Concrete box girder shear lag analysis method for bridge deck slab inclination

Abstract

The invention discloses a method for analyzing shear lag of a concrete box girder with an inclined bridge deck slab. The method comprises the following steps: (1) constructing a shear lag displacement mode of the concrete box girder bridge with the inclined bridge deck slab; based on a traditional symmetrical shear hysteresis warpage displacement distribution mode, the invention provides a concrete box girder bridge shear hysteresis displacement mode capable of objectively revealing inclination of a bridge deck slab. (2) a shear lag control differential equation and boundary conditions of the concrete box girder bridge with the inclined bridge deck slab; an energy variation principle and a potential energy standing value principle are applied to export a concrete box girder shear lag control differential equation set and boundary conditions of bridge deck slab inclination, and solving is carried out through the boundary conditions. (3) solving shear lag of the concrete box girder bridge with an inclined bridge deck slab; according to the shear lag control differential equation, the simply supported box girder bridge shear lag additional deflection, the box girder deflection considering the shear lag effect, the section longitudinal stress and the shear lag coefficient expression are given in combination with the boundary conditions.

Description

Technical Field

[0001] The present invention belongs to the technical field of bridge mechanics analysis, and particularly relates to a method for analyzing shear lag of concrete box girders with inclined deck slabs. Background Art

[0002] In the actual design of box girder bridges, inclined deck slabs are often arranged. The shear lag effect of box girders with inclined deck slabs is an influencing factor that must be considered in the design practice of box girder bridges. It will cause uneven stress distribution and additional deformation of the cross-section, and may even lead to safety accidents in severe cases. From the three-dimensional space finite element calculation results (as Figure 1 shown), the shear lag warping normal stress of box girders with inclined deck slabs is asymmetrically distributed along the cross-section. The longitudinal stress on the side with shorter webs is greater than that on the side with taller webs, and the existing analytical methods for shear lag of box girders are not applicable to the analysis and calculation of the shear lag effect of box girders with inclined deck slabs. In addition, for the existing warping displacement modes for analyzing the shear lag effect of box girders with inclined deck slabs, the traditional symmetric warping displacement mode of box girders is adopted, and this method cannot objectively reveal the asymmetric stress distribution mode of box girders with inclined deck slabs. Therefore, it is necessary to propose an asymmetric shear lag warping displacement mode considering the inclined deck slab. Summary of the Invention

[0003] Based on the traditional symmetric distribution mode of shear lag, the present invention proposes a method for analyzing shear lag of concrete box girders with inclined deck slabs, which can accurately calculate the generalized shear force (shear force) and moment (bending moment) of the cross-section caused by shear lag (elementary beam) of concrete box girder bridges with inclined deck slabs, so as to correspond to the current bridge design code and facilitate the understanding and use by designers.

[0004] The present invention proposes a method for analyzing shear lag of concrete box girders with inclined deck slabs, providing theoretical support for the engineering design of such box girder bridges. The specific scheme is as follows:

[0005] (1) Construction of the shear lag displacement mode of concrete box girder bridges with inclined deck slabs

[0006] The warping displacement mode is the basis for establishing the subsequent shear lag control differential equation and stress calculation. By calculating the bending shear stresses at the ribs of the top and bottom slabs on the same side of the box girder and using the principle of constant ratio of the bending shear stresses at the ribs of the top and bottom slabs, the shear lag displacement mode of box girder bridges with inclined deck slabs can be obtained through conversion.

[0007] (2) Establishment of the shear lag control differential equation of concrete box girder bridges with inclined deck slabs

[0008] The establishment of the shear lag control differential equation is a key step in solving the shear lag effect. The present invention comprehensively uses the energy variational principle, the principle of stationary potential energy, etc. to derive the shear lag control differential equation of concrete box girders with inclined deck slabs, and gives the corresponding boundary constraint conditions in combination with the variational boundary conditions.

[0009] To this end, the present invention adopts the following technical solutions:

[0010] The present invention proposes a calculation method for the shear lag effect of box girders considering the inclination of the bridge deck, providing theoretical support for the engineering design of such box girder bridges. The specific solution is as follows:

[0011] (1) Construction of the shear lag displacement mode of the concrete box girder bridge with an inclined deck.

[0012] Based on the traditional symmetric shear lag warping displacement distribution mode, the present invention proposes a shear lag displacement mode that can objectively reveal the shear lag displacement of the concrete box girder bridge with an inclined deck.

[0013] (2) Shear lag control differential equations and boundary conditions of the concrete box girder bridge with an inclined deck.

[0014] Using the principle of variational energy and the principle of stationary potential energy, the shear lag control differential equation group and boundary conditions of the concrete box girder with an inclined deck are derived and solved through the boundary conditions.

[0015] (3) Shear lag solution of the concrete box girder bridge with an inclined deck.

[0016] According to the shear lag control differential equation, combined with the boundary conditions, expressions for the additional deflection of the simply supported box girder bridge due to shear lag, the deflection of the box girder considering the shear lag effect, the longitudinal stress of the cross-section, and the shear lag coefficient are given.

[0017] The beneficial effects of the present invention are as follows:

[0018] Based on the existing traditional symmetric shear lag distribution mode, the present invention proposes a multi-parameter shear lag displacement mode considering the longitudinal displacement difference between the upper and lower flanges, which can objectively and accurately reveal the spatial stress state of the concrete box girder bridge with an inclined deck, providing theoretical support for the refined design, parameter analysis, and construction measures of such box girder bridges, and also providing technical support for the health monitoring of such bridges. Description of the Drawings

[0019] Figure 1 is the longitudinal stress distribution diagram of the shear lag of the concrete box girder bridge with an inclined deck;

[0020] Figure 2 is the flow block diagram of the analysis method of the present invention;

[0021] Figure 3 is the cross-section and coordinate system of the concrete box girder with an inclined deck;

[0022] Figure 4 is the cross-sectional dimension of the concrete box girder with an inclined deck (unit: mm);

[0023] Figure 5 It is the calculation flow chart of the shear lag effect of a concrete box girder with a tilted deck slab;

[0024] Figure 6 It is the finite element model diagram of a concrete box girder with a tilted deck slab in the embodiment;

[0025] Figure 7 It is the longitudinal stress curve diagram of the mid-span section of a concrete box girder with a tilted deck slab in the embodiment. Specific implementation manners

[0026] The present invention will be further described below in conjunction with the accompanying drawings and specific embodiments:

[0027] ① Basic overview of the example of a concrete box girder with a tilted deck slab:

[0028] A reduced-scale model of a concrete simply supported box girder with a calculated span of 800 mm is selected, and the cross-sectional dimensions and load application methods are as Figure 4 shown. The material is C25 concrete, the elastic modulus E = 2.8 GPa, the shear modulus G = 1.077 GPa, the Poisson's ratio μ = 0.37, and a vertical uniform load q = 0.2 N / mm is symmetrically applied at the rib of the top slab.

[0029] ② Calculation results and analysis of a concrete box girder with a tilted deck slab:

[0030] According to the calculation method of the present invention, the longitudinal stresses of the mid-span section of a concrete box girder with a tilted deck slab are calculated respectively under the modified warping displacement mode and the traditional warping displacement mode, and the calculation process is as Figure 5 shown. A finite element model of the concrete box girder is established using Abaqus - C3D8R elements, with a total of 25,677 nodes and 19,680 elements. The load is symmetrically applied to the nodes at the intersection of the top slab and the web, and the longitudinal stresses of the upper and lower flanges of the mid-span section are analyzed and extracted. The model and the deformation diagram are as Figure 6 shown. The longitudinal stress diagrams of the upper and lower flanges of the mid-span section are plotted with the finite element results and the calculation results of the present invention.

[0031] From Figure 7 it can be seen that the modified warping displacement mode proposed by the present invention can objectively and accurately reveal the shear lag effect of a concrete box girder with a tilted deck slab, that is, there are significant differences in the longitudinal stresses of the upper and lower flanges of the box girder, providing a theoretical support for the refined design and parametric analysis of this type of box girder bridge.

Claims

1. A shear lag analysis method for concrete box girders with inclined bridge decks, characterized in that: The following steps are involved: (1) Construction of shear lag displacement model of concrete box girder with inclined bridge deck ① The shear lag longitudinal displacement u(x,y,z) of the concrete box girder with inclined bridge deck is: The Cartesian coordinate system oxyz is used, where o is the origin, x is the horizontal direction, y is the vertical direction, and z is the longitudinal direction; h l 、h r are the heights of the short side and high side webs, respectively, h u 、h b is the distance from the middle surface of the top and bottom plates to the centroid axis, i.e., the x-axis; b1, b2, and b3 are the half-width of the top and bottom plates and the width of the cantilever plate, respectively; t u ,t b and t w is the thickness of the top and bottom plates and the thickness of the web; combined with the elastic superposition principle, the shear lag longitudinal displacement u(x, y, z) of the concrete box girder with inclined bridge deck is: u(x,y,z) = u0(x,y,z) + u ωi (x,y,z) = -yw′(z) - ω i (x,y)f′(z) (Formula 1) Where: u0 is the longitudinal displacement of the primary concrete box beam, u ω is the shear lag warping longitudinal displacement; w' is the first-order derivative of the box girder deflection, ω i is the shear lag displacement distribution function of the concrete box girder with inclined bridge deck, the subscript i is c and nc, representing the modified and traditional shear lag displacement modes, the same below; f' is the first-order derivative of the shear lag additional deflection; ② Shear lag modified distribution function ω of concrete box girder with inclined bridge deck c (x,y) is: In the formula: α1, α2, β1, β2, Δ1 and Δ2 are the warping displacement correction coefficients of the box beam corresponding to each plate, α1 = b 12 3 / b 11 3 , α2=b 22 2 / b 21 2 , β1=(b2·h b ) / (b1·h u ), β2=b3 / b1, Δ1=(1-α1)h u (1-η1), Δ2=(α2-1)h b (β1-η2); η1 and η2 are the warping self-balancing coefficients, and their values ​​can be determined by the self-balancing condition ∫ A ωdA=0 and ∫ A ωydA=0, A is the cross-sectional area of ​​the box girder; ③ Traditional distribution function of shear lag of concrete box girder with inclined bridge deck ω nc (x,y) is: Where: β1, β2, is the warping displacement correction coefficient of each concrete box beam plate, β1=(b2·h b ) / (b1·h u ), β2=b3 / b1, η is the traditional warping self-balancing correction coefficient, which is calculated by the self-balancing condition ∫ A ωydA=0, A is the cross-sectional area of ​​the box girder; ④ Longitudinal stress σ of the concrete box girder section with inclined bridge deck zi for: Where: σ0 is the normal stress of the primary beam of the inclined concrete box girder section of the bridge deck, σ ωi is the shear hysteresis warping normal stress of the concrete box girder with inclined bridge deck; Mx is the bending moment of the box girder section, and Ix is the moment of inertia of the box girder section, that is, I x =∫ A y 2 dA;M ω i is the shear lag moment of the box girder section, M ωi =∫ A σ ωi ω i dA=-EI ωi f″, E is the elastic modulus of concrete, f” is the second-order derivative of the shear lag additional deflection, I ω i is the shear hysteresis warping moment of inertia, that is, I ωi =∫ A ω i 2 dA; (2) Shear lag control differential equations and boundary conditions for concrete box girders with inclined bridge decks ① The total potential energy Π of shear lag of concrete box girder with inclined bridge deck is: Where: G is the shear modulus of concrete; ε and γ are the longitudinal strain and shear strain of the box girder; q is any vertical symmetrical load; w and f are the primary beam deflection and shear lag additional deflection of the concrete box girder, respectively; A ωi is the shear hysteresis warping area, that is, ② The shear lag control differential equation of the concrete box girder with inclined bridge deck is: According to the principle of stationary potential energy, the total potential energy formula (5) of the concrete box girder is subjected to first-order variation operation, and δΠ is set to obtain the differential equation: EI x w″″-q=0 (Formula 6) EI ωi f″″-GA ωi f″-q=0(Formula 7) Where: w″″ is the fourth-order derivative of the concrete elementary beam deflection, f″″ is the fourth-order derivative of the concrete shear lag additional deflection; By rearranging formula (7), we can obtain the control differential equation of the concrete box girder with shear lag additional deflection as generalized displacement: Where: k i is the shear lag Reissner parameter of the inclined concrete box girder with bridge deck, expressed as ③ The shear lag boundary condition of the concrete box girder with inclined bridge deck is: Fixed end: f = 0, f′ = 0; Simply supported end: f = 0, f″ = 0; Free end: f″=0, f″′-k 2 f′=0; (3) Solution for shear lag of simply supported concrete box girder with inclined deck: ① Analytical solution for shear lag additional deflection of simply supported concrete box girder with inclined bridge deck: According to the additional deflection differential equation (8), the general solution of the differential equation is: Where: C1~C4 are 4 unknown coefficients, which are determined by the simply supported boundary conditions; Combined with the simply supported boundary conditions, the shear lag additional deflection of the simply supported concrete box girder with inclined bridge deck is: Where: l is the span of the box girder; ② The analytical solution for the vertical deflection of a simply supported concrete box girder with an inclined bridge deck is: The vertical deflection of the inclined concrete box girder of the bridge deck considering the shear lag effect is: In formula 11: the first term is the primary concrete beam deflection, the second and third terms are the additional deflections of concrete shear lag; ③ The analytical solution of the longitudinal stress and shear lag coefficient of the concrete simply supported box girder with inclined bridge deck is: According to formula (4), the longitudinal stress of the concrete box girder section with inclined bridge deck is: According to the definition of shear lag coefficient λ The shear lag coefficient of the concrete simply supported box girder with inclined bridge deck is:

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