Method for evaluating beams

Abstract

To provide a beam evaluation method that evaluates the collapse temperature of beams by considering local buckling under high temperatures.SOLUTION: The present invention provides an evaluation method S1 for beams made of H-beams in steel structures that collapse due to heating due to fire. The evaluation method evaluates the collapse temperature of the beams based on the high temperature local buckling yield strength of the beams.SELECTED DRAWING: Figure 4

Description

[Technical Field]

[0001] This invention relates to a method for evaluating beams and to beams themselves. [Background technology]

[0002] In conventional buildings, thin-walled steel members were used in non-fire-resistant buildings such as low-rise houses, where fire resistance performance of the members is not legally required. On the other hand, with the advancement of rational design in recent years, there has been a growing need to use thin-walled steel members even in mid-to-high-rise fire-resistant buildings that require fire resistance, with the aim of reducing the amount of steel used in buildings. One of the problems that arises here is local buckling during a fire. In addition to the decrease in heat capacity due to the thinning and weight reduction of steel members, the Young's modulus of steel members decreases when heated by fire. Therefore, the likelihood of local buckling occurring due to long-term live loads increases. Patent documents 1 and 2 estimate the buckling stress (local buckling stress, shear buckling stress) of H-shaped steel at room temperature. [Prior art documents] [Patent Documents]

[0003] [Patent Document 1] Japanese Patent Publication No. 2021-006791 [Patent Document 2] Japanese Patent Publication No. 2021-006787 [Overview of the project] [Problems that the invention aims to solve]

[0004] However, Patent Documents 1 and 2 do not provide a method for evaluating the local buckling strength of beams under high temperatures. Furthermore, current fire-resistant design does not consider local buckling under high temperatures, and the strength of beams is evaluated based on the full plastic bending strength of the beam or the shear yield strength of the web, as described in the Fire-Resistant Design Guidelines for Steel Structures (edited by the Architectural Institute of Japan, published by Maruzen Publishing Co., Ltd.; hereinafter referred to as Document 1). Consequently, for thin-walled H-shaped steel beams where local buckling occurs, if the local buckling strength is less than the full plastic bending strength or shear yield strength, the beam will collapse due to the local buckling strength. Therefore, there is a possibility of underestimating the beam's safety.

[0005] This invention has been made in view of the above problems, and aims to provide a beam evaluation method for evaluating the collapse temperature of a beam, taking into account local buckling under high temperatures, and a beam that is evaluated and designed using this beam evaluation method. [Means for solving the problem]

[0006] To solve the aforementioned problems, this invention proposes the following means. The present invention provides a method for evaluating a beam formed from H-shaped steel in a steel frame structure when it is heated by a fire and collapses, characterized in that the collapse temperature of the beam is evaluated based on the high-temperature local buckling strength of the beam. In this invention, the collapse temperature of a beam is evaluated based on the high-temperature local buckling strength, which is the local buckling strength under high temperatures. Therefore, the collapse temperature of a beam can be evaluated while taking into account local buckling under high temperatures.

[0007] Furthermore, the beam evaluation method may include a temperature calculation step of determining the temperature of the first flange, second flange, and web of the beam when the beam is heated by experiment or numerical calculation; a buckling strength evaluation step of evaluating the high-temperature local buckling strength using the high-temperature Young's modulus determined based on the temperatures of the first flange, second flange, and web; and a collapse evaluation step of evaluating the collapse temperature based on the high-temperature local buckling strength. In this context, high-temperature Young's modulus refers to the Young's modulus at temperatures higher than room temperature, which is lower than the Young's modulus at room temperature. In this invention, in the temperature calculation step, the temperatures of the first flange, the second flange, and the web are determined by experiment or numerical calculation. Then, in the buckling strength evaluation step, the high-temperature local buckling strength is evaluated using the high-temperature Young's modulus determined based on these temperatures, and in the collapse evaluation step, the collapse temperature can be evaluated based on the high-temperature local buckling strength.

[0008] Furthermore, in the beam evaluation method, the axis in the direction of the material axis through which the beam extends is defined as the x-axis, the axis in the direction in which the first flange and the second flange face each other is defined as the y-axis, the axis in the thickness direction of the web is defined as the z-axis, it is assumed that the first flange is subjected to a tensile force due to a bending moment M acting on the beam around the z-axis, the position of the center of the first flange in the direction along the y-axis is defined as the origin of the y-axis, the direction from the first flange toward the second flange is defined as the positive direction of the y-axis, and the distance between the center of the first flange and the center of the second flange in the direction along the y-axis is defined as b w As defined, and as the web is displaced alternately in a wave-like manner in the positive and negative directions of the z-axis toward the first end of the web in the direction along the x-axis, the half wavelength of the web in the direction along the x-axis is defined as a, then in the buckling strength evaluation step, the high-temperature local buckling strength M is calculated from equation (1). cr The collapse evaluation step evaluates the high-temperature local buckling strength M that satisfies equation (2). cr The collapse temperature may be evaluated based on the bending collapse temperature, which is the temperature corresponding to the aforementioned temperature.

[0009]

number

[0010] However, Z is the section modulus of the beam, and σ cr is the local buckling stress of the beam, N is a natural number greater than or equal to 2, and a0, a n ,b nis an undetermined coefficient, and E f1 is the high-temperature Young's modulus of the first flange, and E w is the high-temperature Young's modulus of the web, and E f2 is the high-temperature Young's modulus of the second flange, ν is the Poisson's ratio of the beam, and t w is the thickness of the web, and t f is the thickness of each of the first flange and the second flange, and b f is the value of half of the width of each of the first flange and the second flange. The local buckling stress σ cr is, using equations (5) to (11), the local buckling stress σ obtained by equation (12) cr is the real number a that gives the minimum positive value to n , b n and is obtained based on the half wavelength a.

[0011]

Equation

[0012] The present invention can evaluate the high-temperature local buckling resistance M In the buckling resistance evaluation step, considering the combined buckling of the first flange, the second flange, and the web of the beam, from equation (1) cr . And in the collapse evaluation step, the collapse temperature can be evaluated based on the bending collapse temperature, which is the temperature corresponding to the high-temperature local buckling resistance M cr that satisfies equation (2).

[0013] Also, in the evaluation method of the beam, it is assumed that a bending moment M acts on the beam such that the first flange receives a tensile force around an axis along the plate thickness direction of the web. In the buckling resistance evaluation step, the high-temperature local buckling resistance M crf obtained by equation (27), and the high-temperature local buckling resistance M crw obtained by equation (28), the smaller one of them is evaluated as the high-temperature local buckling resistance M cr , and in the collapse evaluation step, the high-temperature local buckling resistance M cr that satisfies equation (29)The collapse temperature may be evaluated based on the bending collapse temperature, which is the temperature corresponding to the aforementioned temperature. However, Z is the section modulus of the beam, and E w is the high-temperature Young's modulus of the web, and E f2 ν is the high-temperature Young's modulus of the second flange, ν is the Poisson's ratio of the beam, and t w is the thickness of the web, and t f b is the thickness of the first flange and the second flange, respectively. f b is half the width of the first flange and the second flange, respectively. w This is the distance between the center of the first flange and the center of the second flange in the direction in which the first flange and the second flange face each other.

[0014]

number

[0015] In this invention, in the buckling strength evaluation process, the high-temperature local buckling strength M cr This can be evaluated relatively easily, and in the collapse evaluation process, the high-temperature local buckling strength M cr The collapse temperature can be evaluated based on the bending collapse temperature, which is the temperature corresponding to the collapse.

[0016] Furthermore, in the beam evaluation method, the temperature used to determine the high-temperature Young's modulus of the first flange and the second flange in the buckling strength evaluation step may be the average temperature, maximum temperature, or minimum temperature of the first flange and the second flange determined in the temperature calculation step, respectively, and the temperature used to determine the high-temperature Young's modulus of the web in the buckling strength evaluation step may be the average temperature of the web determined in the temperature calculation step. In this invention, the high-temperature local buckling resistance M cr This allows for a more appropriate evaluation.

[0017] Furthermore, in the beam evaluation method, the axis in the direction of the material axis through which the beam extends is defined as the x-axis, the axis in the direction in which the first flange and the second flange face each other is defined as the y-axis, the axis in the thickness direction of the web is defined as the z-axis, it is assumed that a shear force Q acts on the beam in the direction along the y-axis, the position of the center in the direction along the y-axis on the web is defined as the origin of the y-axis, the direction from the first flange toward the second flange is defined as the positive direction of the y-axis, and the distance between the center of the first flange and the center of the second flange in the direction along the y-axis is defined as b w As defined, and as the web moves toward the first end in the direction along the x-axis, the half wavelength of the web in the direction along the x-axis is defined as a, and as the web moves toward the first end in the direction along the x-axis, it is alternately displaced in a wave-like manner in the positive and negative directions of the z-axis. In the buckling strength evaluation step, the high-temperature local buckling strength Q is calculated from equation (41). cr The collapse evaluation step evaluates the high-temperature local buckling strength Q that satisfies equation (42). cr The collapse temperature may be evaluated based on the shear collapse temperature, which is the temperature corresponding to the above.

[0018]

number

[0019] However, cr is the shear buckling stress of the beam, and A w is the cross-sectional area of ​​the web perpendicular to the material axis direction, N is a natural number greater than or equal to 2, and a0, a n ,b n λ is an undetermined coefficient, and E f1 is the high-temperature Young's modulus of the first flange, and E w is the high-temperature Young's modulus of the web, and E f2 ν is the high-temperature Young's modulus of the second flange, ν is the Poisson's ratio of the beam, and t w is the thickness of the web, and t f b is the thickness of the first flange and the second flange, respectively. f This value is half the width of the first flange and the second flange, respectively. The shear buckling stress τ cr Using equations (45) to (50), the shear buckling stress τ according to equation (51) is obtained. cr The real number a that gives the smallest positive value to is the a n ,b n It is determined based on λ and the aforementioned half-wavelength a.

[0020]

number

[0021] In this invention, in the buckling strength evaluation process, coupled buckling of the first flange, second flange, and web of the beam is taken into consideration, and the high-temperature local buckling strength Q is calculated from equation (41). cr This can be evaluated. And in the collapse evaluation process, the high-temperature local buckling strength Q that satisfies equation (42) can be evaluated. cr The collapse temperature can be evaluated based on the shear collapse temperature, which is the temperature corresponding to the collapse temperature.

[0022] Furthermore, in the beam evaluation method, it is assumed that a shear force Q acts on the beam in the direction in which the first flange and the second flange face each other, and in the buckling strength evaluation step, the high-temperature local buckling strength Q is calculated using equation (61). cr The collapse evaluation process evaluates the high-temperature local buckling strength Q that satisfies equation (62). cr The collapse temperature may be evaluated based on the shear collapse temperature, which is the temperature corresponding to the above. However, A w E is the cross-sectional area perpendicular to the direction of the material axis in which the beam extends in the web, w ν is the Young's modulus of the web, ν is the Poisson's ratio of the beam, and t w b is the thickness of the web, w This is the distance between the center of the first flange and the center of the second flange in the direction in which the first flange and the second flange face each other.

[0023]

number

[0024] In this invention, in the buckling strength evaluation process, the high-temperature local buckling strength Q cr This can be evaluated relatively easily, and in the collapse evaluation process, the high-temperature local buckling resistance Q cr The collapse temperature can be evaluated based on the shear collapse temperature, which is the temperature corresponding to the collapse temperature.

[0025] Furthermore, in the beam evaluation method, the temperature used to determine the high-temperature Young's modulus of the first flange and the second flange in the buckling strength evaluation step is one of the average temperature, maximum temperature, or minimum temperature of the first flange and the second flange determined in the temperature calculation step, respectively, and the temperature used to determine the high-temperature Young's modulus of the web in the buckling strength evaluation step may be the maximum temperature of the web determined in the temperature calculation step. In this invention, the high-temperature local buckling resistance Q cr This allows for a more appropriate evaluation.

[0026] Furthermore, in the beam evaluation method, the temperature of the first flange, the temperature of the second flange, and the temperature of the web may be equal to each other. In this invention, the high-temperature local buckling resistance M cr ,Q cr This makes evaluation easier.

[0027] Furthermore, in the beam evaluation method, in the temperature calculation step, the beam may be heated based on a predetermined heating curve, and in the collapse evaluation step, the collapse time, which is the time when the beam collapses, may be evaluated based on the high-temperature local buckling strength. In this invention, the collapse time when a beam is heated according to a predetermined heating curve can be evaluated based on its high-temperature local buckling resistance.

[0028] Furthermore, in the beam evaluation method, the collapse temperature may be evaluated based on the lower of the bending collapse temperature evaluated by the beam evaluation method described above and the shear collapse temperature evaluated by the beam evaluation method described above. This invention makes it possible to appropriately evaluate the collapse temperature when a bending moment M and a shear force Q act simultaneously on a beam.

[0029] Furthermore, the beam of the present invention is characterized by being evaluated and designed using the beam evaluation method described above. This invention allows for the evaluation of the beam's collapse temperature, taking into account local buckling under high temperatures, and enables the design of the beam accordingly. [Effects of the Invention]

[0030] The beam evaluation method and beam of the present invention allow for the evaluation of the beam's collapse temperature, taking into account local buckling under high temperatures. [Brief explanation of the drawing]

[0031] [Figure 1] This is a cross-sectional view of a floor structure using a beam according to one embodiment of the present invention. [Figure 2] This is a schematic perspective view showing a beam buckling under the influence of a bending moment. [Figure 3] This is a cross-sectional view perpendicular to the axial direction of the beam. [Figure 4] This is a flowchart showing a method for evaluating a beam in one embodiment of the present invention. [Figure 5] This is a schematic perspective view showing a beam buckling under shear force. [Figure 6] This is a schematic perspective view showing the state of the second flange when a bending moment acts on the beam. [Figure 7] This is a schematic perspective view showing the state of the web when a bending moment acts on a beam. [Figure 8] This is a schematic perspective view showing the state of the web when a shear force is applied to a beam. [Figure 9] This is a cross-sectional view illustrating the boundary conditions and temperature measurement locations when determining the temperature of a bed structure using numerical calculations. [Figure 10] This diagram shows the change in the temperature of a beam over time. [Figure 11]This figure shows an example of the temperature distribution of a floor structure when the high-temperature local buckling resistance is set using the method in [2.1]. [Figure 12] This figure shows an example of the temperature distribution of a floor structure when the high-temperature local buckling resistance is set using the method in [2.2]. [Figure 13] This figure shows the temperature history at each measurement point on the first flange. [Figure 14] This diagram shows the temperature history at each measurement point on the web. [Figure 15] This figure shows the temperature history at each measurement point on the second flange. [Figure 16] This diagram shows how temperature changes depending on the height of the web. [Figure 17] This figure shows the change in high-temperature Young's modulus over time. [Figure 18] This diagram shows the loads applied to the fire-resistant beam. [Figure 19] This is a perspective view showing the position of the stiffening members that are joined to the fire-resistant beam when a bending moment is applied. [Figure 20] This is a perspective view showing the position of the stiffening members that are joined to the fire-resistant cladding beam when shear force is applied. [Figure 21] This figure shows the change in deflection over time for a fire-resistant clad beam where local buckling occurs in a region where equal bending moments are applied. [Figure 22] This figure shows the change in deflection over time for a fire-resistant clad beam where local buckling occurs in the region where shear force is applied. [Figure 23] This figure shows the change in high-temperature local buckling strength Mcr over time when a bending moment is applied. [Figure 24] This figure shows an example of the change in high-temperature local buckling resistance Qcr over time when a shear force is applied. [Figure 25] This figure shows an example of the change in high-temperature local buckling strength Mcr over time when a bending moment is applied. [Figure 26] This figure shows another example of the change in high-temperature local buckling strength Mcr over time when a bending moment is applied. [Figure 27] This figure shows an example of the change in high-temperature local buckling resistance Qcr over time when a shear force is applied. [Figure 28] This figure shows another example of the change in high-temperature local buckling strength Qcr over time when a shear force is applied. [Modes for carrying out the invention]

[0032] Hereinafter, an embodiment of the beam and the beam evaluation method according to the present invention will be described with reference to Figures 1 to 28. First, we will describe the floor structure in which the beams of this embodiment are used.

[0033] [1. Floor structure configuration] As shown in Figures 1 and 2, the floor structure 1 comprises fire-resistant coated beams (fire-coated beams) 10 and a floor slab 20. In Figure 2, only beam 11, which will be described later, is shown among the fire-resistant coated beams 10, and the beam 11 is shown in a state where it has buckled due to a bending moment. The fire-resistant clad beam 10 comprises a beam 11 and a fire-resistant clad 12. The beam 11 is formed from an H-shaped steel frame. The beam 11 has 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 formed from steel plates, which are elastic elements. Elastic elements are elements that do not take material nonlinearity into account.

[0034] For example, the beam 11 is positioned such that the axis direction of the beam 11 is aligned with the horizontal plane. The first flange 16 is formed in a flat plate shape and is positioned such that the thickness direction of the first flange 16 is aligned with the vertical direction. The second flange 17 is formed in a flat plate shape and is positioned above the first flange 16. The second flange 17 is positioned such that the thickness direction of the second flange 17 is aligned with the vertical direction. The web 18 is formed in a flat plate shape that exhibits a rectangular shape when viewed in the thickness direction of the web 18. The web 18 is positioned so that its thickness direction aligns with the horizontal plane. The web 18 connects the center in the width direction on the upper surface of the first flange 16 and the center in the width direction on the lower surface of the second flange 17. However, the orientation in which the beam 11 is positioned is not limited to this; the beam 11 may also be positioned so that its axis intersects the horizontal plane.

[0035] The fire-resistant covering 12 is made of insulating material such as rock wool or glass wool. The fire-resistant covering 12 covers all surfaces of the beam 11 except the upper surface of the second flange 17. The thickness of the fire-resistant coating 12 is set in accordance with the "Guidelines for Quality Control of Construction of Sprayed Rock Wool Coated Fire-Resistant Structures (Rock Wool Industry Association, Spraying Division)". If the fire-resistant beam 10 is required to have 1 hour fire resistance, the thickness of the fire-resistant coating 12 shall be 25 mm. Similarly, if the fire-resistant beam 10 is required to have 2 hours fire resistance, the thickness of the fire-resistant coating 12 shall be 45 mm. If the fire-resistant beam 10 is required to have 3 hours fire resistance, the thickness of the fire-resistant coating 12 shall be 60 mm. The ends of the fire-resistant cladding beam 10 in the direction of the material axis are fixed to columns or the like (not shown).

[0036] The floor slab 20 is a concrete slab, a deck slab, or the like. The floor slab 20 is formed in a flat plate shape and is positioned so that its thickness direction is aligned with the vertical direction. The floor slab 20 is supported from below by the second flange 17 of the beam 11. Preferably, a shear connector (not shown) is fixed to the upper surface of the second flange 17. The shear connector is embedded in the floor slab 20. Floor structure 1 is used by installing equipment (not shown) on the floor slab 20, etc.

[0037] Here, as shown in Figure 3, the dimensions of the cross section perpendicular to the axis direction of the beam 11 are defined. Note that for the units of length and other parameters described below, SI units such as "m" are preferably used for length. The thickness of the first flange 16 and the second flange 17 is t f It is stipulated that the thickness of web 18 is t w The following is stipulated: The value of half the width of the first flange 16 and the second flange 17 is set to b f The distance between the center of the first flange 16 and the center of the second flange 17 in the y-axis direction is defined as b w It is stipulated that... The section modulus of beam 11 is defined as Z. The Poisson's ratio of beam 11 is defined as ν.

[0038] [2. Method for Evaluating Beams] The beam evaluation method of this embodiment evaluates the collapse temperature and collapse time of the beam 11 based on the high-temperature local buckling strength of the beam 11, which will be described later. Collapse, as defined in References 2 and 3 below, is when the maximum deflection amount of the beam 11 reaches (L) based on the limit deflection formula of Reference 2. 2 This means that the value exceeds / (400d)). However, L is the span length (length) of beam 11. d is the fault of beam 11. Document 2: "ISO 834-1: Fire-resistance tests - Elements of building construction -", International Organization for Standardization, p23 Reference 3: "Methodology for Fire Resistance Performance Testing and Evaluation," Japan Building Research Institute, p7 The collapse temperature refers to the temperature of beam 11 when it collapses due to being heated by a fire. The collapse time refers to the time when beam 11 collapses.

[0039] As shown in Figure 4, the beam evaluation method S1 includes a temperature calculation step S10, a buckling strength evaluation step S11, and a collapse evaluation step S12. In the temperature calculation step S10, the temperatures of the first flange 16, second flange 17, and web 18 of the beam 11 when the beam 11 is heated are determined by experiment or numerical calculation. For example, numerical calculations may include heat conduction analysis, FEM (Finite Element Method), manual calculation, and linear approximation.

[0040] In the buckling strength evaluation process S11, the high-temperature local buckling strength is evaluated using the high-temperature Young's modulus determined based on the temperature of the first flange 16, the second flange 17, and the web 18, respectively. The high-temperature Young's modulus is defined in the aforementioned reference 1 and reference 4 below, among others. Document 4: "Eurocode 3: Design of steel structures - Part 1-2: General rules - Structural fire design" en. 1993-1-2, 2005, p.22, Table 3.1

[0041] In the collapse evaluation process S12, the collapse temperature and collapse time of the beam 11 are evaluated based on the high-temperature local buckling strength evaluated in the buckling strength evaluation process S11.

[0042] The following describes four methods for performing the buckling strength evaluation process S11 and the collapse evaluation process S12, as described in [2.1] to [2.4]. Sections [2.1] and [2.2] describe an evaluation method that adds the concept of high-temperature Young's modulus to the buckling stress (local buckling stress, shear buckling stress) disclosed in Patent Documents 1 and 2. Patent Documents 1 and 2 consider coupled buckling of the first flange 16, the second flange 17, and the web 18 of the beam 11. Generally, bending moments and shear forces act on beams, but not compressive forces. Therefore, bending moments and shear forces are considered as external forces acting on beams. Specifically, in sections [2.1] and [2.3], bending moments are considered as external forces, while in sections [2.2] and [2.4], shear forces are considered as external forces.

[0043] [2.1. Method considering the high-temperature Young's modulus and coupled buckling of a beam when a bending moment acts on it] The method in [2.1] is a method for evaluating a beam subjected to a bending moment (first evaluation method for beams) when only a bending moment acts on the beam 11. First, we define the axis to be used when evaluating beam 11. As shown in Figures 1 and 2, the axis in the direction of the material axis through which the beam 11 extends is defined as the x-axis. The axis in the direction in which the first flange 16 and the second flange 17 face each other is defined as the y-axis. The axis in the thickness direction of the web 18 is defined as the z-axis. The x, y, and z axes are orthogonal to each other. Viewed in the direction along the z axis (hereinafter referred to as the z-axis direction), the web 18 has edges extending in the direction along the x axis (hereinafter referred to as the x-axis direction) and edges extending in the direction along the y axis (hereinafter referred to as the y-axis direction). The out-of-plane displacement of the web 18 is the displacement of the web 18 in the direction of the z-axis direction.

[0044] The flanges 16 and 17 are positioned to sandwich the web 18 in the y-axis direction. The out-of-plane displacement of flanges 16 and 17 is a displacement in the y-axis direction. The beam 11 is assumed to be sufficiently long in the x-axis direction. Here, "sufficiently long in the x-axis direction" means that the beam 11 has a length such that the boundary conditions of the surfaces 11a that are located at each end of the beam 11 in the x-axis direction and extend in the y-axis direction (hereinafter referred to as the end faces in the x-axis direction) have little effect on the buckling deformation.

[0045] As shown in Figure 2, when bending moments M about the z axis are applied to each end face 11a of the beam 11 in the x-axis direction, the beam 11 may buckle. The bending moments M acting on each end face 11a of the beam 11 in the x-axis direction are external forces of equal magnitude. In this example, the bending moments M acting on the beam 11 cause the first flange 16 to experience a tensile force and the second flange 17 to experience a compressive force, causing the beam 11 to bend downwards in a convex shape. In this case, as you move toward the first end of the web 18 in the x-axis direction (one of the end faces 11a in the x-axis direction), the web 18 is displaced alternately in the positive and negative directions of the z-axis, causing the web 18 as a whole to be displaced in a wave-like manner (hereinafter referred to as wave-like in the x-axis direction) for multiple wavelengths. For one wavelength of web 18 displaced along the x-axis, the second end opposite to the first end in the x-axis direction is defined as the origin of the x-axis, and the direction from this second end toward the first end in the x-axis direction is defined as the positive direction of the x-axis.

[0046] The center of the first flange 16 in the y-axis direction is defined as the origin of the y-axis. The direction from the first flange 16 toward the second flange 17 is defined as the positive direction of the y-axis. The z-axis origin is defined as the center of web 18 in the z-axis direction (center in the thickness direction). The positive direction of the z-axis is defined as the direction that constitutes a right-handed Cartesian coordinate system with respect to the positive directions of the x-axis and y-axis.

[0047] Assume that when the y-axis coordinate of a web 18 displaced in a wave-like manner in the x-axis direction is a certain value, the out-of-plane displacement of the web 18 in the z-axis direction at a certain x-axis coordinate can be expressed by the equation sin(πx / a). In this case, the wavelength of the web 18 in the x-axis direction when it is displaced in a wave-like manner in the x-axis direction is 2a. The half-wavelength (half the length of the wavelength) of the web 18 in the x-axis direction is a.

[0048] As shown in Patent Document 1, the out-of-plane displacement W of the web 18. w This is estimated by equation (71). Out-of-plane displacement W of the first flange 16 f1 , out-of-plane displacement W of the second flange 17 f2 These are estimated by equations (72) and (73), respectively.

[0049]

number

[0050] However, N is a natural number greater than or equal to 2, and a0, a n ,b n This is an undetermined coefficient. Here, based on the energy method, the strain energy U generated in the web 18 due to buckling deformation w This is expressed as in equation (75). The strain energy U of flanges 16 and 17 f This can be expressed as shown in equation (76).

[0051]

number

[0052] However, the plate stiffness D of web 18 w This is expressed as in equation (79). Plate stiffness D of the first flange 16 and the second flange 17. f1 ,D f2 This can be expressed as in equations (80) and (81). Here, E f1 This is the high-temperature Young's modulus of the first flange 16. f2 This is the high-temperature Young's modulus of the second flange 17. w This is the high-temperature Young's modulus of web 18.

[0053]

number

[0054] Stress function σ of web 18 under the bending moment M w The local buckling stress (buckling stress) of beam 11 is σ cr It can be expressed as shown in equation (84), where the stress function σ w Compression is considered positive. Stress function σ f1 ,σ f2 These are the stress functions of flanges 16 and 17, respectively, under the bending moment M. Stress function σ f1 ,σ f2 This is given by equations (85) and (86), where the stress function σ f1 ,σ f2 Compression is considered positive. function δ w This is the displacement in the x-axis direction at a certain coordinate of the y-axis of the web 18 when buckling occurs, and is expressed as equation (87) with δ being the displacement in the x-axis direction that occurs at the center of the second flange 17.

[0055]

number

[0056] Function δ f1 , δ f2 is a function that represents the displacement of the flanges 16 and 17 in the x-axis direction when buckling occurs. Function δ f1 is equal to -δ, and function δ f2 is equal to δ. At this time, equation (76) is expressed as equation (76A).

[0057]

Equation

[0058] Also, the external potential energy V of the web 18 w is expressed as in equation (90), and the external potential energy V of the flanges 16 and 17 f is expressed as in equation (91).

[0059]

Equation

[0060] The total potential energy Π of the beam 11 is expressed as in equation (92) as the sum of the strain energy and the external potential energy.

[0061]

Equation

[0062] In the buckling strength evaluation step S11, the local buckling stress σ of the beam 11 cr is obtained based on the out-of-plane displacement W of the web 18 w , the out-of-plane displacement W of the flanges 16 and 17 f1 , W f2 , and the energy method. That is, in the buckling strength evaluation step S11, the local buckling stress σ according to equation (99) is obtained using equations (95) to (98), and the real numbers a cr that gives the minimum positive value to n , b n and the half wavelength a, and the local buckling stress σcr We seek.

[0063]

number

[0064] Here, in the first term on the right-hand side of equation (99), the Young's modulus E of the H-shaped steel in the first term on the right-hand side of equation (55) in the embodiment of Patent Document 1 is equal to the high-temperature Young's modulus E of the web 18. w It has been replaced by this. The reason for this is that the first term on the right side of equation (99) is the local buckling stress based on the web 18. On the other hand, in the second term on the right side of equation (99), the Young's modulus E of the H-shaped steel in the second term on the right side of equation (55) in the embodiment of Patent Document 1 is replaced by Σ(E fi D i This has been replaced by the second term on the right-hand side of equation (99), which is the local buckling stress based on the i-th flange.

[0065] Next, local buckling stress σ cr I will now explain how to calculate it in detail. With half a wavelength a treated as a constant, the total potential energy Π is given by an undetermined coefficient a. n ,b n We set up a system of equations that show the function obtained by partial differentiation with respect to equal 0, and the real number a n ,b n We find the solution to the system of equations. n ,b n If there are multiple pairs, a n ,b n Among the multiple sets of (99), the local buckling stress σ cr a that gives the smallest positive value n ,b n Based on the set (a n ,b n Substituting the set into equation (99), we obtain the local buckling stress σ cr Next, we determine the half-wavelength a as a variable. n ,b nThe set of values ​​obtained is used to determine the half-wavelength a from an equation that shows that the function obtained by partially differentiating the total potential energy Π with respect to half-wavelength a is equal to 0. n ,b n Local buckling stress σ determined based on the set and half wavelength a cr However, the required local buckling stress σ cr This is the result. The undetermined coefficient a is the solution to the system of equations. n ,b n If there is only one set, a n ,b n The set is the local buckling stress σ according to equation (99). cr When a gives the smallest positive value, n ,b n Based on the set, local buckling stress σ cr Next, we determine the local buckling stress σ, treating half-wavelength a as a variable, as described above. cr We seek.

[0066] Thus, the local buckling stress σ cr Using equations (71) to (73) and (95) to (98), the local buckling stress σ according to equation (99) is calculated. cr a is a real number that gives the smallest positive value to n ,b n And it is determined based on half wavelength a. Then, in the buckling strength evaluation process S11, the high-temperature local buckling strength M is calculated using equation (102). cr The following is evaluated: In the collapse evaluation process S12, the high-temperature local buckling strength M that satisfies equation (103) is evaluated. cr The collapse temperature is evaluated based on the bending collapse temperature, which is the temperature corresponding to the bending moment. For example, when only a bending moment M acts on beam 11, the bending collapse temperature is evaluated as being equal to the collapse temperature. High temperature local buckling strength M cr The method for determining the corresponding temperature will be described later in [2.5].

[0067]

number

[0068] [2.2. Method considering the high-temperature Young's modulus and coupled buckling of a beam when shear force is applied to the beam] The method in [2.2] is a method for evaluating a beam under shear force when only shear force acts on the beam 11 (the first evaluation method for beams). The differences between method [2.2] and method [2.1] are as follows (1) and (2). (1) As shown in Figure 5, assume that a shear force Q acts on the beam 11 in the y-axis direction (the direction in which the first flange 16 and the second flange 17 face each other). (2) In the Cartesian coordinate system, the position of the center in the y-axis direction of the web 18 is defined as the origin of the y-axis. Here, the cross-sectional area of ​​the web 18 perpendicular to the material axis direction is A. w It is stipulated that...

[0069] As shown in Patent Document 2, the out-of-plane displacement W of the web 18. w This is estimated by equation (111). Out-of-plane displacement W of the first flange 16 f1 , out-of-plane displacement W of the second flange 17 f2 These are estimated by equations (112) and (113), respectively.

[0070]

number

[0071] However, N is a natural number greater than or equal to 2, and a0, a n ,b n λ is an undetermined coefficient. Here, based on the energy method, the strain energy U generated in the web 18 due to buckling deformation w This is expressed as in equation (115). The strain energy U of flanges 16 and 17 f This can be expressed as shown in equation (116). Note that δ is the displacement of the web 18 in the x-axis direction when buckling occurs.

[0072]

number

[0073] However, the plate stiffness D of web 18 w , plate rigidity D of the second flange 17 f1 ,D f2 This can be expressed as shown in equations (79), (80), and (81) above. Furthermore, the external force potential energy V of web 18 w It can be expressed as in equation (119). However, τ cr This is the shear buckling stress (buckling stress) of beam 11.

[0074]

number

[0075] The total potential energy Π of beam 11 is expressed as the sum of strain energy and external force potential energy, as shown in equation (120).

[0076]

number

[0077] In the buckling strength evaluation process S11, the shear buckling stress τ of beam 11 is evaluated. cr The out-of-plane displacement W of web 18. w Out-of-plane displacement W of flanges 16, 17 f1 , W f2 , and is determined based on the energy method. That is, in the buckling strength evaluation process S11, equations (123) to (125) are used to determine the shear buckling stress τ according to equation (126). cr a is a real number that gives the smallest positive value to n ,b n Based on λ and half wavelength a, the shear buckling stress τ cr We seek.

[0078]

number

[0079] Here, in the first term on the right-hand side of equation (126), the Young's modulus E of the H-shaped steel in the first term on the right-hand side of equation (54) in the embodiment of Patent Document 2 is equal to the high-temperature Young's modulus E of the web 18. w It has been replaced by this. The reason for this is that the first term on the right side of equation (126) is the local buckling stress based on the web 18. On the other hand, in the second term on the right side of equation (126), the Young's modulus E of the H-shaped steel in the second term on the right side of equation (54) in the embodiment of Patent Document 2 is Σ(E fi C i This has been replaced by the second term on the right-hand side of equation (126), which is the local buckling stress based on the i-th flange.

[0080] Next, local buckling stress σ cr I will now explain how to calculate it in detail. With half-wavelength a and undetermined coefficient λ treated as constants, the total potential energy Π is given by the undetermined coefficient a n ,b n We set up a system of equations that show the function obtained by partial differentiation with respect to equal 0, and the real number a n ,b n We find the solution to the system of equations. n ,b n If there are multiple pairs, a n ,b n Among the multiple sets, the shear buckling stress τ according to equation (126) cr a that gives the smallest positive value n ,b n Based on the set (a n ,b n Substituting the set into equation (126), we obtain the shear buckling stress τ cr Next, we determine the half-wavelength a and the undetermined coefficient λ as variables, and the a n ,b n The set of half wavelengths a and undetermined coefficients λ are determined from the equation that shows that the function obtained by partially differentiating the total potential energy Π with respect to half wavelength a and undetermined coefficient λ is equal to 0. n ,b n The shear buckling stress τ was determined based on the set of half wavelength a and the undetermined coefficient λ. cr However, the required shear buckling stress τ crThis is the result. The undetermined coefficient a is the solution to the system of equations. n ,b n If there is only one set, a n ,b n The set of values ​​is the shear buckling stress τ according to equation (126). cr When a gives the smallest positive value, n ,b n Shear buckling stress τ based on the set cr Next, we determine the shear buckling stress τ, treating the half-wavelength a and the undetermined coefficient λ as variables, as described above. cr We seek.

[0081] Thus, the shear buckling stress τ cr Using equations (111) to (113) and (123) to (125), the shear buckling stress τ according to equation (126) is calculated. cr a is a real number that gives the smallest positive value to n ,b n It is determined based on λ and half wavelength a. Then, in the buckling strength evaluation process S11, the high-temperature local buckling strength Q is calculated using equation (129). cr The following is evaluated: In the collapse evaluation process S12, the high-temperature local buckling strength Q that satisfies equation (130) is evaluated. cr The collapse temperature is evaluated based on the shear collapse temperature, which is the temperature corresponding to the force. For example, when only a shear force Q acts on beam 11, the shear collapse temperature is considered to be equal to the collapse temperature. High temperature local buckling strength Q cr The method for determining the corresponding temperature will be described later in [2.5].

[0082]

number

[0083] [2.3. A relatively simple method for determining the bending moment when a bending moment acts on a beam] The method in [2.3] is a method for evaluating a beam subjected to a bending moment (first evaluation method for beams) when only a bending moment acts on the beam 11. It is assumed that a bending moment M acts on the beam 11 such that the first flange 16 is subjected to tensile force around the z-axis (the axis along the thickness direction of the web 18). Due to this bending moment M, the second flange 17 is subjected to compressive force. Since the first flange 16, which is subjected to tensile force, does not buckle, the elastic buckling stress of the second flange 17 and web 18 of the beam 11 was examined with reference to the following reference 5. Reference 5: Edited by the Architectural Institute of Japan, "Design Guidelines for Buckling of Steel Structures," Maruzen Publishing Co., Ltd. Due to the bending moment M, a uniform compressive force M1 acts on the second flange 17, as shown in Figure 6. It was assumed that the second flange 17 is simply supported on three sides: sides 17a and 17b on the side where the compressive force M1 acts, and the other side, side 17c. The remaining side 17d of the second flange 17 was considered a free end. In this case, the high-temperature local buckling strength M of the second flange 17 crf This can be calculated using equation (135).

[0084]

number

[0085] Due to the bending moment M, a bending moment M2 acts on the web 18, as shown in Figure 7. It is assumed that sides 18a to 18d of the web 18 are simply supported. In this case, the high-temperature local buckling strength M of the web 18 is... crw This can be calculated using equation (136).

[0086]

number

[0087] In the buckling strength evaluation process S11, the high-temperature local buckling strength M crf , and high-temperature local buckling resistance M crw The smaller of the two (the one that is not larger) is the high-temperature local buckling resistance M. cr It is evaluated as follows. Then, in the collapse evaluation process S12, the high-temperature local buckling strength M that satisfies equation (137) is evaluated. crThe collapse temperature is evaluated based on the bending collapse temperature, which is the temperature corresponding to the bending moment. For example, when only a bending moment M acts on beam 11, the bending collapse temperature is evaluated as being equal to the collapse temperature. Note that the high-temperature local buckling resistance M cr This can be calculated using equation (138). High temperature local buckling strength M cr The method for determining the corresponding temperature will be described later in [2.5].

[0088]

number

[0089] [2.4. A relatively simple method for determining the shear force acting on a beam] The method in [2.4] is a method for evaluating a beam under shear force when only shear force acts on the beam 11 (the first evaluation method for beams). The method in [2.4] was also derived by referring to Reference 5. It is assumed that a shear force Q acts on beam 11 in the y-axis direction. In this case, flanges 16 and 17 are not considered to bear the stress caused by the shear force Q. Due to the shear force Q, a shear force Q1 acts on the web 18, as shown in Figure 8. It was assumed that sides 18a to 18d of the web 18 are simply supported. In this case, in the buckling strength evaluation process S11, the high-temperature local buckling strength Q is calculated using equation (141). cr This is evaluated. Then, in the collapse evaluation process S12, the high-temperature local buckling strength Q that satisfies equation (142) is evaluated. cr The collapse temperature is evaluated based on the shear collapse temperature, which is the temperature corresponding to the force. For example, when only a shear force Q acts on beam 11, the shear collapse temperature is considered to be equal to the collapse temperature. High temperature local buckling strength Q cr The method for determining the corresponding temperature will be described later in [2.5].

[0090]

number

[0091] [2.5. Evaluation of decay temperature and decay time] In the collapse evaluation process S12, the high-temperature local buckling strength M was evaluated using one of the methods [2.1] to [2.4]. cr ,Q cr Based on this, the time of collapse of beam 11 is evaluated. In methods [2.1] and [2.2], convergence calculations are required to derive the collapse temperature and collapse time mathematically. For example, in method [2.1], for the temperatures of flanges 16, 17 and web 18 which increase with time due to heating by fire, the local buckling stress σ that satisfies equation (103) is calculated. cr The temperatures at which the value is obtained are determined by convergence calculation. The obtained temperatures of flanges 16, 17 and web 18 are evaluated as the collapse temperatures. The minimum temperature among the obtained temperatures of flanges 16, 17 and web 18 may be used as the collapse temperature. The method in [2.2] is the same as the method in [2.1].

[0092] On the other hand, for example, in the method of [2.3], the high-temperature local buckling strength M is used for the second flange 17 and web 18, whose temperatures increase over time due to heating caused by the fire. crf M crw The smaller of these is the high-temperature local buckling resistance M. cr The temperature of the second flange 17 or web 18 is determined when it equals the acting bending moment M. The obtained temperature of the second flange 17 or web 18 is evaluated as the collapse temperature. The same method applies to [2.4] as to [2.3].

[0093] The collapse time is determined, for example, as the time when any of the flanges 16, 17 and the web 18 reach the collapse temperature. The temperatures of the first flange 16, the second flange 17, and the web 18 calculated in the temperature calculation step S10 may be equal to each other (uniform temperature).

[0094] As explained above, the beam evaluation method S1 is performed by a temperature calculation step S10 and one of the methods [2.1] to [2.4] as the buckling strength evaluation step S11 and collapse evaluation step S12.

[0095] Furthermore, if a bending moment M and a shear force Q act simultaneously on beam 11, the following beam evaluation method (second beam evaluation method) may be used. In other words, the collapse temperature may be evaluated based on the lower of the bending collapse temperature evaluated by the beam evaluation method S1 according to [2.1] or [2.3] when only a bending moment M acts on the beam 11, and the shear collapse temperature evaluated by the beam evaluation method S1 according to [2.2] or [2.4] when only a shear force Q acts on the beam 11. For example, the collapse temperature may be evaluated by assuming that the lower of the bending collapse temperature and the shear collapse temperature is equal. For example, if method [2.1] is selected as the method for evaluating the bending collapse temperature and method [2.2] is selected as the method for evaluating the shear collapse temperature, the lower of the bending collapse temperature evaluated by method [2.1] and the shear collapse temperature evaluated by method [2.2] may be considered equal to the collapse temperature.

[0096] Furthermore, the beam 11 in this embodiment is a beam that was evaluated and designed using the beam evaluation method S1.

[0097] As described above, the characteristics of the methods described in [2.1] to [2.4] are as follows. [2.1] A method for considering coupled buckling when a bending moment M is applied. [2.2] A method for considering coupled buckling when a shear force Q is applied. [2.3] A method that does not consider coupled buckling when a bending moment M is applied. [2.4] A method that does not consider coupled buckling when a shear force Q is applied. For each of the methods described in [2.1] to [2.4], there may be cases where the temperature is uniform or not.

[0098] [3. Examination of methods for evaluating beams] [3.1. Examination of the temperature required in the temperature calculation process] It is assumed that the cross-sectional dimensions of the H-shaped steel beam 11 are H-700 × 175 × 6 × 9, and that the fire-resistant covering 12 is made of 45 mm thick rock wool. The floor slab 20 is assumed to be a concrete slab. In the temperature calculation process S10, the temperatures of the first flange 16, second flange 17, and web 18 of the beam 11 were determined by numerical calculation without applying external forces such as bending moment M and shear force Q to the beam 11. The boundary conditions were assumed to be as shown in (1), (2) below and in Figure 9. (1) It was assumed that the first boundary condition surface B1, which is the upper surface of the floor slab 20, is exposed to an ambient temperature of 20°C. (2) The second boundary condition surface B2 (beam 11), which is the outer surface of the fire-resistant coating 12 and the lower surface of the floor slab 20, was assumed to be heated based on the standard heating curve (predetermined heating curve) specified in ISO 834-11:2014.

[0099] Figure 10 shows the results of determining the temperature by numerical calculation (thermal conduction analysis). In Figure 10, the horizontal axis represents time (min: minutes), and the vertical axis represents temperature (°C). The temperature of the first flange 16 represents the highest temperature of the first flange 16. The temperatures of the second flange 17 and the web 18 are determined in the same way as the temperature of the first flange 16. At the same time, the temperatures are in descending order of decreasing temperature: web 18, first flange 16, and second flange 17.

[0100] Next, we examined the temperature changes in each part of the first flange 16, the second flange 17, and the web 18. As shown in Figure 9, the lower surface of the first flange 16 is divided into four equal parts in the z-axis direction. From the positive side to the negative side in the z-axis direction, measurement point f 11 ,f 12 ,f 13 ,f 14 ,f 15 The upper surface of the second flange 17 is divided into four equal parts in the z-axis direction. From the positive side to the negative side in the z-axis direction, measurement point f 21 ,f 22 ,f23 ,f 24 ,f 25 This defines... The distance between flanges 16 and 17 is defined as h. Measurement points w1, w2, w3, w4, and w5 are defined from the positive side to the negative side in the y-axis direction such that the distance between them is equal to (h / 5).

[0101] Figures 11 and 12 show the temperature distribution in a cross-section of the floor structure 1 perpendicular to the x-axis direction. Figure 11 shows the high-temperature local buckling resistance M. cr Figure 12 shows the temperature distribution at the time of collapse when Q is set using the method in [2.1]. cr This is the temperature distribution at the time of decay when set using the method in [2.2]. In both Figure 11 and Figure 12, it can be seen that the temperature of the first flange 16 is higher than the temperature of the second flange 17.

[0102] Figures 13 to 16 show the temperature differences at each measurement point on beam 11. Figure 13 shows each measurement point f of the first flange 16. 11 ~f 15 Figure 14 shows the temperature history when heated according to the standard heating curve specified in ISO 834-11:2014 for each measurement point w1 to w5 of the web 18 when heated according to the standard heating curve specified in ISO 834-11:2014. Figure 15 shows the temperature history when heated according to the standard heating curve specified in ISO 834-11:2014 for each measurement point f of the second flange 17. 21 ~f 25 The temperature history when heated according to the standard heating curve specified in ISO 834-11:2014 is shown. Figure 16 shows the temperature change with respect to the height of the web 18 at the time of collapse. Figures 13 and 15 show that no significant temperature difference was observed in flanges 16 and 17 depending on the position in the z-axis direction. On the other hand, Figures 14 and 16 show that in web 18, the temperature was highest between measurement point w3 and measurement point w4.

[0103] The high-temperature Young's modulus obtained based on the relationship between temperature and the time shown in FIG. 10 is shown in FIG. 17. FIG. 17 shows the high-temperature Young's modulus of the first flange 16, the high-temperature Young's modulus of the second flange 17, and the high-temperature Young's modulus of the web 18 with respect to time, respectively. The high-temperature Young's modulus was obtained by the method defined in Document 4. At the same time, it was found that the high-temperature Young's modulus of the first flange 16 and the high-temperature Young's modulus of the web 18 are smaller than the high-temperature Young's modulus of the second flange 17, respectively.

[0104] [3.2. Consideration of the collapse time evaluated in the collapse evaluation process] As shown in FIG. 18, for the analytical model of the fire-protected beam 10 (beam 11), both ends were simply supported, and two vertical concentrated loads P were applied. Note that the fire protection 12 is not shown in FIGS. 18 to 20. The concentrated load P (N) was calculated as follows based on Document 3. The yield stress at normal temperature is 295 N / mm 2 , and the Young's modulus at normal temperature is 205000 N / mm 2 . The reduction rates of the yield stress and Young's modulus with the increase in temperature were based on Document 4. Support span L of beam 11: 7200 mm Depth d of beam 11: 700 mm Design reference strength F: 295 N / mm 2 Long-term allowable bending stress f b (N / mm 2 ) Section modulus Z of beam 11 (mm 3 ) Bending moment M (N / m) acting on beam 11 At this time, from Equation (145) to Equation (148), the concentrated load P is obtained as 125 kN.

[0105] [Number]

[0106] Note that the bending moment distribution acting on beam 11 is as shown by line L1. When local buckling occurs in the region where the bending moment M of the beam 11 acts, as shown in Fig. 19, stiffening members 14 of the web 18, which are stiffeners, were joined to both ends in the x-axis direction of the beam 11 (the range from the support point to the part where the concentrated load P acts). The thickness of the stiffening member 14 is 9 mm, and the pitch of the stiffening member in the x-axis direction is 200 mm.

[0107] The part indicated by the dotted line L3 in the second flange 17 is fixed in the z-axis direction movement, and other movements and rotations are free. The part indicated by the broken line L4 in the first flange 16 is fixed in the y-axis and z-axis direction movements, fixed in the x-axis and y-axis rotations, and other movements and rotations are free. The part indicated by the solid line L5 in the second flange 17 is fixed in the x-axis and z-axis direction movements, and other movements and rotations are free.

[0108] When local buckling occurs in the region where the shear force Q of the beam 11 acts, as shown in Fig. 20, the stiffening member 14 was joined to the central part in the x-axis direction of the beam 11 (the range between the two concentrated loads P).

[0109] The results of obtaining the deflection (maximum deflection) of the beam 11 with respect to time when the bending moment M and the shear force Q act on the beam 11 by numerical calculation (thermal stress analysis) are shown in Figs. 21 and 22. The maximum deflection amount (L 2 / (400d)) defined in Document 2 and Document 3 is shown by the straight line L11. The deflection increases with the passage of time. When the deflection reaches the straight line L11, it is determined that the beam 11 has collapsed. When local buckling occurs in the region where the bending moment M of the beam 11 shown in Fig. 22 acts, the beam 11 collapses at the collapse time t1 (112 minutes). When local buckling occurs in the region where the shear force Q of the beam 11 shown in Fig. 23 acts, the beam 11 collapses at the collapse time t2 (112.5 minutes).

[0110] In Fig. 23, in the cases of the methods [2.1] and [2.3] where the bending moment M acts, and in the case of whether the temperature is uniform or not, the high-temperature local buckling resistance M crThe following was calculated. In Figure 23 and Figure 24 described later, the case where coupled buckling is considered is indicated as "coupled," and the case where the temperature is uniform is indicated as "uniform." In Figure 24, the high-temperature local buckling resistance Q is calculated for the cases of [2.2] and [2.4] where a shear force Q is applied, and for the cases where the temperature is not uniform or not. cr They sought it. In Figure 23, the case where temperature is uniform using method [2.3] overlaps with the case of method [2.3]. In Figure 24, the case where temperature is uniform using method [2.4] overlaps with the case of method [2.4].

[0111] Figures 23 and 24 show the total plastic bending strength M using the highest temperature within the cross-section in the conventional method. cr , shear yield strength Q cr The result obtained is M cr (AIJ), Q cr This is shown as (AIJ). Figures 23 and 24 also show the values ​​of bending moment M and shear force Q. Figures 23 and 24 also show the collapse time t calculated numerically. 01 ,t 02 This indicates. For example, in Figure 23, the high-temperature local buckling strength M increases as time progresses. cr The temperature decreases, and the high-temperature local buckling resistance M cr When this coincides with the bending moment M, beam 11 collapses. In the temperature calculation process S10, the highest temperature (maximum temperature) among the temperatures of each part of the first flange 16 was determined, and the high-temperature Young's modulus was calculated based on this assumption. The same procedure was followed for the second flange 17 and the web 18.

[0112] In both Figure 23 and Figure 24, the full plastic bending strength M cr (AIJ) and shear yield strength Q cr (AIJ) appears to be evaluating the collapse time on the riskier side. Methods [2.1] to [2.4] show that the collapse time is evaluated on the safe side. For example, even with method [2.1], it can be seen that the collapse time is evaluated more accurately when coupled buckling is considered than when coupled buckling is not considered. Methods [2.2] to [2.4] are similar to method [2.1].

[0113] Here, we will explain the results of our investigation into the effect of temperature when determining the high-temperature Young's modulus. Figure 25 shows the high-temperature local buckling strength M against time when the maximum temperature, average temperature, and minimum temperature are used as the temperature of the beam 11 in the method [2.1] in which a bending moment M is applied. cr The changes were investigated. For example, the highest temperature of beam 11 refers to the highest temperature value of each part of beam 11. The average temperature and the lowest temperature of beam 11 are defined using the same approach. In this case, the second flange 17 side of the web 18, which is subjected to compressive force, has a significant impact on the buckling resistance, so using the highest temperature, the high-temperature local buckling resistance M cr It is clear that this is being underestimated.

[0114] Figure 26 shows the high-temperature local buckling strength M when a bending moment M is applied in the method described in [2.1], with the web temperature 18 being the average temperature and the flange temperatures 16 and 17 being the average temperature, maximum temperature, and minimum temperature. cr This shows the change. These results indicate that the temperature difference between flanges 16 and 17 is small in the width direction (z-axis direction), and that using the average temperature, maximum temperature, or minimum temperature has little effect on the local buckling resistance. On the other hand, as shown in Figure 14, the web 18 shows a significant temperature difference in the height direction (y-axis direction), and the temperature on the second flange 17 side (the side where local buckling occurs), which is subjected to compression, is low. Therefore, it is preferable to use the average temperature rather than the maximum temperature.

[0115] Therefore, in the buckling strength evaluation step S11, the temperature used to determine the high-temperature Young's modulus of the first flange 16 and the second flange 17 is preferably the average temperature, maximum temperature, or minimum temperature of the first flange 16 and the second flange 17, respectively, determined in the temperature calculation step S10. In the buckling strength evaluation step S11, the temperature used to determine the high-temperature Young's modulus of the web 18 is preferably the average temperature of the web 18, determined in the temperature calculation step S10. Method [2.3] shows a similar tendency to method [2.1].

[0116] Figure 27 shows the high-temperature local buckling resistance Q as a function of time, when the maximum temperature, average temperature, and minimum temperature are used as the temperature of beam 11 in the method described in [2.2] where a shear force Q is applied. cr The changes were investigated. In this case, since it is known that the shear stress is maximum in the central part of the web 18, it is preferable to use the highest temperature, which is the temperature in the central part of the web 18.

[0117] Figure 28 shows the high-temperature local buckling resistance Q in the method described in [2.2] where a shear force Q is applied, with the web temperature 18 being the highest temperature and the average temperature, highest temperature, and lowest temperature being used for the flanges 16 and 17. cr This shows the change. These results indicate that the temperature difference between flanges 16 and 17 is small in the width direction, and that using the average temperature, maximum temperature, or minimum temperature has little effect on the local buckling resistance. On the other hand, as shown in Figure 14, the temperature difference in the height direction is significant in the web 18, and generally the shear stress is greatest at the center of the web 18 in the height direction. Therefore, it is preferable to use the temperature at the center of the web 18 in the height direction, i.e., the maximum temperature.

[0118] Therefore, the temperature used to determine the high-temperature Young's modulus of the first flange 16 and the second flange 17 in the buckling strength evaluation step S11 is preferably the average temperature, maximum temperature, or minimum temperature of the first flange 16 and the second flange 17 determined in the temperature calculation step S10, respectively. The temperature used to determine the high-temperature Young's modulus of the web 18 in the buckling strength evaluation step S11 is preferably the maximum temperature of the web 18 determined in the temperature calculation step S10. Method [2.4] shows a similar tendency to method [2.2].

[0119] [4. Effects of this embodiment] As explained above, in the beam evaluation method S1 of this embodiment, the high-temperature local buckling strength M is the local buckling strength in local buckling under high temperature. cr ,Q cr The collapse temperature of the beam is evaluated based on this. Therefore, the collapse temperature of beam 11 can be evaluated considering local buckling under high temperatures. In beam evaluation method S1, a temperature calculation step S10, a buckling strength evaluation step S11, and a collapse evaluation step S12 are performed. In the temperature calculation step S10, the temperatures of the first flange 16, the second flange 17, and the web 18 are determined by experiment or numerical calculation. Then, in the buckling strength evaluation step S11, the high-temperature local buckling strength M is calculated using the high-temperature Young's modulus determined based on these temperatures. cr ,Q cr The high-temperature local buckling strength M is evaluated in the collapse evaluation process S12. cr ,Q cr The decay temperature can be evaluated based on this.

[0120] In some cases, the collapse temperature may be evaluated using the method described in [2.1]. In this case, in the buckling strength evaluation process S11, the coupled buckling of the first flange 16, second flange 17, and web 18 of the beam 11 is considered, and the high-temperature local buckling strength M is calculated from equation (102). cr This can be evaluated. Then, in the collapse evaluation process S12, the high-temperature local buckling strength M that satisfies equation (103) is evaluated. cr The collapse temperature can be evaluated based on the bending collapse temperature, which is the temperature corresponding to the flexure. In some cases, the collapse temperature may be evaluated using the method described in [2.2]. In this case, in the buckling strength evaluation process S11, the coupled buckling of the first flange 16, second flange 17, and web 18 of the beam 11 is considered, and the high-temperature local buckling strength Q is calculated from equation (129). cr This can be evaluated. Then, in the collapse evaluation process S12, the high-temperature local buckling strength Q that satisfies equation (130) is evaluated. cr The collapse temperature can be evaluated based on the shear collapse temperature, which is the temperature corresponding to the collapse temperature.

[0121] In some cases, the collapse temperature may be evaluated using the method described in [2.3]. In this case, the high-temperature local buckling strength M is evaluated in the buckling strength evaluation process S11. cr This is evaluated relatively easily, and in the collapse evaluation process S12, the high-temperature local buckling strength M cr The collapse temperature can be evaluated based on the bending collapse temperature, which is the temperature corresponding to the flexure. In some cases, the collapse temperature may be evaluated using the method described in [2.4]. In this case, the high-temperature local buckling strength Q is evaluated in the buckling strength evaluation process S11. cr This is evaluated relatively easily, and in the collapse evaluation process S12, the high-temperature local buckling resistance Q cr The collapse temperature can be evaluated based on the shear collapse temperature, which is the temperature corresponding to the collapse temperature.

[0122] In the methods of [2.1] and [2.3] in which a bending moment M is applied, the temperature used to determine the high-temperature Young's modulus of the first flange 16 and the second flange 17 is one of the average temperature, maximum temperature, or minimum temperature of the first flange 16 and the second flange 17 determined in the temperature calculation step S10, respectively, and the temperature used to determine the high-temperature Young's modulus of the web 18 may be the average temperature of the web 18 determined in the temperature calculation step S10. In this case, the high-temperature local buckling strength M cr This allows for a more appropriate evaluation. In the methods of [2.2] and [2.4] where the shear force Q acts, the temperature used to obtain the high-temperature Young's modulus of the first flange 16 and the second flange 17 is either the average temperature, the maximum temperature, or the minimum temperature of the first flange 16 and the second flange 17 obtained in the temperature calculation step S10, and the temperature used to obtain the high-temperature Young's modulus of the web 18 may be the maximum temperature of the web 18 obtained in the temperature calculation step S10. In this case, the high-temperature local buckling resistance Q cr can be more appropriately evaluated.

[0123] The temperatures of the first flange 16, the second flange 17, and the web 18 calculated in the temperature calculation step S10 may be equal to each other. In this case, the high-temperature local buckling resistance M cr , Q cr can be more easily evaluated. In the temperature calculation step S10, the beam 11 is heated based on a standard heating curve, and in the collapse evaluation step S12, the collapse time is evaluated based on the high-temperature local buckling resistances M cr , Q cr . Therefore, based on the high-temperature local buckling resistances M cr , Q cr the collapse time when the beam 11 is heated based on a standard heating curve can be evaluated.

[0124] There may be a case where the collapse temperature is evaluated based on the lower of the bending collapse temperature evaluated by the beam evaluation method S1 by the method of [2.1] or [2.3], and the shear collapse temperature evaluated by the beam evaluation method S1 by the method of [2.2] or [2.4]. In this case, the collapse temperature when the bending moment M and the shear force Q act on the beam 11 simultaneously can be appropriately evaluated. Also, in the beam 11 of the present embodiment, considering local buckling at high temperatures, the collapse temperature of the beam 11 can be evaluated and the beam can be designed.

[0125] As described above, one embodiment of the present invention has been described in detail with reference to the drawings, but the specific configuration is not limited to this embodiment, and modifications, combinations, deletions, etc. of the configuration within the scope not departing from the gist of the present invention are also included. The specified heating curve was assumed to be the standard heating curve specified in ISO 834-11:2014. However, the specified heating curve is not limited to this; for example, a heating curve in which the temperature increases at a rate of 10°C / min may also be used. For example, in the above embodiment, the beam evaluation method S1 does not need to evaluate the time of collapse. The beam evaluation method S1 does not need to perform the temperature calculation step S10, the buckling strength evaluation step S11, and the collapse evaluation step S12. [Explanation of symbols]

[0126] 11 Beam 16. First flange 17. Second flange 18 Web M Bending moment Q Shear force S1 Beam Evaluation Method S10 Temperature calculation process S11 Buckling strength evaluation process S12 Collapse Evaluation Process

Claims

1. A method for evaluating a beam made of H-shaped steel in a steel frame structure when it collapses due to being heated by a fire, The collapse temperature of the beam is evaluated based on the high-temperature local buckling resistance of the beam. A temperature calculation step in which the temperature of the first flange, second flange, and web of the beam when the beam is heated is determined by experiment or numerical calculation, A buckling strength evaluation step in which the high-temperature local buckling strength is evaluated using the high-temperature Young's modulus determined based on the temperature of the first flange, the second flange, and the web, A collapse evaluation step in which the collapse temperature is evaluated based on the high-temperature local buckling resistance, Perform The axis in the direction of the material axis through which the beam extends is defined as the x-axis. The axis in the direction in which the first flange and the second flange face each other is defined as the y-axis. The axis in the thickness direction of the web is defined as the z-axis, Assuming that the first flange is subjected to a tensile force due to a bending moment M acting on the beam around the z-axis, The position of the center in the direction along the y-axis of the first flange is defined as the origin of the y-axis, The direction from the first flange toward the second flange is defined as the positive direction of the y-axis, The distance between the center of the first flange and the center of the second flange in the direction along the y-axis is b. ｗ It is stipulated that, When a is defined as the half wavelength of the web in the direction along the x-axis, which is alternately displaced in a wave-like manner in the positive and negative directions of the z-axis as it approaches the first end of the web in the direction along the x-axis, In the buckling strength evaluation step, the high-temperature local buckling strength M is calculated from equation (1). ｃｒ Evaluate, In the collapse evaluation process described above, the high-temperature local buckling strength M that satisfies equation (2) is ｃｒ A method for evaluating a beam, which evaluates the collapse temperature based on the bending collapse temperature, which is the temperature corresponding to the beam. [Math 1] However, Z is the sectional coefficient of the beam, and σ ｃｒ is the local buckling stress of the beam, N is a natural number of 2 or more, and a ０ , a ｎ , b ｎ are undetermined coefficients, E ｆ１ is the high-temperature Young's modulus of the first flange, E ｗ is the high-temperature Young's modulus of the web, E ｆ２ is the high-temperature Young's modulus of the second flange, ν is the Poisson's ratio of the beam, t ｗ is the thickness of the web, t ｆ is the thickness of each of the first flange and the second flange, and b ｆ is the value of half of the width of each of the first flange and the second flange. The local buckling stress σ ｃｒ The local buckling stress σ is calculated using equations (5) to (11) and equation (12). ｃｒ The real number a that gives the smallest positive value to is the a ｎ , b ｎ And determined based on the aforementioned half-wavelength a. [Math 2]

2. A method for evaluating a beam made of H-shaped steel in a steel frame structure when it collapses due to being heated by a fire, The collapse temperature of the beam is evaluated based on the high-temperature local buckling resistance of the beam. A temperature calculation step in which the temperature of the first flange, second flange, and web of the beam when the beam is heated is determined by experiment or numerical calculation, A buckling strength evaluation step in which the high-temperature local buckling strength is evaluated using the high-temperature Young's modulus determined based on the temperature of the first flange, the second flange, and the web, A collapse evaluation step in which the collapse temperature is evaluated based on the high-temperature local buckling resistance, Perform Assuming that a bending moment M acts on the beam around an axis along the thickness direction of the web, such that the first flange is subjected to a tensile force, In the buckling strength evaluation step, the high-temperature local buckling strength M is determined by equation (27). ｃｒｆ , and the high-temperature local buckling strength M obtained by equation (28) ｃｒｗ The smaller of these is the high-temperature local buckling resistance M. ｃｒ As evaluated, In the collapse evaluation step, the high-temperature local buckling strength M that satisfies equation (29) is ｃｒ A method for evaluating a beam, which evaluates the collapse temperature based on the bending collapse temperature, which is the temperature corresponding to the beam. However, Z is the section modulus of the beam, and E ｗ is the high-temperature Young's modulus of the web, and E ｆ２ ν is the high-temperature Young's modulus of the second flange, ν is the Poisson's ratio of the beam, and t ｗ is the thickness of the web, and t ｆ b is the thickness of the first flange and the second flange, respectively. ｆ b is half the width of the first flange and the second flange, respectively. ｗ This is the distance between the center of the first flange and the center of the second flange in the direction in which the first flange and the second flange face each other. [Math 3]

3. In the buckling strength evaluation step, the temperature used to determine the high-temperature Young's modulus of the first flange and the second flange is one of the average temperature, maximum temperature, or minimum temperature of the first flange and the second flange determined in the temperature calculation step, respectively. The beam evaluation method according to claim 1 or 2, wherein the temperature used to determine the high-temperature Young's modulus of the web in the buckling strength evaluation step is the average temperature of the web obtained in the temperature calculation step.

4. A method for evaluating a beam made of H-shaped steel in a steel frame structure when it collapses due to being heated by a fire, The collapse temperature of the beam is evaluated based on the high-temperature local buckling resistance of the beam. A temperature calculation step in which the temperature of the first flange, second flange, and web of the beam when the beam is heated is determined by experiment or numerical calculation, A buckling strength evaluation step in which the high-temperature local buckling strength is evaluated using the high-temperature Young's modulus determined based on the temperature of the first flange, the second flange, and the web, A collapse evaluation step in which the collapse temperature is evaluated based on the high-temperature local buckling resistance, Perform The axis in the direction of the material axis through which the beam extends is defined as the x-axis. The axis in the direction in which the first flange and the second flange face each other is defined as the y-axis. The axis in the thickness direction of the web is defined as the z-axis, Assuming that a shear force Q acts on the beam in the direction along the y-axis, The position of the center in the direction along the y-axis on the web is defined as the origin of the y-axis, The direction from the first flange toward the second flange is defined as the positive direction of the y-axis, The distance between the center of the first flange and the center of the second flange in the direction along the y-axis is b. ｗ It is stipulated that, When a is defined as the half wavelength of the web in the direction along the x-axis, which is alternately displaced in a wave-like manner in the positive and negative directions of the z-axis as it approaches the first end of the web in the direction along the x-axis, In the buckling strength evaluation step, the high-temperature local buckling strength Q is calculated using equation (41). ｃｒ Evaluate, In the collapse evaluation step, the high-temperature local buckling strength Q that satisfies equation (42) is ｃｒ A method for evaluating a beam, which evaluates the collapse temperature based on the shear collapse temperature, which is the temperature corresponding to the given temperature. [Math 4] However, ｃｒ is the shear buckling stress of the beam, and A ｗ is the cross-sectional area of ​​the web perpendicular to the material axis direction, N is a natural number of 2 or more, and a ０ , a ｎ , b ｎ λ is an undetermined coefficient, E ｆ１ is the high-temperature Young's modulus of the first flange, and E ｗ is the high-temperature Young's modulus of the web, and E ｆ２ ν is the high-temperature Young's modulus of the second flange, ν is the Poisson's ratio of the beam, and t ｗ is the thickness of the web, and t ｆ b is the thickness of the first flange and the second flange, respectively. ｆ This value is half the width of the first flange and the second flange, respectively. The shear buckling stress τ ｃｒ Using equations (45) to (50), the shear buckling stress τ according to equation (51) is obtained. ｃｒ The real number a that gives the smallest positive value to is the a ｎ , b ｎ It is determined based on λ and the half-wavelength a. [Math 5]

5. A method for evaluating a beam made of H-shaped steel in a steel frame structure when it collapses due to being heated by a fire, The collapse temperature of the beam is evaluated based on the high-temperature local buckling resistance of the beam. A temperature calculation step in which the temperature of the first flange, second flange, and web of the beam when the beam is heated is determined by experiment or numerical calculation, A buckling strength evaluation step in which the high-temperature local buckling strength is evaluated using the high-temperature Young's modulus determined based on the temperature of the first flange, the second flange, and the web, A collapse evaluation step in which the collapse temperature is evaluated based on the high-temperature local buckling resistance, Perform Assuming that a shear force Q acts on the beam in the direction in which the first flange and the second flange face each other, In the buckling strength evaluation step, the high-temperature local buckling strength Q is calculated using equation (61). ｃｒ Evaluate, In the collapse evaluation process, the high-temperature local buckling strength Q that satisfies equation (62) is ｃｒ A method for evaluating a beam, which evaluates the collapse temperature based on the shear collapse temperature, which is the temperature corresponding to the given temperature. However, A ｗ E is the cross-sectional area perpendicular to the direction of the material axis in which the beam extends in the web, ｗ ν is the Young's modulus of the web, ν is the Poisson's ratio of the beam, and t ｗ b is the thickness of the web, ｗ This is the distance between the center of the first flange and the center of the second flange in the direction in which the first flange and the second flange face each other. [Math 6]

6. In the buckling strength evaluation step, the temperature used to determine the high-temperature Young's modulus of the first flange and the second flange is one of the average temperature, maximum temperature, or minimum temperature of the first flange and the second flange determined in the temperature calculation step, respectively. The beam evaluation method according to claim 4 or 5, wherein the temperature used to determine the high-temperature Young's modulus of the web in the buckling strength evaluation step is the highest temperature of the web determined in the temperature calculation step.

7. The method for evaluating a beam according to any one of claims 1 to 6, wherein the temperature of the first flange, the temperature of the second flange, and the temperature of the web are equal to each other.

8. In the temperature calculation step, the beam is heated based on a predetermined heating curve. The method for evaluating a beam according to any one of claims 1 to 7, wherein the collapse evaluation step evaluates the collapse time, which is the time at which the beam collapses, based on the high-temperature local buckling strength.

9. A method for evaluating a beam, comprising evaluating the collapse temperature based on the lower of the bending collapse temperature evaluated by the beam evaluation method described in any one of claims 1 to 3 and the shear collapse temperature evaluated by the beam evaluation method described in any one of claims 4 to 6.

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