A method for analyzing the shear lag of a concrete box girder with a deck slab that is skewed

Abstract

The application discloses a concrete box girder shear lag analysis method for bridge deck slab inclination, which comprises the following steps: (1) constructing a shear lag displacement mode of the concrete box girder bridge deck slab inclination; based on the traditional symmetric shear lag warping displacement distribution mode, the application proposes a shear lag displacement mode which can objectively reveal the concrete box girder bridge deck slab inclination; (2) shear lag control differential equations and boundary conditions of the concrete box girder bridge deck slab inclination; by using the energy variation principle and the potential energy stationary value principle, the shear lag control differential equation set and the boundary conditions of the concrete box girder bridge deck slab inclination are derived, and the boundary conditions are solved; (3) solving the shear lag of the concrete box girder bridge deck slab inclination; according to the shear lag control differential equation, the expressions of the additional deflection of the simply-supported box girder bridge shear lag, the box girder deflection considering the shear lag effect, the cross section longitudinal stress and the shear lag coefficient are given in combination with the boundary conditions.

Description

Technical Field

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

[0002] In actual box girder bridge design, inclined deck layout is often seen. The shear lag effect of the box girder with inclined deck is an influencing factor that must be considered in the design of box girder bridges. It will cause uneven stress distribution and additional deformation in the cross section, which may lead to safety accidents in serious cases. From the results of three-dimensional finite element calculation (such as Figure 1 (As shown in the figure), the shear lag warping normal stress of a box girder with an inclined deck is asymmetrically distributed along the cross section, with the longitudinal stress on the shorter web side being greater than on the taller web side. Existing analytical methods for shear lag in box girders are not suitable for analyzing and calculating the shear lag effect of box girders with inclined decks. Furthermore, existing analyses of the shear lag warping displacement model for box girders with inclined decks all employ the traditional symmetrical warping displacement model. This method cannot objectively reveal the asymmetric stress distribution pattern of box girders with inclined decks. Therefore, it is necessary to propose a shear lag warping displacement model for asymmetric box girders that takes into account the tilt of the deck. Summary of the Invention

[0003] Based on the traditional symmetrical distribution pattern of shear lag, the present invention proposes a shear lag analysis method for concrete box girders with inclined bridge decks. This method can accurately calculate the cross-sectional generalized shear force (shear force) and moment (bending moment) caused by shear lag (elementary beams) in concrete box girder bridges with inclined bridge decks, so that it corresponds to the current bridge design specifications and is easier for designers to understand and use.

[0004] This paper proposes a shear lag analysis method for concrete box girders with inclined bridge decks, providing theoretical support for the engineering design of such box girder bridges. The specific solution is as follows:

[0005] (1) Construction of shear lag displacement model for concrete box girder bridge with inclined deck

[0006] The warping displacement model is the basis for the subsequent establishment of the shear lag control differential equation and stress calculation. By calculating the bending shear stress at the top and bottom plate ribs on the same side of the box girder and using the principle that the ratio of the bending shear stress at the top and bottom plate ribs is constant, the shear lag displacement model of the box girder bridge with inclined bridge deck can be converted.

[0007] (2) Establishment of differential equations for shear lag control of concrete box girder bridges with inclined decks

[0008] The establishment of the shear lag control differential equation is a key step in solving the shear lag effect. The present invention comprehensively utilizes the energy variation principle, the potential energy stationary value principle, etc. to derive the shear lag control differential equation of the concrete box girder with inclined bridge deck, and combines the variational boundary conditions to give the corresponding boundary constraints.

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

[0010] This paper proposes a method for calculating the shear lag effect of box girders that takes into account 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 shear lag displacement model for concrete box girder bridge with inclined deck.

[0012] Based on the traditional symmetrical shear lag warping displacement distribution model, the present invention proposes a shear lag displacement model for concrete box girder bridges that can objectively reveal the inclination of the bridge deck.

[0013] (2) Shear lag control differential equations and boundary conditions for concrete box girder bridges with inclined decks.

[0014] By using the energy variation principle and the potential energy stationary value principle, the shear lag control differential equations and boundary conditions of the concrete box girder with inclined bridge deck are derived and solved using the boundary conditions.

[0015] (3) Solution for shear lag in concrete box girder bridges with inclined decks.

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

[0017] A shear lag analysis method for concrete box girders with inclined bridge decks comprises the following steps:

[0018] (1) Construction of shear lag displacement model for concrete box girder with inclined bridge deck

[0019] ① The shear lag longitudinal displacement u(x, y, z) of the concrete box girder with inclined bridge deck is:

[0020] 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, 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 are the thickness of the top and bottom plates and the thickness of the web. Based on the principle of elastic superposition, the shear lag longitudinal displacement u(x, y, z) of the concrete box girder with inclined deck is:

[0021] (Formula 1)

[0022] where u0 is the elementary beam longitudinal displacement of the concrete box girder, u ω is the shear lag warping longitudinal displacement; w' is the first order derivative of the deflection of the box girder, ω i is the shear lag displacement distribution function of the concrete box girder with deck slab inclination, subscript i takes c and nc, representing the correction and traditional shear lag displacement mode, same below; f'is the first order derivative of the shear lag additional deflection;

[0023] 2. The shear lag correction distribution function ω c (x, y) of the concrete box girder with deck slab inclination is:

[0024] (Formula 2)

[0025] where α1, α2, β1, β2, φ, Δ1 and Δ2 are the warping displacement correction coefficients of the box girder 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, φ=1.4, Δ1= (1-α1) h u (1-η1), Δ2= (α2-1) h b (β1-η2); η1, η2 are the warping self-balancing coefficients, the values of which can be determined through the self-balancing condition and , and A is the cross-sectional area of the box girder;

[0026] 3. The traditional distribution function ω nc (x, y) of the shear lag of the concrete box girder with deck slab inclination is:

[0027] (Formula 3)

[0028] where β1, β2, φ are the warping displacement correction coefficients of the concrete box girder, β1= (b2∙h b ) / (b1∙h u ), β2= b3 / b1, φ=1.4, η is the traditional warping self-balancing correction coefficient, which is determined through the self-balancing condition , and A is the cross-sectional area of the box girder;

[0029] 4. The cross-sectional longitudinal stress σ zi of the concrete box girder with deck slab inclination is:

[0030] (4)

[0031] where σ0 is the elementary beam normal stress of the box girder with deck slanting, σ ωi is the shear lag warping normal stress of the box girder with deck slanting; M x is the bending moment of the box girder section, I x is the moment of inertia of the box girder section, i.e. ; M ωi is the shear lag moment of the box girder section, , E is the elastic modulus of concrete, f '' is the second derivative of the shear lag additional deflection, I ωi is the shear lag warping moment of inertia, i.e. ;

[0032] (2) The shear lag control differential equation and boundary conditions of the box girder with deck slanting

[0033] ① The total potential energy Π of the box girder with deck slanting is:

[0034] (5)

[0035] 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 symmetric load; w and f are the elementary beam deflection and shear lag additional deflection of the concrete box girder, respectively, A ωi is the shear lag warping area, i.e. ;

[0036] ② The shear lag control differential equation of the box girder with deck slanting is:

[0037] According to the potential energy stationary value principle, the first-order variation operation is performed on the total potential energy formula (5) of the concrete box girder, and the differential equation is obtained by letting δΠ:

[0038] (6)

[0039] (7)

[0040] where: is the fourth derivative of the elementary beam deflection of concrete, is the fourth derivative of the shear lag additional deflection of concrete;

[0041] The formula (7) can be obtained by rearranging the control differential equation of the concrete box girder with the shear lag additional deflection as the generalized displacement:

[0042] (8)

[0043] where ki The Reissner parameter of the shear lag of the concrete box girder with the bridge deck slant is expressed as ;

[0044] ③ The shear lag boundary condition of the concrete box girder with the bridge deck slant is:

[0045] Fixed end: , ;

[0046] Simply supported end: , ;

[0047] Free end: , ;

[0048] (3) The shear lag solution of the concrete simply supported box girder with the bridge deck slant:

[0049] ① The analytical solution of the shear lag additional deflection of the concrete simply supported box girder with the bridge deck slant:

[0050] According to the differential equation of the additional deflection (8), the general solution of the differential equation is:

[0051] (Formula 9)

[0052] In the formula, C1-C4 are four undetermined coefficients, which are determined by the simply supported boundary condition;

[0053] Combined with the simply supported boundary condition, the shear lag additional deflection of the concrete simply supported box girder with the bridge deck slant is:

[0054] (Formula 10)

[0055] In the formula, l is the span of the box girder;

[0056] ② The analytical solution of the vertical deflection of the concrete simply supported box girder with the bridge deck slant is:

[0057] The vertical deflection of the concrete box girder with the bridge deck slant considering the shear lag effect is:

[0058] (Formula 11)

[0059] In formula 11, the first term is the elementary beam deflection of the concrete, and the second and third terms are the additional deflection of the concrete shear lag;

[0060] ③ The analytical solution of the longitudinal stress and shear lag coefficient of the concrete simply supported box girder with the bridge deck slant is:

[0061] According to formula (4), the longitudinal stress of the cross section of the concrete box girder with the bridge deck slant is:

[0062] Equation 12

[0063] According to the definition of shear lag coefficient λ Shear lag coefficient of the concrete simply supported box girder with deck slab tilt is:

[0064] Equation 13.

[0065] The present application has the beneficial effects in that:

[0066] The present application proposes a multi-parameter shear lag displacement mode considering the difference of longitudinal displacement of upper and lower flanges on the basis of the conventional symmetric shear lag distribution mode, which can objectively and accurately reveal the spatial stress state of the concrete box girder bridge with deck slab tilt, and provide theoretical support for the refined design and parameter analysis of the box girder bridge, the proposal of construction measures, etc., and also provide technical support for the health monitoring of the bridge. BRIEF DESCRIPTION OF DRAWINGS

[0067] Figure 1 is the longitudinal stress distribution diagram of the shear lag of the concrete box girder bridge with deck slab tilt;

[0068] Figure 2 is the flowchart of the analysis method of the present application;

[0069] Figure 3 is the cross section and coordinate system of the concrete box girder with deck slab tilt;

[0070] Figure 4 is the cross section size (unit: mm) of the concrete box girder with deck slab tilt;

[0071] Figure 5 is the calculation flowchart of the shear lag effect of the concrete box girder with deck slab tilt;

[0072] Figure 6 is the finite element model diagram of the concrete box girder with deck slab tilt of the example;

[0073] Figure 7 is the longitudinal stress curve diagram of the mid-span cross section of the concrete box girder with deck slab tilt of the example. DETAILED DESCRIPTION

[0074] The present application will be further described below in combination with the drawings and specific embodiments:

[0075] ① Basic profile of the concrete box girder with deck slab tilt example:

[0076] The concrete simply supported box girder scale model with a calculation span of 800 mm is selected, and the cross section size and load application method are as follows: Figure 4As shown, the material adopts C25 concrete, the elastic modulus E=2.8GPa, the shear modulus G=1.077GPa, and the Poisson's ratio μ=0.37, and a vertical uniform load q=0.2N / mm is symmetrically applied at the top slab rib.

[0077] 2. Calculation results and analysis of the concrete box girder with inclined bridge deck:

[0078] According to the calculation method of the present application, the longitudinal stress of the mid-span section of the concrete box girder with inclined bridge deck is calculated under the modified warping displacement mode and the traditional warping displacement mode respectively, and the calculation flow is as shown in Figure 5 Abaqus-C3D8R unit is used to establish a finite element model of the concrete box girder, 25677 nodes and 19680 units are divided, the load is symmetrically applied on the nodes at the intersection of the top slab and the web, the longitudinal stress of the upper and lower flanges of the mid-span section is analyzed and extracted, and the model and deformation diagram are as shown in Figure 6 The finite element results and the calculation results of the present application are plotted into the longitudinal stress diagram of the upper and lower flanges of the mid-span section.

[0079] As can be seen from Figure 7 It can be seen that the modified warping displacement mode proposed by the present application can objectively and accurately reveal the shear lag effect of the concrete box girder with inclined bridge deck, that is, the longitudinal stress of the upper and lower flanges of the box girder shows great difference, and provides theoretical support for the refined design and parameterized analysis of the 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 for 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, 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 are the thickness of the top and bottom plates and the thickness of the web. Based on the principle of elastic superposition, the shear lag longitudinal displacement u(x, y, z) of the concrete box girder with inclined deck is: (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 beam deflection, ω i is the shear lag displacement distribution function of the concrete box girder with inclined bridge deck, where the subscript i is c and nc, representing the modified and traditional shear lag displacement modes, respectively; 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: (Formula 2) Where: α1, α2, β1, β2, φ, Δ1 and Δ2 are the warping displacement correction coefficients of the box girder 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, φ=1.4, 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 and Determine, A is the cross-sectional area of ​​the box girder; ③ Conventional distribution function of shear lag of concrete box girder with inclined deck ω nc (x,y) is: (Formula 3) Where: β1, β2, φ are the warping displacement correction coefficients of each concrete box beam plate, β1=(b2∙h b ) / (b1∙h u ), β2=b3 / b1, φ=1.4, η is the traditional warping self-balancing correction coefficient, through the self-balancing condition Determine, 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: (Formula 4) Where: σ0 is the primary normal stress of the concrete box girder section with inclined bridge deck, σ ωi M is the shear hysteresis warping normal stress of the concrete box girder with inclined deck; x is the bending moment of the box beam section, I x is the moment of inertia of the box beam section, that is ;M ωi is the shear lag moment of the box girder section, , E is the elastic modulus of concrete, f '' is the second-order derivative of the shear lag additional deflection, I ωi is the shear hysteresis warping moment of inertia, that is ; (2) Shear lag control differential equations and boundary conditions for concrete box girders with inclined decks ① The total potential energy Π of shear lag of concrete box girder with inclined deck is: (Formula 5) Where: G is the shear modulus of concrete; ε and γ are the longitudinal strain and shear strain of the box beam; q is any vertical symmetrical load; w and f are the primary beam deflection and shear lag additional deflection of the concrete box beam, respectively. ωi is the shear hysteresis warping area, that is, ; ② The differential equation governing the shear lag of a concrete box girder with an inclined deck is: According to the principle of stationary potential energy, the total potential energy of the concrete box girder (5) is subjected to first-order variation operation, and the differential equation is obtained by setting δΠ: (Formula 6) (Formula 7) Where: is the fourth-order derivative of the deflection of the elementary concrete beam, is the fourth-order derivative of the additional deflection of concrete shear lag; By rearranging formula (7), we can obtain the governing differential equation of the concrete box girder with shear lag additional deflection as generalized displacement: (Formula 8) Where: k i is the shear lag Reissner parameter of the concrete box girder with inclined bridge deck, which is expressed as ; ③ The shear lag boundary condition of the concrete box girder with inclined deck is: Fixed end: , ; Simply supported end: , ; Free end: , ; (3) Solution for shear lag of simply supported concrete box girder with inclined deck: ① Analytical solution for the shear lag additional deflection of a simply supported concrete box girder with an inclined deck: According to the additional deflection differential equation (8), the general solution of the differential equation is: (Formula 9) 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 deck is: (Formula 10) 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 deck is: The vertical deflection of the inclined concrete box girder of the bridge deck considering the shear lag effect is: (Formula 11) In formula (11), the first term is the primary concrete beam deflection, and the second and third terms are the additional deflections due to concrete shear lag. ③ The analytical solution for the longitudinal stress and shear lag coefficient of the simply supported concrete box girder with an inclined deck is: According to formula (4), the longitudinal stress of the concrete box girder section with inclined bridge deck is: (Formula 12) According to the definition of shear lag coefficient λ , the shear lag coefficient of the simply supported concrete box girder with inclined deck is: (Formula 13).

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