In-service concrete box girder bridge bending dynamic response calculation method considering influence of material degradation

Abstract

The invention discloses an in-service concrete box girder bridge bending dynamic response calculation method considering the influence of material degradation, which comprises the following steps: step 1, determining degradation indexes of two materials, namely reinforcing steel bars and concrete; 2, according to the concept of cross section conversion, the area of the degraded steel bars is converted into the cross section of concrete; 3, constructing a box girder shear lag shear deformation dynamic displacement mode considering material degradation; 4, a concrete box girder dynamic response control differential equation considering material degradation is established; and 5, solving the dynamic response of the concrete box girder considering material degradation. Analytic solutions of shear lag additional dynamic deflection, total section torque and longitudinal dynamic stress can be derived according to a dynamic deflection control differential equation, and the calculation process is visualized by utilizing programming software, so that engineering application is facilitated.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of box girder stress analysis, and particularly relates to a method for calculating the bending dynamic response of a concrete box girder bridge in service considering the influence of material degradation. BACKGROUND

[0002] The shear lag effect is a spatial mechanical behavior that must be considered in the design of a box girder bridge, which can cause the increase of cross-section deflection and stress redistribution. In the design practice of a box girder bridge, the internal force of the cross-section is expanded to 1.15 times in order to fully consider the influence of spatial effects such as bending, torsion and distortion of the box girder. The bending dynamic response is a main parameter index for representing the dynamic characteristics of a box girder bridge. By exploring the influence law of the shear lag shear deformation on the bending dynamic response of the box girder, a scientific basis can be provided for the reasonable parameterized design of the structural details of the box girder bridge. The three-dimensional finite element software can analyze the bending spatial dynamic response of the box girder bridge, but can only output the cross-section stress and deformation caused by the spatial effect, which cannot correspond to the design method based on the internal force in the current bridge design specification, and is not convenient for designers to understand and use.

[0003] At present, the research on the bending dynamic response of a box girder based on the Hamilton principle has achieved rich research results, but there is still a great lack of analysis on the bending dynamic response of a box girder bridge considering the influence of material degradation. The strength, elastic modulus and other mechanical indexes of the concrete and steel bars of a bridge structure located in a corrosive medium decrease year by year, which has a great influence on the external beauty, driving smoothness and safe service performance of the box girder bridge. SUMMARY

[0004] The application proposes a method for calculating the bending dynamic response of a concrete box girder bridge in service considering the influence of material degradation based on the Hamilton principle, which provides technical support for the dynamic parameter evaluation and structural health monitoring of a concrete box girder bridge in a corrosive environment.

[0005] To this end, the application adopts the following technical solutions:

[0006] Step 1: degradation indexes of the steel bars and the concrete

[0007] The time evolution law and degradation formula of the mechanical indexes such as the tensile strength and elastic modulus of the steel bars and the concrete are adopted by using the existing research, and the correctness of the formula is verified by a plurality of test beams.

[0008] Step 2: conversion of the area of the degraded steel bars into the cross-sectional area of the concrete

[0009] The conversion cross-section is one of the key steps for analyzing the bending dynamic response of a degraded concrete box girder bridge, which equivalently converts the degraded steel bars into the equivalent cross-sectional area of the concrete, and further realizes the conversion of the homogeneous cross-section, thereby laying a foundation for the accurate construction of the subsequent dynamic displacement mode.

[0010] Step 3: Constructing box girder shear-lag shear deformation dynamic displacement mode considering material degradation

[0011] According to Reissner theory and elastic thin-walled theory, the longitudinal dynamic displacement mode of the box girder bridge considering shear-lag shear deformation is established by using displacement continuity condition and finite element method, and the calculation formula of the longitudinal dynamic stress of the section is given.

[0012] Step 4: Dynamic response control differential equation of concrete box girder considering material degradation

[0013] Based on Hamilton principle and minimum potential energy stationary value principle, the dynamic response control differential equation of the degraded box girder bridge considering the influence of shear-lag shear deformation is established, and the differential equation of the vertical dynamic deflection is derived.

[0014] Step 5: Dynamic response solution of concrete box girder considering material degradation

[0015] According to the dynamic deflection control differential equation, the analytical solution of the shear-lag additional dynamic deflection, the total moment of the section and the longitudinal dynamic stress can be derived, and the above calculation process can be visualized by using programming software, which is convenient for engineering application.

[0016] The beneficial effects of the present application are that the present application proposes a concrete box girder bridge bending dynamic response calculation method considering the influence of material degradation, which is convenient for designers and construction personnel to use; on the other hand, from the theoretical point of view, the calculation method proposed by the present application corresponds to the method for calculating the stress, deformation and internal force of the box girder section in classical material mechanics, which is convenient for engineering application, and this will provide theoretical support for the health monitoring and real-time dynamic feedback of the in-service box girder bridge. BRIEF DESCRIPTION OF DRAWINGS

[0017] Figure 1 is the algorithm flowchart of the present application;

[0018] Figure 2 is the original section drawing of the reinforced concrete of the present application;

[0019] Figure 3 is the equivalent section drawing of the reinforced concrete of the present application;

[0020] Figure 4 is the bending dynamic response calculation flowchart of the degraded box girder of the embodiment of the present application;

[0021] Figure 5 is the longitudinal stress amplitude value law of the top plate rib of the mid-span section of the concrete box girder under different degradation conditions of the present application;

[0022] Figure 6 is the dynamic deflection amplitude value law of the mid-span section of the concrete box girder under different degradation conditions of the present application. DETAILED DESCRIPTION

[0023] The application will be further described below in connection with the accompanying drawings and specific embodiments:

[0024] Step 1: deterioration index of two materials of steel bar and concrete:

[0025] According to a large number of concrete beam corrosion tests, the degradation model of the related mechanical indexes of concrete and steel bars can be obtained as follows:

[0026] ① Calculation formula of concrete strength and elastic modulus degradation:

[0027]

[0028] In the formula: f c (t) is the strength of the deteriorated concrete; f cd is the design value of the concrete strength (according to the specification JTG3362-2018) ; t is the same as below.

[0029]

[0030] In the formula: E c (t) is the elastic modulus of the deteriorated concrete; E c0 is the original elastic modulus of the concrete (selected according to the specification JTG3362-2018).

[0031] ② Calculation formula of steel bar tensile strength and elastic modulus degradation:

[0032] f s (t) = f sd (1-κ s λ s (t)) (Formula 3)

[0033] A s (t) = A s0 (1-λ s (t)) (Formula 4)

[0034] In the formula: f s (t) is the tensile strength of the deteriorated steel bar; f sd is the design value of the tensile strength of the steel bar before deterioration (selected according to the specification JTG3362-2018) ; A s (t) is the cross-sectional area of the steel bar after corrosion, A s0 is the cross-sectional area of the steel bar before corrosion; κ s is the design strength reduction coefficient of the steel bar, κ s = 1.2-1.5 when the steel bar is corroded by external current in the laboratory, κ s = 1.4-1.5 when the steel bar is naturally exposed to atmospheric environment and corroded, and κ s = 1.0-1.7 when the steel bar is corroded in actual engineering; λ s(t) is the corrosion rate of steel bars.

[0035] E s (t) = E s0 (5)

[0036] Wherein: E s (t) is the elastic modulus of deteriorated steel bars; E s0 is the original elastic modulus of steel bars (selected according to the specification JTG3362-2018); δ e (t) is the corrosion depth of steel bars; d is the diameter of steel bars.

[0037] Step 2: According to the concept of converted section, the area of deteriorated steel bars is converted into the cross-sectional area of concrete.

[0038] According to the definition of converted section, the cross-sectional area of longitudinal steel bars in the upper and lower flange plates of the deteriorated box girder is converted into the equivalent concrete area. The original section of the box girder is shown in Figure 2 , and the converted section is shown in Figure 3 , and the specific idea is as follows:

[0039] ① The converted section of deteriorated steel bars in the upper flange plate of the box girder into concrete is: c α Es A s ′ (6)

[0040] ② The converted section of deteriorated steel bars in the lower flange plate of the box girder into concrete is: c α Es A s (7)

[0041] Wherein: A c ′ and A c are the converted concrete cross-sectional areas; A s ′ and A s are the cross-sectional areas of longitudinal steel bars in the upper and lower flange plates of the box girder; α Es (t) is the elastic modulus ratio of deteriorated steel bars and concrete, that is, α Es (t) = E s0 / E c (t).

[0042] Reference Figure 2 is the original cross-section of reinforced concrete, Figure 3 is the converted cross-section of reinforced concrete.

[0043] Step 3: Considering the material deterioration of the box girder, the shear lag shear deformation dynamic displacement mode is constructed:

[0044] ① The longitudinal dynamic displacement u(x, y, z) of the box girder considering the shear lag shear effect is:

[0045] u(x,y,z)=u0(x,y,z)+u ω (x,y,z)=-yθ c (z)-ξ(x,y)f c ′(z)(Formula 8)

[0046] In the formula: f c To account for the additional deflection due to shear lag caused by material degradation, f c 'For additional deflection angle; θ c The section rotation angle is taken into account for material degradation and Timoshenko effects; ξ(x,y) is the shear hysteresis distribution function.

[0047] ②For example Figure 3 In the Cartesian coordinate system shown, o is the origin, x is the horizontal axis, y is the vertical axis, and z is the vertical axis. Choosing the cosine function as the fundamental function, the shear lag distribution function ξ(x,y) is:

[0048]

[0049] Where: h u h b β1, β2 and β3 are the distances from the mid-surface of the top and bottom plates to the centroidal axis (i.e., the x-axis); 2b1, 2b2, and b3 are the widths of the top and bottom plates and the cantilever plate, respectively; ...β1, β2 and β3 are the distances from the mid-surface of the top and bottom plates to the centroidal axis (i.e., the x-axis); β1, β2 and β3 are the distances For the correction coefficient of the warping displacement function corresponding to each plate, β1=(b2·h b ) / (b1·h u ), β2=b3 / b1, η is the shear warp self-balancing coefficient, and its expression is:

[0050]

[0051] In the formula: I1 is the moment of inertia of the top plate of the box girder about the x-axis, I2 is the moment of inertia of the bottom plate about the x-axis, I3 is the moment of inertia of the cantilever plate about the x-axis, I x Let be the moment of inertia of the entire cross-section about the x-axis.

[0052] ③ Considering the longitudinal dynamic stress amplitude σ of the deteriorated box girder under shear lag and shear deformation c (x,y,z) is:

[0053]

[0054] In the formula: σ 0c To account for the amplitude of the elementary beam normal stress in the box girder section under material degradation, σ ωc To account for the shear warping normal stress amplitude of the box girder due to material degradation; M xc and I x M represents the bending moment amplitude and moment of inertia of the box girder section about the x-axis; ωc and I ωThe amplitude of the shear lag moment and the warping inertia moment of the box girder section, M ωc =∫ A σ ωc ξdA=-E c I ω f c ″,I ω =∫ A ξ 2 dA.

[0055] Step 4: Dynamic response control differential equation of the concrete box girder considering material deterioration

[0056] ① Total strain energy U of the deteriorated box girder considering shear lag and shear deformation c is

[0057]

[0058] In the formula: G c is the shear modulus of the deteriorated concrete; I yω is the section inertia, I yω =∫ A yξdA; w c ' is the flexural rotation angle of the deteriorated concrete box girder; A ω is the shear-warping area, A ω =∫ A ω′ 2 dA; A w is the web section area.

[0059] ② Total potential energy V of the deteriorated box girder considering shear lag and shear deformation c is

[0060]

[0061] In the formula: q is an arbitrary vertical symmetric harmonic load.

[0062] ③ Total kinetic energy T of the deteriorated box girder considering shear lag and shear deformation c is

[0063]

[0064] In the formula: is the self-weight per meter of the box girder, calculated according to the cross section of the box girder and the material density; "·" is the derivative of the variable with respect to time t, and "'" is the derivative of the variable with respect to the longitudinal axis z.

[0065] ④ Dynamic response control differential equation of the deteriorated box girder considering shear lag and shear deformation is

[0066]

[0067] E c Ix θ c "+E c I yω f c "′+G c A w (w c ′-θ c )=0(Formula 17)

[0068] ⑤Simplifying the above three equations, we obtain the differential equation for the vertical deflection w as follows:

[0069]

[0070] In the formula: ω is the circular frequency of the box girder vibration;

[0071] Step 5: Solving the dynamic response of the concrete box girder considering material degradation

[0072] ① According to the differential equation (18), the amplitude of dynamic deflection w of the deteriorated box girder considering shear deformation can be obtained. cs (z) is:

[0073]

[0074] In the formula: C1~C8 are undetermined coefficients, determined by the boundary conditions; λ1~λ4 are the roots of the characteristic equation, determined by the characteristic equation corresponding to the differential equation (18).

[0075] ② Considering the amplitude of dynamic rotation θ of the deteriorated box girder under shear deformation c (z) is:

[0076] θ c (z)=B1(C1cos(λ1z)-C2sin(λ1z))+B2(C3cos(λ2z)-C4sin(λ2z))+B3(C5cosh(λ3z)+C6·

[0077] sinh(λ3z))+B4(C7cosh(λ4z)+C8sinh(λ4z))

[0078] (Formula 20)

[0079] In the formula: B1 to B4 are undetermined coefficients, determined by the principle of identity, i.e. Where i = 1 or 2, j = 3 or 4.

[0080] ③ Considering the shear lag and additional dynamic deflection amplitude f of the box girder due to material degradation c (z) is:

[0081]

[0082] wherein: D1-D4 are undetermined coefficients, which are determined by the identity principle, i.e. i = 1 or 2; j = 3 or 4.

[0083] (4) The total deflection amplitude w of the deteriorated box girder considering shear lag and shear deformation c (z) is:

[0084]

[0085] wherein: the undetermined coefficients C1-C8, D1-D4 are the same as above.

[0086] (5) The longitudinal dynamic stress amplitude σ of the deteriorated box girder considering shear lag and shear deformation c (z) is:

[0087]

[0088] According to the above formula, the dynamic deflection amplitude, dynamic stress amplitude and total moment amplitude of the box girder under different vibration frequencies ω can be calculated, and the process can be visualized by using visual programming software, so as to realize real-time monitoring of the stress and deformation state of the service box girder bridge.

[0089] Taking an actual project as an example, the influence of material deterioration of the concrete box girder on the bending dynamic response is analyzed according to the calculation method of the present scheme, and the calculation and analysis process is as shown in Figure 4 A C50 concrete simply supported box girder with a span of 40 m is selected, and the scale is reduced to 1:0.25. The cross-sectional parameters of the reduced scale are: span l = 10 m, top and bottom plate half width b1 = b2 = 1 m, cantilever plate width b3 = 0.5 m, top plate, bottom plate and cantilever plate thickness t1 = t2 = t3 = 0.08 m, web thickness t w = 0.1 m, beam height h = 0.8 m, concrete mass density ρ = 2500 kg / m 3 , concrete elastic modulus E c = 3.45 × 10 10 Pa, steel elastic modulus E s = 2 × 10 11 Pa, steel reinforcement cross-sectional area A s ' of the upper wing plate of the box girder = 2280 mm 2 , steel reinforcement cross-sectional area A s of the lower wing plate = 5320 mm 2 , under the action of a simple harmonic uniform load, the load intensity q = 25 kN / m; the vibration frequency ω increases from 3 Hz to 15 Hz with a step of 3 Hz; the material deterioration is calculated according to the deterioration of 0 years, 40 years and 80 years, and the corresponding steel reinforcement corrosion rate λ sFor 0%, 5% and 25%, the amplitude of the dynamic deflection of the cross section and the amplitude of the longitudinal stress at the intersection of the roof and the web at each vibration frequency are calculated, and the law is drawn as shown in Figures 5-6 Fig. 4.

Claims

1.A method for calculating the bending dynamic response of a concrete box girder bridge in service considering the influence of material deterioration, characterized in that, The method comprises the following steps: Step 1: deterioration index of two materials of steel bar and concrete ① Calculation formula of concrete strength and elastic modulus degradation: In the formula: f c (t) is the strength of the deteriorated concrete; f cd is the design value of the concrete strength, which is taken according to relevant standards or specification documents; t is the unit of years, which is the same below; In the formula: E c (t) is the elastic modulus of the deteriorated concrete; E c0 is the original elastic modulus of the concrete, which is taken from the relevant standard or specification document; ② Calculation formula of steel bar tensile strength and elastic modulus degradation: f s (t) = f sd (1 - k s λ s (t)) (Equation 3) A s (t) = A s0 (1 - λ s (t))(Equation 4) In the formula: f s (t) is the tensile strength of the deteriorated steel bar; f sd is the design value of the tensile strength of the steel bar before deterioration, which is taken from the relevant standard or specification document; A s (t) is the cross-sectional area of the steel bar after corrosion, A s0 is the cross-sectional area of the steel bar before corrosion; κ s is the design strength reduction coefficient of the steel bar, λ s (t) is the corrosion rate of the steel bar; E s (t) = E s0 (Equation 5) In the formula: E s (t) is the elastic modulus of the deteriorated steel bar; E s0 is the original elastic modulus of the steel bar, which is taken from the relevant standard or specification document; Step 2: according to the concept of conversion section, the deteriorated steel bar area is converted into the concrete section area According to the definition of conversion section, the longitudinal steel bar section area in the deteriorated box girder upper and lower flange plates is converted into the equivalent concrete area, and the specific method is as follows: ① The conversion section of the deteriorated steel bar in the box girder upper flange surface into the concrete section is: A c ′= a Es A s ′(Equation 6) ② The conversion section of the deteriorated steel bar in the box girder lower flange surface into the concrete section is: A c = a Es A s ( Equation 7 ) wherein: A c and A c is the converted concrete cross-sectional area; A s and A s is the longitudinal reinforcement area in the upper and lower flange surfaces of the box girder; a Es (t) is the elastic modulus ratio of the deteriorated reinforcement and concrete, i.e. a Es (t) = E s0 / E c (t); Step 3: construction of box girder shear lag shear deformation dynamic displacement mode considering material deterioration ① The longitudinal dynamic displacement u(x, y, z) of the box girder considering the shear lag shear effect is: u(x, y, z) = u0(x, y, z) + u ω (x, y, z) = -yθ c (z) - ξ(x, y)f c '(z) (Equation 8) where: f c is the first derivative of the shear-lag additional deflection considering material degradation; θ c is the cross-section rotation considering material degradation and Timoshenko effect; ξ(x, y) is the shear-lag distribution function; u0(x, y, z) is the elementary beam longitudinal displacement; u ω (x, y, z) is the shear-lag warping longitudinal displacement; ② A Cartesian coordinate system is established in the box girder, o is the coordinate origin, x is the horizontal direction, y is the vertical direction, and z is the longitudinal direction, and the cosine function is selected as the basic function, then the shear lag distribution function ξ(x, y) is: wherein: h u , h b is the distance from the top, bottom plate midplane to the centroid axis (i.e. x-axis); 2b1, 2b2, b3 are the top, bottom plate width and cantilever plate width; β1, β2 and β3 are the warping displacement function correction coefficients for the respective plates, β1 = (b2·h b ) / (b1·h u ), β2 = b3 / b1, η is the shear warping self-balancing coefficient, whose expression is: wherein: Ii is the moment of inertia of the top slab of the box girder about the x-axis, I2 is the moment of inertia of the bottom slab about the x-axis, I3 is the moment of inertia of the cantilever slab about the x-axis, I x is the moment of inertia of the entire cross section about the x-axis; iii. The longitudinal dynamic stress amplitude σ of the deteriorated box girder considering shear hysteresis shear deformation c (x,y,z) is: where: σ 0c is the elementary beam normal stress amplitude of the box girder cross section considering material deterioration, σ ωc is the box girder shear warping normal stress amplitude considering material deterioration; M xc and I x are the moment amplitude and the moment of inertia of the box girder cross section around the x axis; M ωc and I ω are the shear lag moment amplitude and the warping moment of inertia of the box girder cross section, M ωc =∫ A σ ωc ξdA=-E c I ω f c ″,I ω =∫ A ξ 2 dA; ξ represents the shear lag distribution function ξ(x,y). Step 4: dynamic response control differential equation of concrete box girder considering material deterioration The total strain energy U of the deteriorated box girder considering shear lag and shear deformation c For where: G c is the shear modulus of deteriorated concrete; I yω is the cross-sectional moment of inertia, I yω =∫ A yξdA; w c is the deflection angle of deteriorated concrete box girder; A ω is the shear area, A ω =∫ A ω′ 2 dA; A w is the web area (2) The total potential energy V of the deteriorated box girder considering shear hysteresis shear deformation c For where q is an arbitrary vertically symmetric harmonic load; w c is the vertical deflection of the deteriorated box girder considering the effect of shear deformation, θ c is the cross-section rotation considering material deterioration and Timoshenko effect, θ c ′ is the first derivative of the cross-section rotation considering material deterioration and Timoshenko effect; iii. Total kinetic energy of the deteriorated box girder considering shear-lag shear deformation T c For where: is the self-weight of the box girder per meter, calculated according to the cross section of the box girder and the material density; is the derivative of the variable with respect to time t, and is the derivative of the variable with respect to the longitudinal axis z; ④ The dynamic response control differential equation of the deteriorated box girder considering the shear lag shear deformation is E c I x θ c ″ + E c I yω f c ″′ + G c A w (w c ′ - θ c ) = 0 (Equation 17) ⑤ The differential equation about the vertical deflection w is obtained by arranging the above three formulae 15-17 wherein ω is the circular frequency of the box girder vibration; Step 5: solution of dynamic response of concrete box girder considering material deterioration The dynamic deflection amplitude w of the deteriorated box girder considering shear deformation can be obtained according to the differential equation (18) cs (z) is: In the formula: C1-C8 are undetermined coefficients, which are determined by the boundary conditions; λ1-λ4 are the roots of the characteristic equation, which are determined by the characteristic equation corresponding to the differential equation formula (18); (ii) the amplitude of the dynamic rotation angle Θ of the deteriorated box girder considering shear deformation c (z) is: θ c (z) = B1(C1 cos(λ1z) - C2 sin(λ1z)) + B2(C3 cos(λ2z) - C4 sin(λ2z)) + B3(C5 cosh(λ3z) + C6 sinh(λ3z)) + B4(C7 cosh(λ4z) + C8 sinh(λ4z)) (Equation 20) wherein: B1-B4 are undetermined coefficients determined by the identity principle, i.e. wherein i = 1 or 2, and j = 3 or 4. iii. Consideration of material deterioration of box girder shear lag additional dynamic deflection amplitude f c (z) is: wherein: D1-D4 are undetermined coefficients determined by the identity principle, i.e. i = 1 or 2; j = 3 or 4; (iv) the total deflection amplitude w of the deteriorated box girder considering shear hysteresis shear deformation c (z) is: In the formula: the undetermined coefficients C1-C8, D1-D4 are the same as above; (5) Consider the shear lag shear deformation of the deteriorated box girder longitudinal dynamic stress amplitude σ c (z) is: According to the above formula, the dynamic deflection amplitude, dynamic stress amplitude and total moment amplitude of the box girder under different vibration frequencies ω can be calculated, and the visual programming software is used to make the process, so as to realize the real-time monitoring of the stress and deformation state of the service box girder bridge.

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