0078] Figure 13 This refers to the lateral movement load application position (unit: cm) in Embodiment 1 of the present invention.

[0079] Figure 14 This is a schematic diagram for calculating the lateral distribution coefficient of the reaction load at the support of the No. 1 side main beam in Embodiment 1 of the present invention.

[0080] Figure 15 This is a diagram of the moving load and measurement layout (unit: m) according to Embodiment 1 of the present invention.

[0081] Figure 16 These are the characteristic values ​​of the No. 1 side main beam section in Embodiment 1 of the present invention.

[0082] Figure 17 These are the cross-sectional dimensions of the transverse diaphragm in one embodiment of the present invention.

[0083] Figure 18 These are the characteristic values ​​of the cross-section of the transverse diaphragm in one embodiment of the present invention.

[0084] Figure 19 These are the two load lateral distribution coefficient values ​​calculated using the lever principle method of the No. 1 side main beam in Embodiment 1 of the present invention.

[0085] Figure 20 This is a comparison of the lateral distribution coefficient values ​​of the reaction load at the support of the No. 1 side main beam in Embodiment 1 of the present invention.

[0086] Figure 21 This is a schematic diagram for calculating the lateral distribution coefficient of the load at the mid-span position of the No. 1 side main beam in Embodiment 1 of the present invention.

[0087] Figure 22 These are the calculated values ​​of the lateral load distribution coefficient in Embodiment 1 of the present invention, considering and not considering the stiffness of the end diaphragm beam.

[0088] Figure 23 This is a schematic diagram of the adjustment of the lateral load distribution coefficient in Embodiment 1 of the present invention.

[0089] Figure 24 This is a comparison of the lateral distribution coefficient values ​​of the load on the No. 1 side main beam in Embodiment 1 of the present invention.

[0090] Figure 25This refers to the error in Embodiment 1 of the present invention, which considers the effect of the end crossbeam stiffness versus not.

[0091] Figure 26 This is an analysis of the working conditions with and without a crossbeam in Embodiment 1 of the present invention (unit: m).

[0092] Figure 27 These are the characteristic values ​​of the cross-section of the end transverse diaphragm beam in Embodiment 2 of the present invention.

[0093] Figure 28 This is a comparison of the lateral distribution coefficient values ​​of the reaction load at the No. 1 side main beam support in Embodiment 2 of the present invention.

[0094] Figure 29 This is the calculated value of the load lateral distribution coefficient in Embodiment 2 of the present invention, considering and not considering the stiffness of the end diaphragm beam.

[0095] Figure 30 This is a comparison of the lateral distribution coefficient values ​​of the load on the No. 1 side main beam in Embodiment 2 of the present invention.

[0096] Figure 31 This refers to the error in Embodiment 2 of the present invention, which considers the effect of the end crossbeam stiffness versus not.

[0097] Figure 32 This is the layout diagram of the ramp bridge in Embodiment 3B of the present invention (unit: m).

[0098] Figure 33 This is a cross-sectional view of the double main beam in Embodiment 3 of the present invention (unit: cm).

[0099] Figure 34 This is a comparison of the lateral distribution coefficient values ​​of the reaction load at the No. 1 side main beam support in Embodiment 3 of the present invention.

[0100] Figure 35 This is the calculated value of the load lateral distribution coefficient in Embodiment 3 of the present invention, considering and not considering the stiffness of the end transverse diaphragm.

[0101] Figure 36 This is a comparison of the lateral distribution coefficient values ​​of the load on the No. 1 side main beam in Embodiment 3 of the present invention.

[0102] Figure 37 This refers to the error in Embodiment 3 of the present invention, which considers the effect of the end crossbeam stiffness versus not. Detailed Implementation

[0103] The present invention will be further described below with reference to the accompanying drawings and embodiments. When the following description refers to the drawings, unless otherwise indicated, the same numbers in different drawings represent the same or similar elements.

[0104] Figure 1 The flowchart shows the modified eccentric pressure method for considering the torsional constraint stiffness of the end diaphragm beam in this invention.

[0105] I. Force Analysis of a Simply Supported Beam with Torsional Restraint at Both Ends

[0106] According to mechanics of materials, we have:

[0107] (1)

[0108] Where: φ AB Let l be the torsion angle between the left end point A and the right end point B of the component. AB Let T be the length of the component, G be the torque at both ends of the component, G be the shear modulus of the component material, and J be the torsional moment of inertia of the component section.

[0109] A schematic diagram of the deformation of a simply supported beam under torsional restraint at both ends is shown below. Figure 2 The beam cross-section in the figure is shown as a circular cross-section. Assuming the torsional elastic support stiffness at both ends of the beam is k, a torque T is applied at a distance x from the left end, and the beam span is L, the following system of equations can be listed based on the equilibrium of displacement and force:

[0110] (2)

[0111] In the formula: φ, φ1, and φ2 are the torsional angles at the x-section position of the beam and the two ends of the beam, respectively; T1 and T2 are the torques distributed to the left and right isolators at the x-section position of the beam, respectively. These five quantities are unknowns to be determined.

[0112] Equation (2) can be simplified to:

[0113] (3)

[0114] In equation (3), eliminating φ simplifies to:

[0115] (4)

[0116] (5)

[0117] The angle of twist at the beam end is obtained as follows:

[0118] (6)

[0119] Substituting equation (6) into the first formula in equation (3), the torsional angle at the point where the torque T is applied can be obtained as follows:

[0120] (7)

[0121] In summary, the solution for all 5 unknown variables is:

[0122] (8)

[0123] II. Calculation of the influence line of lateral load distribution using the modified eccentric pressure method for torsional restraint supports at beam ends

[0124] The basic assumption of the eccentric pressure method is that under vehicle load, the central diaphragm can be approximated as a rigid beam with infinite stiffness, and the diaphragm only undergoes rigid body displacement; for example... Figure 3 The diagram illustrates a 5-section T-beam bridge. In the diagram, B is the bridge width, L is the bridge span, and the load P is applied to a certain transverse position at the mid-span of the bridge. Due to the infinite stiffness of the central transverse beam, the transverse beam undergoes rigid displacement, i.e., the deflection curve c'd' after deformation is a straight line.

[0125] Analyze the force R1 borne by main girder No. 1 when a unit load P=1 is applied at any position on the bridge deck (eccentricity e). Analyze the section at a distance x from the left end support of the bridge. The eccentric load P=1 can be replaced by the central load P=1 applied to the bridge axis and the eccentric moment M=1*e, as follows: Figure 4 In the figure, a i I is the distance from the centroid of the section to the i-th main beam. i Let be the moment of inertia of the section of main beam i. Calculate the forces borne by main beam i in both cases, and then superimpose them.

[0126] (1) The effect of the central load P=1

[0127] like Figure 5 Due to the central load, the rigid transverse diaphragm shifts downwards as a whole, resulting in the same deflection in all main beams, i.e.:

[0128] (9)

[0129] In the formula: Let be the deflection of main beam i, and n be the number of main beams. This represents the average deflection.

[0130] According to mechanics of materials, the relationship between the load and deflection acting on a simply supported beam at position x in the longitudinal direction is as follows:

[0131] (10)

[0132] In the formula: E is the elastic modulus of the material, I i Let be the moment of inertia of the section of main beam i, and L be the span of the main beam. The reaction force is the i-th main beam.

[0133] The solution can be obtained from equation (10):

[0134] (11)

[0135] (12)

[0136] From the static equilibrium condition, we get:

[0137] (13)

[0138] Find:

[0139] (14)

[0140] Substituting equation (14) into equation (11), we obtain the load distribution of the central load P=1 between the main beams as follows:

[0141] (15)

[0142] If all main beams have the same cross-section, then:

[0143] (16)

[0144] (2) The effect of eccentric moment M=1*e

[0145] like Figure 6 M in the figure Ti Let the resisting torque of the i-th main beam be obtained from equation (8):

[0146] (17)

[0147] In the formula: k i Let GJ be the torsional restraint stiffness at the end of main beam i. i Let be the torsional stiffness of main beam i.

[0148] Under the action of the eccentric moment M, the cross section of the bridge will rotate around the center point by an angle φ. Therefore, the deflection of each main beam is:

[0149] (18)

[0150] Based on geometric relations:

[0151] (19)

[0152] Based on the load-deflection relationship of the main beam:

[0153] (20)

[0154] Substituting equation (20) into equation (19), we get:

[0155] (twenty one)

[0156] Substituting equation (21) into equation (17) yields:

[0157] (twenty two)

[0158] From equation (21), we can obtain:

[0159] (twenty three)

[0160] Right now:

[0161] (twenty four)

[0162] According to the torque equilibrium condition, we can obtain:

[0163] (25)

[0164] Substituting equations (24) and (22) into equation (25), we get:

[0165] (26)

[0166] Simplifying, we get:

[0167] (27)

[0168] The reaction force can be obtained from equation (27):

[0169] (28)

[0170] The correction factor is:

[0171] (29)

[0172] If the effect of the end diaphragm is not considered, then the stiffness is taken as infinite. Equation (29) degenerates into:

[0173] (30)

[0174] When analyzing the mid-span section, i.e., x=L / 2, equation (30) degenerates into the coefficient of the ordinary corrected eccentric pressure method:

[0175] (31)

[0176] The relationship between the elastic modulus E and the shear modulus G is as follows:

[0177] (32)

[0178] In the formula: ν is Poisson's ratio.

[0179] (3) The effect of eccentric load P=1

[0180] The reaction force of the eccentric load P=1 is the superposition of the reaction forces obtained from the two cases of the central load P=1 on the bridge axis and the eccentric moment M=1*e, as shown in equation (33).

[0181] (33)

[0182] In the formula: η ie This is the influence line value of the lateral load distribution of the main girder i when an eccentric load P is applied at position e in the transverse direction of the bridge. e represents the transverse position of the load application, with a negative sign to the left of the centroid of the section and a positive sign to the right. i This represents the distance from the i-th main beam to the centroid of the section. The main beam to the left of the centroid is given a negative sign, and the main beam to the right is given a positive sign.

[0183] The influence line values ​​of the lateral load distribution at multiple locations in the transverse direction of the main beam i are obtained by equation (33), such as... , Connecting these lines with a straight line yields the influence line η of the lateral load distribution of the main beam i. i .

[0184] (34)

[0185] (35)

[0186] (36)

[0187] In the formula: y is the transverse position of the bridge, the origin of the coordinate axis is the position of the centroid of the cross section, and the sign is negative to the left of the centroid of the cross section and positive to the right.

[0188] The above analysis is for simply supported beam bridges. For continuous beam bridges, the equivalent simply supported beam method can be used to correct the bending stiffness and torsional stiffness.

[0189] III. Calculation of Torsional Stiffness of End Diaphragm Beams

[0190] Based on the lever principle, the end transverse diaphragm beam is simplified to a simply supported beam. Figure 7 To obtain the bending moment diagram of a simply supported beam under unit end bending moment, the rotation angle generated by the beam end bending moment can be calculated. for:

[0191] (37)

[0192] Where: EI h For the bending stiffness of the end diaphragm, The bending moments generated by the unit bending moment and the load bending moment are respectively, l h The span of the end crossbeam.

[0193] Figure 8To obtain the shear force diagram of a simply supported beam under unit end bending moment, the rotation angle caused by the shear force under the beam end bending moment can be calculated. for:

[0194] (38)

[0195] In the formula: A sz The effective shear area of ​​the end diaphragm. These are the shear forces generated by the unit bending moment and the load bending moment, respectively, and κ is the shear coefficient of the beam section.

[0196] (39)

[0197] In the formula: A h The cross-sectional area of ​​the end diaphragm beam is given.

[0198] If elastic supports such as plate rubber bearings are used, under vertical loads, the supports will undergo compressive deformation, causing the end transverse diaphragm beam to rotate. Figure 9 The elastic support rotation angle generated when the beam end bending moment is applied can be obtained. for:

[0199] (40)

[0200] In the formula: k R The elastic support stiffness of the support.

[0201] For steel supports, the following can be adopted: For plate rubber bearings and pot rubber bearings, Calculate using the following formula:

[0202] (41)

[0203] In the formula: E e Let A be the compressive elastic modulus of the rubber bearing. e h is the area of ​​the rubber bearing. e This refers to the total thickness of the rubber layer in the rubber bearing.

[0204] Therefore, the total rotation angle θ generated under the action of the bending moment is:

[0205] (42)

[0206] The stiffness k of the torsional elastic support is:

[0207] (43)

[0208] For the edge main beam, it is connected to only one end transverse diaphragm, and the torsional restraint stiffness at the beam end is taken as k. For the middle main beam, it is connected to two end transverse diaphragms, and the torsional restraint stiffness at the beam end is taken as 2k. Figure 10,Right now:

[0209] (44)

[0210] In the formula: i is the main beam number, n is the number of main beams, and k is the number of main beams. i The torsional constraint stiffness at the end of main beam i is given. Main beams 1 and n are side main beams, and the other numbered main beams are middle main beams.

[0211] The above applies to conventional beam bridges where each main girder has one support. If other support arrangements are used, the torsional stiffness k at the beam ends needs to be derived based on mechanical principles. i .

[0212] IV. Calculation of Lateral Load Distribution Coefficient

[0213] After obtaining the influence line of the lateral distribution of the load on the i-th main beam, the vehicle load is distributed according to the most unfavorable load position, and then the lateral distribution coefficient of the vehicle load is calculated, as shown in equation (45).

[0214] (45)

[0215] Where: m iqki Let ξ be the lateral load distribution coefficient of the vehicle load on main beam i, ξ be the lateral lane distribution coefficient of the vehicle load (i.e., the lateral multi-lane reduction coefficient), N be the number of vehicles loaded, and η be the lateral load distribution coefficient of the vehicle load on main beam i. ij The value of the lateral load distribution influence line of main beam i when the vehicle tires act on position j in the transverse direction of the bridge.

[0216] For the lateral load distribution coefficient m of the crowd load on main beam i... irki Calculate as shown in equation (46).

[0217] (46)

[0218] In the formula: η ir The influence line value of the lateral distribution of the load on the main beam i when the pedestrian load acts on the center point of the resultant force of the sidewalk.

[0219] V. Calculation of Effect

[0220] The general formula for calculating the effect of vehicle load on main beam i is:

[0221] (47)

[0222] In the formula: S q The effect of vehicle load is represented by μ, where μ is the impact coefficient of the vehicle load, and P is the effect of vehicle load. k This represents the concentrated load value in the vehicle load. q represents the coordinate value of the influence line of the longitudinal bridge action effect. kΩ represents the uniformly distributed load value in the vehicle load, Ω represents the area of ​​the influence line of the effect, and x represents the longitudinal bridge position.

[0223] The general formula for calculating the effect of crowd load on main beam i is:

[0224] (48)

[0225] Example 1: 75m span three-main-girder narrow steel box girder composite bridge

[0226] The Yangchun-Xinyi Expressway Civil Engineering Section 10 extends from K102+874 to K141+767.643, with a total length of 38.894km. The main line adopts a six-lane expressway standard with a design speed of 120km / h. The Shuikou South Interchange is a junction between the Yangchun-Xinyi Expressway and the Baomao Expressway. The Baomao Expressway is currently a two-way four-lane expressway with a design speed of 100km / h.

[0227] The Shuikou South Interchange overpass of the Baomao Expressway, including ramps A, B, E, and F, is a composite beam bridge using narrow-span steel box girders and SC composite bridge decks. The ramps have a double main girder structure; ramps A and E have a span of 70m, and ramps B and F have a span of 75m. The overpass itself is a double-span, three-main-girder structure, with a steel support under each narrow-span steel box girder at the beam end. The Yangxin Expressway intersects the Baomao Expressway at an oblique angle; the cross-section of the three-main-girder bridge is as follows... Figure 11 The bridge features narrow steel box girders spaced 5.7m apart, with a box height of 3m and a width of 1.6m. The upper flange is 20mm thick, the lower flange is 56mm thick, and the web is 14mm thick. The steel-concrete (SC) composite bridge deck has an overall thickness of 260mm, a steel bottom plate thickness of 8mm, and crossbeam spacing of 5m. The main steel girders of the composite beam bridge are made of Q420 steel, the crossbeams are made of Q355 steel, and the concrete is C50.

[0228] To analyze the accuracy of the method presented in this paper, finite element method (FEM) simulation was used to model the stress on an actual bridge structure. The specific operation of the FEM method is as follows:

[0229] 1) Finite element model establishment

[0230] A spatial grid finite element model of the composite beam bridge was established using the finite element software Midas Civil. The longitudinal direction of the bridge was divided into elements of 1m. The finite element model is as follows: Figure 12 .

[0231] 2) Calculation principle of lateral load distribution coefficient m

[0232] A lateral moving load is applied, with a unit concentrated load P=1 applied to each of the following locations: Figure 13 At the 37 locations shown (j=1, 2 ... 37 in the figure), the lateral influence lines of the load effect on the structure are calculated sequentially, as shown in Figure B. iLet i be the position number of the main beam (i=1,2,3), and then calculate the vertical value of the load lateral distribution influence line of each main beam according to formula (49).

[0233] (49)

[0234] In the formula: For a moving load acting on position j in the transverse direction of the bridge, the load effects of the i-th main girder, such as support reactions (S=R), bending moment (S=M), deflection (S=w), etc., are given, where n is the number of main girders. To determine the load effect on the i-th main girder when a moving load is applied to the k-th main girder in the transverse direction, The vertical axis value of the influence line of the i-th main beam when the moving load is applied to position j in the transverse direction of the bridge.

[0235] Vertical values ​​of the influence lines where the load acts at each location By connecting the lines, the influence line of the lateral load distribution of the main beam can be obtained. Then, by using the most unfavorable lateral vehicle load distribution, the lateral load distribution coefficient of the main beam is calculated according to formula (50).

[0236] (50)

[0237] In the formula: Let ξ be the lateral distribution coefficient of the load S effect at measuring point c of the i-th main girder, where the lateral moving load P acts on the longitudinal bridge at position z in the longitudinal direction; ξ is the lateral lane distribution coefficient of the vehicle load; and N is the number of vehicles loaded. The vertical axis value of the influence line of the i-th main beam when the vehicle load acts on position k in the transverse direction of the bridge is calculated by linear interpolation of the points adjacent to the influence line.

[0238] The loads are distributed according to 1 to 4 vehicles, and the maximum value is taken as the final lateral load distribution coefficient. When 2 vehicles are placed at the support position, the lateral distribution coefficient of the support reaction force of beam No. 1 is... Maximum, the calculation process is as follows Figure 14 .

[0239] 3) Moving loads and measurement layout

[0240] like Figure 15 The influence line measuring points for the support reaction force are arranged at the left support, i.e., at c=0m. The influence line measuring points for the bending moment and deflection are arranged at mid-span (c=37m), L / 4 (c=19m), L / 8 (c=10m), and L / 16 (c=5m). The longitudinal bridge-direction application points for the lateral moving load are spaced 5m apart, with an additional loading point at mid-span 37m. A total of 16 loading sections are used. Figure 15 The z-position in the equation.

[0241] The operation steps of the method of the present invention are as follows:

[0242] Step (1): Based on the engineering data, obtain the relevant parameters of the simply supported beam bridge: bridge span L = 75m, cross-sectional layout as follows Figure 11 The elastic modulus E of steel and concrete is 2.06 × 10⁻⁶. 5 MPa, 3.45×10 4 MPa, Poisson's ratio ν are 0.31 and 0.2 respectively, shear modulus G of steel and concrete are 0.382 and 0.417 times E respectively, G of composite section is taken as 0.4 times E, bending moment of inertia and torsional moment of inertia of each main beam section, taking the edge main beam as an example, Midas Civil is used to calculate the equivalent section characteristics of composite section (converting concrete to steel), such as Figure 16 The bending moment of inertia and the torsional moment of inertia are 0.882 m respectively. 4 0.180m 4 The cross-sectional dimensions of the end diaphragm are as follows: Figure 17 Cross-sectional features such as Figure 18 .

[0243] Step (2): Calculate the torsional stiffness of the end diaphragm beam on the main beam.

[0244] According to equation (39), the beam section shear coefficient κ is calculated using the following formula:

[0245] (51)

[0246] In the formula: A h Let A be the cross-sectional area of ​​the end diaphragm. sz This represents the effective shear area of ​​the end crossbeam.

[0247] For steel supports, take That is, without considering the deformation of the support, the torsional elastic support stiffness k is calculated by equation (43) as follows (for ease of writing, E=1 is taken in the calculation, since as long as the value of G / E remains unchanged, it will not affect the calculation result):

[0248] (52)

[0249] The span-to-depth ratio of the end diaphragm is 5.7 / 2.6=2.19, which is very small. As can be seen from equation (52), the influence of shear deformation is greater than that of bending deformation. Therefore, for end diaphragms with a small span-to-depth ratio, the influence of shear deformation needs to be considered.

[0250] Therefore, the torsional stiffness of the end diaphragm relative to the main beam can be obtained as follows:

[0251] (53)

[0252] Step (3): Calculate the lateral load distribution coefficient at the fulcrum using the lever principle method:

[0253] A single-beam model refers to a bridge deck that breaks off at the centroid line of the main beam section; a double-beam model refers to a bridge deck that breaks off at the junction of the two webs of the box girder and the bridge deck. The bridge deck is calculated as a multi-span simply supported beam. Figure 19 m calculated by the single beam model 01 =1.632, m calculated using the double-beam model 02 =1.561, as Figure 20 By comparing the results with those of the finite element method, it can be seen that for a three-main-girder bridge, the m calculated using the double-girder model is more efficient. 02 =1.561 is more appropriate.

[0254] Step (4): The steps for calculating the transverse load distribution coefficient at any position in the longitudinal direction of the main girder using the modified eccentric pressure method considering torsional constraints at the beam ends are as follows:

[0255] a) Calculate the influence line η of the lateral distribution of the load. i By using formula (33), the lateral load distribution influence line values ​​of the No. 1 side main beam acting at multiple locations in the transverse direction are obtained, and connected into a straight line, the lateral load distribution influence line η of the No. 1 side main beam is obtained. i ,like Figure 21 The two influence lines of the transverse load distribution in the figure are the influence lines obtained by calculating the correction coefficients according to equations (29) and (31), respectively. , As can be seen, after considering the torsional constraint stiffness of the end diaphragm, the correction coefficient increases, and the value of the influence line is also larger.

[0256] b) Calculate the lateral load distribution factor

[0257] Vehicle loads are distributed according to the most unfavorable load location, such as... Figure 21 The vehicle load is arranged near the left side of the cross-section, with 1 to 4 vehicles. The maximum value calculated under four conditions is taken as the lateral load distribution coefficient of the vehicle load. Figure 22 Except for the support position, the load lateral distribution coefficient considering the torsional constraint stiffness of the end diaphragm is greater than the value when it is not considered (the correction coefficient is calculated by equation (30)). At the support position, both are 1.4825, which is higher than the value of m in the double beam model. 02 =1.561 is small, therefore it needs to be adjusted.

[0258] Step (5): For concentrated loads, consider the variation of the lateral distribution of the load along the longitudinal direction of the bridge. For uniformly distributed loads, take the lateral distribution coefficient of the load at the mid-span position. Reduce the load by the lateral distribution coefficient of the load and then calculate the load effect.

[0259] For a three-girder bridge, the load distribution along the longitudinal direction varies as follows:

[0260] a) Calculate the transition zone length xt

[0261] like Figure 23 Considering the transition region length x t The lateral load distribution coefficient changes linearly within the range, while the mid-span region remains unchanged. The length of the transition zone is calculated using the following formula, based on the principle that the area before and after the transformation is equal:

[0262] (54)

[0263] In the formula: L is the span of the main girder, and x is the longitudinal position of the bridge. Let x be the lateral load distribution factor of the vehicle load at position x. , These are the lateral load distribution coefficients for vehicle loads at the fulcrum and mid-span positions, respectively.

[0264] In practical calculations, discrete summation can be used instead of integration. Figure 23 There are a total of 38 points, with an actual area of ​​43.3480. , Transition region length x t The value is 18.1643m obtained from equation (55).

[0265] (55)

[0266] b) Regarding the transition region length x t Adjust the lateral load distribution coefficient value within the range.

[0267] (56)

[0268] in: The lateral load distribution factor of the vehicle load after adjustment in the transition zone is m. 02 The lateral load distribution factor is calculated for the double-beam model. The adjusted curve is shown below. Figure 23 The adjusted lateral load distribution coefficient at the support point is m. 02 . Figure 24 It is the lateral load distribution coefficient calculated considering the torsional constraint stiffness at the beam ends. A comparison with the lateral load distribution coefficient calculated by finite element software shows that the two are in good agreement.

[0269] In step (5), when calculating the vehicle load effect, the influence of the impact coefficient μ is temporarily ignored, and the method for calculating the support reaction force is as follows:

[0270] (57)

[0271] In the formula: R1 is the most unfavorable support reaction force of the side main beam, P k qk These represent the concentrated load and uniformly distributed load values ​​for Highway Class I loads, respectively. 1.2 is the amplification factor for calculating shear force, and L is the span.

[0272] The calculated value of the finite element model of the No. 1 side main beam is 1271.5kN, with an error of 1.38%, which is a good result.

[0273] The maximum bending moment M1 at each measuring point c of the side main beam is calculated as shown in equation (58).

[0274] (58)

[0275] Error due to bending moment effect with or without considering the stiffness of the end diaphragm: Figure 25 (a) The bending moment error considering the torsional constraint stiffness of the end diaphragm is basically less than 2%, which is significantly better than not considering it.

[0276] The maximum deflection w1 at each measuring point c on the side main beam is calculated as shown in equation (59):

[0277] (59)

[0278] Error due to deflection effect with or without considering the stiffness of the end diaphragm, such as Figure 25 (b) Although the deflection error is smaller if the torsional constraint stiffness of the end diaphragm is not considered, the error is still less than 2% if the torsional constraint stiffness is considered, and the error is also small.

[0279] In addition, it also analyzed, such as Figure 26 In cases with and without crossbeams, when there are no crossbeams, the end crossbeams are calculated as 0.3m thick concrete slabs, and the torsional elastic support stiffness k is:

[0280] (60)

[0281] The span-to-depth ratio of the end crossbeam is 5.7 / 0.3=19, which is relatively large. As can be seen from equation (60), the influence of shear deformation is very small and can be ignored.

[0282] Without the crossbeam, the theoretical value of the lateral load distribution coefficient at the mid-span section of the No. 1 side main beam is 1.1388, and the finite element value is 1.1498, which are quite close. With the crossbeam, the values ​​are 1.0334 and 1.0414, respectively. This indicates that after considering the torsional constraint of the end crossbeam on the main beam, the calculated lateral load distribution coefficient value of the method of the present invention is more consistent with the actual situation.

[0283] Example 2: 35m span three-main-girder narrow steel box girder composite bridge

[0284] Based on Example 1, the span of the bridge was changed to 35m, while the height-to-span ratio of the beam remained unchanged. The impact of the span change on the accuracy of the invention method was analyzed.

[0285] The operation steps of the method of the present invention are as follows:

[0286] Step (1): According to the engineering data, the bridge span L=35m. Taking the side main beam as an example, the equivalent section characteristics of the composite section are calculated using Midas Civil. The bending moment of inertia and the torsional moment of inertia are 0.1824m. 4 0.06636m 4 The cross-sectional characteristics of the end transverse diaphragm are as follows: Figure 27 Other parameters are the same as in Example 1.

[0287] Step (2): Calculate the torsional stiffness of the end diaphragm beam on the main beam.

[0288] According to equation (39), the beam section shear coefficient κ is calculated using the following formula:

[0289] (61)

[0290] The torsional elastic support stiffness k is calculated from equation (43) as follows:

[0291] (62)

[0292] The span-to-depth ratio of the end diaphragm is 5.7 / 1.3=4.38, which is less than the limit value of 5 for deep beams. As can be seen from equation (62), the influence of shear deformation is slightly greater than that of bending deformation. Therefore, for end diaphragms with a small span-to-depth ratio, the influence of shear deformation needs to be considered.

[0293] Therefore, the torsional stiffness of the end diaphragm relative to the main beam can be obtained as follows:

[0294] (63)

[0295] Step (3): Calculate the lateral load distribution coefficient at the fulcrum using the lever principle method:

[0296] like Figure 19 m calculated by the single beam model 01 =1.632, m calculated using the double-beam model 02 =1.561, as Figure 28 By comparing the results with those of the finite element method, it can be seen that for a three-main-girder bridge, the m calculated using the double-girder model is more efficient. 02 =1.561 is more appropriate.

[0297] Step (4): Calculate the transverse load distribution coefficient at any position in the longitudinal direction of the main girder using the modified eccentric pressure method considering torsional constraints at the beam ends, as follows: Figure 29 Except for the support position, the load lateral distribution coefficient considering the torsional constraint stiffness of the end diaphragm is greater than the value when it is not considered (the correction coefficient is calculated by equation (30)). At the support position, both are 1.4825, which is higher than the value of m in the double beam model.02 =1.561 is small, therefore it needs to be adjusted.

[0298] Step (5): For concentrated loads, consider the variation of the lateral distribution of the load along the longitudinal direction of the bridge. For uniformly distributed loads, take the lateral distribution coefficient of the load at the mid-span position. Reduce the load by the lateral distribution coefficient of the load and then calculate the load effect.

[0299] For a three-girder bridge, the adjusted curve is as follows: Figure 30 (a) The adjusted lateral load distribution coefficient at the support point is m 02 It can be seen that the lateral load distribution coefficient calculated considering the torsional constraint stiffness at the beam end... The lateral distribution factor of the bending moment load calculated by the finite element software is in good agreement with that calculated without considering the torsional constraint stiffness at the beam ends. It is noticeably too small. Figure 30 (b) The lateral distribution coefficient of the deflection load calculated by the finite element software and Closer.

[0300] In step (5), when calculating the vehicle load effect, the influence of the impact coefficient μ is temporarily ignored, and the method for calculating the support reaction force is as follows:

[0301] (64)

[0302] The finite element model of the No. 1 side main beam has a calculated value of 901.6 kN, with an error of only 0.38%, which is very accurate.

[0303] Error due to bending moment effect with or without considering the stiffness of the end diaphragm: Figure 31 (a) When considering the torsional constraint stiffness of the end diaphragm, the bending moment error is less than 1%, while when not considering it, more than half of the errors exceed 5%.

[0304] Error due to deflection effect with or without considering the stiffness of the end diaphragm, such as Figure 31 (b) When considering the torsional constraint stiffness of the end diaphragm, the deflection error is less than 2%, while when not considering it, the error is generally greater than 3%.

[0305] Furthermore, considering both cases with and without crossbeams, the theoretical value of the lateral load distribution coefficient at the mid-span section of the main beam on side 1 without crossbeams is 1.2759, and the finite element value is 1.2734, which are almost identical. With crossbeams, the values ​​are 1.2095 and 1.2373 respectively. This indicates that the method of this invention, after considering the torsional constraint of the end crossbeams on the main beam, calculates a lateral load distribution coefficient value that is more consistent with the actual situation.

[0306] Example 3: 75m span double main girder narrow steel box girder composite bridge

[0307] The layout of Yangxin Expressway B ramp bridge is as follows Figure 32The cross-section of the main beam is as follows Figure 33 Other parameters are the same as in Example 1.

[0308] The operation steps of the method of the present invention are as follows:

[0309] Step (1): The parameters are the same as in Example 1.

[0310] Step (2): Calculate the torsional stiffness of the end diaphragm beam on the main beam.

[0311] Referring to Example 1, the torsional stiffness of the end diaphragm beam relative to the main beam is obtained as follows:

[0312] (65)

[0313] Step (3): Calculate the lateral load distribution coefficient at the fulcrum using the lever principle method:

[0314] like Figure 19 m calculated by the single beam model 01 =1.632, m calculated using the double-beam model 02 =1.561, as Figure 34 By comparing the results with those of the finite element method, it can be seen that for a double main girder bridge, the m calculated using the single girder model is more efficient. 01 =1.632 is more appropriate.

[0315] Step (4): Calculate the transverse load distribution coefficient at any position in the longitudinal direction of the main girder using the modified eccentric pressure method considering torsional constraints at the beam ends, as follows: Figure 35 Except for the support position, the load lateral distribution coefficient considering the torsional constraint stiffness of the end diaphragm is greater than the value when it is not considered (the correction coefficient is calculated by equation (30)). At the support position, both are 1.632, which is consistent with the value of m in the single beam model. 01 =1.632 is the same, so no adjustment is needed.

[0316] Step (5): Calculate the load effect:

[0317] For double girder bridges, such as Figure 36 (a) shows that the lateral load distribution coefficient calculated considering the torsional constraint stiffness at the beam end is... The result is in good agreement with the lateral distribution coefficient of the bending moment load calculated by finite element software. Figure 36 (b) The lateral distribution coefficient of the deflection load calculated by the finite element software and Closer.

[0318] In step (5), when calculating the vehicle load effect, the influence of the impact coefficient μ is temporarily ignored, and the method for calculating the support reaction force is as follows:

[0319] (66)

[0320] The calculated value of the finite element model of the No. 1 side main beam is 1345.7kN, with an error of only 0.14%, which is very accurate.

[0321] Error due to bending moment effect with or without considering the stiffness of the end diaphragm: Figure 37 (a) The bending moment error considering the torsional constraint stiffness of the end diaphragm is basically less than 2%, while the error exceeds 5% when it is not considered.

[0322] Error due to deflection effect with or without considering the stiffness of the end diaphragm, such as Figure 37 (b) The deflection error considering the torsional constraint stiffness of the end diaphragm is also slightly smaller than that when it is not considered. The error is less than 1.5% in both cases.

[0323] Furthermore, considering both cases with and without crossbeams, the theoretical value of the lateral load distribution coefficient at the mid-span section of the main beam on side 1 without crossbeams is 1.2260, and the finite element value is 1.22577, showing good agreement. With crossbeams, the values ​​are 1.1880 and 1.19468 respectively. This indicates that the method of this invention, after considering the torsional constraint of the end crossbeams on the main beam, calculates the lateral load distribution coefficient value that is more consistent with the actual situation.

[0324] The above descriptions are merely three embodiments of the present invention. All equivalent changes and modifications made within the scope of the claims of the present invention are within the scope of the present invention.

Claims

1. A modified eccentric pressure method considering the torsional constraint stiffness of the end transverse diaphragm, characterized in that, The steps include: (1) Based on the engineering data, obtain the relevant parameters of the simply supported beam bridge: bridge span L, cross-sectional layout, bending moment of inertia and torsional moment of inertia of each main beam section, cross-sectional dimensions of the end diaphragm, material elastic modulus E, shear modulus G, Poisson's ratio ν; (2) Calculate the torsional stiffness of the end diaphragm beam to the main beam; (3) Calculate the lateral load distribution coefficient at the fulcrum using the lever principle method: a) For T-shaped and I-shaped single-web section main beams, a single-beam model is used for calculation; b) For small box girders, hollow slab type double web section main beams, and double main beam bridges, a single beam model is used for calculation; for multi-main beam bridges with three or more main beams, a double beam model is used for calculation. The lateral load distribution factor calculated using the single beam model is denoted as m. 01 The lateral load distribution coefficient calculated using the double-beam model is denoted as m. 02 ; (4) The load distribution coefficient at any position in the longitudinal direction of the main beam is calculated by the modified eccentric pressure method considering the torsional constraint at the beam end. (5) For concentrated loads, consider the variation of the lateral distribution of the load along the longitudinal direction of the bridge. For uniformly distributed loads, take the lateral distribution coefficient of the load at the mid-span position. Reduce the load by the lateral distribution coefficient of the load and then calculate the load effect. In step (2), according to the lever principle, the end diaphragm beam is simplified to a simply supported beam. For the main beam on the adjacent side, its torsional elastic support stiffness k is: ； Among them: EI h For the bending stiffness of the end diaphragm, l h Let G be the span of the end diaphragm, G be the shear modulus of the material, and A be the length of the end diaphragm. sz κ is the effective shear area of ​​the end diaphragm, and k is the shear coefficient of the beam section. R The elastic support stiffness of the support; For the edge main beam, it is connected to only one end transverse diaphragm, and the torsional restraint stiffness at the beam end is taken as k. For the middle main beam, it is connected to two end transverse diaphragms, and the torsional restraint stiffness at the beam end is taken as 2k, that is: ； Where: i is the main beam number, n is the number of main beams, and k is the main beam number. i The torsional constraint stiffness at the end of main beam i is given. Main beams 1 and n are side main beams, and the other numbered main beams are middle main beams. In step (4), the steps for calculating the transverse load distribution coefficient at any position in the longitudinal direction of the main girder using the modified eccentric pressure method considering torsional constraints at the beam ends are as follows: a) Calculate the influence line η of the lateral load distribution i ； ； Where: η ie This is the influence line value of the lateral load distribution of the main girder i when an eccentric load is applied at position e in the transverse direction of the bridge. e represents the transverse position of the load application, with a negative sign to the left of the centroid of the section and a positive sign to the right. i Let be the moment of inertia of the section of main beam i, n be the number of main beams, and a be the moment of inertia of the section of main beam i. i Let be the distance from the i-th main beam to the centroid of the cross-section. The main beam to the left of the centroid is considered negative, and the main beam to the right is considered positive. E is the elastic modulus of the material, L is the span of the main beam, x is the position of the eccentric load acting in the longitudinal direction of the bridge, with the left end support of the beam as the origin and the right end as the positive direction. k i Let GJ be the torsional restraint stiffness at the end of main beam i. i Let i be the torsional stiffness of the main beam. pass The influence line values ​​of the lateral load distribution at multiple locations along the transverse direction of the main girder i were obtained, such as... , Connecting these lines with a straight line yields the influence line η of the lateral load distribution of the main beam i. i ; ； ； ； Where: y represents the transverse bridge position, the origin of the coordinate axis is the centroid of the cross section, and negative signs are taken to the left of the centroid of the cross section and positive signs are taken to the right; b) Calculate the lateral load distribution factor The influence line η of the lateral load distribution of the main beam i is obtained. i Then, the vehicle load is distributed according to the most unfavorable load position, and the lateral load distribution coefficient of the vehicle load is calculated: ； Where: m iqki Let ξ be the lateral load distribution coefficient of the vehicle load on main beam i, ξ be the lateral lane distribution coefficient of the vehicle load (i.e., the lateral multi-lane reduction coefficient), N be the number of vehicles loaded, and η be the lateral load distribution coefficient of the vehicle load on main beam i. ij The value of the lateral load distribution influence line of main beam i when the vehicle tires act on position j in the transverse direction of the bridge. For the lateral load distribution coefficient m of the crowd load on main beam i... irki The calculation is as follows: ； Where: η ir The influence line value of the lateral distribution of the load on the main beam i when the crowd load acts on the center point of the resultant force of the sidewalk; In step (5), for concentrated loads, when it is a double main girder bridge, the value is directly taken as... For multi-main-girder bridges with three or more main girders, the load lateral distribution along the longitudinal direction varies as follows: a) Calculate the transition zone length x t The length of the transition zone is calculated using the following formula: ； Where: L is the span of the main girder, and x is the longitudinal position of the bridge. Let x be the lateral load distribution factor of the vehicle load at position x. , These are the lateral load distribution coefficients for vehicle loads at the fulcrum and mid-span positions, respectively. b) Regarding the transition region length x t Adjust the lateral load distribution coefficient value within the range. ； in: The lateral load distribution factor of the vehicle load after adjustment in the transition zone is m. 02 The lateral load distribution coefficient is calculated for the double-beam model.

2. The modified eccentric pressure method considering the torsional constraint stiffness of the end transverse diaphragm as described in claim 1, characterized in that: In step (1), the bending moment of inertia and torsional moment of inertia of the main beam section are calculated by finite element software, such as Midas Civil, Bridge Doctor, and ANSYS, or by AutoCAD software, or by theoretical mechanics methods.

3. The modified eccentric pressure method considering the torsional constraint stiffness of the end diaphragm beam according to claim 1, characterized in that: In step (1), the shear modulus G is calculated by the following formula: 。 4. The modified eccentric pressure method considering the torsional constraint stiffness of the end transverse diaphragm as described in claim 1, characterized in that: In step (1), if the beam bridge is a continuous beam bridge or a continuous rigid frame bridge, the equivalent simply supported beam method is used to correct the bending stiffness and torsional stiffness of the main beam section.

5. The modified eccentric pressure method considering the torsional constraint stiffness of the end diaphragm beam according to claim 1, characterized in that: In step (2), the beam section shear coefficient κ is calculated using the following formula: ； Among them: A h Let A be the cross-sectional area of ​​the end diaphragm. sz This represents the effective shear area of ​​the end crossbeam.

6. The modified eccentric pressure method considering the torsional constraint stiffness of the end transverse diaphragm as described in claim 1, characterized in that: In step (2), when the span-to-depth ratio of the end transverse diaphragm is greater than 10, take... That is, the effect of shear deformation is not considered.

7. The modified eccentric pressure method considering the torsional constraint stiffness of the end diaphragm beam according to claim 1, characterized in that: In step (2), for the steel support, take That is, without considering the deformation of the support, for plate rubber bearings and pot rubber bearings, Calculate using the following formula: ； Where: E e Let A be the compressive elastic modulus of the rubber bearing. e h is the area of ​​the rubber bearing. e This refers to the total thickness of the rubber layer in the rubber bearing.

8. The modified eccentric pressure method considering the torsional constraint stiffness of the end diaphragm beam according to claim 1, characterized in that: Step (2) is for a conventional beam bridge with one support under each main girder. If other support arrangements are used, the torsional stiffness k at the beam end of the main girder is derived based on mechanical principles. i .

9. The modified eccentric pressure method considering the torsional constraint stiffness of the end diaphragm beam according to claim 1, characterized in that: In step (3), the single beam model refers to the bridge deck being broken at the centroid line of the main beam section, and the double beam model refers to the bridge deck being broken at the junction of the two webs of the box-section main beam and the bridge deck. The bridge deck is calculated as a multi-span simply supported beam.

10. The modified eccentric pressure method considering the torsional constraint stiffness of the end diaphragm beam according to claim 1, characterized in that: In step (5), the calculation method for the load effect is as follows: The general formula for calculating the effect of vehicle load on main beam i is: ； Wherein: S q The effect of vehicle load is represented by μ, where μ is the impact coefficient of the vehicle load, and P is the effect of vehicle load. k This represents the concentrated load value in the vehicle load. q represents the coordinate value of the influence line of the longitudinal bridge action effect. k Ω represents the uniformly distributed load value in the vehicle load, x represents the area of ​​the influence line of the effect, and x represents the longitudinal bridge position. The general formula for calculating the effect of crowd load on main beam i is: ； Wherein: S r For the effect of crowd load, q r Ω represents the population load value, and Ω represents the area of ​​the influence line of the effect.

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