Method for rapidly calculating interface slippage of steel-concrete composite beam under action of vehicle dynamic load

Abstract

The invention discloses a rapid calculation method for interface slippage of a steel-concrete composite beam under the action of a vehicle dynamic load, and relates to the technical field of bridge construction. The method sequentially comprises the following steps: calculating the first n-order natural vibration frequency of the simply supported steel-concrete composite beam according to basic parameters of a bridge; calculating the dynamic load circular frequency of the vehicle; calculating bridge natural vibration frequency and critical damping frequency considering damping influence; calculating introduction parameters; calculating interface slippage under the vehicle dynamic load action considering the first n orders of vibration modes; judging convergence; and calculating the interface slippage under the dynamic load action of the plurality of vehicles. According to the method, a calculation formula of the interface slip of the simply supported steel-concrete composite beam under the dynamic load action of the vehicle is provided by considering the rigidity of the material, the section and the connecting piece on the basis of a theoretical model of the Timoshinesson beam, and rapid and accurate calculation of the interface slip can be realized under the condition of not depending on numerical simulation and field test.

Description

Technical Field

[0001] The invention relates to the technical field of bridge construction, and in particular to a method for quickly calculating the interface slip of a steel-concrete composite beam under the action of a vehicle dynamic load. Background Art

[0002] Steel has high strength and good toughness and can withstand large tensile forces, while concrete has high compressive strength and good durability. The steel-concrete composite beam is a structure composed of a steel beam and a concrete slab through shear connectors. The shear connectors enable the steel beam and concrete slab to work together and effectively utilize the characteristics of the two materials when bearing loads. The steel-concrete composite beam combines the advantages of both steel and concrete to improve mechanical properties.

[0003] At present, the calculation of relative slip of steel-concrete composite beams is mainly based on numerical simulation and experimental methods. Among them: numerical simulation mainly uses finite element software such as ABAQUS to establish simulation models, steel beams and concrete slabs are simulated with solid units, connectors are simulated with beams or springs, and slip distribution is obtained by applying loads. However, the numerical simulation method takes a lot of time, and the accuracy and precision of modeling depend on the professional level of the modelers. Therefore, it is difficult to promote it on a large scale for engineering applications; the experimental method mainly focuses on making scaled or full-scale models. The former selects materials and arranges connectors according to similarity ratios, loads with hydraulic jacks, and measures slip with displacement sensors. The latter sets boundaries and loads according to reality. Both require data analysis to provide a basis for interface slip prediction. The experimental method can accurately obtain the interface slip of steel-concrete composite beams, but it is costly, has high requirements for sites and equipment, is difficult to load and measure, and is also time-consuming, which is not conducive to quickly obtaining results. Summary of the invention

[0004] In order to overcome the shortcomings of the background technology, the present invention provides a method for quickly calculating the interface slip of a steel-concrete composite beam under vehicle dynamic load. Based on the Timoshenko beam theoretical model and taking into account the material, cross-section and connector stiffness, a calculation formula for the interface slip of a simply supported steel-concrete composite beam under vehicle dynamic load is proposed. The method can realize fast and accurate calculation of the interface slip without relying on numerical simulation and field tests.

[0005] To achieve the above object, the present invention adopts the following technical solution: a method for quickly calculating the interface slip of a steel-concrete composite beam under vehicle dynamic load, comprising the following steps:

[0006] Step 1: Calculate the first n-order natural frequencies of the simply supported steel-concrete composite beam according to the basic parameters of the bridge

[0007] Definition i is the natural frequency of the i-th vibration mode, i=1,2,...,n-1,n, then the first n-order natural frequency of the simply supported steel-concrete composite beam is expressed as follows:

[0008] W=[ω 1 ,ω 2 ,…,ω n-1 ,ω n ]

[0009] in,

[0010]

[0011] EI F =EI B +EI c

[0012] Where, L is the calculated span of the bridge, EI c is the sum of the bending stiffness of the concrete slab and the steel beam around the neutral axis of the composite section, EI B is the sum of the bending stiffness of the concrete slab and the steel beam around their own cross-section neutral axis, EI F is the bending stiffness of the composite beam without considering the stiffness of the connector, m is the mass of the composite beam per unit length, h is the distance between the neutral axis of the steel beam and the neutral axis of the concrete slab, K d is the shear stiffness of the shear stud per unit length, α and β are introduced parameters;

[0013] Step 2: Calculate the vehicle dynamic load circular frequency

[0014] The calculation formula of vehicle dynamic load circular frequency is as follows:

[0015]

[0016] Where V is the vehicle moving speed;

[0017] Step 3: Calculate the natural frequency and critical damping frequency of the bridge considering the influence of damping

[0018] The calculation formula of the bridge natural frequency considering the damping effect is as follows:

[0019] W D =[ω D1 ,ω D2 ,…,ω Dn-1 ,ω Dn ]

[0020] The calculation formula of the critical damping frequency of the bridge is as follows:

[0021] W b =[ω b1 ,ω b2 ,…,ω bn-1 ,ω bn ]

[0022] in,

[0023] ω bi =ω i ξ

[0024] In the formula, ω Di is the natural frequency of the i-th order considering the damping effect, ω bi is the i-th order critical damping frequency, ξ is the structural damping ratio;

[0025] Step 4: Calculate the introduced parameters

[0026] Introducing parameter G s , G c , K s , K c and EA, and calculated as follows:

[0027]

[0028] in,

[0029]

[0030] In the formula, (GA) s =k s G s A s and (GA) c =k c G c A c is the introduced parameter, k s and k c are the cross-sectional shear shape coefficients of the steel main beam and the concrete slab, G s and G c are the shear moduli of the steel main beam and the concrete slab, A s and A c The cross-sectional area of ​​the steel main beam and the concrete slab, h s and h c They represent the distances from the neutral axis of the steel main beam and the concrete slab to the steel-concrete interface, E s and E c are the elastic modulus of the steel main beam and the concrete slab, I s and I c are the principal moments of inertia of the steel main beam and the concrete slab respectively;

[0031] Step 5: Calculate the interface slip under the vehicle dynamic load considering the first n vibration modes

[0032] Interface slip u under vehicle dynamic load considering the first n vibration modes n (x, t) is obtained by the following formula:

[0033]

[0034] in,

[0035]

[0036] A i =(iω y +ω Di )siniω y t+ω bi cosiω y t

[0037] B i =(iω y +ω Di )sinω Di t-ω bi cosω Di t

[0038] C i =(iω y -ω Di )siniω y t+ω bi cosiω y t

[0039] D i =(iω y -ω Di )sinω Di t+ω bi cosω Di t

[0040] In the formula, C i (t) is the coefficient to be determined, x is the vertical coordinate of the bridge, and t is the time of vehicle dynamic load action;

[0041] Step 6: Determine convergence

[0042] Repeat steps 1 to 5 to calculate the interface slip u under the vehicle dynamic load considering the first n+1 vibration modes n+1 (x, t), the bridge ordinate x and the vehicle dynamic load action time t are scattered into x j and t k , j = 1, 2, ..., g, k = 1, 2, ..., l, we get:

[0043] X=(x 1 ,x 2 ,…x g )

[0044] T=(t 1 ,t 2 ,…t l )

[0045] The relative error RE is calculated as follows:

[0046]

[0047] If the relative error RE is less than the error standard ε, it is considered to be converged, and the calculated u n+1 (x, t) is available, proceed to the next step; if the relative error RE is greater than or equal to the error standard ε, increase the vibration mode order n, repeat steps 1 to 4, and calculate u n+1 (x,t) and u n+2 (x, t), until it meets the convergence condition, and then proceed to the next step;

[0048] Step 7: Calculate the interface slip under multiple vehicle dynamic loads

[0049] Introduce multiple vehicle dynamic loads as needed, calculate according to steps 1 to 6, and then superimpose to obtain the interface slip of the simply supported steel-concrete composite beam under the action of multiple vehicle dynamic loads.

[0050] Compared with the prior art, the present invention has the following beneficial effects:

[0051] 1. The interface slip calculation of the present invention is more streamlined. Compared with the traditional complex numerical simulation method, it can greatly reduce the calculation time and quickly obtain the approximate result of the interface slip of the simply supported steel-concrete composite beam without spending too much energy on tedious calculations, thereby improving work efficiency;

[0052] 2. The method of the present invention closely meets the needs of the engineering site and is applicable to simply supported steel-concrete composite beam structures of various specifications. It can obtain interface slip data with acceptable accuracy for the engineering, and provide a powerful decision-making basis for engineering design, construction and operation and maintenance;

[0053] 3. The principle of the method of the present invention is concise and easy to understand, the operation process is simple, and the theoretical level requirements for professional and technical personnel are relatively low. Engineers and technicians in related fields can quickly understand and master it, and it is easy to be widely disseminated and applied in the industry. BRIEF DESCRIPTION OF THE DRAWINGS

[0054] Figure 1 It is a flowchart of the method of the present invention;

[0055] Figure 2 The interface slip under the vehicle dynamic load of the first 10 vibration modes is considered in the embodiment;

[0056] Figure 3 The interface slip under the vehicle dynamic load of the first 11 vibration modes is considered in the embodiment. DETAILED DESCRIPTION

[0057] The technical solution of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the invention, rather than all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative work are within the scope of protection of the present invention.

[0058] like Figure 1 As shown, a fast calculation method for the interface slip of a steel-concrete composite beam under vehicle dynamic load includes the following steps:

[0059] Step 1: Calculate the first n-order natural frequencies of the simply supported steel-concrete composite beam according to the basic parameters of the bridge

[0060] Definition i is the natural frequency of the i-th vibration mode, i=1,2,...,n-1,n, then the first n-order natural frequency of the simply supported steel-concrete composite beam is expressed as follows:

[0061] W=[ω 1 ,ω 2 ,…,ω n-1 ,ω n ]

[0062] in,

[0063]

[0064] EI F =EI B +EI c

[0065] Where, L is the calculated span of the bridge, EI c is the sum of the bending stiffness of the concrete slab and the steel beam around the neutral axis of the composite section, EI B is the sum of the bending stiffness of the concrete slab and the steel beam around their own cross-section neutral axis, EI F is the bending stiffness of the composite beam without considering the stiffness of the connector, m is the mass of the composite beam per unit length, h is the distance between the neutral axis of the steel beam and the neutral axis of the concrete slab, K d is the shear stiffness of the shear stud per unit length, α and β are introduced parameters.

[0066] Step 2: Calculate the vehicle dynamic load circular frequency

[0067] The calculation formula of vehicle dynamic load circular frequency is as follows:

[0068]

[0069] Where V is the vehicle moving speed.

[0070] Step 3: Calculate the natural frequency and critical damping frequency of the bridge considering the influence of damping

[0071] The calculation formula of the bridge natural frequency considering the damping effect is as follows:

[0072] W D =[ω D1 ,ω D2 ,…,ω Dn-1 ,ω Dn ]

[0073] The calculation formula of the critical damping frequency of the bridge is as follows:

[0074] W b =[ω b1 ,ω b2 ,…,ω bn-1 ,ω bn ]

[0075] in,

[0076] ω bi =ω i ξ

[0077] In the formula, ω Di is the natural frequency of the i-th order considering the damping effect, ω bi is the i-th order critical damping frequency, and ξ is the structural damping ratio.

[0078] Step 4: Calculate the introduced parameters

[0079] Introducing parameter G s , G c , K s , K c and EA, and calculated as follows:

[0080]

[0081] in,

[0082]

[0083] In the formula, (GA) s =k s G s A s and (GA) c =k c G c A c is the introduced parameter, k s and k c are the cross-sectional shear shape coefficients of the steel main beam and the concrete slab, G s and G care the shear moduli of the steel main beam and the concrete slab, A s and A c The cross-sectional area of ​​the steel main beam and the concrete slab, h s and h c They represent the distances from the neutral axis of the steel main beam and the concrete slab to the steel-concrete interface, E s and E c are the elastic modulus of the steel main beam and the concrete slab, I s and I c are the principal moments of inertia of the steel main beam and the concrete slab, respectively.

[0084] Step 5: Calculate the interface slip under the vehicle dynamic load considering the first n vibration modes

[0085] Interface slip u under vehicle dynamic load considering the first n vibration modes n (x, t) is obtained by the following formula:

[0086]

[0087] in,

[0088]

[0089] A i =(iω y +ω Di )siniω y t+ω bi cosiω y t

[0090] B i =(iω y +ω Di )sinω Di t-ω bi cosω Di t

[0091] C i =(iω y -ω Di )siniω y t+ω bi cosiω y t

[0092] D i =(iω y -ω Di )sinω Di t+ω bi cosω Di t

[0093] In the formula, C i(t) is the unknown coefficient, x is the vertical coordinate of the bridge, and t is the time of vehicle dynamic load action.

[0094] Step 6: Determine convergence

[0095] Repeat steps 1 to 5 to calculate the interface slip u under the vehicle dynamic load considering the first n+1 vibration modes n+1 (x, t), the bridge ordinate x and the vehicle dynamic load action time t are scattered into x j and t k , j = 1, 2, ..., g, k = 1, 2, ..., l, we get:

[0096] X=(x 1 ,x 2 ,…x g )

[0097] T=(t 1 ,t 2 ,…t l )

[0098] The relative error RE is calculated as follows:

[0099]

[0100] If the relative error RE is less than the error standard ε, it is considered to be converged, and the calculated u n+1 (x, t) is available, proceed to the next step; if the relative error RE is greater than or equal to the error standard ε, increase the vibration mode order n, repeat steps 1 to 4, and calculate u n+1 (x,t) and u n+2 (x, t) until it meets the convergence condition and then proceed to the next step.

[0101] Step 7: Calculate the interface slip under multiple vehicle dynamic loads

[0102] As needed, multiple vehicle dynamic loads are introduced, calculated according to steps one to six, and then the interface slip of the simply supported steel-concrete composite beam under the action of multiple vehicle dynamic loads can be obtained by superposition.

[0103] Example

[0104] This embodiment applies the method of the present invention to the Juba River Bridge in Somalia, as follows:

[0105] S1. Calculate the first 10 natural frequencies of simply supported steel-concrete composite beams

[0106] Take the calculated span of the bridge as L = 30m, the bending stiffness of the concrete slab and steel beam around the neutral axis of the combined section and EI c =160900.830×10 5 N·m2 , the bending stiffness of the concrete slab and the steel beam around their own cross-section neutral axis and EI B =39648.698×10 5 N·m 2 , then the bending stiffness of the composite beam without considering the stiffness of the connecting key is EI F =200549.528×10 5 N·m 2 , the mass of the composite beam per unit length is m = 10330 kg, the distance between the neutral axis of the steel beam and the neutral axis of the concrete slab is h = 1.215 m, and the shear stiffness of the shear nail per unit length is K d =9.5×10 13 N / m, and the calculation results show that the first 10 natural frequencies of the simply supported steel-concrete composite beam are W=[0.254, 0.877, 1.91, 3.35, 5.21, 7.48, 10.2, 13.3, 16.8, 2.07].

[0107] S2. Calculate the vehicle dynamic load circular frequency

[0108] Take the vehicle moving speed V = 10m / s, and calculate the vehicle dynamic load circular frequency ω y =1.046667.

[0109] S3. Calculate the natural frequency and critical damping frequency of the bridge considering the influence of damping

[0110] Take the structural damping ratio ξ = 0.2, substitute the first 10 natural frequencies W, and calculate the bridge natural frequency W considering the damping effect. D =[0.249, 0.859, 1.87, 3.29, 5.11, 7.33, 9.69, 13.0, 16.3, 20.3] and the critical damping frequency of the bridge W b =[0.05,0.18,0.38,0,67,1.04,1.50,2.03,2.65,3.35,4.14].

[0111] S4. Calculate the introduced parameters

[0112] Take the cross-sectional shear shape factor k of the steel main beam as s =0.26, the cross-sectional shear shape factor k of the concrete slab c =0.75, shear modulus of steel main beam G s =8.08×10 10 N / m 2 , the shear modulus of the concrete slab G c =1.48×10 10 N / m 2 , the cross-sectional area of ​​the steel main beam is A s =0.052328m2 、Cross-sectional area of ​​concrete slab A c =1m 2 , the distance h from the neutral axis of the concrete slab to the steel-concrete interface s = 0.125m, distance from the neutral axis of the steel main beam to the steel-concrete interface h c =1.456m, elastic modulus of steel main beam E s =2.06×10 11 N / m 2 , the elastic modulus E of the concrete slab c =3.45×10 10 N / m 2 , the principal moment of inertia of the steel main beam I s =1.8374×10 -2 m 4 , the principal moment of inertia of the concrete slab I c =5.2083×10 -3 m 4 , the introduced parameters can be calculated:

[0113]

[0114] EA=8.21×10 9

[0115] S5. Calculate the interface slip u under the vehicle dynamic load considering the first 10 vibration modes 10 (x, t), combined Figure 2 shown.

[0116] S6. Determine convergence

[0117] Repeat steps 1 to 5 to calculate the interface slip u under the vehicle dynamic load considering the first 11 vibration modes. 11 (x, t), combined Figure 3 As shown. The bridge ordinate x and vehicle dynamic load action time t are scattered into x j and t k , j = 1, 2, ..., 11, k = 1, 2, ..., 11, we get:

[0118] X=(0,3,6,9,12,15,18,21,24,27,30)

[0119] T=(t 1 ,t 2 ,…t l )

[0120] The relative error RE = 0.02. Since the error standard ε = 0.05, RE < ε, it is considered that the accuracy requirement is met.

[0121] It will be apparent to those skilled in the art that the invention is not limited to the details of the exemplary embodiments described above and that the invention can be implemented in other forms of assembly without departing from the spirit or essential features of the invention. Therefore, the embodiments should be considered in all respects as exemplary and non-restrictive, and the scope of the invention is defined by the appended claims rather than the foregoing description, and it is intended that all variations within the meaning and range of equivalents of the claims be included in the invention. Any reference numeral in a claim should not be considered as limiting the claim to which it relates.

[0122] In addition, it should be understood that although the present specification is described according to implementation modes, not every implementation mode contains only one independent technical solution. This description of the specification is only for the sake of clarity. Those skilled in the art should regard the specification as a whole. The technical solutions in each embodiment may also be appropriately combined to form other implementation modes that can be understood by those skilled in the art.

Claims

1. A fast calculation method for the interface slip of a steel-concrete composite beam under vehicle dynamic load, characterized by: The following steps are involved: Step 1: Calculate the first n-order natural frequencies of the simply supported steel-concrete composite beam according to the basic parameters of the bridge Definition i is the natural frequency of the i-th vibration mode, i=1,2,...,n-1,n, then the first n-order natural frequency of the simply supported steel-concrete composite beam is expressed as follows: W=[ω1,ω2, … ,oh n-1 ,oh n ] in, NO F =NO B +NO c Where, L is the calculated span of the bridge, EI c is the sum of the bending stiffness of the concrete slab and the steel beam around the neutral axis of the composite section, EI B is the sum of the bending stiffness of the concrete slab and the steel beam around their own cross-section neutral axis, EI F is the bending stiffness of the composite beam without considering the stiffness of the connector, m is the mass of the composite beam per unit length, h is the distance between the neutral axis of the steel beam and the neutral axis of the concrete slab, K d is the shear stiffness of the shear stud per unit length, α and β are introduced parameters; Step 2: Calculate the vehicle dynamic load circular frequency The calculation formula of vehicle dynamic load circular frequency is as follows: Where V is the vehicle moving speed; Step 3: Calculate the natural frequency and critical damping frequency of the bridge considering the influence of damping The calculation formula of the bridge natural frequency considering the damping effect is as follows: W D =[ω D1 ,oh D2 , … ,oh Dn-1 ,oh Dn ] The calculation formula of the critical damping frequency of the bridge is as follows: W b =[ω b1 ,oh b2 , … ,oh bn-1 ,oh bn ] in, oh bi =ω i x In the formula, ω Di is the natural frequency of the i-th order considering the damping effect, ω bi is the i-th order critical damping frequency, ξ is the structural damping ratio; Step 4: Calculate the introduced parameters Introducing parameter G s , G c , K s , K c and EA, and calculated as follows: in, In the formula, (GA) s =k s G s A s and (GA) c =k c G c A c is the introduced parameter, k s and k c are the cross-sectional shear shape coefficients of the steel main beam and the concrete slab, G s and G c are the shear moduli of the steel main beam and the concrete slab, A s and A c The cross-sectional area of ​​the steel main beam and the concrete slab, h s and h c They represent the distances from the neutral axis of the steel main beam and the concrete slab to the steel-concrete interface, E s and E c are the elastic modulus of the steel main beam and the concrete slab, I s and I c are the principal moments of inertia of the steel main beam and the concrete slab respectively; Step 5: Calculate the interface slip under the vehicle dynamic load considering the first n vibration modes Interface slip u under vehicle dynamic load considering the first n vibration modes n (x, t) is obtained by the following formula: in, A i =(iω y +oh Di )siniω y t+ω bi I sew y t B i =(iω y +oh Di )sinω Di t-w bi I'm sorry. Di t C i =(iω y -oh Di )siniω y t+ω bi I sew y t D i =(iω y -oh Di )sinω Di t+ω bi I'm sorry. Di t In the formula, C i (t) is the coefficient to be determined, x is the vertical coordinate of the bridge, and t is the time of vehicle dynamic load action; Step 6: Determine convergence Repeat steps 1 to 5 to calculate the interface slip u under the vehicle dynamic load considering the first n+1 vibration modes n+1 (x, t), the bridge ordinate x and the vehicle dynamic load action time t are scattered into x j and t k , j = 1, 2, ..., g, k = 1, 2, ..., l, we get: X=(x1,x2, … x g ) T=(t1,t2, … t l ) The relative error RE is calculated as follows: If the relative error RE is less than the error standard ε, it is considered to be converged, and the calculated u n+1 (x, t) is available, proceed to the next step; if the relative error RE is greater than or equal to the error standard ε, increase the vibration mode order n, repeat steps 1 to 4, and calculate u n+1 (x,t) and u n+2 (x, t), until it meets the convergence condition, and then proceed to the next step; Step 7: Calculate the interface slip under multiple vehicle dynamic loads Introduce multiple vehicle dynamic loads as needed, calculate according to steps 1 to 6, and then superimpose to obtain the interface slip of the simply supported steel-concrete composite beam under the action of multiple vehicle dynamic loads.

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