Evaluation method for rotational rigidity, design method for composite beam end, and joint structure of composite beam end

Abstract

To provide an evaluation method of rotational stiffness capable of precisely evaluating the rotational stiffness of a joint part.SOLUTION: A joint structure of a composite beam end comprises a steel beam 25, a slab 35 supported on the steel beam, and a shear connector 45 connecting the steel beam and the slab to each other discretely or continuously in the longitudinal X of the steel beam, at least one end in the longitudinal direction is supported respectively by a support member 15 through a joint member 33, and in a method for evaluating the rotational stiffness of a joint 47a connected to a support member formed at least in one end in the longitudinal direction in a composite beam 47 subjected to a vertical load, in which the rotational stiffness is evaluated using the bending and axial stiffness of a slab at the joint, the bending and axial stiffness of a steel beam and the joint member, and the shear stiffness of a shear connector.SELECTED DRAWING: Figure 2

Description

[Technical Field]

[0001] The present invention relates to a method for evaluating rotational rigidity, a method for designing the end of a composite beam, and a joint structure of the end of a composite beam. [Background technology]

[0002] Conventionally, a composite beam is constructed by joining a steel beam and a slab (floor) with a shear connector. This steel beam is used as a sub-girder or a main girder. Usually, the web of the sub-girder is joined with a bolt to a gusset plate (fin plate) attached to the main girder, and the upper and lower flanges of the sub-girder are not joined to the main girder. With this configuration, joints are formed at both ends of the sub-girder using pin joints. The sub-girder supports the slab from below. The sub-girder is a non-earthquake-resistant component that is not required to withstand horizontal external forces such as those caused by earthquakes.

[0003] On the other hand, the ends of the sub-beams can be rigidly connected to the main beams by welding, bolts, etc. Alternatively, a concrete slab appropriately reinforced with rebar can be placed on top and fastened to the steel beams with shear connectors, or, as is well known, the ends of the sub-beams can be semi-rigidly connected to the main beams. When the ends of the sub-beams are rigidly or semi-rigidly connected in this way, the deflection and moment at the center of the sub-beam can be suppressed compared to when the ends of the sub-beams are pin-connected. Furthermore, the cross-sectional area of ​​the sub-beams can be reduced, making them lighter. Small beams constructed in this way are used in long span structures.

[0004] However, conventional methods for evaluating the stiffness of these joints do not adequately take into account the bending resistance of the slab at the joint or the bending resistance of steel beams and other steel frame parts. On the other hand, for example, Non-Patent Documents 1 to 3 evaluate the rigidity before bending cracks or the like occur in the slab. [Prior art documents] [Non-patent literature]

[0005] [Non-Patent Document 1] Written by Masaki Arita, Yuichi Nishida, Satoshi Kitaoka, Ryoichi Kanno, JY Richard Liew, Jun Iyama and Koji Hanya, “Semi-rigid beam-to-beam joint model for long span composite floor beams”, 12th Pacific Structural Steel Conference Tokyo, Japan, November 9-11, 2019 [Non-patent document 2] Yuichi Nishida, Masaki Arita, Satoshi Kitaoka, Kazuben Suzuki, "Structural Performance of Joints of Continuous Beams with Composite Slabs Part 1 Experimental Plan", Proceedings of the Annual Meeting of the Architectural Institute of Japan (Hokuriku), September 2019, pp.1493-1494 [Non-patent document 3] Arita, Masaki, Nishida, Yuichi, Kitaoka, Satoshi, and Suzuki, Kazuben, "Structural Performance of Joints of Continuous Beams with Composite Slabs, Part 2: Experimental Results," Proceedings of the Annual Meeting of the Architectural Institute of Japan (Hokuriku), September 2019, pp. 1495-1496 Summary of the Invention [Problem to be solved by the invention]

[0006] However, Non-Patent Documents 1 to 3 do not take into account the bending resistance of steel beams and other steel frame components, and therefore may underestimate the rotational rigidity, especially before bending cracks occur in the slab. Therefore, it is not possible to precisely evaluate the rigidity of the joints and suppress the deflection and moment at the center of the composite beam. As a result, the cross-sectional area of ​​the sub-beams perpendicular to the longitudinal direction becomes large, resulting in an uneconomical design. In addition, the negative bending moment occurring near the joint was underestimated, and even when the actual negative bending moment exceeded the joint's strength or lateral buckling strength, the design was judged to have sufficient strength (a design that was on the dangerous side).

[0007] The present invention has been made in consideration of these problems, and aims to provide a method for evaluating rotational rigidity that can precisely evaluate the rotational rigidity of a joint, a method for designing the end of a composite beam using this method for evaluating rotational rigidity, and a joint structure for the end of a composite beam. [Means for solving the problem]

[0008] In order to solve the above problems, the present invention proposes the following means. The method for evaluating rotational rigidity of the present invention is a composite beam comprising a steel beam, a slab supported on the steel beam, and shear connectors that discretely or continuously join the steel beam and the slab to each other in the longitudinal direction of the steel beam, wherein at least one end in the longitudinal direction is supported by a support member via a joint member and subjected to a vertical load. The method evaluates the rotational rigidity of a joint with the support member formed at at least one end in the longitudinal direction, and is characterized in that the rotational rigidity is evaluated using the bending rigidity and axial rigidity of the slab at the joint, the bending rigidity and axial rigidity of the steel beam and the joint member, and the shear rigidity of the shear connector.

[0009] In this invention, the rotational rigidity of a joint between a composite beam and a support member is evaluated. The composite beam has a steel beam and a slab joined together via shear connectors. The inventors conducted extensive research and found the following: The rotational rigidity of the joint can be precisely evaluated using the bending rigidity and axial rigidity of the slab at the joint, the bending rigidity and axial rigidity of the steel beam and the joint member, and the shear rigidity of the shear connector. Therefore, the rotational rigidity of the joint can be precisely evaluated using these bending rigidity, axial rigidity, and shear rigidity.

[0010] In addition, in the method for evaluating rotational stiffness, a plurality of the shear connectors are provided, and the plurality of shear connectors are arranged in the longitudinal direction at intervals s (m), and the rotational stiffness S is calculated using equations (1) to (13). j may be evaluated. where L is the length of the composite beam or the distance (m) between the pair of joints of the composite beam when both ends of the composite beam in the length direction are supported by the support members via the joint members to form a pair of joints, k is the shear stiffness (kN / m) per shear connector, and l j is twice the length (m) of each of the joints in the longitudinal direction. (E s I s ) j is the bending stiffness (kNm 2 ) and (E s A s ) j is the axial stiffness of the slab at each of the joints (kN), and (E b I b ) j is the bending rigidity (kNm 2 ) and (E b A b ) j is the axial stiffness (kN) of the steel beam and the connection member at each of the joints. E s I s is the bending stiffness of the slab (kNm 2 ) and E s A s is the axial stiffness of the slab (kN), and c s is the distance (m) from the interface between the slab and the steel beam to the axis of the slab, and E b I b is the bending rigidity of the steel beam (kNm 2 ) and E b A b is the axial stiffness of the steel beam (kN), and c b is the distance (m) from the interface to the axis of the steel beam.

[0011]

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[0012] In this invention, the rotational rigidity of the joint can be evaluated more precisely using equations (1) to (13).

[0013] Furthermore, the design method for the end of a composite beam of the present invention is a method for evaluating the rotational rigidity S j , and the rotational stiffness S so that the acting moment acting on the joint in response to the vertical load is less than the moment resistance capacity of the joint. j The moment resistance is also adjusted.

[0014] In this invention, for example, when a building is constructed using composite beams, the acting moment acting on the joint is less than the moment resistance of the joint, so it is possible to prevent both ends of the composite beam from reaching the so-called service limit state, which affects the usability or habitability of the building. The service limit state here refers to a state in which, in the case of a composite beam, the acting moment acting on the joint is equal to the moment resistance of the joint, and a hinge is formed at the end of the composite beam, causing excessive irreversible deformation (deflection of the beam).

[0015] In addition, another method for designing the end of a composite beam of the present invention is a method for designing the end of a composite beam, in which both ends in the length direction of the composite beam are supported by the support members via the joint members, and the rotational rigidity S evaluated using the evaluation method for rotational rigidity described above is j , and the absolute value M of the acting moment acting on the joint in response to the vertical load j,Ed The moment resistance M of the joint j,Rd When the moment resistance M of the positively bent part of the composite beam is bs,Rd is set to satisfy equation (16). However, M bs,pin is the absolute value (kNm) of the maximum moment acting on the positively bent portion of the reference composite beam, assuming that each of the joints in the reference composite beam is a pin joint.

[0016]

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[0017] In this invention, when a building is constructed using composite beams, the absolute value of the acting moment M j,Ed is the moment resistance of the joint M j,Rd Even if a load exceeding the serviceability limit state is applied, by satisfying equation (16), the absolute value of the maximum moment acting on the positive bending part of the composite beam will be equal to the moment resistance M bs,Rd The ultimate limit state here refers to a state in which hinges are formed at three locations in a composite beam, both ends of which are supported, and at the center, causing the entire composite beam to form a mechanism (the composite beam collapses). Therefore, even if a load exceeding the service limit state at which hinges occur at both ends of the composite beam is applied, it is possible to prevent the composite beam from reaching the ultimate limit state and collapsing.

[0018] The joint structure of the composite beam end of the present invention comprises at least one end of the composite beam and the support member supporting the at least one end, and the plurality of shear connectors are arranged in the longitudinal direction at intervals s (m), and the rotational stiffness S evaluated using equations (21) to (33) based on the above-described method for evaluating rotational stiffness is j , and the rotational stiffness S so that the acting moment acting on the joint in response to the vertical load is less than the moment resistance capacity of the joint. j The moment resistance is adjusted. where L is the length of the composite beam or the distance (m) between the pair of joints of the composite beam when both ends of the composite beam in the length direction are supported by the support members via the joint members to form a pair of joints, k is the shear stiffness (kN / m) per shear connector, and l j is twice the length (m) of each of the joints in the longitudinal direction. (E s I s )j is the bending stiffness (kNm 2 ) and (E s A s ) j is the axial stiffness of the slab at each of the joints (kN), and (E b I b ) j is the bending rigidity (kNm 2 ) and (E b A b ) j is the axial stiffness (kN) of the steel beam and the connection member at each of the joints. E s I s is the bending stiffness of the slab (kNm 2 ) and E s A s is the axial stiffness of the slab (kN), and c s is the distance (m) from the interface between the slab and the steel beam to the axis of the slab, and E b I b is the bending rigidity of the steel beam (kNm 2 ) and E b A b is the axial stiffness of the steel beam (kN), and c b is the distance (m) from the interface to the axis of the steel beam.

[0019]

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[0020] In this invention, for example, when a building is constructed using a joint structure at the end of a composite beam, the acting moment acting on the joint is less than the moment resistance strength of the joint, preventing both ends of the composite beam from reaching a so-called service limit state, which affects the usability or livability of the building.

[0021] In addition, in the joint structure of the composite beam end of the present invention, the slab has concrete and a reinforcing bar embedded in the concrete, and the moment resistance Mj,Rd (kNm), and the absolute value of the acting moment M j,Ed (kNm) satisfies the formulas (41) to (44). However, σ ry is the yield stress of the reinforcing bar (kN / m 2 ) and A r,eff is the total cross-sectional area of ​​the reinforcing bars within the effective width of the slab (m 2 ) and x h is the length (m) in the longitudinal direction from the joint to the position where the bending moment acting on the composite beam becomes zero.

[0022]

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[0023] In this invention, when a slab contains concrete and reinforcing bars, the joint structure of the end of a composite beam used in a building can be precisely evaluated using equations (41) to (44).

[0024] Further, another joint structure of a composite beam end of the present invention comprises at least one end of the composite beam and the support member that supports the at least one end, and the rotational rigidity S evaluated using the evaluation method for rotational rigidity described above j , and the absolute value M of the acting moment acting on the joint in response to the vertical load j,Ed The moment resistance M of the joint j,Rd When the moment resistance M of the positively bent part of the composite beam is bs,Rd is characterized in that it is configured to satisfy equation (45). However, M bs,pin is the absolute value (kNm) of the maximum moment acting on the positively bent portion of the reference composite beam, assuming that each of the joints in the reference composite beam is a pin joint.

[0025]

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[0026] In this invention, for example, when a building is constructed using a joint structure of a composite beam end, the absolute value of the acting moment M j,Ed is the moment resistance of the joint M j,Rd Even if a load exceeding the serviceability limit state is applied, by satisfying equation (45), the absolute value of the maximum moment acting on the positive bending part of the composite beam will be equal to the moment resistance M bs,Rd Therefore, even if a load exceeding the service limit state at which hinges occur at both ends of the composite beam is applied, it is possible to prevent the composite beam from reaching the ultimate limit state. Therefore, it is possible to prevent the composite beam from reaching its ultimate limit state and collapsing. [Effects of the Invention]

[0027] The method for evaluating rotational rigidity, the method for designing the end of a composite beam, and the joint structure of the end of a composite beam of the present invention enable precise evaluation of the rotational rigidity of the joint. [Brief explanation of the drawings]

[0028] [Figure 1] 1 is a perspective view showing a schematic view of a building in which a joint structure of a composite beam end according to one embodiment of the present invention is used. [Figure 2] This is a cross-sectional view of a main part of the building. [Figure 3] A diagram illustrating Aribert's work. [Figure 4] FIG. 10 is a diagram illustrating the boundary conditions of a modeled composite beam. [Figure 5] FIG. 10 is a diagram showing the bending moment M(x) acting on a composite beam. [Figure 6] FIG. 1 is a diagram showing an outline of a composite beam on which a bending moment acts. [Figure 7] FIG. 10 is a diagram showing the curvature and strain distribution in the cross section of a composite beam. [Figure 8] FIG. 10 is a diagram showing the rotation angle and displacement at the joint of a composite beam. [Figure 9] FIG. 10 is a diagram showing the rotation angle and displacement at the joint of a composite beam. [Figure 10] FIG. 1 is a plan view of a floor structure from which a test specimen used in a conventional evaluation of rotational rigidity is extracted. [Figure 11] FIG. 1 is a diagram showing the shape and dimensions of a test specimen. [Figure 12] FIG. 10 is a diagram showing the change in joint moment Mj with respect to the rotation angle φj of the joint. [Figure 13] FIG. 10 is a diagram showing the variation of non-dimensional deflection with respect to non-dimensional coordinates. [Figure 14] FIG. 10 is a diagram showing the change in the non-dimensional bending moment with respect to the non-dimensional coordinates. DETAILED DESCRIPTION OF THE INVENTION

[0029] Hereinafter, a first embodiment of a joint structure for an end portion of a composite beam according to the present invention will be described with reference to FIGS.

[0030] [1. Structure of buildings using composite beam end joint structures] This composite beam end joint structure 48 is used in the building 1 shown in Figures 1 and 2. The building 1 includes a plurality of columns 10, a plurality of main girders (support members) 15, a plurality of steel beams 25 which are sub-girders, a floor slab (slab) 35, and a shear connector 45. In FIG. 1, the floor slab 35 is shown in a see-through manner, and the shear connectors 45 are not shown.

[0031] The pillars 10 extend in the vertical direction. The pillars 10 are arranged at intervals from one another. The pillars 10 are made of steel, RC (Reinforced Concrete), SRC (Steel Reinforced Concrete), CFT (Concrete Filled Steel Tube), or the like. As shown in Figs. 1 and 2, for example, the girder 15 is made of H-shaped steel. The girder 15 includes a first flange 16, a second flange 17, and a web 18. The first flange 16, the second flange 17, and the web 18 are each formed of a steel plate. The first flange 16 and the second flange 17 are each disposed along a horizontal plane and face each other in the up-down direction. The first flange 16 is disposed higher than the second flange 17. The web 18 is disposed between the first flange 16 and the second flange 17. The web 18 joins the center of the first flange 16 in the width direction and the center of the second flange 17 in the width direction to each other.

[0032] As shown in Fig. 2, a gusset plate (fin plate) 21 is joined by welding or the like to the first flange 16 and web 18 of the girder 15. The upper end of the gusset plate 21 and the lower surface of the first flange 16 are positioned at the same level in the vertical direction. The lower end of the gusset plate 21 is disposed above the upper surface of the second flange 17. In other words, a gap is formed between the gusset plate 21 and the second flange 17. A horizontal rib 22 is fixed to the web 18 at a position corresponding to the lower end of the gusset plate 21 by welding or the like.

[0033] 1, the girder 15 is placed between a pair of adjacent columns 10 and extends in a direction along the horizontal plane. Both ends of the girder 15 are joined to the columns 10 by welding or the like. The girder 15 may be made of reinforced concrete or steel reinforced concrete.

[0034] 1 and 2, the steel beam 25 is made of, for example, an H-shaped steel. The steel beam 25 includes a first flange 26, a second flange 27, and a web 28. The first flange 26, the second flange 27, and the web 28 are each formed of a steel plate. The first flange 26 and the second flange 27 are each disposed along a horizontal plane and face each other in the vertical direction. The first flange 26 is disposed higher than the second flange 27. The web 28 is disposed between the first flange 26 and the second flange 27. The web 28 joins the center of the first flange 26 in the width direction and the center of the second flange 27 in the width direction to each other.

[0035] 2, the first flange 26 of the steel beam 25 and the first flange 16 of the girder 15 are disposed at the same position in the vertical direction. The second flange 27 of the steel beam 25 is disposed above the second flange 17 of the girder 15. The steel beam 25 is placed between a pair of adjacent girders 15 and extends in a direction along the horizontal plane. Specifically, the gusset plates 21 provided on the girders 15 and the webs 28 of the steel beam 25 are joined to each other by fastening members 31 such as high-strength bolts.

[0036] The horizontal rib 22 and the second flange 27 of the steel beam 25 are joined to each other by a welded portion 32 formed by welding. The gusset plate 21 and horizontal rib 22 provided on the main girder 15, the fastening member 31, and the welded portion 32 form a joint member 33. In this example, the first flange 26 of the steel beam 25 and the first flange 16 of the main girder 15 are not joined to each other, and a gap is formed between them. The horizontal rib 22 and the second flange 27 of the steel beam 25 do not have to be joined by the weld 32. The horizontal rib 22 and the second flange 27 of the steel beam 25 may be joined by bolts or metal-to-metal. The second flange 27 does not have to be joined to the girder 15 or its joint member 33, in which case the horizontal rib 22 may be omitted.

[0037] The floor slab 35 is a composite deck slab. The floor slab 35 includes a deck plate 36, concrete 37, and reinforcing bars 38. The deck plate 36 is formed by bending a steel plate or the like. The deck plate 36 is disposed on the first flange 26 of the steel beam 25 and on the first flange 16 of the main girder 15. For example, the deck plate 36 and the first flanges 16, 26 are joined to each other by joints (not shown) such as by burn-out plug welding. The reinforcing bars 38 are arranged reinforcement. The floor slab 35 is provided with a plurality of reinforcing bars 38. A first reinforcing bar 41, which is a part of the plurality of reinforcing bars 38, extends in the longitudinal direction X of the steel beam 25. A second reinforcing bar 42, which is the remainder of the plurality of reinforcing bars 38, extends along the horizontal plane and in a direction perpendicular to the first reinforcing bar 41. The first reinforcing bar 41 and the second reinforcing bar 42 are embedded in the concrete 37. The floor slab 35 constructed as described above is supported on the girders 15 and the steel beams 25 .

[0038] The building 1 is provided with a plurality of shear connectors 45, which are headed studs. The steel beams 25 between the pitches of the pair of girders 15, the floor slab 35, and the plurality of shear connectors 45 form a composite beam 47. The lower ends of the multiple shear connectors 45 are joined by welding or the like to the first flanges 16 of the main girder 15 and the first flanges 26 of the steel beams 25. The multiple shear connectors 45 extend upward from the first flanges 16, 26 and are embedded in the concrete 37. The multiple shear connectors 45 discretely (intermittently) connect the steel beams 25 and the floor slabs 35 to each other in the longitudinal direction X. The multiple shear connectors 45 provided on the steel beams 25 are arranged side by side in the longitudinal direction X at intervals (s (m), see Figure 2). The shear connectors are not limited to headed studs, and may be, for example, L-angles. The shear connectors may join the steel beams 25 and the floor slabs 35 to each other continuously in the length direction X.

[0039] In the composite beam 47, both ends in the longitudinal direction X are supported by the main girders 15 via joint members 33, thereby forming a pair of joints. The joints include the ends in the longitudinal direction X of the steel beams 25, the ends in the longitudinal direction X of the floor slabs 35, and multiple shear connectors 45. Note that one end of the composite beam 47 in the length direction X does not have to be supported by the main girder 15. In this case, the joint is formed only at one end of the composite beam 47 in the length direction X. The composite beam 47 is subjected to vertical loads such as downward static loads. The vertical loads are determined, for example, by the Architectural Institute of Japan, Building Load Guidelines and Commentary (2015), 5th Edition. In the composite beam 47, joints 47a with the girder 15 are formed at both ends in the longitudinal direction X. The girder 15 and the ends of a pair of composite beams 47 that sandwich the girder 15 in the longitudinal direction X form a joint structure 48 of the composite beam end. Note that the joint structure of the composite beam end may be formed by the end of the composite beam 47 and the girder 15 that supports this end, or the joint structure of the composite beam end may be formed by both ends of the composite beam 47 and the pair of girders 15.

[0040] In this example, the joint 47a is a semi-rigid joint as defined in the European design standards described below. The configuration of the joint 47a of the composite beam 47 is not limited to this configuration. The floor slab 35 may be an RC slab that does not use the deck plate 36, or may be an RC slab that uses a deck with trusses. The floor slab 35 may be a precast concrete slab.

[0041] Here, the dimensions of the joint structure 48 at the end of the composite beam are specified as follows. The distance between a pair of joints 47a of the composite beam 47 (the span of the composite beam 47) is defined as L (m). The distance L is calculated by multiplying the distance 1 m between the centers of the girders 15 by the length 1 m of the joints 47a in the longitudinal direction X, which will be described later. j (m) is excluded. For example, if the center distance of the main beam 15 is 8m, and j= 400 mm, L = 7.6 m. The length of the composite beam 47 in the longitudinal direction X may be L (m). The shear stiffness of each shear connector 45 is k (kN / m). However, if the shear connectors 45 are arranged in multiple rows, the shear stiffness is the value obtained by multiplying k by the number of rows. The distance between adjacent shear connectors 45 in the longitudinal direction X provided on the steel beam 25 is defined as s (m). j (m). The subscript "j" means joint. That is, the length of each joint 47a in the longitudinal direction is (l j / 2)(m). Length l j When a pair of steel beams 25 are joined to both sides of a main girder 15, the distance between the axes of the shear connectors 45 (hereinafter also referred to as shear connector 45a) closest to the main girder 15 among the shear connectors 45 joined to each steel beam 25 is defined as the distance between the axis of the shear connector 45 (hereinafter also referred to as shear connector 45a).

[0042] The bending stiffness of the floor slab 35 at each joint 47a is expressed as (E s I s ) j (kNm 2 ) The axial stiffness of the floor slab 35 at each joint 47a is defined as (E s A s ) j The bending rigidity of the steel beam 25 and the joint member 33 at each joint 47a is expressed as (E b I b ) j (kNm 2 The axial rigidity of the steel beam 25 and the joint member 33 at each joint 47a is expressed as (E b A b ) j (kN).

[0043] The bending stiffness of the floor slab 35 is E s I s (kNm 2 ) The axial stiffness of the floor slab 35 is E s A sThe distance from the interface (boundary surface, see Figure 2) 50 between the floor slab 35 and the steel beam 25 to the axis of the floor slab 35 is defined as c s (m). The bending rigidity of the steel beam 25 is E b I b (kNm 2 ) The axial stiffness of the steel beam 25 is E b A b (kN). The distance from the interface 50 to the axis of the steel beam 25 is c b (m). distance c s When the floor slab 35 is subjected to a compressive force, the distance c may be the distance from the center of the thickness of the concrete 37 of the floor slab 35 to the interface 50, or the distance from the neutral axis of the net cross section of the floor slab 35 to the interface 50, taking into account the height of the first reinforcing bars 41 and the deck plate 36. s In the case where the floor slab 35 is subjected to a tensile force, the effective width of the floor slab 35 may be the distance from the center of gravity of the plurality of first reinforcing bars 41 included within the effective width of the floor slab 35 to the interface 50. For example, the effective width of the floor slab 35 is determined according to Non-Patent Document 5 (Architectural Institute of Japan, "Guidelines and Commentary on Design of Various Composite Structures," 2nd Edition, 2010) etc. Next, a method for evaluating the rotational rigidity of the joint portion 47a of the joint structure 48 of the composite beam end configured as above will be described.

[0044] [2. Evaluation method for rotational rigidity] 2.1 Introduction Composite beams, in which a floor slab and a steel beam are connected by shear connectors, deflect at the interface between the slab and the steel beam, causing longitudinal displacement at the interface due to shear deformation of the shear connector. For this reason, these composite beams are called imperfect composite beams, in which the assumption of plane retention does not hold. Research into this type of composite beam was first conducted by Newmark, focusing on simply supported imperfect composite beams. This research is exemplified by Non-Patent Document 6 (Newmark, NM, SIESS, CP and VIEST, IM, "Test and analysis of composite beams with incomplete interaction", Proc. Soc. Exp. Stress Anal., 9(1), pp. 75-92, 1951).

[0045] Subsequently, Aribert applied the above research to semi-rigid connections at the ends of composite beams, taking into account the anchorage of rebars in the floor slab and deformation of the shear connectors. Examples of this research include Non-Patent Document 7 (Aribert JM, "INFLUENCE OF SLIP OF THE SHEAR CONNECTION ON COMPOSITE JOINT BEHAVIOUR, Connections in Steel Structure III", pp. 11-22, Pergamon, Trento, 1996) and Non-Patent Document 8 (Aribert JM, "THEORETICAL SOLUTIONS RELATING TO PARTIAL SHEAR CONNECTION OF STEEL-CONCRETE COMPOSITE BEAMS AND JOINTS", Steel and Composite Structures, Delft, 1999). An outline of this research is shown in Figure 3 and equation (51).

[0046]

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[0047] However, K scis the equivalent axial stiffness of the stud for shear deformation. K s,r is the axial rigidity of the reinforcing bar itself. α and β are constants determined by the cross-sectional dimensions of the composite beam 47. In Non-Patent Documents 7 and 8, the bending moment (moment bearing rate) M a (φ j ) is 0 and tanhβ is 1, and equation (51) is solved. sc , K. s,r The calculation method and the evaluation method using Eq. (51) are incorporated into the European design standard (CEN “EN 1994-1-1:2004 Eurocode 4: Design of composite steel and concrete structures - Part 1-1: General rules and rules for buildings”, European Committee for Standardization, Brussels, Belgium, 2004).

[0048] However, Non-Patent Documents 7 and 8 do not take into account the bending rigidity of each component of the joint. In addition, the explicit formula for determining the joint rigidity relies on an approximate formula with a limited range of application, and for other reasons, it is difficult to say that the formula can be sufficiently generalized to be applied to semi-rigid joints. According to a separate study conducted by the inventors, the assumption that tanhβ is 1 in Non-Patent Documents 7 and 8 is valid for the structure of a common joint in a composite beam. On the other hand, according to the inventors' investigations, when the web of an H-shaped steel is friction-jointed with a high-strength bolt, the steel beam portion of the joint actually has rotational resistance. Therefore, the bending moment M a (φ j ) is 0 is not correct. In this invention, by taking into account the rotational resistance of the steel frame part of the joint, it is possible to avoid overestimating the deflection of the composite beam and to achieve economical design. In addition, it is thought that damage to the joint (hinge formation) can be prevented by avoiding underestimating the moment actually acting on the joint, and the bending moment M of the steel frame beam 25 is a (φ j) is not zero, a formula for evaluating the stiffness of the joint is derived.

[0049] In Japan, Non-Patent Document 5 is often referred to when designing composite beams. However, it does not describe designs that take into account the influence of adjacent spans, such as continuous beams, or semi-rigid joints where the joint state of the beam ends has rigidity between that of a rigid joint and a pin joint. Furthermore, Non-Patent Document 5 describes a deflection evaluation formula for imperfect composite beams, but it uses an experimental formula, and there are issues with the range of application and the accuracy of deflection evaluation.

[0050] [2.2. Evaluation method for incomplete composite beams] [2.2.1. Basic formula] In the following, the composite beam 47 is evaluated as an incomplete composite beam. First, the x-axis is taken to be the longitudinal direction X of the steel beam 25. Using the force balance condition, we derive the relational expression between the slip displacement γ(x) (m) of the interface 50 between the floor slab 35 and the steel beam 25 and the internal force F(x) (kN) in the longitudinal direction X. A model of the target composite beam 47 is shown in Figures 4 to 6. Here, the shear force acting on the shear connector 45 is Q(x) (kN), and the shear force distribution per unit length at the interface 50 between the floor slab 35 and the steel beam 25 is q(x) (kN / m). Then, equation (58) can be derived from equation (56).

[0051]

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[0052] Substituting equation (58) into equation (57), equation (59) is derived.

[0053]

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[0054] Differentiating both sides of equation (59) with respect to x yields equation (60).

[0055]

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[0056] Next, from the deformation compatibility conditions and governing equations, we derive the condition equation that the slip displacement γ(x) and the internal force F(x) in the longitudinal direction X satisfy. The rate of change (first-order derivative) of the slip displacement γ(x) can be expressed by equation (63), with tensile strain being positive (+).

[0057]

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[0058] Figure 7 shows the curvature and cross-sectional strain distribution of the composite beam 47. In Figure 7, the left diagram shows the composite beam 47 near the joint 47a, and the right diagram shows the strain distribution of the floor slab 35 and the steel beam 25 within the cross section of the composite beam 47. In the strain distribution, the strain on the right side represents tension, and the strain on the left side represents compression. The dotted line L1 represents the strain distribution when only a bending moment acts on each of the individual steel beams 25 and floor slab 35 that make up the composite beam 47. The solid line L2 represents the strain distribution when, in addition to a bending moment, a shear force transmitted by the shear connector 45 acts on the steel beams 25 and floor slab 35 that make up the composite beam 47 in the longitudinal direction X (X direction). The shear force transmitted by the shear connector 45 acts as a tension force in the length direction X on the floor slab 35. The shear force transmitted by the shear connector 45 acts as a compression force in the length direction X on the steel beam 25.

[0059] Since the strain distribution shown in FIG. 7 can be expressed by the resultant force of the bending component and the axial force component in the length direction X, equations (64) and (65) are derived for the strain at the interface 50 of the composite beam 47.

[0060]

number

[0061] where εs is the strain in the floor slab 35 at the interface 50. M s (x) is the bending moment component (kNm) of the force acting on the floor slab 35. ε b is the strain in the steel beam 25 at the interface 50. M b (x) is the bending moment component (kNm) of the force acting on the steel beam 25. By substituting equations (64) and (65) into equation (63), equation (66) is derived.

[0062]

number

[0063] Here, equation (66) is an equation that aggregates the axial stiffness of the floor slab 35 and the steel beam 25 into an equivalent axial stiffness using the definition of equation (72). If the curvature κ(x) of the floor slab 35 and the steel beam 25 at any position x on the X axis is equal to each other, then equation (67) is derived. Here, z is defined by equation (67a), and the bending rigidity of the floor slab 35 and the steel beam 25 is summarized as a simple sum of the bending rigidity using equation (73) described later.

[0064]

number

[0065] In equation (66), when equation (67) is taken into consideration, where the curvature of the floor slab 35 and the curvature of the steel beam 25 are equal to each other, equation (68) is derived.

[0066]

number

[0067] Substituting equation (67) into equation (68), equation (69) is derived. Here, the bending rigidity of the floor slab 35 and the steel beam 25 and the bending resistance due to the couple and axial rigidity acting on them are aggregated into the equivalent bending rigidity of the composite beam 47 using equation (74) described later.

[0068]

number

[0069] By substituting equation (69) into equation (60), differential equations (70) and (71) for the internal force F(x) in the longitudinal direction X can be derived.

[0070]

number

[0071] Here, the axial rigidity and bending rigidity of the composite beam 47 are summarized as in equations (72) to (74).

[0072]

number

[0073] Furthermore, we derive a differential equation for the shear displacement γ(x) as follows: Differentiating both sides of equation (69) with respect to x yields equation (75).

[0074]

number

[0075] Using equation (59), equations (76) and (77) are derived.

[0076]

number

[0077] [2.2.2. General solution of the basic equation for cantilever type semi-rigid connections] The differential equation (71) for the internal force F(x) in the longitudinal direction X and the differential equation (77) for the slip displacement γ(x), which are the basic equations obtained in [2.2.1], are solved for x, taking into account the moment distribution and boundary conditions of the cantilever type semi-rigid connection subjected to a single point concentrated load, as shown in Figures 4 to 6. Here, the concentrated load is P (kN) and the length of the cantilever beam (the distance from the point of application of the concentrated load P to the joint) is L (m). In this case, the moment distribution M(x) can be expressed by equation (80).

[0078]

number

[0079] Therefore, by substituting equation (80) into equations (71) and (77), the basic equations for a cantilever beam are derived as equations (81) and (82).

[0080]

number

[0081] Here, the formulas are defined as in (83) and (84).

[0082]

number

[0083] Then, equations (81) and (82) can be rearranged as equations (85) and (86).

[0084]

number

[0085] Both equations (85) and (86) are second-order inhomogeneous linear differential equations, and the general solutions of equations (85) and (86) can be expressed as the sum of a particular solution and a homogeneous general solution. The characteristic equations of equations (85) and (86) both satisfy equation (87).

[0086]

number

[0087] Therefore, the general solutions of the homogeneous form are derived as shown in equations (88) and (89), respectively.

[0088]

number

[0089] Here, C1 to C4 represent integral constants, and λ1 and λ2 are expressed by equation (90).

[0090]

number

[0091] The particular solutions are derived as shown in equations (91) and (92).

[0092]

number

[0093] From the above, the general solutions of equations (85) and (86), taking equation (97) into consideration, are as shown in equations (95) and (96).

[0094]

number

[0095] By using the boundary condition of equation (98) for F(x) in the cantilever beam in equation (95), equation (99) is obtained, and C2 can be expressed in terms of C1 as in equation (100).

[0096]

number

[0097] Therefore, equation (101) is derived from equations (95) and (100).

[0098]

number

[0099] On the other hand, the boundary condition for the slip displacement γ(x) is 0 at x = 0 in the case of a rigid joint. However, the boundary condition for the slip displacement γ(x) cannot be obtained as a given condition at either x = 0 or x = L in the case of a semi-rigid joint. On the other hand, by substituting equation (101) into equation (59), equation (102) is obtained.

[0100]

number

[0101] Comparing equations (96) and (102), equations (103) and (104) are derived.

[0102]

number

[0103] [2.3. Evaluation method for incomplete composite beams] [2.3.1. Slippage and internal force distribution] In [2.2.], the internal force F(x) in the longitudinal direction X and the shear displacement γ(x) were derived as functions of x as in equations (101) and (102). Here, the rotation angle φ of the semi-rigid joint 47a at x=0 j C1 is calculated under the boundary conditions. When x=0, equations (101) and (102) become equations (111) and (112).

[0104]

number

[0105] On the other hand, from Figure 8, the displacement γ(x) at x=0 and the rotation angle φ j The relationship between these can be expressed by equation (113).

[0106]

number

[0107] where x ns represents the distance from the neutral axis of the floor slab 35 (the point where the strain is zero) to the interface 50 between the floor slab 35 and the steel beam 25. nb represents the distance from the neutral axis of the steel beam 25 (the point where the strain is zero) to the interface 50. Since equations (112) and (113) are equivalent, equation (114) is obtained, and when equation (114) is solved for C1, C1 can be expressed as equation (115).

[0108]

number

[0109] FIG. 9 shows the displacement of the cross section of the floor slab 35 and the cross section of the steel beam 25 at x=0. Next, from the displacement shown in Figure 9, the distance x ns ,x nb The length l in the longitudinal direction X where the pair of joints 47a sandwiching the girder 15 are present is calculated. j In this region (hereinafter referred to as the "joint 47a region"), it is assumed that the curvature of the joint 47a is constant and equal to the value at x = 0. In this case, the curvature at x = 0 is determined by the rotation angle φ of the joint 47a. j Using this, it can be expressed as equation (116).

[0110]

number

[0111] Here, the length l jAn example is shown in Figure 2. Here, the target is the end joint of the steel beam 25, which is a sub-beam that is continuous across the main girder 15. In this case, the distance between the axes of the shear connectors 45 installed on the steel beam 25 that are closest to the main girder 15 is calculated by multiplying the length of the joint 47a (l j / 2) twice the l j Assuming that the joints 47a on both sides of the girder 15 are rotated with a curvature in mirror symmetry, the curvature and (l j / 2) is the rotation angle φ of the joint 47a at the position of the shear connector 45a. j Therefore, equation (116) is obtained. The axis of the shear connector 45a is taken as the origin and the x-axis is taken along the length direction X. Next, the magnitude of displacement of the cross section of the floor slab 35 at the position of the joint 47a (x=0) of the interface 50 between the floor slab 35 and the steel beam 25 is calculated as △ s At the same position, the magnitude of the virtual displacement when the floor slab 35 is assumed to be subjected to only bending deformation of the same rotation angle is defined as △m s The distance from the neutral axis of displacement in the cross section of the floor slab 35 at x=0 to the interface 50 is defined as x ns The thickness of the floor slab 35 is 1 / 2 of the thickness of the floor slab 35. s If there is a deck, the dimension is c, which is the sum of 1 / 2 of the thickness of the net floor slab 35 excluding the deck height. s It is stipulated that: At this time, the relationship of equation (117) holds from the geometric relationship shown in FIG.

[0112]

number

[0113] Furthermore, △ s is the magnitude of the virtual displacement △m s and the tensile displacement in the longitudinal direction X caused by the axial force component F. Note that the axial force component F means the internal force F(x) in the longitudinal direction X at x=0. Considering the positive and negative of each, △ s can be expressed by equation (118).

[0114]

number

[0115] The axial rigidity (E s A s ) j is the axial stiffness of the floor slab 35 in the area of ​​the joint 47a. Substituting equation (118) into equation (117) gives x ns By solving this, we derive equation (119), which is written as equation (119a).

[0116]

number

[0117] Next, the magnitude of the displacement of the cross section of the steel beam 25 at the position of the joint 47a (x=0) of the interface 50 between the floor slab 35 and the steel beam 25 is calculated as △ b The magnitude of the hypothetical displacement when the steel beam 25 is assumed to be subjected to only rotational deformation of the same rotation angle at the same position is defined as △ mb The distance from the neutral axis of displacement of the cross section of the steel beam 25 at x=0 to the interface 50 is defined as x nb The height of the cross section of the steel beam 25 (due to the steel beam 25) is 1 / 2, c b It is stipulated that: When the gusset plate 21 and the fastening member 31 constituting the steel frame part of the joint 47a are arranged asymmetrically above and below the center of the height of the cross section of the steel beam 25, the distance from the center of gravity of the steel frame part of the joint 47a considering the eccentricity of the steel frame part to the interface 50 is b It is stipulated that: In this case, equation (122) holds true from the geometric relationship shown in FIG.

[0118]

number

[0119] Furthermore, △ b is the tensile displacement △ caused by the rotational deformation acting on the steel beam 25 at x=0 mb and the compressive displacement in the longitudinal direction X caused by the axial force component F. Considering the positive and negative of each, △ b can be expressed as equation (123).

[0120]

number

[0121] Here, the axial stiffness (E b A b ) j represents the axial stiffness of the steel frame part in the joint 47a region. Substituting equation (123) into equation (122), x nb By solving for (124), we derive equation (124).

[0122]

number

[0123] From the above, by substituting equations (119) and (124) into equation (113) and eliminating F using equation (111), equation (125) is obtained.

[0124]

number

[0125] However, as shown in Fig. 9, equation (67a) is satisfied. In addition, the axial stiffness of the floor slab 35 and the axial stiffness of the steel frame part at the joint 47a are integrated into the equivalent axial stiffness defined by equation (126).

[0126]

number

[0127] Furthermore, since equations (112) and (125) are equal, if we set up an equation and solve for C1, C1 is determined as in equation (127).

[0128]

number

[0129] Also, (φ j =M j / S j ) and (M j Using the relationship (=PL), C1 can also be expressed by equations (128) and (129). However, φ j is the rotation angle of the joint 47a, and M j is the bending moment (joint moment) of the joint 47a, and S j is the rotational stiffness of the joint 47a.

[0130]

number

[0131] Furthermore, equations (101) and (102) can also be expressed as equations (130) and (131) by dividing F(x) and the deviation displacement γ(x) by PL.

[0132]

number

[0133] [2.3.2. Rotational rigidity of joints] When the external force of the joint 47a shown in FIG. 8 is used, the joint moment M j can be expressed as equation (134).

[0134]

number

[0135] where κ jis the curvature in the joint 47a region of the elements (floor slab 35, steel beam 25, gusset plate 21) that make up the joint 47a. In the joint 47a region, the curvature κ j Assuming that is constant, the curvature κ j is the rotation angle φ of the joint 47a j and are related by equation (135).

[0136]

number

[0137] Substituting equations (111) and (135) into equation (134), (M j Using the relation (=PL), equation (136) is derived.

[0138]

number

[0139] However, the bending rigidity E of the floor slab 35 at the joint 47a s I s and the bending rigidity E of the steel beam 25 and the gusset plate 21 b I b is summarized into the equivalent bending stiffness as shown in equation (137).

[0140]

number

[0141] Substituting equation (129) into equation (136), (M j =PL), equation (138) is derived.

[0142]

number

[0143] (138) on both sides of the equation (S j / M j), equation (139) is derived.

[0144]

number

[0145] (139) is the rotational stiffness S j By solving (140), equation (142) is derived from equation (140).

[0146]

number

[0147] Furthermore, when the terms representing the stiffness of each element of the joint 47a are summarized by symbols, it can be simply expressed as in equation (145).

[0148]

number

[0149] Here, each symbol is defined as in equations (146) to (155).

[0150]

number

[0151] Furthermore, equation (156) is derived from equations (84) and (155).

[0152]

number

[0153] Therefore, the rotational stiffness S j Equation (157) for is derived.

[0154]

number

[0155] [2.4. Summary of rotational stiffness evaluation methods] As described above, in the method for evaluating rotational rigidity of this embodiment, the rotational rigidity S of the joint 47a is calculated using the bending rigidity and axial rigidity of the floor slab 35 at the joint 47a, the bending rigidity and axial rigidity of the steel beam 25 and the joint member 33, and the shear rigidity of the shear connector 45. j Evaluate (calculate). More specifically, in this method for evaluating rotational stiffness, the rotational stiffness S is calculated using equations (67a), (126), (137), (146) to (151), (153) to (155), and (157). j Evaluate.

[0156] [3. Application of the rotational stiffness evaluation method] [3.1. Design method for composite beam ends using rotational rigidity evaluation method] Using the method for evaluating rotational rigidity of this embodiment, a method for designing the end of the composite beam 47 can be carried out. This design method for composite beam ends uses the concepts of the so-called serviceability limit state and ultimate limit state. The serviceability limit state is a state in which, when a building 1 is constructed using a composite beam 47, both ends of the composite beam 47 begin to impair the usability or habitability of the building 1. On the other hand, the ultimate limit state is a state in which, when a building 1 is constructed using a composite beam 47, the safety of the building 1 begins to be impeded.

[0157] In this evaluation method of rotational stiffness, when considering the service limit state, the rotational stiffness S j , and the rotational stiffness S so that the acting moment acting on the joint 47a in response to the vertical load is less than the moment resistance of the joint 47a. j and adjust moment resistance. The acting moment is, for example, the value obtained by modifying equation (70) disclosed in JP 2021-82152 A (evaluation method and evaluation program for continuous beams) into equation (161) where x = 0, L.

[0158]

number

[0159] Here, L is the distance (m) between the pair of joints 47a, and w is the vertical load. jl is the absolute value of the bending moment acting on the end of the composite beam 47 where x=0. M jr is the absolute value of the bending moment acting on the end of the composite beam 47 where x=L. M jl and M jr is S shown in, for example, claim 4 and FIG. 4 of JP 2021-82152 A. jl and S jr can be obtained by substituting the rotational stiffness obtained by this rotational stiffness evaluation method into and performing a convergence calculation.

[0160] On the other hand, in this evaluation method of rotational stiffness, the rotational stiffness S j , and the absolute value M of the acting moment acting on the joint 47a in response to the vertical load j,Ed The moment resistance M of the joint 47a j,Rd The safety verification is carried out taking into account the ultimate limit state when the moment resistance M of the composite beam 47 is positively bent (bent downwards). bs,Rd is set so as to satisfy equation (164).

[0161]

number

[0162] However, a standard composite beam is defined assuming that each joint 47a of the composite beam 47 is a pin joint. The absolute value of the maximum moment acting on the positively bent part of the standard composite beam is M bs,pin (kNm). Generally, in composite beams (steel beams) in which both ends in the longitudinal direction are rigidly or semi-rigidly connected and subjected to vertical loads, the center portion in the longitudinal direction is bent positively, and both ends in the longitudinal direction are bent negatively (convexly upward). For example, if the vertical load is a uniformly distributed load w and the length of the composite beam 47 (steel beam) is l, the absolute value of the maximum moment M bs,pin is (wl 2 / 8).

[0163] Here, equation (164) is derived as follows. First, regardless of the rotational rigidity of the joint, the magnitude of the negative bending moment acting on the beam (absolute value of the acting moment) M j,Ed and the magnitude of the positive bending moment M bs,Ed The sum of these is the magnitude of the positive bending moment acting on the pin-jointed beam (absolute value of the maximum moment) M bs,pin and satisfies equation (167).

[0164]

number

[0165] Next, when the rotational stiffness of the joint is greater than 0, the magnitude of the negative bending moment acting on the joint increases as the rotational stiffness increases. However, the moment (joint) strength M j,Rd Since the moment acting on the joint cannot bear a moment greater than the moment resistance capacity M j,Rd Once this value is reached, the moment acting on the joint does not increase even if the rotational stiffness is increased. On the other hand, if the rotational stiffness is kept constant and the load acting on the beam is increased, the negative bending moment of the joint and the positive bending moment of the beam increase in a ratio that depends on the rotational stiffness of the joint. However, the magnitude of the negative bending moment at the joint M j,Ed is the moment resistance of the joint M j,Rd reached (M j,Ed =M j,Rd ) after this, the moment acting on the joint does not increase, and only the positive bending moment of the beam increases. Whether the moment at the joint reaches its strength or not, as mentioned above, the magnitude of the negative bending moment acting on the beam M j,Ed and the magnitude of the positive bending moment M bs,EdThe sum of these is the magnitude of the positive bending moment M acting on the pin-jointed beam. bs,pin Therefore, the magnitude of the positive bending moment of the beam M bs,Ed satisfies equation (168).

[0166]

number

[0167] Magnitude of the positive bending moment of the beam M bs,Ed Moment strength M of the beam joint j,Rd Since the beam will not collapse unless it reaches the ultimate limit state, the condition equation (169) can be obtained to prevent the beam from reaching the ultimate limit state.

[0168]

number

[0169] By transforming equation (169), we obtain equation (164).

[0170] [3.2. Joint structure of composite beam end using rotational rigidity evaluation method] The rotational rigidity evaluation method of this embodiment can be used to construct a joint structure 48 at the end of a composite beam. In the joint structure 48 of this composite beam end, when considering the serviceability limit state, the rotational stiffness S evaluated using the rotational stiffness evaluation method is j , and the rotational stiffness S so that the acting moment acting on the joint 47a in response to the vertical load is less than the moment resistance of the joint 47a. j and moment resistance is adjusted. More specifically, in the joint structure 48 of the composite beam end, the moment resistance M j,Rd (kNm), and the absolute value of the acting moment M j,Ed (kNm) is configured to satisfy the formulas (126), (152), (175), and (176).

[0171]

number

[0172] However, σ ry is the yield stress of the reinforcing bar 38 (kN / m 2 ) A r,eff is the total cross-sectional area (m 2 The total cross-sectional area of ​​the reinforcing bars 38 referred to here means, for example, the cross-sectional area of ​​the reinforcing bars 38 in a cross section perpendicular to the longitudinal direction X of the steel beam 25. x h is the length (m) in the longitudinal direction X from the joint 47a to the position where the bending moment acting on the composite beam 47 becomes 0. The position where this bending moment becomes 0 is the end point of the region where the composite beam 47 is subjected to negative bending, which starts from the end (X=0 or X=L) of the longitudinal direction X of the composite beam 47. In addition, the moment resistance M j,Rd is calculated based on the assumption that multiple rebars 38 yield at the same time.

[0173] On the other hand, in the joint structure 48 of the composite beam end, when the ultimate limit state is considered, the rotational stiffness S evaluated using the rotational stiffness evaluation method is j , and the absolute value M of the acting moment acting on the joint 47a in response to the vertical load j,Ed The moment resistance M of the joint 47a j,Rd When the moment resistance M of the positive bending part of the composite beam 47 is bs,Rd is configured to satisfy equation (164).

[0174] Here, equation (176) is derived as follows. In equation (91), M j Considering that = PL, L for a cantilever beam is the length of the negative bending region including the joint of the beam in a double-supported beam x h By substituting the above equation, we can obtain equation (180), which can also be applied to a beam supported at both ends.

[0175]

number

[0176] Here, the shear force at the interface between the floor slab and the steel beam at the joint is the shear strength F Rd When it is equal to the joint strength M j,Rd When this is defined, the shear strength F Rd is the yield strength of the reinforcing bars in the floor slab 35, or the shear strength P of the shear connector 45 in the negative bending region including the beam joint. Rd We consider that it is determined by the smaller of the sums of (181) and (182).

[0177]

number

[0178] where P Rd can be obtained from "Architectural Institute of Japan: Composite Structure Design Guidelines and Commentary (2010) 2nd Edition" and "Eurocode 4: Design of composite steel and concrete structures - Part 1-1: General rules and rules for buildings" December 2004, Authority: The European Union Per Regulation 305 / 2011, Directive 98 / 34 / EC, Directive 2004 / 18 / EC". n is the bending moment in one negative bending region x h Within(0≦x≦x h ) is the total number of shear connectors 45 arranged in

[0179] Substituting equations (128) and (181) into equation (180), M j is the joint strength M j,Rd Therefore, equation (182) is obtained.

[0180]

number

[0181] (182) is applied to the joint strength M j,Rd By solving this, we obtain equation (176).

[0182] [4. Evaluation results of rotational rigidity] 4.1. Experimental results of conventional rotational stiffness First, the results of conventional experiments on rotational rigidity will be outlined below. In the above Non-Patent Documents 1 to 3, a test specimen was prepared that had steel-framed girders, sub-girders, and a composite slab, as shown in Figure 10. The test specimen was a full-scale specimen extracted from a floor frame with a span of 8.4 m and a beam spacing of 2.8 m. The length of the sub-girder was set as the area that would be subjected to negative bending when both ends of the sub-girder were fixed. Table 1 shows the specifications of the test specimen, and Fig. 11 shows the shape and dimensions of the test specimen.

[0183] [Table 1]

[0184] The main and secondary girders are welded H-shaped steel beams and are connected to the floor slab with shear studs (shear connectors). The shear studs are φ19 and are arranged in a row at 200mm intervals. The test specimen constructed in this way was attached to a loading device. The test specimen was set between the loading frames of the loading device, 1500 mm away from the center of the main girder. The test specimen was lifted by a hydraulic jack placed under the main girder, experimentally reproducing the negative bending region of a composite sub-girder subjected to vertical load. The experimental results are shown by line L6 in Figure 12. In Figure 12, the horizontal axis represents the rotation angle φ of the joint. j (rad), and the vertical axis represents the joint moment M j (kNm). Repeated loading was performed to confirm the unloading rigidity of the joint of the test specimen. Therefore, the actual test results are the part above line L6 excluding the unloaded part.

[0185] 4.2. Evaluation results of rotational rigidity evaluation method of the example As described above, the moment resistance M of the joint 47aj,Rd is calculated based on the assumption that multiple reinforcing bars 38 yield at the same time. However, in reality, it is considered that the reinforcing bars 38 yield in order, starting from the reinforcing bars 38 closest to the center of the joint 47a, and eventually all the reinforcing bars 38 yield. Also, the moment resistance M j,Rd is the yield stress σ of rebar 38 ry The moment resistance M is proportional to j,Rd Therefore, the elastic limit is defined as the limit of elasticity (2M j,Rd / 3) (see Figure 12). In addition, when a vertical load that causes deformation of the composite beam 47 exceeding its elastic deformation acts on the composite beam 47, the reinforcing bars 38 yield, and the rotational rigidity S obtained by the evaluation method of the rotational rigidity of the embodiment is j Therefore, in this case, the rotational stiffness S j is multiplied by (1 / 10) and used.

[0186] In addition, the straight line L7 in FIG. 12 passes through the origin and has a slope equal to the rotational stiffness S obtained by the evaluation method of the rotational stiffness of the embodiment. j This is the line. Moment resistance M obtained by the rotational rigidity evaluation method of the example j,Rd is shown by line L8. Rotational stiffness S j and moment resistance M j,Rd It was found that the trilinear model expressed by the following equation was able to accurately reproduce the experimental results shown by line L6, generally on the safe side.

[0187] Furthermore, the rotational stiffness S obtained by the evaluation method of the rotational stiffness of the embodiment j Using the same rotational stiffness S j The deflection and bending moment distribution of an 8.4m span composite beam supported by semi-rigid joints with a 47a were calculated. The deflection reduction rate was compared with that when each joint 47a was designed as a pin joint. Here, the calculated value of the deflection at the center of the composite beam 47, in which each joint 47a is supported by a pin joint, is δ b,pinThe deflection of the composite beam 47, in which each joint 47a is supported by a semi-rigid joint, is defined as the deflection δ b,pin The non-dimensional deflection divided by is shown in Figure 13.

[0188] In Figure 13, the horizontal axis represents the non-dimensional coordinate (x / L), and the vertical axis represents the non-dimensional deflection. The solid line L11 represents the non-dimensional deflection for the experimental results of [4.1.]. The dotted line L12 represents the non-dimensional deflection obtained by this rotational stiffness evaluation method. The dashed-dotted line L13 represents the deflection at the center of the composite beam 47, whose joints 47a are supported by pin joints, as a deflection δ b,pin represents the non-dimensional deflection divided by It was found that the maximum values ​​of the non-dimensional deflection shown by lines L11 and L12 were reduced to about 40% of the maximum value of the non-dimensional deflection of the pin joint shown by line L13.

[0189] Here, the calculated value of the bending moment at the center of the composite beam 47, in which each joint 47a is supported by a pin joint, is M b,pin The calculated moment distribution of the composite beam 47, in which each joint 47a is supported by a semi-rigid joint, is defined as the bending moment M b,pin The non-dimensional bending moment divided by is shown in Figure 14. In Figure 14, the horizontal axis represents the non-dimensional coordinate (x / L), and the vertical axis represents the non-dimensional bending moment. The solid line L16 represents the non-dimensional bending moment for the experimental results of [4.1.]. The dotted line L17 represents the non-dimensional bending moment obtained by this rotational rigidity evaluation method. The dashed-dotted line L18 represents the bending moment M of the composite beam 47, in which each joint 47a is supported by a pin joint. b,pin represents the non-dimensional bending moment divided by

[0190] It was found that the maximum value of the non-dimensional bending moment shown by lines L16 and L17 was reduced to about 54% of the maximum value of the non-dimensional bending moment of the pin joint shown by line L18. It was also found that the reduced bending moment (about 46% of the maximum value of the non-dimensional bending moment of the pin joint) acts on joint 47a at the end of composite beam 47. The moment acting on the joint 47a is the elastic limit of the joint 47a (2M j,Rd / 3), the design load is about 21.6kN / m. For example, if the control width, which is the pitch of the steel beam 25, is 2.8m, the design load per control width is 790kg / m. 2 This design load has a sufficient margin compared to the design load of a typical office or residential room (according to the Architectural Institute of Japan, Building Load Guidelines and Commentary (2015), 5th Edition, etc.). From the above, by using the rotational rigidity evaluation method, composite beam end design method, and composite beam end joint structure 48 of this embodiment, which use the concept of semi-rigid joints, the deflection of the composite beam 47 can be reduced to about half compared to conventional pin joints, making it possible to design economically.

[0191] [5. Effects of the rotational rigidity evaluation method, etc.] As described above, in the method for evaluating rotational rigidity of this embodiment, the rotational rigidity S of the joint 47a of the composite beam 47, which includes the steel beam 25 and the floor slab 35 joined together via the shear connector 45, is evaluated. j In this case, the inventors have found the following as a result of extensive research: the rotational rigidity S of the joint 47a can be calculated by using the bending rigidity and axial rigidity of the floor slab 35 at the joint 47a, the bending rigidity and axial rigidity of the steel beam 25 and the joint member 33, and the shear rigidity of the shear connector 45. j can be precisely evaluated. Therefore, the rotational stiffness S of the joint 47a is calculated using these bending stiffness, axial stiffness, and shear stiffness. j can be precisely evaluated.

[0192] In addition, in the method for evaluating rotational stiffness, a plurality of shear connectors 45 are provided, and the rotational stiffness S is calculated using equations (67a), (126), (137), (146) to (151), (153) to (155), and (157). j Therefore, using these equations, the rotational stiffness S of the joint 47a is evaluated. j can be evaluated more precisely.

[0193] Furthermore, in the method for evaluating rotational rigidity of this embodiment, when the building 1 is constructed using the composite beam 47, the acting moment acting on the joint 47a may fall below the moment resistance of the joint 47a. In this case, the ends of the composite beam 47 do not reach a service limit state that affects the usability or habitability of the building 1, and usability and habitability are maintained.

[0194] In addition, in the method for evaluating rotational rigidity of this embodiment, the absolute value of the acting moment M j,Ed is the moment resistance M of the joint 47a j,Rd When the moment resistance of the positively bent part of the composite beam 47 is bs,Rd may be set to satisfy equation (164). In this case, when the building 1 is constructed using the composite beam 47, both ends of the composite beam 47 will reach a service limit state that will impair the usability or habitability of the building, but the ultimate limit state regarding safety will not be reached, and the beam will not collapse and safety will be maintained.

[0195] In addition, in the joint structure 48 of the composite beam end of this embodiment, the rotational rigidity S evaluated using the rotational rigidity evaluation method j , and the rotational stiffness S so that the acting moment acting on the joint 47a in response to the vertical load is less than the moment resistance of the joint 47a. j In this case, when the building 1 is constructed using the joint structure 48 at the end of the composite beam, the ends of the composite beam 47 will not reach the service limit state that affects the usability or habitability of the building 1, and usability and habitability will be maintained. In addition, in the joint structure 48 of the composite beam end, the moment resistance M j,Rd(kNm), and the absolute value of the acting moment M j,Ed (kNm) satisfies the formulas (126), (152), (175), and (176). In this case, when the floor slab 35 has concrete 37 and reinforcing bars 38, the joint structure 48 of the composite beam end used in the building 1 can be precisely evaluated using the formulas (152), (175), and (176).

[0196] In addition, in the joint structure 48 of the composite beam end of this embodiment, the absolute value M of the acting moment acting on the joint 47a j,Ed is the moment resistance M of the joint 47a j,Rd When this is the case, the moment resistance M of the positive bending part of the composite beam 47 bs,Rd may be configured to satisfy equation (164). In this case, both ends of the composite beam 47 will reach a serviceability limit state that will impair the usability or habitability of the building 1, but the ultimate limit state for safety will not be reached, and the beam will not collapse and safety will be maintained.

[0197] Although one embodiment of the present invention has been described in detail above with reference to the drawings, the specific configuration is not limited to this embodiment, and configuration changes, combinations, deletions, etc. are also included within the scope that does not deviate from the gist of the present invention. For example, in the method for evaluating the rotational stiffness of the above embodiment, the rotational stiffness S is calculated without using the formulas (67a), (126), (137), (146) to (151), (153) to (155), and (157). j may be evaluated. In the joint structure 48 of the composite beam end, the moment resistance M j,Rd (kNm), and the absolute value of the acting moment M j,Ed (kNm) may be configured not to satisfy the formulas (152), (175), and (176).

[0198] The floor slab 35 may not have the reinforcing bars 38. The number of shear connectors 45 provided in the composite beam 47 may be one. In the above description, the steel beams are the minor beams and the supporting members are the major girders 15. However, the steel beams may be the major girders and the supporting members may be the columns. [Explanation of symbols]

[0199] 15 Large beam (supporting member) 25 Steel beams 33 Joints 35 Floor slab (slab) 37 Concrete 38 Reinforced concrete 45 shear connector 47 Composite beam 47a Joint 48 Joint structure of composite beam end 50 Interface X length direction

Claims

1. Steel beams and a slab supported on the steel beam; a shear connector that connects the steel beam and the slab to each other discretely or continuously in the longitudinal direction of the steel beam; Equipped with A method for evaluating rotational rigidity of a composite beam in which at least one end in the length direction is supported by a support member via a joint member and subjected to a vertical load, the method comprising: A method for evaluating rotational rigidity, which evaluates the rotational rigidity using the bending rigidity and axial rigidity of the slab at the joint, the bending rigidity and axial rigidity of the steel beam and the joint member, and the shear rigidity of the shear connector.

2. A plurality of the shear connectors are provided, The plurality of shear connectors are arranged side by side at intervals s (m) in the longitudinal direction, Using equations (1) to (13), the rotational stiffness S ｊ The method for evaluating rotational rigidity according to claim 1 , further comprising: evaluating the rotational rigidity of the rotational rigidity. where L is the length of the composite beam or the distance (m) between the pair of joints of the composite beam when both ends of the composite beam in the length direction are supported by the support members via the joint members to form a pair of joints, k is the shear stiffness (kN / m) per shear connector, and l ｊ is twice the length (m) of each of the joints in the longitudinal direction. (E ｓ I ｓ ) ｊ is the bending stiffness (kNm) of the slab at each of the joints ２ ) and (E ｓ A ｓ ) ｊ is the axial stiffness of the slab at each of the joints (kN), and (E ｂ I ｂ ) ｊ is the bending rigidity (kNm ２ ) and (E ｂ A ｂ ) ｊ is the axial stiffness (kN) of the steel beam and the joint member at each of the joints. E ｓ I ｓ is the bending rigidity of the slab (kNm ２ ) and E ｓ A ｓ is the axial stiffness of the slab (kN), and c ｓ is the distance (m) from the interface between the slab and the steel beam to the axis of the slab, and E ｂ I ｂ is the bending rigidity of the steel beam (kNm ２ ) and E ｂ A ｂ is the axial stiffness of the steel beam (kN), and c ｂ is the distance (m) from the interface to the axis of the steel beam. [Equation 1]

3. The rotational stiffness S evaluated using the method for evaluating rotational stiffness according to claim 1 or 2. ｊ , and the rotational stiffness S so that the acting moment acting on the joint in response to the vertical load is less than the moment resistance capacity of the joint. ｊ and a design method for the end of a composite beam that adjusts the moment resistance.

4. In the composite beam, both ends in the length direction are supported by the support members via the joint members, The rotational stiffness S evaluated using the method for evaluating rotational stiffness according to claim 1 or 2. ｊ , and the absolute value M of the acting moment acting on the joint in response to the vertical load ｊ，Ｅｄ The moment resistance M of the joint ｊ，Ｒｄ If it is more than this, Moment resistance M of the positively bent part of the composite beam ｂｓ，Ｒｄ A design method for the end of a composite beam, in which the value is set so as to satisfy equation (16). However, M ｂｓ，ｐｉｎ is the absolute value (kNm) of the maximum moment acting on the positively bent portion of the reference composite beam, assuming that each of the joints in the reference composite beam is a pin joint. [Equation 2]

5. the at least one end of the composite beam; the support member supporting the at least one end by a semi-rigid joint; At least one end of a second composite beam that is the composite beam and is arranged so as to sandwich the support member and is supported by the support member by a semi-rigid connection; Equipped with The plurality of shear connectors are arranged side by side at intervals s (m) in the longitudinal direction, The rotational stiffness S evaluated using equations (21) to (33) based on the method for evaluating rotational stiffness according to claim 1 or 2 ｊ and a joint structure of a composite beam end using a semi-rigid joint, in which the acting moment acting on the joint in response to the vertical load is smaller than the moment resistance of the joint. where L is the length of the composite beam or the distance (m) between the pair of joints of the composite beam when both ends of the composite beam in the length direction are supported by the support members via the joint members to form a pair of joints, k is the shear stiffness (kN / m) per shear connector, and l ｊ is twice the length (m) of each of the joints in the longitudinal direction. (E ｓ I ｓ ) ｊ is the bending stiffness (kNm) of the slab at each of the joints ２ ) and (E ｓ A ｓ ) ｊ is the axial stiffness of the slab at each of the joints (kN), and (E ｂ I ｂ ) ｊ is the bending rigidity (kNm ２ ) and (E ｂ A ｂ ) ｊ is the axial stiffness (kN) of the steel beam and the joint member at each of the joints. E ｓ I ｓ is the bending rigidity of the slab (kNm ２ ) and E ｓ A ｓ is the axial stiffness of the slab (kN), and c ｓ is the distance (m) from the interface between the slab and the steel beam to the axis of the slab, and E ｂ I ｂ is the bending rigidity of the steel beam (kNm ２ ) and E ｂ A ｂ is the axial stiffness of the steel beam (kN), and c ｂ is the distance (m) from the interface to the axis of the steel beam. [Equation 3]

6. The slab is Concrete and a reinforcing bar embedded in the concrete, The moment resistance M ｊ，Ｒｄ (kNm), and the absolute value of the acting moment M ｊ，Ｅｄ (kNm) is the moment resistance M calculated by equations (42) to (44). ｊ，Ｒｄ The composite beam end joint structure according to claim 5, wherein the joint structure is smaller than the above. However, σ ｒｙ is the yield stress of the reinforcing bar (kN / m ２ ) and A ｒ，ｅｆｆ is the total cross-sectional area of ​​the reinforcing bars within the effective width of the slab (m ２ ) and x ｈ is the length (m) in the longitudinal direction from the joint to the position where the bending moment acting on the composite beam becomes 0, and P Ｒｄ is the shear strength (kN) per shear connector, and n is the number of shear connectors arranged between the joint and the position where the bending moment acting on the composite beam becomes zero. [Equation 4]

7. the at least one end of the composite beam; the support member supporting the at least one end by a semi-rigid joint; At least one end of a second composite beam that is the composite beam and is arranged so as to sandwich the support member and is supported by the support member by a semi-rigid connection; Equipped with The rotational stiffness S evaluated using the method for evaluating rotational stiffness according to claim 1 or 2. ｊ , and the absolute value M of the acting moment acting on the joint in response to the vertical load ｊ，Ｅｄ The moment resistance M of the joint ｊ，Ｒｄ If it is more than this, Moment resistance M of the positively bent part of the composite beam ｂｓ，Ｒｄ The joint structure of the composite beam end is a semi-rigid joint that satisfies equation (45). However, M ｂｓ，ｐｉｎ is the absolute value (kNm) of the maximum moment acting on the positively bent portion of the reference composite beam, assuming that each of the joints in the reference composite beam is a pin joint. [Equation 5]

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