Analysis method for bearing capacity of two-span concrete floor under column end restraint

Abstract

The application discloses a method for analyzing bearing capacity of two-span concrete floor slabs under column end constraints, comprising: obtaining position parameters of the concrete floor slabs, calculating displacement coefficients of different beams based on the position parameters; establishing slab plane equations of a yield line load stage based on the displacement coefficients, obtaining slab rotation angles of each slab along x and y coordinate axes based on the slab plane equations; calculating yield line load values according to the virtual work principle based on the slab rotation angles; calculating membrane force parameters based on force balance equations, moment balance equations and elliptical regions of each slab; establishing plane equations of each slab under a limit state, calculating bearing capacity increase coefficients according to the plane equations; obtaining a total increase coefficient based on the force equivalence principle, and obtaining a limit bearing capacity based on the total increase coefficient and the yield line load values. The application can make the bearing capacity of the reinforced concrete floor slabs under column end constraints have a smaller error compared with the actual situation.

Description

Technical Field

[0001] This invention belongs to the field of reinforced concrete floor slab bearing capacity analysis, and in particular relates to a method for analyzing the bearing capacity of a two-span concrete floor slab under column end constraint. Background Technology

[0002] With the diversification of building forms, the restraint forms of reinforced concrete floor slabs also change, and column end restraint is an important part of the restraint of reinforced concrete floor slabs, which has a great influence on the load-bearing capacity of the floor slab.

[0003] The analysis of ultimate bearing capacity is essential for reinforced concrete structures. Based on the analysis of the bearing capacity of two-span concrete floor slabs under column end constraints, scholars have never stopped their theoretical research on it, such as the classical plastic strand theory, Bailey theory, and the simply supported ellipse method. However, there are always large errors between the principles and calculation results of these methods and the experimental results. Summary of the Invention

[0004] The purpose of this invention is to provide a method for analyzing the bearing capacity of a two-span concrete floor slab under column end constraints, so as to solve the problems existing in the prior art.

[0005] To achieve the above objectives, this invention provides a method for analyzing the bearing capacity of a two-span concrete floor slab under column end constraints, comprising:

[0006] Obtain the position parameters of the concrete floor slab, and calculate the displacement coefficients of different beams based on the position parameters, wherein the displacement coefficients include: displacement coefficients of long beams, displacement coefficients of short beams, and displacement coefficients of intermediate beams;

[0007] Based on the displacement coefficient, the plate plane equations for the yield line load stage are established. Based on the plate plane equations, the plate rotation angles along the x and y coordinate axes of each plate are obtained. Based on the plate rotation angles, the yield line load value is calculated according to the principle of virtual work.

[0008] Based on the force balance equation, moment balance equation and elliptical region of each plate, the membrane force parameters are calculated; and the plane equation of each plate under the ultimate state is established, and the bearing capacity amplification factor is calculated according to the plane equation.

[0009] Based on the principle of force equivalence, the total amplification factor is obtained. Based on the total amplification factor and the yield line load value, the ultimate bearing capacity of the reinforced concrete floor slab under support constraint is obtained.

[0010] Preferably, the process of obtaining the location parameters of the concrete floor slab includes:

[0011] The single-span reinforced concrete slab is divided into several regions, and several slabs are selected. Based on the length, width, and reinforcement parameters of a single slab, the positional parameters of the concrete floor slab are obtained.

[0012] Preferably, the plurality of plates includes: two right trapezoidal plates with different constraints and two triangular plates with different constraints.

[0013] Preferably, the process of calculating the displacement coefficients of different beams includes:

[0014] Based on the location parameters, and according to the load distribution method, constraint form, and length parameters, the mid-span displacement of different beams and the support negative bending moment of beams with fixed ends are calculated. Based on the mid-span displacement, the displacement coefficients of different beams are obtained.

[0015] Preferably, the process of obtaining the rotation angle of each plate along the x and y coordinate axes includes:

[0016] Based on the plane equation of the plate, the normal vector of the plate is calculated. According to the angle between the normal vector and the direction vectors of the x and y coordinate axes, the plate rotation angle along the x and y coordinate axes of each plate is obtained.

[0017] Preferably, the process of calculating the yield line load value includes:

[0018] Based on the plate rotation angle, and according to the principle of virtual work, the external work and internal force work generated under the load are obtained. Based on the fact that the external work is equal to the internal force work, the yield line load value is calculated.

[0019] Preferably, the elliptical region is a region divided into four equal parts based on the length and width parameters of the floor slab.

[0020] Preferably, the process of calculating the bearing capacity amplification factor includes:

[0021] Based on the coordinates of each known point, establish the plane equation for each plate under the limit state. Based on the negative bending moment of the support, substitute the x and y coordinate values ​​of the force on each plate into the plane equation to obtain the vertical displacement value of each force. Based on the vertical displacement value, obtain the bearing capacity amplification factor, wherein the bearing capacity amplification factor includes: the bearing capacity amplification factor caused by bending and the bearing capacity amplification factor caused by axial force.

[0022] The technical effects of this invention are as follows:

[0023] This invention provides a method for analyzing the bearing capacity of a two-span reinforced concrete floor slab considering the influence of column end constraints. This method not only considers the influence of the vertical deformation of the edge beams on the floor slab's bearing capacity, but also the influence of column end and beam end constraints on the floor slab's ultimate bearing capacity and vertical displacement, etc. By calculating the position parameters of the yield line of the reinforced concrete floor slab, the vertical displacement coefficients of the long and short beams, the yield line load value of the reinforced concrete slab, the negative bending moment at the constraint, and the amplification factor, the bearing capacity of the reinforced concrete floor slab under column end constraints can be obtained. This method establishes force balance equations and moment balance equations, and the ultimate bearing capacity of the two-span reinforced concrete floor slab analyzed by this method has a smaller error compared with the data obtained from actual experiments. Attached Figure Description

[0024] The accompanying drawings, which form part of this application, are used to provide a further understanding of this application. The illustrative embodiments and descriptions of this application are used to explain this application and do not constitute an undue limitation of this application. In the drawings:

[0025] Figure 1 This is a schematic diagram of plate division and a diagram of thin film effect region division in an embodiment of the present invention;

[0026] Figure 2 This is a diagram showing the constraint and stress distribution of the reinforced concrete beams and slabs in an embodiment of the present invention, wherein... Figure 2 (a) is the force diagram of the side beam in the long span direction. Figure 2 (b) is the force diagram of the beam in the short span direction. Figure 2 (c) is the force diagram of the intermediate beam. Figure 2 (d)- Figure 2 (f) Equivalent force diagram of vertical displacement at mid-span of floor slab;

[0027] Figure 3 These are floor slab deformation diagrams under the yield and ultimate states in embodiments of the present invention, wherein... Figure 3 (a) is a diagram of the floor slab deformation under yielding conditions. Figure 3 (b) shows the floor slab deformation under the ultimate condition;

[0028] Figure 4 This is a diagram showing the internal force distribution of each reinforced concrete slab in an embodiment of the present invention, wherein... Figure 4 (a)- Figure 4 (d) Internal force diagrams of plates ①”, ②”, ③” and ④” under extreme conditions, respectively;

[0029] Figure 5 This is a flowchart of the method in an embodiment of the present invention. Detailed Implementation

[0030] It should be noted that, unless otherwise specified, the embodiments and features described in this application can be combined with each other. This application will now be described in detail with reference to the accompanying drawings and embodiments.

[0031] It should be noted that the steps shown in the flowchart in the accompanying drawings can be executed in a computer system such as a set of computer-executable instructions, and although a logical order is shown in the flowchart, in some cases the steps shown or described may be executed in a different order than that shown here.

[0032] Example 1

[0033] like Figure 5 As shown, this embodiment provides a method for analyzing the bearing capacity of a two-span concrete floor slab under column end constraints, including:

[0034] The reinforced concrete floor slab supported by reinforced concrete beams is a rectangular two-way slab, and the beams include long-side beams, short-side beams, and intermediate beams, and are supported by columns. The method for analyzing the bearing capacity of the reinforced concrete floor slab includes the following steps:

[0035] S1. Based on the actual deformation state of the reinforced concrete floor slab under vertical load and considering the vertical deformation of the beam and the constraint problem at the end of the intermediate column, the single-span reinforced concrete slab is divided into 8 regions. Since the 8 slabs have dimensional and constraint symmetry along the x-direction, only the four slabs on one side are analyzed, namely two right trapezoidal slabs with different constraints and two triangular slabs with different constraints.

[0036] S2. Ignoring the torsion and horizontal displacement of the beam, and based on the length L, width l, and reinforcement parameters of the floor slab, determine the position parameter n. The boundary condition of the reinforced concrete slab is simply supported on both sides, the boundary condition of the long side beam is simply supported at one end and fixed at the other end, and the boundary condition of the short side beam is simply supported at both ends.

[0037] S3. Reference Figure 2 Based on the load distribution method, constraint form, and length parameters, calculate the mid-span displacement w of each beam and the support negative bending moment M of beams with fixed ends, and determine the displacement coefficient λ of long beams, short beams, and intermediate beams. B’ , λ H’ , λ E’ ;

[0038] S4. Reference Figure 3(a) Based on the assumption of rigid floor slabs (the floor slab changes only the vertical displacement z coordinate value during deformation, and does not change its x and y coordinate values), the plane equations of slabs ①', ②', ③' and ④' in the yield line load stage are established. The normal vector n of each slab is obtained according to the plane equations. The rotation angle β of each slab along the x and y coordinate axes is obtained according to the angle between the normal vector and the x and y coordinate axis direction vectors s.

[0039] S5. According to the principle of virtual work, the external work W produced under the load is equal to the work D done by the internal forces. Therefore, the yield line load q can be calculated. y,m The calculation formula;

[0040] S6. Reference Figure 1 Based on the length L and width l of the floor slab, a four-part elliptical region is established to distinguish the tension and compression membrane effect regions. That is, the ellipse passes through the four corner points of the four-part inner grid of the floor slab.

[0041] S7. Based on the force balance equation, moment balance equation and elliptical region of each plate, the relevant film force parameters are obtained;

[0042] S8. Reference Figure 3 (b) and Figure 4 Based on the coordinates of each known point, calculate the plane equation of each plate under the limit state. Substitute the x and y coordinate values ​​of the force on each plate into the plane equation to obtain the vertical displacement value of each force. Based on the negative bending moment value at the fixed support end under the limit state, calculate the bearing capacity increase factor caused by bending and the bearing capacity increase factor caused by axial force.

[0043] S9. Based on the principle of force equivalence, the total increase factor is calculated, and combined with the yield line load value, the ultimate bearing capacity of the reinforced concrete floor slab considering support constraints is then calculated.

[0044] In the two-span reinforced concrete floor slab bearing capacity analysis method considering the influence of column end constraints provided in the embodiments of the present invention, the support form of the long and short beams in step S2 is affected by the column end constraints. Since the constraint of the middle column is stronger and the constraint of the side column is weaker, the constraint method of the long side beam is equivalent to simply supported at one end and fixed at the other end, the constraint method of the short side beam is equivalent to simply supported at both ends, and the constraint form of the middle beam is fixed at both ends.

[0045] In the method for analyzing the bearing capacity of a two-span reinforced concrete floor slab considering the influence of column end constraints provided in the embodiments of the present invention, reference is made to... Figure 2 The mid-span displacement w of each beam determined in step S3 and the negative bending moment M at the support of the beam with fixed end support are as follows:

[0046]

[0047] In the formula w B’ wH’ w E’ w C’ q1 represents the mid-span displacement of the long-side beam, short-side beam, intermediate beam, and slab; L(l) represents the length of the slab's long (short) span; q1 represents the calculated load of the beam in the long span direction; q2 represents the calculated load of the beam in the short span direction; q y,m B is the yield line load; n is the location parameter; S,i E represents the stiffness of edge beams, intermediate beams, slabs, or columns (hereinafter the same); S,i The elastic modulus of the reinforcement in edge beams, intermediate beams, slabs, or columns; A S,i h is the cross-sectional area of ​​the longitudinal tensile reinforcement. 0,i α is the effective height of the cross section; ψ: the strain non-uniformity coefficient of the longitudinal tensile reinforcement between cracks (for members directly subjected to repeated loads, ψ = 1.0); α E,i ρ is the ratio of the elastic modulus of the steel reinforcement to the elastic modulus of the concrete. i γ' is the reinforcement ratio of longitudinal tensile reinforcement; f,i M' is the ratio of the cross-sectional area of ​​the compression flange to the effective cross-sectional area of ​​the web. Ax F' is the negative bending moment at the fixed support end of the long-side beam; A M' is the vertical support reaction at the fixed end of the long-side beam; Ay λ represents the negative bending moment at the fixed support end of the short-side beam; B’ , λ H’ , λ E’ represents the displacement coefficients for the long side beam, short side beam, and middle beam.

[0048] In the method for analyzing the bearing capacity of a two-span reinforced concrete floor slab considering the influence of column end constraints provided in the embodiments of the present invention, reference is made to... Figure 3 (a) The plane equation of plate ①', the normal vector n, and the rotation angle β in step S4 are:

[0049]

[0050] s x = (1, 0, 0); s y = (0, 1, 0)

[0051] The plane equation of plate ②', as well as the normal vector n and rotation angle β, are:

[0052]

[0053] s x =(1, 0, 0),s y = (0, 1, 0)

[0054] The plane equation of plate ③', as well as the normal vector n and rotation angle β, are:

[0055]

[0056]

[0057] The plane equation of plate ④', as well as the normal vector n and rotation angle β, are:

[0058]

[0059]

[0060]

[0061] In the formula s x s y The direction vectors are the positive directions of the x and y coordinate axes.

[0062] In the method for analyzing the bearing capacity of a two-span reinforced concrete floor slab considering the influence of column end constraints provided in this embodiment of the invention, the external work W and internal force work D under the load in step S5 are as follows:

[0063]

[0064] Yield line load q y,m The calculation formula is:

[0065]

[0066] In the formula M x M y The yield moment of the floor slab in the x and y directions per unit length.

[0067] In the method for analyzing the bearing capacity of a two-span reinforced concrete floor slab considering the influence of column end constraints provided in the embodiments of the present invention, reference is made to... Figure 1 In step S6, the four-part grid region is a grid formed by connecting the four points of the two longer sides and the four points of the two shorter sides. The ellipse passes through the four corner points of the inner grid. The area inside the ellipse is the tensile film effect region, and the area outside the ellipse is the compressive film effect region. The equation of the ellipse is:

[0068]

[0069] In the formula L CJ (l CK ) is the length of the major (minor) semi-axis of the ellipse.

[0070] In the method for analyzing the bearing capacity of a two-span reinforced concrete floor slab considering the influence of column end constraints provided in this embodiment of the invention, the membrane force parameters and internal forces in step S7 are as follows:

[0071]

[0072]

[0073] In the formula, C represents concrete pressure; T represents steel bar tension; k, b, k', and k" are all membrane force parameters; K is the ratio of the tension per unit length of steel bar in the x-direction to the tension per unit length of steel bar in the y-direction; x c The width of the concrete under pressure at point B of section ① is denoted as .

[0074] In the method for analyzing the bearing capacity of a two-span reinforced concrete floor slab considering the influence of column end constraints provided in the embodiments of the present invention, reference is made to... Figure 3 and Figure 4 In step S8, the vertical displacement h corresponding to each force in plate ① is:

[0075]

[0076] In the formula, w B” w C” These are the vertical displacement values ​​of points B and C under the extreme condition.

[0077] The vertical displacement h corresponding to each force in "segment ②" is:

[0078]

[0079] In the formula, w E” w D” These are the vertical displacement values ​​of points B and C under the extreme condition.

[0080] The vertical displacement h corresponding to each force in "plate ③" is:

[0081]

[0082] The vertical displacement h corresponding to each force in "segment ④" is:

[0083]

[0084] The bearing capacity increase factor caused by bending is:

[0085]

[0086] In the formula:

[0087]

[0088]

[0089]

[0090]

[0091]

[0092]

[0093]

[0094]

[0095] In the formula, M im,x M im,y θ represents the resultant moment in the x and y directions of the floor slab caused by bending; θ is the angle between the oblique yield line and the negative y-axis; S is the in-plane shear force of the slab; h is the vertical displacement corresponding to each force on the four slabs; α1, α2, and α3 are the ratios of the stiffness of the long-span side beam, the short-span side beam, and the column to the stiffness of the floor slab, respectively; M” Ax M” Ay The negative support moment in the x and y directions at end A (fixed support end) under the ultimate condition.

[0096] The bearing capacity increase factor caused by axial force is:

[0097] e 1b,x =2(Zsinθ+R+U) / le 1b,y =2(Zcosθ+Y) / L

[0098] e 2b,x =2Zsinθ / le 2b,y = (Zcosθ + J + O) / nL

[0099] In the formula:

[0100]

[0101]

[0102] In the formula, Z, Y, R, and U are the parameters contributing to the increase in bearing capacity in the x and y directions under axial force; u1 and v1 are proportional parameters.

[0103] In the method for analyzing the bearing capacity of a two-span reinforced concrete floor slab considering the influence of column end constraints provided in this embodiment of the invention, the total increase factor in step S9 is:

[0104]

[0105] Ultimate bearing capacity q limit for:

[0106] q limit =eq y,m

[0107] In the formula, e is the total increase factor of the floor slab.

[0108] Beneficial effects: Compared with the prior art, the advantages of the present invention include:

[0109] This embodiment presents a method for analyzing the bearing capacity of a two-span reinforced concrete floor slab that considers the influence of column end constraints. This method not only considers the influence of the vertical deformation of the edge beams on the floor slab's bearing capacity, but also the influence of column end and beam end constraints on the floor slab's ultimate bearing capacity and vertical displacement, etc. By calculating the position parameters of the yield line of the reinforced concrete floor slab, the vertical displacement coefficients of the long and short beams, the yield line load value of the reinforced concrete slab, the negative bending moment at the constraint, and the amplification factor, the bearing capacity of the reinforced concrete floor slab under column end constraints can be obtained. This method establishes force balance equations and moment balance equations, and the ultimate bearing capacity of the two-span reinforced concrete floor slab analyzed by this method has a smaller error compared with the data obtained from actual experiments.

[0110] The above description is merely a preferred embodiment of this application, but the scope of protection of this application is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in this application should be included within the scope of protection of this application. Therefore, the scope of protection of this application should be determined by the scope of the claims.

Claims

1. A method for analyzing the bearing capacity of a two-span concrete floor slab under column end constraint, characterized in that, Includes the following steps: Obtain the position parameters of the concrete floor slab, and calculate the displacement coefficients of different beams based on the position parameters, wherein the displacement coefficients include: displacement coefficients of long beams, displacement coefficients of short beams, and displacement coefficients of intermediate beams; Based on the displacement coefficient, the plate plane equations for the yield line load stage are established. Based on the plate plane equations, the plate rotation angles along the x and y coordinate axes of each plate are obtained. Based on the plate rotation angles, the yield line load value is calculated according to the principle of virtual work. Based on the force balance equation, moment balance equation and elliptical region of each plate, the membrane force parameters are calculated; and the plane equation of each plate under the ultimate state is established, and the bearing capacity amplification factor is calculated according to the plane equation. Based on the principle of force equivalence, the total amplification factor is obtained. Based on the total amplification factor and the yield line load value, the ultimate bearing capacity of the reinforced concrete floor slab under support constraint is obtained.

2. The method for analyzing the bearing capacity of a two-span concrete floor slab under column end constraint according to claim 1, characterized in that, The process of obtaining the location parameters of the concrete floor slab includes: The single-span reinforced concrete slab is divided into several regions, and several slabs are selected. Based on the length, width, and reinforcement parameters of a single slab, the positional parameters of the concrete floor slab are obtained.

3. The method for analyzing the bearing capacity of a two-span concrete floor slab under column end constraint according to claim 2, characterized in that, The components include: two right-angled trapezoidal components with different constraints and two triangular components with different constraints.

4. The method for analyzing the bearing capacity of a two-span concrete floor slab under column end constraint according to claim 2, characterized in that, The process of calculating the displacement coefficients of different beams includes: Based on the location parameters, and according to the load distribution method, constraint form, and length parameters, the mid-span displacement of different beams and the support negative bending moment of beams with fixed ends are calculated. Based on the mid-span displacement, the displacement coefficients of different beams are obtained.

5. The method for analyzing the bearing capacity of a two-span concrete floor slab under column end constraint according to claim 1, characterized in that, The process of obtaining the rotation angle of each plate along the x and y coordinate axes includes: Based on the plane equation of the plate, the normal vector of the plate is calculated. According to the angle between the normal vector and the direction vectors of the x and y coordinate axes, the plate rotation angle along the x and y coordinate axes of each plate is obtained.

6. The method for analyzing the bearing capacity of a two-span concrete floor slab under column end constraint according to claim 1, characterized in that, The process of calculating the yield line load value includes: Based on the plate rotation angle, and according to the principle of virtual work, the external work and internal force work generated under the load are obtained. Based on the fact that the external work is equal to the internal force work, the yield line load value is calculated.

7. The method for analyzing the bearing capacity of a two-span concrete floor slab under column end constraint according to claim 1, characterized in that, The elliptical region is a quadrature region established based on the length and width parameters of the floor slab.

8. The method for analyzing the bearing capacity of a two-span concrete floor slab under column end constraint according to claim 4, characterized in that, The process of calculating the bearing capacity amplification factor includes: Based on the coordinates of each known point, establish the plane equation for each plate under the limit state. Based on the negative bending moment of the support, substitute the x and y coordinate values ​​of the force on each plate into the plane equation to obtain the vertical displacement value of each force. Based on the vertical displacement value, obtain the bearing capacity amplification factor, wherein the bearing capacity amplification factor includes: the bearing capacity amplification factor caused by bending and the bearing capacity amplification factor caused by axial force.

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