A fast calculation method for interface slip of steel-concrete composite beams under vehicle dynamic loads

Abstract

A method for rapidly calculating the interface slip of a steel-concrete composite beam under vehicle dynamic loads relates to the technical field of bridge construction. The method comprises the following steps: calculating the first n natural frequencies of a simply supported steel-concrete composite beam based on basic bridge parameters; calculating the circular frequency of vehicle dynamic loads; calculating the bridge's natural frequencies and critical damping frequencies taking into account the influence of damping; calculating the introduction parameters; calculating the interface slip under vehicle dynamic loads taking into account the first n vibration modes; determining convergence; and calculating the interface slip under multiple vehicle dynamic loads. Based on the Timoshenko beam theoretical model, and taking into account the material, cross-section, and connector stiffness, a formula for calculating the interface slip of a simply supported steel-concrete composite beam under vehicle dynamic loads is proposed. This formula enables rapid and accurate calculation of interface slip without relying on numerical simulation or field testing.

Description

Technical Field

[0001] The present 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 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 steel beams and concrete slabs through shear connectors. The shear connectors enable the steel beams and concrete slabs to work together, effectively utilizing the properties of both materials when bearing loads. The steel-concrete composite beam combines the advantages of both steel and concrete to improve mechanical properties.

[0003] Currently, 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, simulates steel beams and concrete slabs with solid units, simulates connectors with beams or springs, and applies loads to obtain slip distribution. However, the numerical simulation method is time-consuming, and the accuracy and precision of modeling depend on the professional level of the modelers. Therefore, it is difficult to promote it for large-scale engineering applications. The experimental method mainly relies on the production of 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 expensive, has high site and equipment requirements, is difficult to load and measure, and is also time-consuming, which is not conducive to quickly obtaining results. Summary of the Invention

[0004] To address the shortcomings of the background technology, the present invention provides a method for quickly calculating the interface slip of steel-concrete composite beams under vehicle dynamic loads. 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 simply supported steel-concrete composite beams under vehicle dynamic loads is proposed. This method can achieve rapid and accurate calculation of interface slip without relying on numerical simulations and field tests.

[0005] To achieve the above objectives, the present invention adopts the following technical solution: a method for quickly calculating the interfacial 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 based on the basic parameters of the bridge

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

[0008] W=[ω1,ω2,…,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 steel beam around the neutral axis of the combined 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's 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 bridge's natural frequency and critical damping frequency considering the influence of damping

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

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

[0020] The calculation formula for the critical damping frequency of a 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] Where, 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 introduction parameters

[0026] Introducing parameter G s , G c , K s , K c and EA, and calculate 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 concrete slab, h s and h c They represent the distances from the neutral axis of the steel beam and concrete slab to the steel-concrete interface, E s and E c are the elastic moduli 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] Where 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 vertical coordinate x and the vehicle dynamic load action time t are equidistantly scattered into x j and t k , j=1,2,...,g,k=1,2,...,l, we get:

[0043] X=(x1,x2,…x g )

[0044] T=(t1,t2,…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 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 them 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 significantly reduce the calculation time and quickly obtain the approximate result of the interface slip of the simply supported steel-concrete composite beam without consuming 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, providing a strong 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 embodiment considers the interface slip under the vehicle dynamic load of the first 10 vibration modes;

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

[0057] The technical solutions 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 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 based on the basic parameters of the bridge

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

[0061] W=[ω1,ω2,…,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 steel beam around the neutral axis of the combined 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, and α and β are introduced parameters.

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

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

[0068]

[0069] Where V is the vehicle speed.

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

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

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

[0073] The calculation formula for the critical damping frequency of a 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] Where, 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 introduction parameters

[0079] Introducing parameter G s , G c , K s , K c and EA, and calculate 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 c are the shear moduli of the steel main beam and the concrete slab, A s and Ac The cross-sectional area of ​​the steel main beam and concrete slab, h s and h c They represent the distances from the neutral axis of the steel beam and concrete slab to the steel-concrete interface, E s and E c are the elastic moduli 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] Where 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 vertical coordinate x and the vehicle dynamic load action time t are equidistantly scattered into x j and t k , j=1,2,...,g,k=1,2,...,l, we get:

[0096] X=(x1,x2,…x g )

[0097] T=(t1,t2,…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 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 proceeds 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, and calculations are performed according to steps 1 to 6. The interfacial 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, specifically 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·m 2 , the bending stiffness of the concrete slab and steel beam around their own 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 connector is EI F =200549.528×10 5 N·m 2 , the mass of the composite beam per unit length m = 10330 kg, the distance between the neutral axis of the steel beam and the neutral axis of the concrete slab h = 1.215 m, and the shear stiffness of the shear stud per unit length K d =9.5×10 13 N / m, the first 10 natural frequencies of the simply supported steel-concrete composite beam can be calculated as 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] Taking the structural damping ratio ξ=0.2, and substituting the first 10 natural frequencies W, the natural frequency of the bridge considering the damping effect can be calculated as W 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 coefficient k of the steel main beam 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 G of the concrete slab c =1.48×10 10 N / m 2 , the cross-sectional area of ​​the steel main beam is A s =0.052328m 2 , cross-sectional area A of the concrete slab c =1m 2 , the distance h from the neutral axis of the concrete slab to the steel-concrete interface s= 0.125m, distance h from the neutral axis of the steel main beam to the steel-concrete interface 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 , principal moment of inertia of the concrete slab I c =5.2083×10 -3 m 4 , the parameters introduced 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 vertical coordinate x and vehicle dynamic load action time t are equidistantly 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=(t1,t2,…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 present invention is not limited to the details of the exemplary embodiments described above and that the invention can be implemented in other configurations without departing from the spirit or essential characteristics of the invention. Therefore, the embodiments should be considered in all respects as illustrative and non-restrictive, and the scope of the invention is defined by the appended claims, not the foregoing description, and all variations coming within the meaning and range of equivalents of the claims are intended to be embraced therein. Any reference sign in a claim should not be construed as limiting the claim to which it relates.

[0122] In addition, it should be understood that although this specification is described in terms of implementation methods, not every implementation method contains only one independent technical solution. This narrative method 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 can also be appropriately combined to form other implementation methods that can be understood by those skilled in the art.

Claims

1. A method for rapid calculation of interfacial slip of steel-concrete composite beams under vehicle dynamic loads, characterized by: The following steps are involved: Step 1: Calculate the front of the simply supported steel-concrete composite beam according to the basic parameters of the bridge n order natural frequency; definition ω i For the i The natural frequency of the order mode, , then the front of the simply supported steel-concrete composite beam n The natural frequency of the order W is expressed as follows: in, ， ， ， Where, L Calculate spans for bridges, EI c is the sum of the bending stiffness of the concrete slab and steel beam around the neutral axis of the combined section, EI B is the sum of the bending stiffness of the concrete slab and the steel beam around their own 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 is the introduced parameter, represents the simply supported steel-concrete composite beam without considering the interface slip. i order natural frequency; Step 2: Calculate the vehicle's 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 for the natural frequency of a bridge considering the influence of damping is as follows: The calculation formula for the critical damping frequency of a bridge is as follows: in, ， Where, ω Di For the i The natural frequency considering the damping effect, ω bi For the i The critical damping frequency, is the structural damping ratio; Step 4: Calculate the introduced parameters; Introducing parameters 、 、 、 and EA , and calculated as follows: ， ， in, ， ， Where, and 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 areas of the steel main beam and concrete slab, h s and h c Respectively 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 moduli 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: Before calculation n Interface slip under step-mode vehicle dynamic load; Before consideration n Interface slip under vehicle dynamic load of order vibration mode Obtained by the following formula: in, Where, is the unknown coefficient, x is the vertical coordinate of the bridge, t is the vehicle dynamic load action time, f is the vehicle dynamic load size; Step 6: Determine convergence; Repeat steps 1 to 5 to calculate the n Interface slip under vehicle dynamic load of +1-order vibration mode , the vertical coordinate of the bridge x and vehicle dynamic load action time t They are equidistantly dispersed into x j and t k , , ,get: Relative error RE The calculation is as follows: If the relative error RE Less than the error standard ɛ , then it is considered to be convergent, and the calculated Available, proceed to the next step; if the relative error RE Greater than or equal to the error standard ɛ , then increase the mode order n , repeat steps 1 to 4 to calculate and , 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 them 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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