Method for determining elastic buckling coefficient of web of steel plate composite beam with two elastically constrained sides

Abstract

The invention discloses a method for determining the elastic buckling coefficient of a web of a steel plate composite beam with two-side elastic constraints, which comprises the following steps of: establishing a displacement function equation with an elastic rotation constraint boundary and boundary conditions; deriving an explicit solution of elastic buckling of the rectangular plate under the combined action of uniform compression, non-uniform compression, shearing and compression-shearing based on a Rayleigh-Ritz energy method; obtaining a calculation method of elastic rotation constraint coefficients of upper and lower edges of a main beam web by considering upper and lower flanges of a composite beam, concrete, steel beam spacing, cantilever end length and stud positions; the composite beam web elastic rotation constraint coefficient is substituted into a composite beam web upper and lower edge elastic buckling coefficient calculation formula, and the composite beam web elastic buckling coefficient under the elastic rotation constraint boundary condition is obtained; according to the method, the boundary conditions of the upper flange and the lower flange of the composite beam are considered as elastic rotation constraints, and the elastic buckling coefficient can be calculated more accurately.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of bridge engineering, and particularly relates to a method for determining elastic buckling coefficients of a steel plate composite beam web with elastic constraints on two sides. BACKGROUND

[0002] The local stability design of a steel-concrete composite beam in China is performed according to the current standard GB50917-2013 Steel-Concrete Composite Bridge Design Standard. The standard indicates that the boundary conditions for local stability calculation are considered as elastic constraints on the flanges and simple supports on the vertical stiffening ribs and the horizontal stiffening ribs of the thin plate. The high-thickness ratio of the web is mainly limited and the stiffening ribs are set to make the web meet the local stability requirements.

[0003] The standard considers the elastic constraints of the boundary by introducing the elastic constraint coefficients, but does not provide a calculation method for the elastic constraint coefficients. In actual application, the four-side simply supported plate is still calculated.

[0004] The concrete bridge deck in the steel-concrete composite beam provides strong elastic constraints for the steel web. The calculation according to the four-side simply supported plate will underestimate the critical buckling stress of the web, resulting in material waste. The actual boundary conditions are considered by using the rotational constraint method, which can more accurately calculate the buckling strength and is helpful to reduce the plate thickness in the design, thereby reducing the cost and CO2 emission. Therefore, it is necessary to consider the actual boundary conditions and accurately calculate the buckling strength of the plate under the action of the typical load. SUMMARY

[0005] To solve the problems in the prior art, the purpose of the present application is to provide a method for determining elastic buckling coefficients of a steel plate composite beam web with elastic constraints on two sides. The present application considers the boundary conditions of the upper and lower flanges of the composite beam as elastic rotational constraints, which can more accurately calculate the elastic buckling coefficients.

[0006] To achieve the above-mentioned purpose, the technical scheme adopted by the present application is as follows: a method for determining elastic buckling coefficients of a steel plate composite beam web with elastic constraints on two sides, comprising the following steps:

[0007] Step 1, establishing a displacement function equation and boundary conditions with elastic rotational constraint boundaries, and deriving an explicit solution of elastic buckling of a rectangular plate under the action of uniform compression, non-uniform compression, shear and compression-shear combination based on the Rayleigh-Ritz energy method;

[0008] Step 2, obtaining a calculation method for the elastic rotational constraint coefficients of the upper and lower edges of the web of the main beam by considering the upper and lower flanges of the composite beam, the concrete, the distance between the steel beams, the length of the cantilever end, and the bolt position;

[0009] Step 3, the elastic rotation restraint coefficient of the combined beam web is brought into the elastic buckling coefficient calculation formula of the upper and lower edges of the combined beam web to obtain the elastic buckling coefficient of the combined beam web under the elastic rotation restraint boundary condition.

[0010] As a further improvement of the present application, the step 1 is specifically as follows:

[0011] The displacement function adopts the form of the product of a polynomial and a trigonometric function:

[0012]

[0013] Wherein, a and b are the width and height of the web, ψ1, ψ2 and ψ3 are unknown coefficients to be determined, a m is an unknown coefficient, and m is the half-wave number of buckling in the width direction of the web;

[0014] Considering that the upper and lower edges of the steel-concrete composite beam web are elastically rotationally restrained and the other two edges are simply supported, the boundary conditions are as follows:

[0015] w(x,0)=0;

[0016] w(x,b)=0;

[0017] w(0,y)=0;

[0018] w(a,y)=0;

[0019]

[0020]

[0021] Wherein: is the bending stiffness of the unit width plate, t is the thickness of the web, E is the elastic modulus; v is the Poisson's ratio; K0, K b are the equivalent elastic rotation restraint stiffnesses at y=0 and b.

[0022] By bringing the displacement function into the boundary conditions, the displacement function obtained is:

[0023]

[0024] Wherein: χ0, χ b are the dimensionless rotation restraint coefficients at y=0 and b;

[0025] The buckling coefficient is calculated: according to the Rayleigh-Ritz method, the boundary equivalent strain energy U S and the bending elastic strain energy U e are respectively:

[0026]

[0027] When the plate is subjected to uniform compression, non-uniform compression, shear and bending-shear combined action, the external work is:

[0028]

[0029] Among them, N x is the pressure per unit length when uniformly compressed, is the compression coefficient, which indicates the degree of non-uniform distribution of non-uniform pressure, N0,N b are the unit length pressure at y=0 and point b, N xy is the shear size per unit length on all edges;

[0030] The total potential energy π is:

[0031] Π=U e +U s -W is the first-order variation of the total potential energy, and using the principle of minimum potential energy, we have

[0032] δΠ=δU e +δU s -δW=0

[0033] Substitute the displacement function and solve them separately:

[0034] The buckling coefficient under uniform compression is:

[0035]

[0036] Where, α = a / b is the aspect ratio of the web,

[0037]

[0038] The buckling coefficient under non-uniform compression is:

[0039]

[0040] in,

[0041]

[0042] The buckling coefficient under shear is:

[0043]

[0044] in,

[0045]

[0046] The buckling coefficient under combined compression and shear is:

[0047]

[0048] Wherein, μ = N xy / N0

[0049]

[0050] As a further improvement of the application, the step 2 specifically comprises:

[0051] The calculation method of the elastic rotation restraint coefficient of the upper and lower flanges of the web plate of the main beam: in the design, the upper and lower flanges, the concrete, the steel beam spacing, the cantilever end length and the bolt position of the composite beam will all affect the buckling coefficient, which mainly changes the buckling coefficient by affecting the rotation restraint coefficient of the web plate, and the middle beam and the side beam in the single-beam model and the multi-beam model are analyzed respectively.

[0052] (1) Single beam calculation

[0053] At this time, the effect of the upper and lower flanges of the concrete plate and the steel beam on the rotation restraint is taken into account, and according to the Timoshenko elastic stability theory, the equivalent bending moment can be obtained:

[0054]

[0055] Wherein, I p is the polar moment of inertia, G is the shear modulus, b1 and b2 are the upper flange widths of the concrete plate and the steel beam respectively, and t1 and t2 are the thicknesses of the upper flanges of the concrete plate and the steel beam respectively;

[0056] According to the elastic boundary condition

[0057]

[0058] That is

[0059]

[0060] Then the upper side of the web plate

[0061]

[0062] Similarly, the elastic rotation restraint stiffness of the lower flange is obtained:

[0063]

[0064] Wherein b3 and t3 are the width and thickness of the lower flange of the steel beam;

[0065] According to the law of half-wave number and the length-width ratio of the web plate, the elastic rotation restraint stiffness can be approximately:

[0066]

[0067] (2) Side beam calculation: for the side beam with multiple longitudinal beams, one side is cantilevered, and the other side is connected with adjacent longitudinal beams through a concrete slab, the longitudinal beam provides a simply supported constraint, and a partial rotational constraint is generated through a stud, which is simplified for calculation, and the total equivalent bending moment is calculated as:

[0068]

[0069] Wherein, M1, M2, M3 are the equivalent bending moments provided by the inter-beam concrete, the cantilever end concrete and the upper flange of the steel beam respectively, l is the distance between the two longitudinal beams, d is the distance of the stud from the center of the main beam, and l1 is the width of the cantilever end of the concrete;

[0070] The elastic boundary condition is obtained as

[0071]

[0072] The elastic rotational constraint stiffness of the lower flange is obtained as

[0073]

[0074] (3) Middle beam calculation: for the middle beam, i.e. the longitudinal beam on both sides is connected with adjacent longitudinal beams through a concrete slab, which is simplified as two simply supported models, and the equivalent bending moment is:

[0075]

[0076] For the upper flange of the steel beam, the torsional stiffness is taken into account, and the total equivalent bending moment is:

[0077]

[0078] The elastic boundary condition is obtained as

[0079]

[0080] The elastic rotational constraint stiffness of the lower flange is obtained as

[0081]

[0082] As a further improvement of the present application, the step 3 specifically comprises:

[0083] The elastic rotational constraint coefficients of the upper and lower flanges of the main beam web calculated by step 2 are brought into the elastic rotational constraint boundary combined beam web elastic buckling coefficient expression under the load condition obtained in step 1, and the corresponding buckling coefficient is obtained.

[0084] The beneficial effects of the present application are:

[0085] 1、The present application considers the boundary conditions of the upper and lower flanges of the composite beam as elastic rotational constraints, which can more accurately calculate the elastic buckling coefficient; more comprehensively calculate the buckling coefficient calculation formula of the plate under various load cases; for the calculation method of the rotational constraint of the upper and lower flanges of the web of the composite beam, the upper and lower flanges of the main beam, the distance between the concrete and the steel beam, the length of the cantilever end, the distance between the bolts and the like are considered, more factors are considered, more comprehensive, and the actual situation can be accurately simulated.

[0086] 2、The method is based on energy method, introduces displacement function, establishes a kind of elastic buckling coefficient solving method of steel plate composite beam web with two elastic constraints. The method is simple, and the error of the finite element analysis result is not more than 5%. BRIEF DESCRIPTION OF DRAWINGS

[0087] Figure 1 It is the schematic diagram of boundary condition in the embodiment of the present application.

[0088] Figure 2 It is the schematic diagram of composite beam section in the embodiment of the present application.

[0089] Figure 3 It is the schematic diagram of composite beam multi-beam section in the embodiment of the present application. DETAILED DESCRIPTION

[0090] The embodiments of the present application will be described in detail below with reference to the drawings.

[0091] EMBODIMENT

[0092] A kind of elastic buckling coefficient solving method of steel plate composite beam web with two elastic constraints, first, establish displacement function equation and boundary condition with elastic rotational constraint boundary, based on Rayleigh-Ritz energy method, the explicit solution of elastic buckling of rectangular plate under uniform compression, non-uniform compression, shear and compression-shear combined action is deduced;Then, by considering the upper and lower flanges of the composite beam, the distance between the concrete and the steel beam, the length of the cantilever end, the bolt position, the calculation method of the elastic rotational constraint coefficient of the upper and lower flanges of the web of the main beam is obtained;Finally, the elastic rotational constraint coefficient of the web of the composite beam is brought into the elastic buckling coefficient calculation formula of the upper and lower flanges of the web of the composite beam, to obtain the elastic buckling coefficient of the web of the composite beam under the boundary condition of elastic rotational constraint.

[0093] 1, elastic buckling coefficient calculation method of composite beam web under elastic rotational constraint boundary

[0094] (1) boundary condition

[0095] As shown in the formula, the upper and lower flanges of the web of the steel-concrete composite beam are considered to be elastically rotationally constrained, and the other two sides are simply supported, i.e. the boundary conditions are as follows: Figure 1

[0096] w (x, 0) = 0; ​

[0097] w(x,b)=0;

[0098] w(0,y)=0;

[0099] w(a,y)=0;

[0100]

[0101] in: is the bending stiffness of the plate per unit width, t is the web thickness, E is the elastic modulus; ν is the Poisson's ratio; K0, K b is the equivalent elastic rotational constraint stiffness at y = 0 and b.

[0102] (2) Displacement function

[0103] The displacement function takes the form of a product of a polynomial and a trigonometric function:

[0104]

[0105] Among them, a and b are the width and height of the web, ψ1, ψ2 and ψ3 are unknown coefficients, and a m is the unknown coefficient, m is the buckling half-wave number in the width direction of the web;

[0106] By substituting the boundary conditions, we can obtain the unknown coefficients ψ1, ψ2 and ψ3. Substituting the unknown coefficients into the displacement function expression, the displacement function obtained is:

[0107]

[0108] Among them: χ0, χ b is the dimensionless rotation constraint coefficient;

[0109] (3) Calculation of buckling coefficient

[0110] According to the Rayleigh-Ritz method, the boundary equivalent strain energy and bending elastic strain energy are:

[0111]

[0112] When the plate is subjected to uniform compression, non-uniform compression, shear and bending-shear combined action, the external work is:

[0113]

[0114] Among them, N x is the pressure per unit length when uniformly compressed, is the compression coefficient, which indicates the degree of non-uniform distribution of non-uniform pressure, N0,N b are the unit length pressure at y=0 and point b, Nxy For all sides of the unit length of the shear size

[0115] The total potential energy Π is:

[0116] Π = U e + U s - W

[0117] The first variation of the total potential energy, and by using the principle of minimum potential energy, we have

[0118] δΠ = δU e + δU s - δW = 0

[0119] Substituting the displacement function, we can solve respectively:

[0120] The buckling coefficient under uniform compression is:

[0121]

[0122] Where, α = a / b is the length-width ratio of the web,

[0123]

[0124] The buckling coefficient under non-uniform compression is:

[0125]

[0126] Where,

[0127]

[0128] The buckling coefficient under shear action is

[0129]

[0130] Where,

[0131]

[0132] The buckling coefficient under the combined action of compression and shear is

[0133]

[0134] Where, μ = N xy / N0

[0135]

[0136] 2、The calculation method of the elastic rotational restraint coefficient of the upper and lower edges of the web of the main beam:

[0137] In the design, the upper and lower flanges of the composite beam, concrete, steel beam spacing, cantilever end length, and stud position all have an impact on the buckling coefficient, mainly by affecting the web rotation constraint coefficient. The buckling coefficient changes differently for single-beam models and multi-beam models. The impact behavior of the center beam and side beams in the multi-beam model is also inconsistent, so the center beam and side beams are analyzed separately.

[0138] (1) Calculation of single beam rotation constraint coefficient

[0139] like Figure 2 The figure shows a single beam model. In this case, the effects of the concrete slab and the upper and lower flanges of the steel beam on the rotational restraint are taken into account. According to Timoshenko's elastic stability theory, the rate of change of the torque per unit length is equal to the value of the web edge bending moment per unit length.

[0140] The rotation angle of the upper edge of the web is

[0141] The rate of change of the rotation angle along the longitudinal direction of the bridge is

[0142] From the knowledge of material mechanics, we know that

[0143] The rate of change of torque along the x direction is

[0144] From the displacement function and boundary conditions set previously, we can see that

[0145]

[0146] The equivalent bending moment can be obtained:

[0147]

[0148] Among them, I p is the polar moment of inertia, T is the torque, G is the shear modulus, b1 and b2 are the widths of the upper flange of the concrete slab and the steel beam, respectively, and t1 and t2 are the thicknesses of the upper flange of the concrete slab and the steel beam, respectively;

[0149] According to the elastic boundary conditions,

[0150]

[0151] Right now:

[0152]

[0153] Then the elastic rotation constraint stiffness of the web side is:

[0154]

[0155] Similarly, the elastic rotational restraint stiffness of the lower flange is:

[0156]

[0157] where b3, t3 are the width and thickness of the lower flange of the steel beam;

[0158] According to the law of half-wave number and the length-width ratio of the web, the elastic rotational restraint stiffness can be approximated as:

[0159]

[0160] (2) Side beam calculation

[0161] For a side beam with multiple longitudinal beams, one side is cantilevered, and the other side is connected to the adjacent longitudinal beam through a concrete slab, which provides simple support restraint and partial rotational restraint through studs. However, due to the difficulty in quantifying the rotational restraint at this point and the possibility of repeated calculation of rotational restraint when the adjacent longitudinal beam buckles, this point is simplified as a simple support restraint for calculation:

[0162] As shown in Figure 3 , the concrete slab from the edge to the center of the adjacent longitudinal beam is analyzed separately, and the cantilevered section and the simply supported model are simplified. For the simply supported model, a bending moment is applied to the top edge of the web, and the deflection curve under this bending moment is obtained. Conversely, given the deflection curve, the bending moment value at this time can also be obtained.

[0163] The deflection curve under the bending moment M is

[0164] where l is the distance between the two longitudinal beams, and I is the moment of inertia of the concrete to the unit width of the bridge transverse plate;

[0165] From (1), the rotation angle near the top edge is

[0166] Assuming that the stud is completely connected with the concrete slab, the deflection at the stud position is:

[0167]

[0168] where d is the distance from the stud to the center of the main beam;

[0169] Thus, the equivalent bending moment provided by the inter-beam concrete is calculated as

[0170]

[0171] For the cantilevered section, the equivalent bending moment caused by the torsional stiffness is directly calculated as

[0172]

[0173] where l1 is the width of the concrete cantilevered end.

[0174] For the top flange, the equivalent bending moment caused by torsional stiffness is taken into account

[0175]

[0176] The total equivalent bending moment is:

[0177]

[0178] From the elastic boundary condition, we have:

[0179]

[0180] That is

[0181] Then

[0182] The elastic rotational restraint stiffness of the bottom flange is obtained by the previous calculation:

[0183]

[0184] (3) Middle beam calculation

[0185] For the middle beam, that is, the longitudinal beam on both sides is connected with the adjacent longitudinal beam through the concrete slab, which is simplified as two two-point simply supported models, and the equivalent bending moment is:

[0186]

[0187] For the top flange of the steel beam, the total equivalent bending moment is:

[0188]

[0189] From the elastic boundary condition, we have:

[0190]

[0191] The elastic rotational restraint stiffness of the bottom flange is obtained by the previous calculation:

[0192]

[0193] 3. The elastic rotational restraint coefficients of the top and bottom flanges of the web of the main beam obtained by calculation 2 are brought into the elastic rotational restraint boundary combined web elastic buckling coefficient expression obtained in 1 under the load condition, that is, the corresponding buckling coefficient is obtained.

[0194] The above embodiments only express the specific implementation of the present application, which is described in more detail and in more detail, but cannot be understood as a limitation on the scope of the patent of the present application. It should be noted that for ordinary skilled in the art, without departing from the concept of the present application, a number of modifications and improvements can be made, which are within the scope of protection of the present application.

Claims

1. A method for determining the elastic buckling coefficient of the web of a steel plate composite beam elastically constrained on both sides, characterized by: The following steps are involved: Step 1: Establish the displacement function equation and boundary conditions with elastic rotation constraint boundaries, and derive the explicit solution of the elastic buckling of rectangular plates under uniform compression, non-uniform compression, shear, and combined compression-shear based on the Rayleigh-Ritz energy method; Step 2: By considering the upper and lower flanges of the composite beam, concrete, the spacing between steel beams, the length of the cantilever end, and the position of the studs, the calculation method of the elastic rotation constraint coefficient of the upper and lower edges of the main beam web is obtained; Step 3: Substitute the elastic rotation constraint coefficient of the composite beam web into the calculation formula of the elastic buckling coefficient of the upper and lower edges of the composite beam web to obtain the elastic buckling coefficient of the composite beam web under the elastic rotation constraint boundary condition.

2. The method for determining the elastic buckling coefficient of the web of a steel plate composite beam with elastic constraints on both sides according to claim 1, characterized in that: The step 1 is specifically as follows: The displacement function takes the form of a product of a polynomial and a trigonometric function: Among them, a and b are the width and height of the web, ψ1, ψ2 and ψ3 are unknown coefficients, and a m is the unknown coefficient, m is the buckling half-wave number in the width direction of the web; The boundary conditions are expressed as the following equations: w(x,0)=0; w(x,b)=0; w(0,y)=0; w(a,y)=0; in: is the bending stiffness of the plate per unit width, t is the web thickness, E is the elastic modulus; ν is the Poisson's ratio; K0, K b is the equivalent elastic rotation constraint stiffness at y = 0 and b; By substituting the displacement function into the boundary conditions, the displacement function is obtained: Among them: χ0, χ b is the dimensionless rotation constraint coefficient at y = 0 and b; Calculation of buckling coefficient: According to the Rayleigh-Ritz method, the boundary equivalent strain energy U S and bending elastic strain energy U e They are: When the plate is subjected to uniform compression, non-uniform compression, shear and combined bending-shear action, the external work W is: Among them, N x is the pressure per unit length when uniformly compressed, is the compression coefficient, which indicates the degree of non-uniform distribution of non-uniform pressure, N0,N b are the unit length pressure at y=0 and point b, N xy is the shear size per unit length on all edges; The total potential energy π is: Π=U e +U s -IN The first-order variation of the total potential energy and the application of the principle of minimum potential energy give us δΠ=δU e +δU s -δW=0 Substitute the displacement function and solve them separately: The buckling coefficient under uniform compression is: Where, α = a / b is the aspect ratio of the web, The buckling coefficient under non-uniform compression is: in, The buckling coefficient under shear is: in, The buckling coefficient under combined compression and shear is: Where μ = N xy / N0 3. The method for determining the elastic buckling coefficient of the web of a steel plate composite beam with elastic constraints on both sides according to claim 1, characterized in that: The step 2 specifically includes: (1) Single beam calculation: Taking into account the effects of the concrete slab and the upper and lower flanges of the steel beam on the rotational restraint, according to Timoshenko's elastic stability theory, the rate of change of the torque per unit length is equal to the value of the web edge bending moment per unit length; The calculated equivalent bending moment is: Among them, I p is the polar moment of inertia, G is the shear modulus, b1 and b2 are the widths of the upper flange of the concrete slab and the steel beam, respectively, and t1 and t2 are the thicknesses of the upper flange of the concrete slab and the steel beam, respectively; From the elastic boundary conditions, we know that: Right now: Then the upper side of the belly plate: Similarly, the elastic rotation constraint stiffness of the lower flange is obtained as: Among them, b3 and t3 are the width and thickness of the lower flange of the steel beam; According to the law of half-wave number and web aspect ratio, the elastic rotation constraint stiffness can be approximated as: (2) Calculation of side beams: For side beams with multiple longitudinal beams, one side is cantilevered and the other side is connected to the adjacent longitudinal beams through a concrete slab. The longitudinal beams provide simple support constraints and generate partial rotational constraints through studs. The calculation is simplified to simple support constraints. The total equivalent bending moment is calculated as: Where M1, M2, and M3 are the equivalent bending moments provided by the concrete between beams, the concrete at the cantilever end, and the upper flange of the steel beam, respectively; l is the spacing between the two longitudinal beams; d is the distance between the stud and the center of the main beam; and l1 is the width of the concrete cantilever end. According to the elastic boundary conditions, we can get: The elastic rotation constraint stiffness of the lower flange is calculated as follows: (3) Middle beam calculation: For the middle beam, both sides of the longitudinal beam are connected to the adjacent longitudinal beams through concrete slabs, which is simplified to two simply supported models. The equivalent bending moment at this time is: For the upper flange of the steel beam, taking into account the torsional stiffness, the total equivalent bending moment is: According to the elastic boundary conditions, we can get: The elastic rotation constraint stiffness of the lower flange is calculated as follows:

4. The method for determining the elastic buckling coefficient of the web of a steel plate composite beam with elastic constraints on both sides according to claim 3, characterized in that: The step 3 specifically includes: Substitute the elastic rotation constraint coefficient of the upper and lower edges of the main beam web calculated in step 2 into the elastic buckling coefficient expression of the composite beam web under the elastic rotation constraint boundary under the load condition obtained in step 1 to obtain the corresponding buckling coefficient.

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