A method for calculating the bearing capacity of concrete floor considering the interaction between beams and slabs

Abstract

The present invention relates to the field of calculation methods for the ultimate load of a floor slab, and discloses a method for calculating the bearing capacity of a concrete floor slab taking into account the interaction between a beam and a slab. The present invention takes into account the influence of the torsional deformation and vertical deformation of the beam on the bearing capacity of the floor slab, quantifies the restraining effect of the beam on the slab, and applies the method to the calculation formula for the ultimate bearing capacity of the floor slab. The method has the advantages of sufficient theoretical basis, simple process, and reliable results. The calculation results can meet the needs of actual projects and have wide adaptability.

Description

Technical Field

[0001] The invention relates to the field of calculation of slab ultimate load, and in particular to a method for calculating the bearing capacity of a concrete floor slab taking into account the interaction between beams and slabs. Background Art

[0002] As an important load-bearing component of the structure, the load-bearing capacity of the floor is of vital importance to the building, so the calculation method of the load-bearing capacity of the floor is also very important. Although domestic and foreign experts and scholars have never stopped studying this theory, there is always a certain error between the research results and the experimental results.

[0003] When the beam is under different torsional deformation and vertical deformation conditions, the bearing capacity of the floor will certainly be affected differently. Therefore, it is very necessary to establish a method for calculating the bearing capacity of the floor under the influence of the interaction between beams and slabs. Summary of the invention

[0004] In view of the above problems, the present invention provides a method for calculating the ultimate bearing capacity of a floor slab taking into account the influence of beam deformation on the floor slab under the ultimate load condition. The specific scheme is as follows:

[0005] A method for calculating the bearing capacity of a concrete floor considering the interaction between beams and slabs, wherein the floor comprises a slab and a beam supporting the slab, and the slab is a rectangular slab, and comprises the following steps:

[0006] Step 1, determine the yield line diagram formed by the floor under uniform load conditions, consider the interaction between beams and slabs, and assume the position and variation parameters of the yield line;

[0007] Step 2, using the linear difference method to solve the yield line bending moment of each plate;

[0008] Step 3, determine the length, width and height parameters of the beam and plate;

[0009] Step 4, calculating the negative bending moment ratio M′1 / M1 at the spring boundary and the fixed boundary according to the parameters and formulas in steps 1 to 3;

[0010] Step 5: Calculate the negative bending moment ratio M′ of the plate in the y-axis direction under the elastic support constraint and the fixed support constraint 1y / M 1y , the negative bending moment ratio of the plate in the x-axis direction under elastic support constraint and fixed support constraint M′ 1x / M 1x ;

[0011] Step 6: Calculate the short-term stiffness B of the beam bending member s and the mid-span displacement w of the beam under different boundary conditions max ;

[0012] Step 7: Calculate the external work W done by the uniformly distributed load q when considering the beam deformation, the internal work D done by the upper line of each plate boundary, and the height parameter h of each plate i and the area parameter S of each plate i Where i = 1, 2, 3, 4, ...;

[0013] Step 8, calculate the ultimate load q of the floor when the beam deforms.

[0014] Furthermore, in step 4, the negative bending moment ratio M′1 / M1 under the spring boundary and the fixed boundary has the following relationship:

[0015]

[0016] Among them, k and k' are parameters.

[0017] Further, in step 5, M′ 1y / M 1y The ratio relationship is:

[0018]

[0019]

[0020] M′ 1x With M 1x The ratio relationship is:

[0021]

[0022] R′2=k x Gβhb 3 ;

[0023] Among them, l is the length of the side of the plate in the y-axis direction, l1 is the length of the main beam in the y-axis direction; L is the length of the side of the plate in the x-axis direction, L1 is the length of the main beam, G is the shear modulus of concrete, E is the elastic modulus of concrete, D is the bending stiffness of the plate, β is the torsion coefficient of the rectangular section member, h and b are the height and width of the beam section, h′ is the thickness of the concrete plate; μ is Poisson's ratio, R1, R2, R′1, R′2, k x , k y All are parameters.

[0024] Further, in step 6, B s and w max for:

[0025]

[0026] Among them, E s is the elastic modulus of the reinforcement of the beam, A sis the cross-sectional area of ​​the longitudinal reinforcement in the tension zone of the beam, h0 is the effective cross-sectional height of the beam, ψ is the strain non-uniformity coefficient of the longitudinal tension reinforcement between the cracks of the beam, a E =E s / E c , E c is the elastic modulus of concrete, ρ is the longitudinal reinforcement ratio of the tensile zone of the beam; r′ f It is the ratio of the cross-sectional area of ​​the tension flange of the beam to the effective cross-sectional area of ​​the web of the beam;

[0027] w max The maximum mid-span displacement w of the beam under the simply supported trapezoidal load on both sides max(a) :

[0028]

[0029] in: q is the ultimate load that the plate can withstand, and n is the ratio of the length of the trapezoidal hypotenuse projected on the horizontal plane to L1;

[0030] The maximum mid-span displacement w of the beam under the action of a simply supported triangular load on both sides max(b) :

[0031]

[0032] Where: q2 = qnl;

[0033] Maximum mid-span displacement w of the beam under trapezoidal load with one side fixed and the other simply supported max(c) :

[0034]

[0035]

[0036] Maximum mid-span displacement w of a beam under a triangular load with one side fixed and the other simply supported max(d) :

[0037]

[0038]

[0039] The maximum mid-span displacement w of the beam under the action of a trapezoidal load with two sides fixed max(e) :

[0040]

[0041]

[0042] The maximum mid-span displacement w of the beam under the action of a simply supported triangular load with two sides fixed max(f) :

[0043]

[0044]

[0045] Furthermore, the expression of W in step 7 is:

[0046]

[0047] Among them: λ = L / l, ξ, ξ′ are the variation parameters of each plate in the x-axis direction.

[0048] Furthermore, the expression of D in step 7 is:

[0049]

[0050] in:

[0051] β0=M x / M y ; β1=M′ y / M y ; β′1=M″ y / M y ; β2=M′ x / M x ; β′1=M″ x / M x

[0052] M x , M′ x , M″ x 、M y , M′ y , M″ y is the bending capacity per unit width at different positions of the plate.

[0053] Furthermore, the expression of q in step 8 is:

[0054]

[0055] in:

[0056]

[0057] A2=[6λ-(ξ+ξ′)]l 2 +4(h1S1+h2S2+h3S3+h4S4)

[0058] Among them, x1 and x2 are the ratios of the ultimate displacement of the beam in the x-axis direction to the ultimate displacement of the plate; y1 and y2 are the ratios of the ultimate displacement of the beam in the y-axis direction to the ultimate displacement of the plate, and η is the variation parameter of each plate in the y-axis direction.

[0059] Compared with the prior art, the advantages of the present invention are as follows:

[0060] The present invention considers the influence of the torsional deformation and vertical deformation of the beam on the bearing capacity of the floor, and establishes a calculation method for the bearing capacity of the concrete floor considering the interaction between the beam and the slab. The present invention quantifies the constraint effect of the beam on the slab, and can more accurately calculate the ultimate bearing capacity of the floor compared to the existing theory. The calculation method has sufficient theoretical basis, simple process, reliable results, and the calculation results can meet the needs of actual projects, and has wide adaptability. BRIEF DESCRIPTION OF THE DRAWINGS

[0061] Figure 1 is the yield line diagram of the plate considering the deformation of the beam;

[0062] Figure 2 It is a schematic diagram of plate deformation (1 / 4 structure);

[0063] Figure 3 yes Figure 1 Deformation diagram of the cd section of the middle floor slab;

[0064] Figure 4 yes Figure 1 Deformation diagram of the ab section of the middle floor slab;

[0065] Figure 5 It is the location diagram of the primary and secondary beams of the floor;

[0066] Figure 6 This is a simplified diagram of the calculation of negative bending moment at a fixed boundary.

[0067] Figure 7 This is a simplified diagram for calculating the negative bending moment at the elastic boundary;

[0068] Figure 8 It is the plate strip division diagram in the y-axis direction;

[0069] Fig. 9 It is the plate division diagram in the x-axis direction;

[0070] Fig.10 It is the trapezoidal load diagram of the beam under the simple support at both ends;

[0071] Fig.11 It is the triangular load diagram of the beam under the simple support constraints at both ends;

[0072] Fig.12 It is a trapezoidal load on a beam with one end fixed and the other end simply supported;

[0073] Fig.13 It is the triangular load of the beam under the constraint that one end is fixed and the other end is simply supported;

[0074] Fig.14 It is the trapezoidal load of the beam under the fixed support at both ends;

[0075] Fig.15 It is the triangular load of the beam under the constraints of fixed supports at both ends.

[0076] Fig.16 It is a calculation flow chart of the present invention.

[0077] Reference numerals:

[0078] 1. Plate; 2. Beam; 21. Main beam; 22. Secondary beam; 3. Column; 4. Plate strip in the y-axis direction; 5. Plate strip in the x-axis direction. DETAILED DESCRIPTION

[0079] The specific scheme of this scheme is described in detail below with reference to the attached drawings

[0080] Example 1

[0081] A method for calculating the bearing capacity of a concrete floor considering the interaction between beams and slabs, wherein the floor comprises a slab and a beam supporting the slab, the slab is a rectangular slab, and the rectangular slab in this embodiment is a bidirectional slab. The specific calculation steps are as follows:

[0082] Step 1: Determine the yield line diagram of the floor slab under uniform load conditions, consider the interaction between beams and slabs, and assume the position and variation parameters of the yield line.

[0083] Step 2, using the linear difference method to solve the yield line bending moment of each plate;

[0084] Step 3, determine the length, width and height parameters of the beam and plate;

[0085] Step 4, calculating the negative bending moment ratio M′1 / M1 at the fixed boundary and the spring boundary according to the parameters and formulas in steps 1 to 3;

[0086] Step 5: Calculate the negative bending moment ratio M′ of the plate in the y-axis direction under the elastic support constraint and the fixed support constraint 1y / M 1y , the negative bending moment ratio of the plate in the x-axis direction under elastic support constraint and fixed support constraint M′ 1x / M 1x ;

[0087] Step 6: Calculate the short-term stiffness B of the beam bending member s and the mid-span displacement w of the beam under different boundary conditions max ;

[0088] Step 7: Calculate the external work W done by the uniformly distributed load q when considering the beam deformation, the internal work D done by the upper line of each plate boundary, and the height parameter h of each plate i and the area parameter S of each plate i Where i = 1, 2, 3, 4, ...;

[0089] Step 8, calculate the ultimate load q of the floor when the beam deforms.

[0090] The following is a detailed description of each step:

[0091] Step 1 to Step 3:

[0092] 1. Under the limit state, when the beam is deformed, it is assumed that the position of the boundary line does not change, but the value of the bending moment on the boundary line changes. Considering the different effects of different stiffness conditions on the bending moment of the plate when the beam is subjected to the limit load, the limit bending moment of each plate in the yield line diagram is M when the beam stiffness is large enough (no deformation or very small deformation). u , the ultimate bending moment of each plate in the yield line diagram when the beam stiffness is finite is (1-x f )M u , where x f is the ratio of the mid-span displacement of the beam to the mid-span displacement of the plate, and Mu is the ultimate bending moment on the boundary line.

[0093] 2. If Figure 1 As shown in the figure, when the deformation of the beam is considered, the block situation of the rectangular plate is considered. When the beam is deformed, it is considered that the position of the boundary does not change, but the change of the bending moment value on the boundary is considered. In the figure, the change parameters of each plate in the x-axis direction are ξ and ξ′, and the change parameter of each plate in the y-axis direction is η; the value when the deformation of the beam is not considered is taken, and it does not change with the deformation of the beam.

[0094] 3. Such as Figure 1 As shown, x1 and x2 are the ratios of the limit displacement of the beam in the x-axis direction to the limit displacement of the plate; y1 and y2 are the ratios of the limit displacement of the beam in the y-axis direction to the limit displacement of the plate.

[0095] 4. The internal work of each plate due to the deformation of the beam is not considered.

[0096] Step 4:

[0097] The beams include four main beams located at the outer edges of the plate and a secondary beam arranged along the y-axis direction and located between two main beams arranged along the x-axis direction, and the negative bending moment ratio at the spring boundary and the fixed boundary at the connection position between the main beam and the secondary beam is considered. Figure 5 As shown in the figure, when the torsional stiffness K2 of the main beam is relatively large, the boundary condition at the connection position between the secondary beam and the main beam can be simplified as follows: Figure 6 For the fixed support shown in FIG. 1 , when the torsional stiffness of the main beam is relatively small, the boundary condition at the connection between the secondary beam and the main beam can be simplified as follows: Figure 7 For the spring support shown in the figure, under the same load, the negative bending moment ratio at the spring boundary and the fixed boundary has the following relationship:

[0098]

[0099]

[0100] l1 is the length of the secondary beam; L1 is the length of the main beam; E is the elastic modulus of concrete; I is the section moment of inertia of the concrete beam; x is the distance between the middle point of the secondary beam and the main beam; G is the shear modulus of concrete; β is the torsion coefficient of the rectangular section member, and the specific parameters are shown in Table 1; h and b are the height and width of the beam section, respectively.

[0101] Table 1 Torsion coefficient β of rectangular cross-section members

[0102]

[0103] Step 5:

[0104] Taking the y-axis strip as an example, when considering the influence of the ratio of the torsional stiffness of the beam and the bending stiffness of the plate on the negative bending moment at the plate support, Figure 8 As shown in the figure, the plate is first divided into a large number of plate strips perpendicular to the length of the beam, and the negative bending moment of each plate strip at the support is calculated separately (such as Figure 7 ), and then sum to get the total negative bending moment M′ 1y At the same time, the same method is used to calculate the negative bending moment of each plate strip under fixed boundary conditions (such as Figure 6 ), and then sum to get the total negative bending moment M 1y , and finally get M′ 1y With M 1y ratio relationship.

[0105] Negative bending moment at the plate edge at the elastic support:

[0106]

[0107] The negative bending moment of the plate edge at the fixed support is:

[0108]

[0109]

[0110] in:

[0111] R2=k y Gβhb 3 ;

[0112] Similarly, for the plate strip in the x-axis direction:

[0113]

[0114] in:

[0115]

[0116] Among them, l is the length of the side of the plate in the y-axis direction, l1 is the length of the main beam in the y-axis direction; L is the length of the side of the plate in the x-axis direction, L1 is the length of the main beam in the x-axis direction, G is the shear modulus of concrete, E is the elastic modulus of concrete, D is the bending stiffness of the plate, β is the torsion coefficient of the rectangular section member, h and b are the height and width of the beam section respectively, and h′ is the thickness of the plate; μ is the Poisson's ratio, which is 0.2 for concrete, and Δ 2P is the displacement in the bending direction of the simply supported beam at the beam end under the ultimate load of the floor slab, R1, R2, R′1, R′2, k, k′, k x , k′ x , k y , k′ y All are parameters.

[0117] Step 6:

[0118] The main actual boundary conditions for calculating beam deformation in frame structures include three situations: fixed support on both sides, fixed support on one side and simple support on the other side, and simple support on both sides. According to the load form transferred from the plate to the beam, it can be divided into two load distribution forms: trapezoidal load and triangular load. This step will calculate the mid-span deformation of the beam under these three boundary conditions and two load conditions respectively.

[0119] According to the relevant provisions of the "Concrete Structure Design Code", the short-term stiffness B of the reinforced concrete flexural member in step 5 is s for:

[0120]

[0121] Among them, E s is the elastic modulus of the reinforcement of the beam, A s is the cross-sectional area of ​​the longitudinal reinforcement in the tension zone of the beam, h0 is the effective cross-sectional height of the beam, ψ is the strain non-uniformity coefficient of the longitudinal tension reinforcement between the cracks of the beam, a E =E s / E c , E c is the elastic modulus of concrete, ρ is the longitudinal reinforcement ratio of the tensile zone of the beam; r′ f It is the ratio of the cross-sectional area of ​​the tension flange of the beam to the effective cross-sectional area of ​​the web of the beam;

[0122] (1) Simply supported on both sides

[0123] like Fig.10 As shown in the figure, the maximum mid-span displacement w of the beam under the action of simply supported trapezoidal loads on both sides max(a) for:

[0124]

[0125] in: q is the ultimate load that the plate can withstand, and n is the ratio of the length of the trapezoidal hypotenuse projection on the horizontal plane to L1;.

[0126] like Fig.11 As shown in the figure, the maximum mid-span displacement w of the beam under the simply supported triangular load on both sides max(b) for:

[0127]

[0128] Among them: q2=qnl, q is the ultimate load that the plate can withstand.

[0129] (2) One side is fixedly supported while the other side is simply supported

[0130] like Fig.12 , 13 As shown in the figure, it is the load distribution transferred to the beam by the plate under the boundary condition of one side being fixed and the other side being simply supported. In this case, the maximum displacement of the beam is not at the mid-span. Studies have shown that the maximum displacement of the beam under the action of uniformly distributed load is at a distance of 0.422L from the simply supported side. In order to simplify the calculation, this paper takes the deflection in the middle of the beam as the maximum deflection of the beam.

[0131] like Fig.12 As shown in Figure 2, the maximum mid-span displacement w of the beam under the trapezoidal load with one side fixed and the other simply supported is max(c) for:

[0132]

[0133] in:

[0134]

[0135] like Fig.13 As shown in the figure, the maximum mid-span displacement w of the beam under the triangular load with one side fixed and the other side simply supported max(d) for:

[0136]

[0137] in:

[0138]

[0139] (3) Support on both sides

[0140] like Fig.14 As shown in the figure, the maximum mid-span displacement w of the beam under the trapezoidal load with fixed supports on both sides max(e) for:

[0141]

[0142] in:

[0143]

[0144] like Fig.15 As shown in Figure 2, the maximum mid-span displacement of the beam under the triangular load

[0145] w max(f) for:

[0146]

[0147] in:

[0148]

[0149] Step 7:

[0150] Assuming the virtual displacement of the midpoint of the plate is 1, the external work done by the uniformly distributed load q when the beam is deformed in step 5 is:

[0151]

[0152] in:

[0153] λ = L / l;

[0154]

[0155]

[0156]

[0157]

[0158] When considering the deformation of the beam, the internal work done on the boundary lines of each plate is:

[0159]

[0160] in:

[0161] β0=M x / M y ; β1=M′ y / M y ; β′1=M″ y / M y ; β2=M′ x / M x ; β′1=M″ x / M x

[0162] Among them, M x , M′ x , M″ x 、M y , M′ y , M″ yare the bending capacity per unit width at different positions of the plate.

[0163] Step 8:

[0164] According to the principle of virtual work D = W, we get:

[0165]

[0166] in:

[0167]

[0168] A2=[6λ-(ξ+ξ′)]l 2 +4(h1S1+h2S2+h3S3+h4S4)

[0169] Example 2

[0170] If the rectangular plate is a one-way plate, then x1, x2, y1, y2 in step 3 of Example 1 are changed to make x1=x2=1 or y1=y2=1, that is, the rectangular plate is a one-way plate, and the other calculation steps are consistent with Example 1.

[0171] Example 3

[0172] Two groups of floor specimens (numbered S1 and S2) were selected to verify the accuracy and efficiency of this calculation method. The parameters of the specimens are as follows: S1 and S2 both use floor dimensions of 4800mm×2300mm×70mm, beam cross-section dimensions of 100mm×180mm, and column cross-section dimensions of 200mm×200mm; S1 uses a double-layer bidirectional Steel bar arrangement, S2 adopts separate type Reinforcement arrangement: HRB400 is used for the steel bars in the slab, beam and column, with diameters of 6mm, 10mm and 12mm, respectively, yield strengths of 424.6MPa, 417.0MPa and 415.8MPa, ultimate strengths of 605.9MPa, 565.0MPa and 553.2MPa, respectively, and stirrup diameter of 6mm. The specimens are made of C30 commercial concrete, with a 28d cubic compressive strength of 30.9MPa and a concrete cover thickness of 15mm.

[0173] Tables 2 and 3 give the calculated and tested values ​​of the ultimate bearing capacity of S1 and S2, where Table 2 shows the ultimate load of the floor calculated using the beam failure theory, and Table 3 shows the ultimate load of the floor calculated using the beam deformation theory, where Ps (Pf, Psp) are the ultimate bearing capacities of the floor under simply supported (fixed support, elastic) boundary conditions.

[0174] Table 2 Comparison between calculated and tested values ​​of ultimate bearing capacity of concrete slab when beam is damaged

[0175]

[0176] P1: Crushing of the side beam in the middle span; P2: Crushing of the middle beam in the middle span; P3: Punching at the plate loading point; P4: Crushing failure of the plate corner; P5: Shearing of the side beam.

[0177] Table 3 Comparison between calculated and tested values ​​of ultimate bearing capacity of concrete slab when beam is deformed

[0178]

[0179] In summary, the bearing capacity of the fixed-support plate is twice that of the simply-supported plate calculated using the traditional yield line theory. The present invention quantifies the torsion and vertical restraint effects of the beam on the plate. The calculation method has sufficient theoretical basis, simple process, reliable results, and can accurately calculate the ultimate bearing capacity of the plate when considering the torsion deformation and vertical deformation of the beam, which can provide strong theoretical guidance for engineering applications. It is suitable for the design and analysis of concrete floor slabs when considering the torsion deformation and vertical deformation of the beam.

[0180] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or replace some or all of the technical features therein with equivalents. However, these modifications or replacements do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for calculating the bearing capacity of a concrete floor considering the interaction between beams and slabs, wherein the floor comprises a slab and a beam supporting the slab, and the slab is a rectangular slab, characterized in that: The following steps are involved: Step 1, determine the yield line diagram formed by the floor slab under uniform load conditions, consider the interaction between beams and slabs, and assume the position and variation parameters of the yield line; Step 2, using the linear difference method to solve the yield line bending moment of each plate; Step 3, determine the length, width and height parameters of the beam and plate; Step 4, calculating the negative bending moment ratio M′1 / M1 at the spring boundary and the fixed boundary according to the parameters and formulas in steps 1 to 3; The negative bending moment ratio M′1 / M1 under the spring boundary and fixed boundary has the following relationship: Among them, k and k' are parameters; Step 5: Calculate the negative bending moment ratio M′ of the plate in the y-axis direction under the elastic support constraint and the fixed support constraint 1y / M 1y , the negative bending moment ratio of the plate in the x-axis direction under elastic support constraint and fixed support constraint M′ 1x / M 1x ; M′ 1y / M 1y The ratio relationship is: R2=k y Gβhb 3 ; M′ 1x With M 1x The ratio relationship is: R2=k x Gβhb 3 ; Among them, l is the length of the side of the plate in the y-axis direction, l1 is the length of the main beam in the y-axis direction; L is the length of the side of the plate in the x-axis direction, L1 is the length of the main beam in the x-axis direction, G is the shear modulus of concrete, E is the elastic modulus of concrete, D is the bending stiffness of the plate, β is the torsion coefficient of the rectangular cross-section member, h and b are the height and width of the beam section, h′ is the thickness of the plate, μ is the Poisson's ratio, R1, R2, R1′, R2′, k x , k y All are parameters; Step 6: Calculate the short-term stiffness B of the beam bending member s and the mid-span displacement w of the beam under different boundary conditions max ; B s and w max for: Among them, E s is the elastic modulus of the reinforcement of the beam, A s is the cross-sectional area of ​​the longitudinal reinforcement in the tension zone of the beam, h0 is the effective cross-sectional height of the beam, ψ is the strain non-uniformity coefficient of the longitudinal tension reinforcement between the cracks of the beam, a E =E s / E c , E c is the elastic modulus of concrete, ρ is the longitudinal reinforcement ratio of the tensile zone of the beam; r f ′ is the ratio of the cross-sectional area of ​​the tension flange of the beam to the effective cross-sectional area of ​​the web of the beam: w max The maximum mid-span displacement w of the beam under the simply supported trapezoidal load on both sides max(a) : in: q is the ultimate load that the plate can withstand, and n is the ratio of the length of the trapezoidal hypotenuse projection on the horizontal plane to L1; The maximum mid-span displacement w of the beam under the action of a simply supported triangular load on both sides max(b) : Where: q2 = qnl; Maximum mid-span displacement w of the beam under trapezoidal load with one side fixed and the other simply supported max(c) : Maximum mid-span displacement w of a beam under a triangular load with one side fixed and the other simply supported max(d) : The maximum mid-span displacement w of the beam under the action of a trapezoidal load with two sides fixed max(e) : The maximum mid-span displacement w of the beam under the action of a simply supported triangular load with two sides fixed max(f) : Step 7: Calculate the external work W done by the uniformly distributed load q when considering the beam deformation, the internal work D done by the upper line of each plate boundary, and the height parameter h of each plate i and the area parameter S of each plate i Where i = 1, 2, 3, 4, ...; the expression of W is: Among them, λ = L / l, ξ, ξ′ are the variation parameters of each plate in the x-axis direction; The expression of D is: in: β0=M x / M y ; β1=M′ y / M y ; β′1=M″ y / M y ; β2=M′ x / M x ; β″1=M″ x / M x M x , M′ x , M″ x 、M y , M′ y , M″ y is the bending bearing capacity per unit width at different positions of the plate; Step 8, calculate the ultimate load q of the floor when the beam deforms. The expression of q is: in: A2=[6λ-(ξ+ξ′)]l 2 +4(h1S1+h2S2+h3S3+h4S4) Among them, x1 and x2 are the ratios of the ultimate displacement of the beam in the x-axis direction to the ultimate displacement of the plate; y1 and y2 are the ratios of the ultimate displacement of the beam in the y-axis direction to the ultimate displacement of the plate, and η is the variation parameter of each plate in the y-axis direction.

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