Calculation method and application of bearing capacity of composite beam with web opening under negative bending moment

Abstract

The present invention discloses a method for calculating the bearing capacity of a composite beam with an opening in the web under negative bending moment. The key points are as follows: the composite beam with an opening in the web includes a concrete slab located on the upper side and an I-shaped steel section located on the lower side; a rectangular opening is provided in the web of the I-shaped steel section; the bearing capacity is the flexural bearing capacity M under negative bending moment. u . By adopting the design method of a steel-concrete composite beam with an opening in the web of the present application, the design work of a steel-concrete composite beam with an opening in the web can be guided.

Description

Technical Field

[0001] The present invention relates to the technical field of building structures, and particularly 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 building engineering and bridge engineering. Existing research results show that: after holes are provided in the web, the mechanical properties of steel-concrete composite beams change greatly, the stiffness and bearing capacity in 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 a composite beam with openings, it is not only necessary to determine the bearing capacity at the maximum bending moment or maximum shear force, but more importantly, to calculate the ultimate bearing capacity of the hole area.

[0003] However, existing literature (for example: Literature 1: K.F Chung, R.M Lawson. Simplified design of composite beams with large web openings to Eurocode 4[J]. Journal of Constructional Steel Research, 2001, 57(2); Literature 2: Pattamad Panedpojaman. Simplified equations for Vierendeel design calculations of composite beams with web openings[J]. Steel and Composite Structures, 2018, 27(4).) derived a calculation method for bearing capacity when hole parameters change according to the Vierendeel model.

[0004] However, there are few calculation methods for steel-concrete composite beams with web openings in the negative bending moment area. In actual engineering, there will also be cases where holes are provided in the negative bending moment area of continuous composite beams. Therefore, 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 purpose of the present invention is to adopt the application of the method for calculating the bearing capacity of a composite beam with web openings under negative bending moment in the design of composite beams with web openings.

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

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

[0009] A rectangular opening is provided in the web of the I-shaped steel section;

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

[0011] The bearing capacity is the flexural bearing capacity M under 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 represents the secondary moment when the neutral axis of corner point 1 of the opening is located in the concrete flange;

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

[0016] M 21 represents the secondary moment when the neutral axis of corner point 2 of the opening is located within the upper flange of the steel beam;

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

[0018] M 31 represents the secondary moment when the neutral axis of corner point 3 of the opening is located within the lower flange of the steel beam;

[0019] M 41 represents the secondary moment when the neutral axis of corner point 4 of the opening is located within the lower flange of the steel beam;

[0020] M 42 represents the secondary moment when the neutral axis of corner point 4 of the opening is located within the web of the steel beam.

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

[0022] The conditions for the I-beam are as follows: The materials of the upper flange and the lower flange are the same. The width of the upper flange is b f , the height of the upper flange is t f , the cross-sectional area of the upper flange is A f , and the yield strength of the upper flange is σ yf ; the width of the lower flange is b fb ; the height of the lower flange is t f , the cross-sectional area of the lower flange is A fb , and the yield strength of the lower flange is also σ yf ; There is an opening in the web. The height from the upper surface of the opening to the lower surface of the upper flange is s t , the height of the opening is h0, and the height from the lower surface of the opening to the upper surface of the lower flange is s b , the thickness of the web is t w , the yield strength is σ yw , the cross-sectional area of the web above the opening is A w , and the cross-sectional area of the web below the opening is A wb .

[0023] Furthermore, M 11 is solved using the following formula:

[0024]

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

[0026] Furthermore, M 12 is solved using the following formula:

[0027]

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

[0029] Furthermore, M 21 is solved using the following formula:

[0030]

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

[0032] Furthermore, M 22 is solved using the following formula:

[0033]

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

[0035] Furthermore, M 31 is solved using the following formula:

[0036]

[0037] where y t , y b , y s , y sb , y 31 , y 32 , m, and a are all calculation parameters. Furthermore, M 41 is solved using the following formula:

[0038]

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

[0040] Furthermore, M 42 is solved using the following formula:

[0041]

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

[0043] Symbol Explanation:

[0044] M u represents the ultimate bearing capacity.

[0045] M 11 represents the secondary moment when the neutral axis NA of the opening corner point 1 is located in the concrete slab.

[0046] M 12Represents the secondary moment when the neutral axis NA of the opening corner point 1 is located in the steel bar area.

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

[0048] M 22 Represents the secondary moment when the neutral axis NA of the opening corner point 2 is located within the web of the steel beam;

[0049] M 31 Represents the secondary moment when the neutral axis NA of the opening corner point 3 is located within the lower flange of the steel beam.

[0050] M 41 Represents the secondary moment when the neutral axis NA of the opening corner point 4 is located within the lower flange of the steel beam.

[0051] M 42 Represents the secondary moment when the neutral axis NA of the opening corner point 4 is located within the web of the steel beam.

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

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

[0054] m represents the distance between the centroid axis of the section and the neutral axis.

[0055] y s Represents converting the maximum plastic axial force of the steel beam sections above and below the opening into the reduced height of the steel beam flange. y c Represents the distance from the centroid axis of the section to the steel bar area.

[0056] y sb Represents the distance from the centroid axis of the section to the lower part of the lower flange of the I-shaped steel.

[0057] y f Represents converting the maximum plastic axial force of the steel beam sections above and below the opening into the reduced height of the steel beam flange. c represents the height of the steel bar area, and its size is a represents the reduced section height corresponding to the axial force in the converted stress diagram.

[0058] A f Represents the cross-sectional area of the upper flange, and its size is b f t f ;

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

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

[0061] A wb It represents the web area below the opening of the steel beam:

[0062] The advantages of the present invention are as follows:

[0063] First, after the web is provided with holes, the mechanical properties of the steel-concrete composite beam change greatly. The stiffness and bearing capacity in 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 open-hole composite beam, it is not only necessary to determine the bearing capacity at the maximum bending moment or maximum shear force, but more importantly, to calculate the ultimate bearing capacity (flexural bearing capacity) of the hole area.

[0064] Second, based on the open-web truss model, the present application establishes a mechanical model of the composite beam with web openings in the negative moment area. According to the cross-sectional plastic stress distribution in the hole area under the ultimate state, multiple local bending moment functions are established, and a calculation method for the ultimate bearing capacity of the steel-concrete composite beam with web openings under negative bending moment is given.

[0065] Third, the calculation method of the present application is also applicable to the case where the sizes of the upper and lower flanges are asymmetric. Description of the Drawings

[0066] The following further elaborates on the present invention with reference to the embodiments in the drawings, but it does not constitute any limitation to the present invention.

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

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

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

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

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

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

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

[0074] Figure 8 It is the solution schematic diagram of M 31 ​​​​​

[0075] Figure 9 is M 41 The solution schematic diagram of

[0076] Figure 10 is M 42 The solution schematic diagram of Specific implementation manners

[0077] Example 1: A method for calculating the bearing capacity of a web-opened composite beam under negative 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, the failure of the open web is the most likely failure mode of the web-opened composite beam. Based on the open web truss model, a mechanical model of the web-opened composite beam in the negative moment area is established, as Figure 1 shown, and theoretical analysis is carried out on this basis.

[0081] Figure 1 where:

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

[0083] M1, M2, M3, and M4 are the local 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, and their values are the product of the corresponding shear force and the length of the hole;

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

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

[0086] z is the distance between the centroid axes of the upper and lower T-shaped sections.

[0087] 1.2 Basic assumptions

[0088] Under the action of negative moment, the stress in the hole area of the composite beam is relatively complex, with the combined action of axial force, shear force and local moment, as Figure 1 , Figure 2 shown. For simplicity of calculation, the following basic assumptions are made:

[0089] 1) The composite beam is fully shear-connected.

[0090] 2) The steel beam at the hole will not undergo local buckling.

[0091] 3) The steel beam web 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 ultimate state, plastic hinges can form at the four corners of the hole.

[0093] 5) Consider that part of the concrete slab in the hole area participates in bending and shear resistance, that is, when calculating the local bending moment M1, consider the effect of the compressed concrete at hole corner 1, and when calculating M2, consider the effect of the tensioned steel bars at hole corner 2.

[0094] Figure 2 The distributions of shear force, bending moment on the full cross-section in the hole area, and axial force, shear force and local bending moment in the upper and lower cross-sections of the hole are given. Additionally, it can also be seen that: the total bending moment (M z ) on the full cross-section can be decomposed into the sum of the principal bending moment (M p ) and the local bending moment (M 1～4 ). The principal bending moment is the product of the axial force in the upper and lower T-shaped cross-sections and its moment arm.

[0095] 2 Solution

[0096] 2.1M 11 Solution of

[0097] The cross-section stress distribution of hole corner 1 under negative bending moment is as shown in Figure 4 . When the NA axis moves within the concrete slab, part of the concrete slab is in tension and part is in compression. In the calculation, only the compressed concrete in the compression zone and the tensioned 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 consists of two segments of functions.

[0098] M 11 is solved by the following method:

[0099] (1)

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

[0101] (3) Solve for the minimum value of M 11 -c] as the result within the range of [0, y 11 .

[0102] 2.2M 12 Solution of

[0103] M 12 is solved 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 of

[0107] M 21 Solve using the following formula:

[0108]

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

[0110] 2.4M 22 Solution of

[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 of

[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 of

[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 of

[0123] M 42 is solved using the following formula:

[0124]

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

[0126] Symbol description:

[0127] M u represents the ultimate bearing capacity.

[0128] M 11 represents the secondary moment when the neutral axis NA of the opening corner point 1 is located in the concrete slab.

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

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

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

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

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

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

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

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

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

[0138] m represents the distance between the centroidal axis of the section and the plastic neutral axis.

[0139] y s It means converting the maximum plastic axial forces of the steel beam sections above and below the opening into the reduced height of the steel beam flange.

[0140] y c It represents the distance between the centroidal axis of the section and the reinforcement area.

[0141] y sb It represents the distance between the centroidal axis of the section and the lower part of the lower flange of the I-shaped steel.

[0142] y f It means converting the maximum plastic axial forces of the steel beam sections above and below the opening into the reduced height of the steel beam flange.

[0143]

[0144] The above-described embodiments are the preferred embodiments of the present invention, which are only used to conveniently illustrate the present invention and do not impose any form of limitation on the present invention. Any person with ordinary knowledge in the relevant technical field, without departing from the technical features of the present invention, making local modifications or equivalent embodiments by using the technical content disclosed in the present invention and without departing from the technical feature content of the present invention, still fall within the scope of the technical features of the present invention.

Claims

1. A calculation method for the bearing capacity of a composite beam with openings in the web under negative bending moment. The composite beam with openings in the web includes: The concrete slab located on the upper side and the I-shaped steel located on the lower side; A rectangular opening is provided in the web of the I-shaped steel; The hole corner points 1, 2, 3, and 4 respectively represent the upper left corner, upper right corner, lower left corner, and lower right corner of the opening; The conditions of the concrete slab are as follows: the height is h c , the width is b e , the cross-sectional area is A c , the compressive strength is σ c , the yield strength of the steel bar is σ s , and the cross-sectional area of the tension steel bar is A s ; The conditions for the I-shaped steel are as follows: the materials of the upper flange and the lower flange are the same, the width of the upper flange is b f , the height of the upper flange is t f , the cross-sectional area of the upper flange is A f , and the yield strength of the upper flange is σ yf ; the width of the lower flange is b fb ; the height of the lower flange is also t f , and the cross-sectional area of the lower flange is A fb ; there is an opening in the web, the height from the upper surface of the opening to the lower surface of the upper flange is s t , the height of the opening is h0, and the height from the lower surface of the opening to the upper surface of the lower flange is s b , the thickness of the web is t w , and the yield strength is σ yw , the cross-sectional area of the web above the opening is A w , and the cross-sectional area of the web below the opening is A wb ; It is characterized in that the bearing capacity is the flexural bearing capacity M under the action of negative bending moment u ; It is calculated by the following formula; M u = min{M 11 , M 12 , M 21 , M 22 , M 31 , M 41 , M 42} Among them, M 11 represents the secondary moment when the neutral axis of the corner point 1 of the opening is located in the concrete slab M 12 Denotes the secondary moment when the neutral axis of the corner point 1 of the opening is located in the reinforcement area; M 21 Denotes the secondary moment where the neutral axis of the opening corner point 2 is located within the upper flange of the steel beam; M 22 Denotes the secondary moment when the neutral axis of the corner point 2 of the opening lies within the web of the steel beam; M 31 Denotes the secondary moment where the neutral axis of the corner point 3 of the opening lies within the lower flange of the steel beam; M 41 Represents the secondary moment where the neutral axis of the corner point 4 of the opening lies within the lower flange of the steel beam; M 42 Indicates the secondary moment where the neutral axis of the corner point 4 of the opening is located within the web of the steel beam; M 11 Solve using the following formula: M 12 Solve using the following formula: M 21 Solve using the following formula: M 22 Solve using the following formula: M 31 Solve using the following formula: M 41 Solve using the following formula: M 42 Solve using the following formula: where y t represents the distance from the centroidal axis of the cross-section to the upper edge of the concrete slab; y b represents the distance from the centroidal axis of the cross-section to the lower edge of the concrete slab; a is the converted section height corresponding to the axial force in the converted stress diagram; m represents the distance between the centroidal axis of the cross-section and the plastic neutral axis; y s represents converting the maximum plastic axial forces of the steel beam sections above and below the opening into the converted height of the steel beam flange; y sb represents the distance from the centroidal axis of the cross-section to the lower part of the lower flange of the I-shaped steel; y f represents converting the maximum plastic axial forces of the steel beam sections above and below the opening into the converted height of the steel beam flange.

2. A design method for a composite beam with web openings, characterized in that The design method is calculated by using the bearing capacity calculation method of the composite beam with web opening under negative bending moment described in claim 1.

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