Method for calculating bearing capacity of web perforated composite beam under action of hogging moment and application

Abstract

The invention discloses a method for calculating the bearing capacity of a web perforated composite beam under the action of hogging moment, which is characterized in that the web perforated composite beam comprises a concrete slab positioned on the upper side and I-shaped steel positioned on the lower side; a rectangular hole is formed in a web plate of the I-shaped steel; and the bearing capacity is flexural bearing capacity Mu under the action of hogging moment. By the adoption of the design method of the steel-concrete composite beam with the holes formed in the web, the design work of the steel-concrete composite beam with the holes formed in the web can be guided.

Description

Technical Field

[0001] The invention relates to the technical field of building structures, and in particular to a method for calculating the bearing capacity of a composite beam with web openings under negative bending moment and its application. Background Art

[0002] Steel-concrete composite beams are widely used in construction and bridge engineering. Existing research results show that after holes are set in the web, the mechanical properties of steel-concrete composite beams have changed greatly, the stiffness and bearing capacity of the hole area are significantly reduced, and the hole area becomes a weak link. At this time, the bearing capacity of the composite beam is often determined by the ultimate bearing capacity of the hole area. Therefore, for composite beams with holes, it is necessary not only to determine the bearing capacity at the maximum bending moment or maximum shear force, but also to calculate the ultimate bearing capacity of the hole area.

[0003] However, existing literature (for example: Literature 1: KF Chung, RM Lawson. Simplified design of composite beams with large web openings to Eurocode 4 [J]. Journal of Constructional Steel Research, 2001, 57 (2); Literature 2: Pattama d Panedpojaman. Simplified equations for Vierendeel design calculations of composite beams with web openings [J]. Steel and Composite Structures, 2018, 27 (4).) derives the bearing capacity calculation method when the hole parameters change based on the Vierendeel model.

[0004] However, there are few calculation methods for steel-concrete composite beams with web openings in the negative moment zone. In actual engineering, holes are also set in the negative moment zone of continuous composite beams, so it is necessary to study and expand the corresponding calculation methods. Summary of the invention

[0005] The purpose of the present invention is to overcome the deficiencies of the prior art and provide a method for calculating the bearing capacity of a composite beam with web openings under negative bending moment.

[0006] Another object of the present invention is to apply the method for calculating the bearing capacity of composite beams with openings in webs under negative bending moments to the design of composite beams with openings in webs.

[0007] The scheme of this application is as follows:

[0008] A method for calculating the bearing capacity of a composite beam with a web opening under negative bending moment, the composite beam with a web opening comprising: a concrete slab located on an upper side and an I-shaped steel located on a lower side;

[0009] The web of the I-beam is provided with a rectangular opening;

[0010] The opening corner points 1, 2, 3, and 4 represent the upper left corner, upper right corner, lower left corner, and lower right corner of the opening respectively;

[0011] The bearing capacity is the bending bearing capacity M under the action of negative bending moment u ;

[0012] It is calculated using the following formula;

[0013] M u =min{M 11 ,M 12 ,M 21 ,M 22 ,M 31 ,M 41 ,M 42}

[0014] Among them, M 11 It represents the secondary bending moment when the neutral axis of the opening corner point 1 is located on the concrete wing plate;

[0015] M 12 It represents the secondary bending moment when the neutral axis of the opening corner point 1 is located in the reinforcement area;

[0016] M 21 It indicates the secondary bending moment when the neutral axis of the opening corner 2 is located inside the upper flange of the steel beam;

[0017] M 22 It represents the secondary bending moment when the neutral axis of the opening corner point 2 is located inside the web of the steel beam;

[0018] M 31 It indicates the secondary bending moment when the neutral axis of the opening corner point 3 is located inside the lower flange of the steel beam;

[0019] M 41 It indicates the secondary bending moment when the neutral axis of the opening corner 4 is located inside the lower flange of the steel beam;

[0020] M 42 It indicates the secondary bending moment when the neutral axis of the opening corner point 4 is located inside the web of the steel beam.

[0021] Furthermore, the condition of the concrete slab is: the height is h c , width is b e The cross-sectional area is A c , the compressive strength is σ c , the yield strength of steel bar is σ s The tensile reinforcement area is As ;

[0022] The conditions of the I-beam are: the upper flange and the lower flange are made of the same material, the upper flange width b f , the upper flange height is t f , the cross-sectional area of ​​the upper flange is A f , the upper flange yield strength is σ yf ; Lower flange width b fb ; The height of the lower flange is t f , the cross-sectional area of ​​the lower flange is A fb , the yield strength of the lower flange is also σ yf ; A hole is opened on the web, and the height of the upper part of the hole from the lower surface of the upper flange is s t , the height of the opening is h0, and the height of the lower part of the opening from the upper surface of the lower flange is s b , the web thickness is t w , the yield strength is σ yw The cross-sectional area of ​​the web above the opening is A w The cross-sectional area of ​​the web below the opening is A wb .

[0023] Further, M 11 Solve using the following formula:

[0024]

[0025] Among them, y t ,y b ,y 11 ,y 12 , m, a, and c are all calculation parameters.

[0026] Further, M 12 Solve using the following formula:

[0027]

[0028] Among them, y t ,y b ,y 13 ,y 14 , m, a, and c are all calculation parameters.

[0029] Further, M 21 Solve using the following formula:

[0030]

[0031] Among them, y t ,y b ,y 21 ,y 22 ,m,a,y s are all calculation parameters.

[0032] Further, M 22 Solve using the following formula:

[0033]

[0034] Among them, y t ,y b ,y 23 ,y 24 , m, and a are all calculation parameters.

[0035] Further, M 31 Solve using the following formula:

[0036]

[0037] Among them, y t ,y b ,y s ,y sb ,y 31 ,y 32 , m, and a are all calculation parameters. 41 Solve using the following formula:

[0038]

[0039] Among them, y t ,y b ,y sb ,y 41 ,y 42 , m, and a are all calculation parameters.

[0040] Further, M 42 Solve using the following formula:

[0041]

[0042] Among them, y t ,y b ,y sb ,y f ,y 43 ,y 44 , m, and a are all calculation parameters.

[0043] Explanation of symbols:

[0044] M u Indicates the ultimate bearing capacity.

[0045] M 11 It represents the secondary bending moment when the neutral axis NA of the opening corner point 1 is located on the concrete wing plate.

[0046] M 12It represents the secondary bending moment when the neutral axis NA of the opening corner point 1 is located in the reinforcement area.

[0047] M 21 It represents the secondary bending moment when the neutral axis NA of the opening corner point 2 is located inside the upper flange of the steel beam.

[0048] M 22 It represents the secondary bending moment when the neutral axis NA of the opening corner point 2 is located inside the web of the steel beam;.

[0049] M 31 It indicates the secondary bending moment when the neutral axis NA at the corner point 3 of the opening is located inside the lower flange of the steel beam.

[0050] M 41 It indicates the secondary bending moment when the neutral axis NA at the corner point 4 of the opening is located inside the lower flange of the steel beam.

[0051] M 42 It represents the secondary bending moment when the neutral axis NA at the corner point 4 of the opening is located inside the web of the steel beam.

[0052] y t Indicates the distance from the centroid of the section to the upper edge of the concrete slab.

[0053] y b Indicates the distance from the centroid of the section to the lower edge of the concrete slab.

[0054] m represents the distance from the centroidal axis to the neutral axis of the cross section.

[0055] y s Indicates that the maximum plastic axial force of the steel beam section above and below the opening is converted into the steel beam flange reduction height. c Indicates the distance from the cross-section centroid to the reinforcement area.

[0056] y sb Indicates the distance from the centroid of the section to the lower part of the lower flange of the I-beam.

[0057] y f It means that the maximum plastic axial force of the steel beam section above and below the opening is converted into the converted height of the steel beam flange. c represents the height of the steel bar area, which is a represents the converted section height corresponding to the axial force in the converted stress diagram.

[0058] A f represents the cross-sectional area of ​​the upper flange, whose size is b f t f ;

[0059] A fb Represents the cross-sectional area of ​​the lower flange, whose size is b fb t f ;

[0060] Aw It represents the web area above the opening of the steel beam;

[0061] A wb Indicates the web area below the steel beam opening:

[0062] The advantages of the present invention are:

[0063] First, after holes are set in the web, the stress performance of the steel-concrete composite beam changes greatly, and the stiffness and bearing capacity of the hole area are significantly reduced. At this time, the bearing capacity of the composite beam is often determined by the ultimate bearing capacity of the hole area. Therefore, for the composite beam with holes, it is necessary not only to determine the bearing capacity at the maximum bending moment or maximum shear force, but also to calculate the ultimate bearing capacity (bending bearing capacity) of the hole area.

[0064] Second, based on the hollow truss model, this application established a mechanical model of the composite beam with web openings in the negative bending moment zone, established multiple local bending moment functions according to the cross-sectional plastic stress distribution in the hole zone under the limit state, and gave a method for calculating the ultimate bearing capacity of the steel-concrete composite beam with web openings under the action of negative bending moment.

[0065] Third, the calculation method of the present application is also applicable to the case where the upper and lower flange sizes are asymmetric. BRIEF DESCRIPTION OF THE DRAWINGS

[0066] The present invention will be further described in detail below in conjunction with the embodiments in the accompanying drawings, but this does not constitute any limitation to the present invention.

[0067] Figure 1 It is a diagram of the mechanical model of a composite beam with web openings in the negative moment zone.

[0068] Figure 2 It is the force diagram of the hole area.

[0069] Figure 3 It is a schematic diagram of the dimensions of the model of this application.

[0070] Figure 4 It is M 11 Schematic diagram of the solution.

[0071] Figure 5 It is M 12 Schematic diagram of the solution.

[0072] Figure 6 It is M 21 Schematic diagram of the solution.

[0073] Figure 7 It is M 22 Schematic diagram of the solution.

[0074] Figure 8 It is M 31 Schematic diagram of the solution.

[0075] Fig. 9 It is M 41 Schematic diagram of the solution.

[0076] Fig.10 It is M 42 Schematic diagram of the solution. DETAILED DESCRIPTION

[0077] Example 1: A method for calculating the bearing capacity of a composite beam with web openings under negative bending moment

[0078] 1 Theoretical basis

[0079] 1.1 Mechanical model of composite beam with web opening in negative moment zone

[0080] When the opening is large, hollow failure is the most likely failure form of composite beams with web openings. Based on the hollow truss model, this application establishes a mechanical model of composite beams with web openings in the negative moment zone, such as Figure 1 As shown, a theoretical analysis is carried out on this basis.

[0081] Figure 1 middle:

[0082] M z L 、M z R are the overall bending moments at the left and right ends of the hole, respectively;

[0083] M1, M2, M3, and M4 are the local bending moments at the four corners of the hole, which are generated by the shear force of the upper and lower T-shaped sections transmitted along the length direction of the hole. Their values ​​are the product of the corresponding shear force and the length of the hole;

[0084] V b 、N b and V t 、N t are the shear force and axial force of the lower T-shaped and upper T-shaped sections of the hole, respectively;

[0085] l0 and h0 are the length and height of the opening respectively;

[0086] z is the distance between the upper and lower T-section centroids.

[0087] 1.2 Basic Assumptions

[0088] Under the action of negative bending moment, the stress in the hole area of ​​the composite beam is more complicated, with the combined action of axial force, shear force and local bending moment, such as Figure 1 , Figure 2 To simplify the calculation, the following basic assumptions are made:

[0089] 1) The composite beam is a full shear connection.

[0090] 2) There will be no local buckling of the steel beam at the hole.

[0091] 3) The web of the steel beam obeys the Von Mises yield criterion under the combined action of local bending moment and shear force.

[0092] 4) When the composite beam reaches the limit state, plastic hinges can be formed at the four corners of the hole.

[0093] 5) Consider the role of part of the concrete slab in the hole area in resisting bending and shearing, that is, consider the effect of the compressive concrete at hole corner 1 when calculating the local bending moment M1, and consider the effect of the tensile steel bars at hole corner 2 when calculating M2.

[0094] Figure 2 The distribution of shear force and bending moment on the whole section of the hole area and the axial force, shear force and local bending moment in the upper and lower sections of the hole are given in the figure. In addition, it can be seen that: the total bending moment on the whole section (M z ) can be decomposed into the main bending moment (M p ) and local bending moment (M 1～4 ), the principal bending moment is the product of the axial force in the upper and lower T-shaped sections and their lever arms.

[0095] 2 Solution

[0096] 2.1M 11 Solution

[0097] The cross-sectional stress distribution of hole angle 1 under negative bending moment is as follows: Figure 4 As shown in the figure, when the NA axis moves in the concrete slab, the concrete slab is partially tensile and partially compressed. In the calculation, only the concrete in the compression zone and the steel bars in the tension zone are considered to participate in the work, and the concrete in the tension zone does not participate in the work. According to the different positions of the neutral axis NA, the local bending moment function M 1j It consists of two functions.

[0098] M 11 The solution is:

[0099] (1)

[0100] (2) For a, it satisfies: 0≤a≤y 11 -c

[0101] (3) Solve for a in [0, y 11 -c] range, M 11 The minimum value of is taken as the result.

[0102] 2.2M 12 Solution

[0103] M 12 Solve using the following formula:

[0104]

[0105] Among them, y t ,y b ,y 13 ,y 14 , m, a, and c are all calculation parameters.

[0106] 2.3M 21 Solution

[0107] M 21 Solve using the following formula:

[0108]

[0109] Among them, y t ,y b ,y 21 ,y 22 ,m,a,y s are all calculation parameters.

[0110] 2.4M 22 Solution

[0111] M 22 Solve using the following formula:

[0112]

[0113] Among them, y t ,y b ,y 23 ,y 24 , m, and a are all calculation parameters.

[0114] 2.5M 31 Solution

[0115] M 31 Solve using the following formula:

[0116]

[0117] Among them, y t ,y b ,y s ,y sb ,y 31 ,y 32 , m, and a are all calculation parameters.

[0118] 2.6M 41 Solution

[0119] M 41 Solve using the following formula:

[0120]

[0121] Among them, yt ,y b ,y sb ,y 41 ,y 42 , m, and a are all calculation parameters.

[0122] 2.7M 42 Solution

[0123] M 42 Solve using the following formula:

[0124]

[0125] Among them, y t ,y b ,y sb ,y f ,y 43 ,y 44 , m, and a are all calculation parameters.

[0126] Explanation of symbols:

[0127] M u Indicates the ultimate bearing capacity.

[0128] M 11 It represents the secondary bending moment when the neutral axis NA of the opening corner point 1 is located on the concrete wing plate.

[0129] M 12 It represents the secondary bending moment when the neutral axis NA of the opening corner point 1 is located in the reinforcement area.

[0130] M 21 It represents the secondary bending moment when the neutral axis NA of the opening corner point 2 is located inside the upper flange of the steel beam.

[0131] M 22 It represents the secondary bending moment when the neutral axis NA of the opening corner point 2 is located inside the web of the steel beam.

[0132] M 31 It represents the secondary bending moment when the neutral axis NA at the corner point 3 of the opening is located inside the lower flange of the steel beam.

[0133] M 41 It represents the secondary bending moment when the neutral axis NA at the corner point 4 of the opening is located inside the lower flange of the steel beam.

[0134] M 42 It represents the secondary bending moment when the neutral axis NA at the corner point 4 of the opening is located inside the web of the steel beam.

[0135] y t Indicates the distance from the centroid of the section to the upper edge of the concrete slab.

[0136] y b Indicates the distance from the centroid of the section to the lower edge of the concrete slab.

[0137] a is the converted section height corresponding to the axial force in the converted stress diagram.

[0138] m represents the distance between the cross-section centroid and the plastic neutral axis.

[0139] y s It means converting the maximum plastic axial force of the steel beam section above and below the opening into the converted height of the steel beam flange.

[0140] y c Indicates the distance from the cross-section centroid to the reinforcement area.

[0141] y sb Indicates the distance from the centroid of the section to the lower part of the lower flange of the I-beam.

[0142] y f It means converting the maximum plastic axial force of the steel beam section above and below the opening into the converted height of the steel beam flange.

[0143]

[0144] The above embodiments are preferred implementation modes of the present invention and are only used to facilitate the description of the present invention. They are not intended to limit the present invention in any form. Any person with ordinary knowledge in the relevant technical field, if they do not depart from the scope of the technical features of the present invention, can make equivalent embodiments by partial changes or modifications to the technical contents disclosed in the present invention, and they still fall within the scope of the technical features of the present invention without departing from the technical features of the present invention.

Claims

1. A method for calculating the bearing capacity of a composite beam with a web opening under negative bending moment, the composite beam with a web opening comprising: A concrete slab on the upper side and an I-beam on the lower side; The web of the I-beam is provided with a rectangular opening; The opening corner points 1, 2, 3, and 4 represent the upper left corner, upper right corner, lower left corner, and lower right corner of the opening respectively; It is characterized in that the bearing capacity is the bending bearing capacity M under the action of negative bending moment u ; It is calculated using the following formula; M u =min{M 11 ,M 12 ,M 21 ,M 22 ,M 31 ,M 41 ,M 42 } Among them, M 11 It represents the secondary bending moment when the neutral axis of the opening corner point 1 is located on the concrete wing plate; M 12 It represents the secondary bending moment when the neutral axis of the opening corner point 1 is located in the reinforcement area; M 21 It indicates the secondary bending moment when the neutral axis of the opening corner 2 is located inside the upper flange of the steel beam; M 22 It represents the secondary bending moment when the neutral axis of the opening corner point 2 is located inside the web of the steel beam; M 31 It indicates the secondary bending moment when the neutral axis of the opening corner point 3 is located inside the lower flange of the steel beam; M 41 It indicates the secondary bending moment when the neutral axis of the opening corner 4 is located inside the lower flange of the steel beam; M 42 It indicates the secondary bending moment when the neutral axis of the opening corner point 4 is located inside the web of the steel beam.

2. The method for calculating the bearing capacity of composite beams with web openings under negative bending moment according to claim 1 is characterized in that: The conditions of the concrete slab are: the height is h c , width is b e The cross-sectional area is A c , the compressive strength is σ c , the yield strength of steel bar is σ s The tensile reinforcement area is A s ; The conditions of the I-beam are: the upper flange and the lower flange are made of the same material, the upper flange width b f , the upper flange height is t f , the cross-sectional area of ​​the upper flange is A f , the upper flange yield strength is σ yf ; Lower flange width b fb ; The height of the lower flange is also t f , the cross-sectional area of ​​the lower flange is A fb ; A hole is opened on the web, and the height of the upper part of the hole from the lower surface of the upper flange is s t , the height of the opening is h0, and the height of the lower part of the opening from the upper surface of the lower flange is s b , the web thickness is t w , the yield strength is σ yw The cross-sectional area of ​​the web above the opening is A w The cross-sectional area of ​​the web below the opening is A wb .

3. The method for calculating the bearing capacity of composite beams with web openings under negative bending moment according to claim 1 is characterized in that: M 11 Solve using the following formula: Among them, y t ,y b ,y 11 ,y 12 , m, a, and c are all calculation parameters.

4. The method for calculating the bearing capacity of composite beams with web openings under negative bending moment according to claim 1 is characterized in that: M 12 Solve using the following formula: Among them, y t ,y b ,y 13 ,y 14 , m, a, and c are all calculation parameters.

5. The method for calculating the bearing capacity of composite beams with web openings under negative bending moment according to claim 1 is characterized in that: M 21 Solve using the following formula: Among them, y t ,y b ,y s ,y 21 ,y 22 , m, and a are all calculation parameters.

6. The method for calculating the bearing capacity of composite beams with web openings under negative bending moment according to claim 1 is characterized in that: M 22 Solve using the following formula: Among them, y t ,y b ,y 23 ,y 24 , m, and a are all calculation parameters.

7. The method for calculating the bearing capacity of composite beams with web openings under negative bending moment according to claim 1 is characterized in that: M 31 Solve using the following formula: Among them, y t ,y b ,y s ,y sb ,y 31 ,y 32 , m, and a are all calculation parameters.

8. The method for calculating the bearing capacity of composite beams with web openings under negative bending moment according to claim 1 is characterized in that: Among them, y t ,y b ,y sb ,y 41 ,y 42 , m, and a are all calculation parameters.

9. The method for calculating the bearing capacity of composite beams with web openings under negative bending moment according to claim 1 is characterized in that: M 42 Solve using the following formula: Among them, y t ,y b ,y sb ,y f ,y 43 ,y 44 , m, and a are all calculation parameters.

10. Application of the method for calculating the bearing capacity of composite beams with web openings under negative bending moment as claimed in claim 1 in the design of composite beams with web openings.

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