JP7575675B2 - Method for evaluating the seismic performance of beams - Google Patents

Description

The present invention relates to a method for evaluating the seismic performance of beams.

Conventionally, in the Ministry of Construction Notification No. 1792 of 1980, based on the provision of Article 82-3, item 2 of the Building Standards Act Enforcement Order (Cabinet Order No. 338 of 1950) (hereinafter referred to as Notification), for beams made of H-shaped steel, width-thickness ratio classifications of the web and flange constituting the H-shaped steel are established. And, based on the width-thickness ratio classifications of the web and flange, the type of beam is determined. The width-thickness ratio classification is determined based on the width-thickness ratio derived for each of the web and flange.
Here, the local buckling of the H-shaped steel is caused by the interaction between the web and the flange, and therefore both the web and the flange exhibit complex periodic behavior. Therefore, in the previous knowledge such as the elastic local buckling strength formula, the web and the flange are considered as individual plate elements, and the out-of-plane displacement _Ww due to the local buckling of the web and the flange is estimated by using the Fourier series expressed by the formula (1), and the elastic local buckling strength of the plate element is derived based on the out-of-plane displacement _Ww (see, for example, Non-Patent Document 1).

Stephen P. Timoshenko and James M. Gere, “Theory of Elastic Stability” Second Edition

However, the local buckling deformation of the H-shaped steel occurs when the web and flange are integrated together, with the web and flange mutually restraining the local buckling deformation, so the elastic buckling strength formula for the plate element underestimates the elastic buckling strength of the H-shaped steel. Therefore, the seismic performance of the beam evaluated by the width-thickness ratio classification constructed based on the elastic buckling strength formula for the plate element is also underestimated.

The present invention was made in consideration of these problems, and aims to provide a method for evaluating the seismic performance of beams that can more appropriately evaluate the type by deriving an elastic local buckling strength formula that takes into account the interacting web and flange.

In order to solve the above problems, the present invention proposes the following means.
The method for evaluating the seismic performance of a beam of the present invention is characterized in that, when evaluating the type of seismic performance of a beam made of H-shaped steel having a web and a pair of flanges, the type of the beam is evaluated by constructing an elastic local buckling strength equation that takes into account the coupled buckling of the web and the pair of flanges of the beam.
According to this invention, the inventors have found, as a result of intensive research, that in H-shaped steel, the web and a pair of flanges do not buckle independently, but buckle in a coupled manner while interacting with each other, and have derived an elastic local buckling strength formula that can accurately predict the coupled buckling of the web and a pair of flanges of a beam. By evaluating the type of beam using this elastic local buckling strength formula, the type of beam can be more appropriately evaluated.

In addition, in the method for evaluating the seismic performance of the beam, the type of the beam may be evaluated using as indicators a first buckling stress τ _{cr that} takes into account the coupled buckling when a shear force acts on the beam and the beam buckles, and a second buckling stress σ _cr that takes into account the coupled buckling when a uniform bending moment acts on the beam and the beam buckles.
In addition, in the method for evaluating the seismic performance of a beam, the first buckling stress intensity τ _cr and the second buckling stress intensity σ _cr may be calculated based on an energy method from an out-of-plane displacement of the web and an out-of-plane displacement of the pair of flanges.

In addition, in the method for evaluating the seismic performance of the beam, the material axis of the beam is defined as the x-axis, the axis extending in the direction in which the pair of flanges sandwich the web is defined as the y-axis, the axis extending in the plate thickness direction of the web is defined as the z-axis, the distance between the centers of the pair of flanges in the direction along the y-axis is defined as _bw , and the half wavelength of the web in the direction along the x-axis that is displaced alternately in a wave-like manner in the direction along the z-axis as it moves toward a first end of the web in the direction along the x-axis is defined as a. When a shear force acts on the beam, the center position of the web in the direction along the y-axis is defined as the origin of the y-axis, and the direction from the origin of the y-axis toward one of the pair of flanges is defined as the positive direction of the y-axis. The first buckling stress τ _cr is calculated by using equations (11) to (16) to obtain the first buckling stress τ cr according to equation (17). When _an equal bending moment acts on _{the beam} , the position of the center of one of the pair of flanges in the direction along the y-axis may be defined as the origin of the y-axis, and the direction from the _origin of the y-axis toward the other of the pair of flanges may be defined as the positive direction of the y-axis. The second buckling stress σ _cr may be determined using equations (18) to (24) based on the _an , _bn , and the half wavelength a, which are real numbers that give the second buckling stress σ _cr calculated by equation (25) a minimum positive value.
where N is a natural number of 2 or greater, _a0 , _an , _bn , and λ are undetermined coefficients, E is the Young's modulus of the H-shaped steel, ν is the Poisson's ratio of the H-shaped steel, _tw is the thickness of the web, _tf is the thickness of each of the pair of flanges, _bw is the center-to-center distance in thickness of the pair of flanges, _bf is half the width of each of the pair of flanges, _Ww is the out-of-plane displacement of the web in the direction along the z-axis, and _Wf1 and _Wf2 are the out-of-plane displacement of the pair of flanges in the direction along the y-axis.

According to the present invention, the first buckling stress τ _cr when a shear force acts on a beam, and the second buckling stress σ _cr when a uniform bending moment acts on the beam can be accurately calculated using equations.

In addition, in the method for evaluating the seismic performance of a beam, the depth H of the H-shaped steel constituting the beam and the width W of the pair of flanges are each a predetermined value, the reference strength of the H-shaped steel is defined as F, and in the H-shaped steel of the type of FA of the beam given by equations (26) and (27), the condition of the width-thickness ratio of the pair of flanges and the web in the beam when the equality of equations (26) and (27) holds is defined as a reference condition, the second buckling stress intensity σ _cr of the H-shaped steel under the reference condition is defined as a second reference buckling stress intensity in the type of FA, and the ratio of the second reference buckling stress intensity in the type of FA to the reference strength F is defined as a second reference buckling stress intensity ratio in the type of FA. The beam in which the ratio of _cr is equal to or greater than the second reference buckling stress ratio in the type of the FA and satisfies equation (26) may be evaluated as the beam in the type of the FA.
where _tf is the thickness of each of the pair of flanges, and _tw is the thickness of the web. In addition, the center-to-center distance in thickness direction of the pair of flanges and the width W of the flanges when calculating the second buckling stress _σcr are equal to the center-to-center distance in thickness direction of the pair of flanges and the width W of the flanges of the H-shaped steel when calculating the second reference buckling stress.

According to this invention, in the case of a beam whose classification is FA, the seismic performance of the beam is evaluated taking into consideration the coupled buckling of the web and a pair of flanges. This makes it possible to expand the cross-sectional shapes of beams whose seismic performance can be evaluated as FA to beams whose ratio of the second buckling stress σ _cr to the reference strength F is equal to or greater than the second reference buckling stress ratio and which satisfy formula (26). Therefore, for example, a beam whose web width-thickness ratio is larger than that of the FA classification in the current evaluation method can be evaluated as a beam having the seismic performance of the FA classification.

In addition, in the method for evaluating the seismic performance of a beam, the depth H of the H-shaped steel constituting the beam and the width W of the pair of flanges are each a predetermined value, the reference strength of the H-shaped steel is defined as F, and in the H-shaped steel of the type of FB of the beam given by equations (28) and (29), the condition of the width-thickness ratio of the pair of flanges and the web in the beam when the equality of equations (28) and (29) holds is defined as a reference condition, the second buckling stress σ _cr of the H-shaped steel under the reference condition is defined as a second reference buckling stress in the type of FB, and the ratio of the second reference buckling stress in the type of FB to the reference strength F is defined as a second reference buckling stress ratio in the type of FB. The beam in which the ratio of _cr is equal to or greater than the second reference buckling stress ratio in the type of the FB and satisfies equation (28) may be evaluated as the beam in the type of the FB.
where _tf is the thickness of each of the pair of flanges, and _tw is the thickness of the web. In addition, the center-to-center distance in thickness direction of the pair of flanges and the width W of the flanges when calculating the second buckling stress _σcr are equal to the center-to-center distance in thickness direction of the pair of flanges and the width W of the flanges of the H-shaped steel when calculating the second reference buckling stress.

According to this invention, in the case of a beam of type FB, the seismic performance of the beam is evaluated by considering the coupled buckling of the web and a pair of flanges. This makes it possible to expand the cross-sectional shape of a beam whose seismic performance can be evaluated as type FB to a beam whose ratio of the second buckling stress σ _cr to the reference strength F is equal to or greater than the second reference buckling stress ratio and which satisfies formula (28). Therefore, for example, a beam whose web width-thickness ratio is larger than that of the type FB in the current evaluation method can be evaluated as a beam having seismic performance of type FB.

In addition, in the method for evaluating the seismic performance of the beam, the yield shear strength of the H-shaped steel may be defined as F', and the beam in which the ratio of the first buckling stress intensity τ _cr to the yield shear strength F' satisfies formula (30) and formula (31) may be evaluated as the beam in the type of FC.
Here, F' is F/√3.

According to this invention, in the case of a beam whose classification is FC, the seismic performance of the beam is evaluated by considering the coupled buckling of the web and a pair of flanges. This makes it possible to expand the cross-sectional shape of a beam whose seismic performance can be evaluated as FC to beams whose ratio of the first buckling stress intensity τ _cr to the yield shear strength F' satisfies formula (30) and formula (31). Therefore, for example, a beam whose web width-thickness ratio is larger than that of the FC classification in the current evaluation method can be evaluated as a beam having the seismic performance of the FC classification.

The method for evaluating the seismic performance of beams of the present invention allows for a more appropriate evaluation of the type by deriving an elastic local buckling strength formula that takes into account the interacting web and flange.

1 is a perspective view of a building having a beam to which a method for evaluating the seismic performance of a beam according to one embodiment of the present invention is applied. FIG. 13 is a perspective view showing a state in which a beam is buckled when a shear force is applied to the beam when the beam is sufficiently long in the direction along the x-axis. FIG. 2 is a cross-sectional view perpendicular to the longitudinal direction of the beam. 3 is a perspective view showing an enlargement of a half wavelength in the direction along the x-axis of the beam in FIG. 2. FIG. 5 is a cross-sectional view taken along line A1-A1 in FIG. 4. FIG. 13 is a perspective view showing a state in which a beam is buckled when a uniform bending moment is applied to the beam when the beam is sufficiently long in the direction along the x-axis. FIG. 2 is a cross-sectional view perpendicular to the longitudinal direction of the beam. FIG. 7 is a perspective view showing an enlargement of a half wavelength in the direction along the x-axis of the beam in FIG. 6 . 9 is a cross-sectional view taken along line A2-A2 in FIG. 8. FIG. 1 is a diagram explaining the width-thickness ratio classification of beams based on the Building Standards Act notification. FIG. 13 is a diagram illustrating the elastic buckling strength of a web that divides FA types. FIG. 13 is a diagram explaining the new width-thickness ratio division of a beam. FIG. 13 is a diagram illustrating the part of the new width-thickness ratio classification of the beam that corresponds to the FA type. FIG. 13 is a diagram illustrating the part of the new beam width-thickness ratio classification corresponding to the FB type. FIG. 13 is a diagram explaining the part of the new width-thickness ratio classification of beams corresponding to the FC type. 1 is a diagram illustrating an analytical model of a beam to which a method for evaluating the seismic performance of a beam according to an embodiment is applied. FIG. 17 is a cross-sectional view taken along line A4-A4 in FIG. 16.

Below, one embodiment of the method for evaluating the seismic performance of a beam according to the present invention will be described with reference to Figures 1 to 17.

[1. Beam structure]
The method for evaluating the seismic performance of a beam according to this embodiment (hereinafter, abbreviated as the evaluation method) is used to evaluate a beam 10 made of an H-shaped steel, which is used in a building 1 shown in Fig. 1. The beam 10 has a web 11, a first flange (flange) 12, and a second flange (flange) 13. In Fig. 1, a floor slab 20, which will be described later, is indicated by a two-dot chain line.

The beam 10 extends in a direction along a horizontal plane, for example. The first flange 12 is formed in a flat plate shape and is disposed so that the thickness direction of the first flange 12 is along the up-down direction. The second flange 13 is formed in a flat plate shape and is disposed below the second flange 13. The second flange 13 is disposed so that the thickness direction of the second flange 13 is along the up-down direction.
The web 11 is formed in a flat plate shape that has a rectangular shape when viewed in the thickness direction of the web 11. The web 11 is disposed so that the thickness direction of the web 11 is along a horizontal plane. The web 11 connects the center of the width direction of the lower surface of the first flange 12 and the center of the width direction of the upper surface of the second flange 13.

The longitudinal ends of the beam 10 are fixed to columns 15 or the like. The beam 10 supports a floor slab 20 from below the floor slab 20. A shear connector 21 such as a headed stud is provided on the first flange 12 of the beam 10. The shear connector 21 is embedded in the floor slab 20.
The building 1 is used by installing equipment (not shown) on the floor slab 20 .
The first buckling stress and the second buckling stress used in this evaluation method are described below. When deriving the first buckling stress and the second buckling stress, it is assumed that the web 11 and the flanges 12 and 13 of the beam 10 are made of steel plates, which are elastic elements. The elastic elements are elements that do not take into account material nonlinearity.

[2. First buckling stress when shear force acts on a beam]
In the evaluation method when a shear force acts on the beam 10, the position coordinates of the beam 10 are recognized based on a right-handed Cartesian coordinate system consisting of an x-axis, a y-axis, and a z-axis, as shown in Fig. 2. Fig. 2 shows a state in which the beam 10 is buckled.
The material axis of the beam 10 (the axis extending in the direction in which the beam 10 extends) is defined as the x-axis. The axis extending in the direction in which the flanges 12, 13 sandwich the web 11 is defined as the y-axis. The axis extending in the plate thickness direction of the web 11 is defined as the z-axis. The x-axis, y-axis, and z-axis are mutually perpendicular. When viewed in a direction along the z-axis (hereinafter referred to as the z-axis direction), the web 11 has a side extending in a direction along the x-axis (hereinafter referred to as the x-axis direction) and a side extending in a direction along the y-axis (hereinafter referred to as the y-axis direction). The out-of-plane displacement of the web 11 is a displacement of the web 11 in the z-axis direction.
The out-of-plane displacement of the flanges 12 and 13 is a displacement in the y-axis direction.

The beam 10 is assumed to be sufficiently long in the x-axis direction. Here, "the beam 10 is sufficiently long in the x-axis direction" means that the beam 10 has a length such that the effect on buckling deformation of the boundary conditions of the surfaces 10a (hereinafter referred to as the x-axis end faces) located at each end of the beam 10 in the x-axis direction and extending in the y-axis direction can be ignored.

When a shear force F1 acts in the y-axis direction on each of the end faces 10a of the beam 10 in the x-axis direction, the beam 10 may buckle. The shear force F1 is transmitted over the entire length of the beam 10 in the x-axis direction.
The shear forces F1 acting on each end face 10a of the beam 10 in the x-axis direction are external forces of equal magnitude but opposite directions. For example, a shear force F1 acting in a positive direction in the y-axis direction acts on the end face 10a on the negative side in the x-axis direction, and a shear force F1 acting in a negative direction in the y-axis direction acts on the end face 10a on the positive side in the x-axis direction. Alternatively, a shear force F1 acting in a negative direction in the y-axis direction may act on the end face 10a on the negative side in the x-axis direction, and a shear force F1 acting in a positive direction in the y-axis direction may act on the end face 10a on the positive side in the x-axis direction.

In this case, as the web 11 moves toward the first end in the x-axis direction (one of the end faces 10a in the x-axis direction), the web 11 is displaced alternately in the positive direction of the z-axis and the negative direction of the z-axis in the z-axis direction, so that the web 11 as a whole is displaced in a wavy shape of a plurality of wavelengths (hereinafter, referred to as wavy in the x-axis direction). Corresponding to the web 11, the flanges 12 and 13 are displaced in a wavy shape.
In one wavelength of the web 11 displaced along the x-axis, the second end opposite 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.

2 and 3, the center position of the web 11 in the y-axis direction is defined as the origin of the y-axis. The direction from the origin of the y-axis toward the first flange 12 (one of the flanges 12 and 13) is defined as the positive direction of the y-axis.
The origin of the z-axis is the center in the z-axis direction (center in the thickness direction) of the web 11. The positive direction of the z-axis is a direction that constitutes a right-handed Cartesian coordinate system with respect to the positive directions of the x-axis and y-axis.

3, dimensions are defined in a cross section perpendicular to the longitudinal direction of the beam 10. Note that, for units of length and the like described below, SI units such as "m" are preferably used for length.
The thickness (length in the z-axis direction) of the web 11 is defined as _tw . The distance between the center of thickness of the first flange 12 and the center of thickness of the second flange 13 in the y-axis direction is defined as _bw . The width of the beam 10 is defined as H.
The width (length in the z-axis direction) of the first flange 12 and the width of the second flange 13 are equal to each other, and half the value of each of the widths of the first flange 12 and the second flange 13 is defined as _bf . Note that each of the widths of the first flange 12 and the second flange 13 is defined as W.
The thickness of the first flange 12 and the thickness of the second flange 13 are equal to each other, and each thickness of the first flange 12 and the second flange 13 is represented as _tf .
The Young's modulus of the beam 10 (H-shaped steel; web 11 and flanges 12, 13) is defined as E, and the Poisson's ratio of the beam 10 is defined as ν.

In the evaluation method of this embodiment, a first buckling stress (buckling stress) of the beam 10 is estimated when a shear force F1 acts on the beam 10 and the beam 10 buckles. In the evaluation method, the following assumptions 1 to 6 are made when estimating the first buckling stress of the beam 10.
1. The thickness of the web 11 is thin, and is shorter than the length of the web 11 in the x-axis direction and the length of the web 11 in the y-axis direction.
2. The deflection of the web 11 (out-of-plane displacement due to buckling) is small, less than the thickness of the web 11.
3. The central plane of the web 11 in the thickness direction does not expand or contract when the web 11 is bent, but remains a neutral plane.
4. The cross section of the beam 10 is subject to the assumption of plane retention against bending.
5. The material of the beam 10 is homogeneous and isotropic.
6. When an external force acts on beam 10, the displacement follows Hooke's law.

As shown in Figure 2, when a shear force F1 acts on the beam 10, the web 11 of the beam 10 may be displaced in a wavy manner in the x-axis direction. When the y-axis coordinate of the web 11 displaced in a wavy manner in the x-axis direction is a certain value, it is assumed that the out-of-plane displacement of the web 11 in the z-axis direction at a certain coordinate on the x-axis is expressed by the formula sin(π(x+λsin(πy/ _bw ))/a). In this case, the wavelength in the x-axis direction of the web 11 displaced in a wavy manner in the x-axis direction of the web 11 is 2a. The half wavelength (half the length of the wavelength) of the web 11 in the x-axis direction is a.
FIG. 4 is a diagram showing the out-of-plane displacement _Ww in the entire region of the beam 10 where the length of the beam 10 in the x-axis direction is half the wavelength a.

In Figure 5, the solid line shows the state when shear force F1 acts on beam 10. The dotted line in Figure 5 shows the state when shear force F1 does not act on beam 10.

When a shear force F1 acts on the web 11, the out-of-plane displacement of the web 11 has conventionally been estimated using a Fourier series.
The inventors have investigated a number of functions that can estimate the out-of-plane displacement (first out-of-plane displacement) _w of the web 11 in the z-axis direction at a certain coordinate on the y-axis when the coordinate on the x-axis is a certain value, while using trigonometric functions and with fewer terms than a Fourier series. Note that the out-of-plane displacement _w is a function of the y-axis coordinate, not a function of the x-axis coordinate (it is a function at a certain coordinate on the x-axis).
As a result, it was found that when a shear force F1 acts on the beam 10, the out-of-plane displacement _ww of the web 11 in the z-axis direction at a certain y-axis coordinate when the x-axis coordinate is a certain value can be estimated by equation (40) with fewer terms than the Fourier series, where N is a natural number of 2 or more, and _a0 , _an , and _bn are undetermined coefficients.
Equation (40) includes terms based on trigonometric functions using power functions of the y-axis coordinate. cosπ(2y/ _bw ) ⁿ and cos(π/2)(2y/ _bw ) ⁿ are the basis. Equation (40) can be preferably used to estimate the out-of-plane displacement of a plate element such as the web 11.

On the other hand, when the x-axis coordinate is an arbitrary value, the out-of-plane displacement (second out-of-plane displacement) _Ww of the web 11 in the z-axis direction at a certain coordinate on the y-axis is estimated by formula (41). As described above, formula (41) is based on the assumption that when the y-axis coordinate of the web 11 is a certain value, the out-of-plane displacement of the web 11 in the z-axis direction at a certain coordinate on the x-axis is expressed by the formula sin(π(x+λsin(πy/ _bw ))/a). However, the undetermined coefficient λ in formula (41) is a real number representing the phase shift in the y-axis direction.
The out-of-plane displacement _Ww in Fig. 2 is obtained by repeating the out-of-plane displacement _Ww in Fig. 4 in the x-axis direction. Since estimating the out-of-plane displacement _Ww of the web 11 in Fig. 2 is synonymous with estimating the out-of-plane displacement _Ww of the web 11 in Fig. 4, the above formula (41) is an estimate of the out-of-plane displacement _Ww in Fig. 4. The out-of-plane displacement _Ww is a function of both the y-axis coordinate and the x-axis coordinate, and is used when estimating the first buckling stress of the web 11.
By substituting equation (40) into equation (41), equation (42) is obtained. When N is 2 in equation (42), it can be transformed into equation (43).

Furthermore, from the above assumptions, the out-of-plane displacement _Wf1 of the first flange 12 at a predetermined coordinate on the z axis is estimated by equation (44), and the out-of-plane displacement _Wf2 of the second flange 13 at a predetermined coordinate on the z axis is estimated by equation (45).
Equations (44) and (45) include terms based on trigonometric functions using power functions of the y-axis coordinate.

For example, before the shear force F1 acts on the beam 10 in Fig. 4, the z-axis coordinate of a portion of the web 11 whose x-axis coordinate is _x0 and whose y-axis coordinate is _y0 (hereinafter referred to as the first estimation target portion) is 0. The out-of-plane displacement _Ww in this first estimation target portion is estimated by equation (42-1). The z-axis coordinate of the first estimation target portion after the shear force F1 acts on the beam 10 in Fig. 4 is a value obtained by adding the out-of-plane displacement _Ww estimated by equation (42-1) to 0. In other words, after the shear force F1 acts on the beam ₁₀ in Fig. 4, the first estimation target portion is estimated to be located at a position whose x-axis coordinate is x0, whose y-axis coordinate is _y0 , and whose z-axis coordinate is _Ww .
For example, before the shear force F1 acts on the beam 10 in Fig. 4, the y-axis coordinate of the portion of the second flange 13 whose x-axis coordinate is _x0 and whose z-axis coordinate is _z0 (hereinafter referred to as the second estimation target portion) is _bw /2. The out-of-plane displacement _Wf2 in this second estimation target portion is estimated by equation (45-1). The z-axis coordinate of the second estimation target portion after the shear force F1 acts on the beam 10 in Fig. 4 is a value obtained by adding the out-of-plane displacement _Wf2 estimated by equation (45-1) to _bw /2. In other words, after the shear force F1 acts on the beam 10 in Fig. 4, the second estimation target portion is estimated to be located at a position whose x-axis coordinate is _x0 , whose y-axis coordinate is ( _bw + _Wf2 ), and whose z-axis coordinate is _z0 .

In the evaluation method, when a shear force F1 acts on the beam 10, the out-of-plane displacement _Ww of the web 11 is estimated by equation (42), the out-of-plane displacement _Wf1 of the first flange 12 is estimated by equation (44), and the out-of-plane displacement _Wf2 of the second flange 13 is estimated by equation (45).
Here, based on the energy method, the strain energy _Uw generated in the web 11 due to buckling deformation is expressed as in equation (47), and the strain energy _Uf of the flanges 12, 13 is expressed as in equation (48).

Here, the plate stiffness _Dw of the web 11 is expressed by equation (49). The plate stiffness _Df of the flanges 12, 13 is expressed by equation (50). The first buckling stress of the beam 10 is represented as τ _cr .
The function _δw is a function that expresses the deformation of the web 11 in the x-axis direction when buckling occurs, and is expressed as in equation (51), where δ is the displacement in the x-axis direction that occurs at the center of the first flange 12 relative to the center of the second flange 13.

The external force potential energy _Vw of the web 11 is expressed by equation (58), and the external force potential energy _Vf of the flanges 12, 13 is expressed by equation (59).

The total potential energy Π of the beam 10 is expressed as the sum of the strain energy and the external force potential energy, as shown in equation (60).

In the evaluation method, the first buckling stress τ _cr of the beam 10 is obtained based on the energy method from the out-of-plane displacement W _w of the web 11 and the out-of-plane displacements W _f1 and W _f2 of the flanges 12 and 13. That is, in the evaluation method, the first buckling stress τ _cr is obtained using equations (61) to (63) based on a _n , b _n , λ, and half wavelength a, which are real numbers that give the first buckling stress τ _cr by equation (64) a minimum positive value.
Equations (61) to (64) are the elastic local buckling strength equations when a shear force F1 acts on the beam 10.

Specifically, while treating the half wavelength a and the undetermined coefficient λ as constants, simultaneous equations are set up that express that the function obtained by partially differentiating the total potential energy Π with the undetermined coefficients _an and _bn is equal to 0, and real numbers _an and _bn are obtained. When there are multiple sets of _an and _bn that are solutions to the simultaneous equations, the first buckling stress τ cr is obtained based on the set of _an and _bn that gives the smallest positive value to the first buckling stress τ _cr according to equation (64) among the multiple sets of _an and _bn (by substituting the set of _an and _bn into equation (64)). Next, treating the half wavelength a and the undetermined coefficient λ as variables, the half wavelength a and the undetermined coefficient _λ are obtained from the equation that expresses that the function obtained by partially differentiating the total potential energy Π obtained by the set of _an and _bn with the half wavelength a and the undetermined coefficient λ is equal to 0. The first buckling stress τ _cr obtained based on the set of a _n and b _n obtained as described above, the half wavelength a, and the undetermined coefficient λ becomes the desired first buckling stress τ _cr .
When there is only one set of undetermined coefficients a _n , b _n that is a solution to the simultaneous equations, the first buckling stress τ _cr is found based on the set of a _n , b _n if the set of a _n , b _n gives the first buckling stress τ _cr according to equation (64) a minimum positive value. Next, the half wavelength a and the undetermined coefficient λ are treated as variables, and the first buckling stress τ _cr is found as described above.
The first buckling stress intensity τ _cr is an index that takes into consideration coupled buckling when a shear force acts on the beam 10 and the beam buckles.

[3. Second buckling stress when a uniform bending moment acts on a beam]
Next, a method for determining the second buckling stress (evaluation method) when a uniform bending moment acts on the beam 10 will be described, with reference to differences from a method for determining the first buckling stress when a shear force acts on the beam 10. In the evaluation method in this case, the position coordinates of the beam 10 are recognized based on a right-handed Cartesian coordinate system consisting of the x-axis, y-axis, and z-axis, as shown in FIG.
6 and 7, the center position of the second flange 13 (one of the pair of flanges) in the y-axis direction is defined as the origin of the y-axis. The direction from the origin of the y-axis toward the first flange 12 (the other of the pair of flanges) is defined as the positive direction of the y-axis.

As shown in Fig. 6, when a bending moment F2 about the z-axis acts on each end face 10a of the beam 10 in the x-axis direction, the beam 10 may buckle. The bending moment F2 is transmitted over the entire length of the beam 10 in the x-axis direction. The transmitted bending moment F2 acts about an axis that passes through the neutral axis of the beam 10 and is parallel to the z-axis (hereinafter referred to as the rotation axis of the bending moment F2).
The bending moments F2 acting on each end surface 10a of the beam 10 in the x-axis direction are external forces of equal magnitude. A pair of bending moments F2 of equal magnitude constitute an equal bending moment.
In this example, a bending moment F2 acts on the beam 10 such that the second flange 13 is subjected to a tensile force due to the bending moment F2 and the first flange 12 is subjected to a compressive force due to the bending moment F2, causing the beam 10 to bend convexly downward.

In this case, as the web 11 moves toward the first end in the x-axis direction (one of the end faces 10a in the x-axis direction), the web 11 is displaced alternately in the positive direction of the z-axis and the negative direction of the z-axis, and the web 11 as a whole is displaced in a wavy manner in the x-axis direction. In response to the web 11, the second flange 13 that is subjected to a compressive force is displaced in a wavy manner, but the second flange 13 that is subjected to a tensile force is hardly displaced.

In the evaluation method of this embodiment, the above assumptions 1 to 6 are made when estimating the second buckling stress (buckling stress) when a uniform bending moment acts on the beam 10.

As shown in Fig. 6, when a bending moment F2 acts on the beam 10, the web 11 of the beam 10 may be displaced in a wavy manner in the x-axis direction. When the y-axis coordinate of the web 11 displaced in a wavy manner in the x-axis direction is a certain value, it is assumed that the out-of-plane displacement of the web 11 in the z-axis direction at a certain coordinate on the x-axis is expressed by the formula sin(πx/a). In this case, the wavelength in the x-axis direction of the web 11 displaced in a wavy manner in the x-axis direction of the web 11 is 2a. The half wavelength (half the length of the wavelength) of the web 11 in the x-axis direction is a.
FIG. 8 is a diagram showing the out-of-plane displacement _Ww in the entire region of the beam 10 where the length of the beam 10 in the x-axis direction is half the wavelength a.

In Figure 9, the solid line shows the state when bending moment F2 acts on beam 10. The dotted line in Figure 9 shows the state when bending moment F2 does not act on beam 10.

When the bending moment F2 acts on the web 11, the out-of-plane displacement of the web 11 has conventionally been estimated using a Fourier series.
The inventors have considered a number of functions that can estimate the out-of-plane displacement (first out-of-plane displacement) _w of the web 11 in the z-axis direction at a certain coordinate on the y-axis when the coordinate on the x-axis is a certain value, while using trigonometric functions and with fewer terms than a Fourier series.
As a result, it was found that when bending moment F2 acts on beam 10, the out-of-plane displacement _ww of web 11 in the z-axis direction at a certain y-axis coordinate when the x-axis coordinate is a certain value can be estimated by equation (80) with fewer terms than the Fourier series, where N is a natural number of 2 or more, and _a0 , _an , and _bn are undetermined coefficients.
Equation (80) includes terms based on trigonometric functions using power functions of the y-axis coordinate. ^cos ( ^2πyn / _bwn ) and sin( ^πyn / _bwn ⁾ are the basis. Equation (80) can be preferably used to estimate the out-of-plane displacement of a plate element such as the web 11.

On the other hand, when the x-axis coordinate is an arbitrary value, the out-of-plane displacement (second out-of-plane displacement) _Ww of the web 11 in the z-axis direction at a certain coordinate on the y-axis is estimated by formula (81). As described above, formula (81) is based on the assumption that when the y-axis coordinate of the web 11 is a certain value, the out-of-plane displacement of the web 11 in the z-axis direction at a certain coordinate on the x-axis is expressed by the formula sin(πx/a).
The out-of-plane displacement _Ww in Fig. 6 is obtained by repeating the out-of-plane displacement _Ww in Fig. 8 in the x-axis direction. The out-of-plane displacement _Ww is a function of the y-axis coordinate and the x-axis coordinate, and is used when estimating the second buckling stress of the web 11.
By substituting equation (80) into equation (81), equation (82) is obtained. When N is 2 in equation (82), it can be transformed into equation (83).

Furthermore, from the above assumptions, the out-of-plane displacement _Wf1 of the first flange 12 at a predetermined coordinate on the z axis is estimated by equation (84), and the out-of-plane displacement _Wf2 of the second flange 13 at a predetermined coordinate on the z axis is estimated by equation (85).
Equations (84) and (85) include terms based on trigonometric functions using power functions of the y-axis coordinate.

In the evaluation method, when a uniform bending moment acts on the beam 10, the out-of-plane displacement _Ww of the web 11 is estimated by equation (82), the out-of-plane displacement _Wf1 of the first flange 12 is estimated by equation (84), and the out-of-plane displacement _Wf2 of the second flange 13 is estimated by equation (85).
Here, based on the energy method, the strain energy _Uw generated in the web 11 due to buckling deformation is expressed as in equation (87), and the strain energy _Uf of the flanges 12, 13 is expressed as in equation (88).

Here, the plate stiffness _Dw of the web 11 is expressed as in equation (89). The plate stiffness _Df of the flanges 12, 13 is expressed as in equation (90). The stress function _σw of the web 11 on which the bending moment F2 acts is expressed as in equation (91), where _σcr is the second buckling stress of the beam 10. Here, the stress function _σw is positive in compression.
The function _δw is the displacement in the x-axis direction at a certain coordinate of the y-axis of the web 11 when buckling occurs, and is expressed as in equation (92), where δ is the displacement in the x-axis direction occurring at the center of the second flange 13.

The stress functions σ _f1 and σ _f2 are the stress functions of the flanges 12 and 13 on which the bending moment F2 acts. The stress function σ _f1 is equal to −σ _cr , and the stress function σ _f2 is equal to σ _cr . However, the stress functions σ _f1 and σ _f2 are positive when they indicate compression.
The functions δ _f1 and δ _f2 are functions that express the displacement in the x-axis direction of the flanges 12 and 13 when buckling occurs. The function δ _f1 is equal to -δ, and the function δ _f2 is equal to δ.

The external force potential energy _Vw of the web 11 is expressed as in equation (98), and the external force potential energy _Vf of the flanges 12, 13 is expressed as in equation (99).

The total potential energy Π of the beam 10 is expressed as the sum of the strain energy and the external force potential energy, as shown in equation (100).

In the evaluation method, the second buckling stress σ _cr of the beam 10 is obtained based on the energy method from the out-of-plane displacement W _w of the web 11 and the out-of-plane displacements W _f1 and W _f2 of the flanges 12, 13. That is, in the evaluation method, the second buckling stress σ _cr is obtained using equations (101) to (104) based on a _n , b _{n ,} and half wavelength a, which are real numbers that give the second buckling stress σ _cr a minimum positive value according to equation (105).
It should be noted that equations (101) to (104) are equations for the elastic local buckling strength when a bending moment F2 acts on the beam 10.

Specifically, with the half wavelength a treated as a constant, a simultaneous equation is set up that expresses that the function obtained by partially differentiating the total potential energy Π with undetermined coefficients a _n and b _n is equal to 0, and real numbers a _n and b _n are obtained. If there are multiple sets of a _n and b _n that are solutions to the simultaneous equations, the second buckling stress intensity σ _cr is obtained based on the set of a _n and b _n that gives the smallest positive value to the second buckling stress intensity σ _cr according to equation (105) among the multiple sets of a _n and b _n (by substituting the set of a _n and b _n into equation (105)). Next, the half wavelength a is treated as a variable, and the half wavelength a is obtained from the equation that expresses that the function obtained by partially differentiating the total potential energy Π obtained by the set of a _n and b _n with the half wavelength a is equal to 0. The second buckling stress intensity σ _cr obtained based on the set of a _n and b _n and the half wavelength a obtained as described above becomes the desired second buckling stress intensity σ _cr .
When there is only one set of undetermined coefficients a _n , b _n that is a solution to the simultaneous equations, the second buckling stress σ _cr is found based on the set of a _n , b _n if the set of a _n , b _n gives the second buckling stress σ _cr according to equation (105) a minimum positive value. Next, the half wavelength a is treated as a variable and the second buckling stress σ _cr is found as described above.
The second buckling stress intensity σ _cr is an index taking into consideration coupled buckling when a uniform bending moment acts on the beam 10 and causes it to buckle.

[4. Width-thickness ratio classification considering coupled buckling]
Figure 10 shows the width-thickness ratio classification of beams based on the notification. In Figure 10, the horizontal axis represents the width-thickness ratio of the web defined by ((H- _2tf )/ _tw ), and the vertical axis represents the width-thickness ratio of the flange defined by (W/( _2tf )). The width-thickness ratio of the web increases toward the right, and the width-thickness ratio of the flange increases toward the top. There are four types (classes) of earthquake resistance performance of beams based on the notification: FA, FB, FC, and FD. These earthquake resistance performance types are displayed as areas on the graph shown in Figure 10. From the area close to the origin O, the types are FA, FB, FC, and FD.
FA and FB types of H-shaped steel are H-shaped steel that can resist earthquake external forces and fully absorb energy even after plasticization. FC type of H-shaped steel is H-shaped steel that does not experience local buckling until the cross section is in a fully plastic state. FD type of H-shaped steel is H-shaped steel that elastically buckles.

For example, lines L1 to L3 at which the web width-thickness ratios that divide the types FA to FC are constant are determined based on the elastic buckling strength of only the web 11 that constitutes the beam 10. In other words, lines L1 to L3 are determined without taking into consideration the influence of the flanges 12 and 13 that the beam 10 has.
Specifically, as shown in Fig. 11, the flanges 12, 13 are removed from the web 11, and the portions of the web 11 to which the flanges 12, 13 were joined are simply supported. Lines L1 to L3 are determined based on the elastic buckling strength of the web 11 when a shear force F1 acts on the web 11. Therefore, even if the width-thickness ratio of the flanges changes, the width-thickness ratio of the web does not change, and the types FA to FC of the web 11 are evaluated independently of the width-thickness ratio of the flanges.

The type of FA is determined by the inequality (120) for the flanges 12 and 13, and by the inequality (121) for the web 11. The type of FB is determined by the inequality (122) for the flanges 12 and 13, and by the inequality (123) for the web 11. The type of FC is determined by the inequality (124) for the flanges 12 and 13, and by the inequality (125) for the web 11.

Here, F is the standard strength of the beam 10. Here, the standard strength is the strength of steel material stipulated in Notification No. 2464 of the Ministry of Construction of 2000.
That is, the formulas (120) and (121) give the H-shaped steel of the beam 10 type FA. The formulas (122) and (123) give the H-shaped steel of the beam 10 type FB. The formulas (124) and (125) give the H-shaped steel of the beam 10 type FC. The H-shaped steel of the beam 10 that does not fall into any of the types is given the type FD.

In contrast, the width-thickness ratio classification of a beam that takes into account coupled buckling proposed in this embodiment (hereinafter referred to as new width-thickness ratio classification) will be explained using Figures 12 to 15. Note that since the shapes of the curves L4, L6, and L8 in Figures 13 to 15 change depending on the value of (H- _tf )/W, these figures show an example for an H-shaped steel with a constant (H- _tf )/W.
As shown in Fig. 12, in the new width-thickness ratio classification, the vertical axis represents the same values as the vertical axis in Fig. 10, and the horizontal axis represents the width-thickness ratio of the web defined by ((H- _tf )/ _tw ). Here, since the shapes of the curves L4, L6, and L8 change depending on the value of (H- _tf )/W as described above, the horizontal axis is interpreted using (H _{- tf} ) as an index instead of (H-2tf).
It should be noted that the value of ((Ht _f )/t _w ) is equal to the value of (b _w /t _w ).
In FIG. 12, the definition of the width-thickness ratio of the web on the horizontal axis is slightly different from that in FIG. 10. However, since each solid line in FIG. 12 is defined based on the formulas (120) to (125) used in defining each solid line in FIG. 10, the cross-sectional shape of the actual H-shaped steel is the same.

In the evaluation method of the present embodiment, when evaluating the type of the seismic performance of the beam 10, the type of the beam 10 is evaluated taking into consideration the coupled buckling of the web 11 and the pair of flanges 12, 13 of the beam 10. In the evaluation method, the type of the beam 10 is evaluated using the first buckling stress intensity τ _cr and the second buckling stress intensity σ _cr as indices.
The new width-thickness ratio classification will be explained separately for the FA and FB types and the FC type. Regarding the FA and FB types, the FA type will be mainly explained.

First, the types of FA will be described.
Fig. 13 is a diagram in which the types of FA are extracted from Fig. 12. In the following description, the depth H of the H-shaped steel constituting the beam 10 and the width W of the flanges 12, 13 are each assumed to be known values.
When the equality between equations (120) and (121) holds, equations (120-1) and (121-1) are obtained.

Here, some rules will be set. In the H-shaped steel of the current FA type of the beam 10, the conditions of the width-thickness ratio of the pair of flanges 12, 13 and the web 11 in the beam 10 when formulas (120-1) and (121-1) are satisfied are set as the standard conditions. The standard condition _PFA is shown in FIG. 13.
The second buckling stress σ _cr of the H-shaped steel under the reference condition P _FA is defined as the second reference buckling stress in the current FA type.
The ratio of the second reference buckling stress in the current FA type to the reference strength F is defined as the second reference buckling stress ratio in the current FA type. Curve L4 in Fig. 13 is a curve connecting points in Fig. 13 corresponding to H-shaped steels in which the ratio of the second buckling stress σ _cr to the reference strength F is the same as the second reference buckling stress ratio. Note that the value of (b w /W), which is the ratio of the plate thickness center distance b _w of the flanges 12, 13 to the width W of the flanges 12, 13 when calculating the second buckling stress σ _cr , is equal to the value of ( _{b w /} W) in the H-shaped steel for which the second reference buckling stress is calculated.

At this time, the beam 10 (hereinafter also referred to as the beam 10 belonging to the extended region of the FA type) in which the ratio of the second buckling stress σ _cr to the reference strength F is equal to or greater than the second reference buckling stress ratio in the current FA type and satisfies the formula (120) is evaluated as the beam 10 in the current FA type. In other words, the beam 10 in which the second buckling stress σ _cr is equal to or greater than the second reference buckling stress and satisfies the formula (120) is evaluated as the beam 10 in the FA type.
The ratio of the second buckling stress σ _cr to the reference strength F being equal to or greater than the second reference buckling stress ratio in the current FA type means that the second buckling stress σ _cr of the H-shaped steel is equal to or greater than the second buckling stress σ _cr of the H-shaped steel under the reference condition P _FA . Satisfying formula (120) means that each width-thickness ratio of the beam 10 falls on the line L5 or in a region closer to the origin O than the line L5. Therefore, the beam 10 satisfying the above condition means a beam 10 whose each width-thickness ratio falls in the region R _FA shown by hatching in FIG. 13.

In the region away from the origin O in the horizontal direction, the flanges 12, 13 are thick and the web 11 is thin, and such H-shaped steel has excellent cross-sectional performance relative to its weight. On the other hand, in the region away from the origin O than the line L5, the flanges 12, 13 are thin and the web 11 is thick, and such H-shaped steel has poor cross-sectional performance relative to its weight.

Next, the FB type will be described in terms of differences from the FA type.
FIG. 14 is a diagram showing the types of FBs extracted from FIG.
When the equality between equations (122) and (123) holds, equations (122-1) and (123-1) are obtained.

Here, some rules will be set. In the H-shaped steel of the current FB type of the beam 10, the conditions of the width-thickness ratio of the pair of flanges 12, 13 and the web 11 in the beam 10 when formulas (122-1) and (123-1) are satisfied are set as the standard conditions. The standard condition P _FB is shown in FIG. 14.
The second buckling stress σ _cr of the H-shaped steel under the reference condition P _FB is defined as the second reference buckling stress in the current FB type.
The ratio of the second reference buckling stress in the current FB type to the reference strength F is defined as the second reference buckling stress ratio in the current FB type. Curve L6 in Fig. 14 is a curve connecting points in Fig. 14 corresponding to H-shaped steels in which the ratio of the second buckling stress σ _cr to the reference strength F is the same as the second reference buckling stress ratio. Note that b w /W, which is the ratio of the plate thickness center distance b _w of the flanges 12, 13 to the width W of the flanges 12, 13 when calculating the second buckling stress _{σ cr} , is equal to the value of the H-shaped steel for calculating the second reference buckling stress.

At this time, the beam 10 (hereinafter also referred to as the beam 10 belonging to the extended region of the FB type) in which the ratio of the second buckling stress σ _cr to the reference strength F is equal to or greater than the second reference buckling stress ratio in the current FB type and satisfies the formula (122) is evaluated as the beam 10 in the current FB type. In other words, the beam 10 in which the second buckling stress σ _cr is equal to or greater than the second reference buckling stress and satisfies the formula (122) is evaluated as the beam 10 in the current FB type.
Satisfying equation (122) means that each width-thickness ratio of the beam 10 falls on the line L7 or in a region closer to the origin O than the line L7. Therefore, the beam 10 satisfying the above condition means the beam 10 whose each width-thickness ratio falls in the region _RFB shown by hatching in FIG.

Next, the types of FC will be described.
FIG. 15 is a diagram showing FC types extracted from FIG.

The yield shear strength of the H-section steel is defined as F'.
In this case, a beam 10 (hereinafter also referred to as a beam 10 belonging to the extended region of the FC type) whose ratio of the first buckling stress intensity τ _cr to the yield shear strength F′ satisfies equation (128) and equation (124) is evaluated as a beam 10 in the current FC type.

15 is a line where (τ _cr /F') is 1.0. Curve L8 passes through a region to the right of the upper right corner of the conventional FC type region (where the web width-thickness ratio is large) in the figure.
The ratio of the first buckling stress τ _cr to the yield shear strength F' satisfies formula (128), which means that the first buckling stress τ _cr is equal to or greater than the yield shear strength F'. Satisfying formula (128) means that each width-thickness ratio of the beam 10 falls on the line L9 or in a region closer to the origin O than the line L9. Therefore, the beam 10 that satisfies the above condition means a beam 10 whose each width-thickness ratio falls in the region _RFC shown by hatching in FIG. 15.

As described above, according to the evaluation method of this embodiment, the inventors have found, as a result of intensive research, that in an H-shaped steel, the web 11 and the flanges 12, 13 do not buckle independently, but buckle in a coupled manner while interacting with each other. Therefore, by constructing the elastic local buckling strength equations (61) to (64) and (101) to (104) that take into account the coupled buckling of the web 11 and the flanges 12, 13 of the beam 10, the type of the beam 10 can be evaluated more appropriately.
When a shear force acts on the beam 10, the first buckling stress τ _cr is calculated using equations (61) to (64), and when a uniform bending moment acts on the beam 10, the second buckling stress σ _cr is calculated using equations (101) to (105). Therefore, the first buckling stress τ _cr when a shear force acts on the beam 10 and the second buckling stress σ _cr when a uniform bending moment acts on the beam 10 can be accurately calculated using the equations.

A beam 10 in which the ratio of the second buckling stress σ _cr to the reference strength F is equal to or greater than the second reference buckling stress ratio in the FA type and satisfies formula (120) is evaluated as a beam 10 in the FA type. In the case of a beam 10 whose type is FA, the seismic performance of the beam 10 is evaluated taking into consideration the coupled buckling of the web 11 and the flanges 12, 13. This allows the cross-sectional shape of the beam 10 whose seismic performance can be evaluated as the FA type to be expanded to beams 10 whose ratio of the second buckling stress σ _cr to the reference strength F is equal to or greater than the second reference buckling stress ratio and satisfies formula (120). Therefore, for example, a beam 10 in which the width-thickness ratio of the web 11 is larger than that of the FA type in the current evaluation method can be evaluated as a beam 10 having the seismic performance of the FA type.

A beam 10 in which the ratio of the second buckling stress σ _cr to the reference strength F is equal to or greater than the second reference buckling stress ratio in the FB type and satisfies formula (122) is evaluated as a beam 10 in the FB type. In the case of a beam 10 whose type is FB, the seismic performance of the beam 10 is evaluated taking into consideration the coupled buckling of the web 11 and the flanges 12, 13. This allows the cross-sectional shape of the beam 10 whose seismic performance can be evaluated as the FB type to be expanded to beams 10 whose ratio of the second buckling stress σ _cr to the reference strength F is equal to or greater than the second reference buckling stress ratio and satisfies formula (122). Therefore, for example, a beam 10 whose web 11 has a larger width-thickness ratio than the FB type in the current evaluation method can be evaluated as a beam 10 having seismic performance of the FB type.

A beam 10 in which the ratio of the first buckling stress intensity τ _cr to the yield shear strength F' satisfies the formula (128) and also satisfies the formula (124) is evaluated as a beam 10 in the FC type. In the case of a beam 10 whose type is FC, the seismic performance of the beam is evaluated taking into consideration the coupled buckling of the web 11 and the flanges 12, 13. This allows the cross-sectional shape of the beam 10 whose seismic performance can be evaluated as the FC type to be expanded to beams 10 whose ratio of the first buckling stress intensity τ _cr to the yield shear strength F' satisfies the formula (128) and also satisfies the formula (124). Therefore, for example, a beam 10 whose web width-thickness ratio is larger than that of the FC type in the current evaluation method can be evaluated as a beam 10 having the seismic performance of the FC type.

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 changes, combinations, deletions, etc. of the configuration are also included within the scope that does not deviate from the gist of the present invention.
For example, in the evaluation method of the above embodiment, it is not necessary to evaluate the type of the beam 10 using the first buckling stress intensity τ _cr and the second buckling stress intensity σ _cr as indices.
When determining the first buckling stress intensity τ _cr and the second buckling stress intensity σ _cr , it is not necessary to determine them from the out-of-plane displacement W _w of the web 11 and the out-of-plane displacements W _f1 and W _f2 of the flanges 12 and 13 based on the energy method.
It is not necessary to use the equations (61) to (64) when calculating the first buckling stress intensity τ _cr , and it is not necessary to use the equations (101) to (105) when calculating the second buckling stress intensity σ _cr .
For example, when determining that the beam 10 is a beam of type FA or FB, the second reference buckling stress ratio, which is used to compare the magnitude relationship with the ratio of the second buckling stress σ _cr to the reference strength F, may be a value obtained by multiplying the second reference buckling stress ratio by a coefficient α (>1.0).

(Example)
EXAMPLES Hereinafter, the present invention will be described in more detail by showing specific examples, but the present invention is not limited to the following examples.
An FEM analysis was performed using the FEM analysis models shown in FIGS.
A first end of the beam 10 in the longitudinal direction is fixed to a support member 15A such as a pillar. The beam 10 extends in a direction along a horizontal plane and is supported in a cantilever manner. A forced displacement F4 is applied downward to a second end (free end) of the beam 10 in the longitudinal direction.

The analysis conditions are as follows:
(1) The analytical model was modeled using shell elements.
(2) The stress-strain relationship was modeled as a multi-linear model with a fixed tensile strength of 490 MPa. The reference strength F was set to two values: reference strength 1 (357.5 MPa) and reference strength 2 (380 MPa).
(3) The longitudinal length L11 of the beam 10 is four times the center-to-center distance _bw of the flange plate thickness.
(4) The first mode of buckling eigenvalue analysis was given as an initial imperfection so that the maximum out-of-plane deformation amount was (b _w /2000).

In the following, a comparison was made between the beam 10 under the standard condition and the beam 10 belonging to the expansion region for those with the same value of ( _bw /W). In order to make it easier to understand, the plate thickness center distance _bw and the flange width W were also set to the same value.
In addition, in order to evaluate the FA and FB types, an index η was used to determine the superiority or inferiority of performance. The index η was defined as the plastic deformation ratio at which the full plastic moment can be maintained in the load-deformation relationship of the analysis model.
The plastic deformation ratio is defined as the plastic deformation angle at which the full plastic moment can be maintained divided by _θp , where _θp is the elastic rotation angle corresponding to the full plastic moment _Mp .
First, the results of trial calculations and evaluations of the beam 10 belonging to the expansion region of the FA type are shown in Tables 1 and 2. Table 1 shows the results in the case of standard strength 1, and Table 2 shows the results in the case of standard strength 2.

For example, the beam 10 in Case 1 in Table 1 is an H-shaped steel under the standard condition P _FA in the current FA type, as shown in Fig. 13. The beam 10 in Case 2, which has the same value of (b _w /W) as the beam 10 in Case 1, is classified as FB under the evaluation method in the current notification.
In the evaluation method of this embodiment, the second buckling stress σ _cr is calculated using equations (101) to (104). Then, it is evaluated whether the ratio of the second buckling stress σ _cr to the yield strength σ _y is equal to or greater than the second reference buckling stress ratio in the FA type and satisfies equation (120).

In Table 1, if the index η of Case 2 is equal to or greater than the index η of the H-shaped steel of the same standard condition P _FA (index η of Case 1) with the (b _w /W) value, then as a result of carrying out the evaluation method of this embodiment, the beam 10 of Case 2 can be evaluated as a beam 10 having seismic performance of the FA type in the current evaluation method. As can be seen from Table 1, the beam 10 of Case 2, which is of the FB type in the current evaluation method, can be evaluated as a beam having seismic performance of the FA type in the current evaluation method.
In the case of beam 10 in Case 2, even when plasticized, H-shaped steel has traditionally been evaluated as being of the lower FB type, which is capable of resisting external earthquake forces and absorbing energy sufficiently. However, this beam can be evaluated as being of the higher FA type.

The beams 10 in cases 4, 5, 7, 9, 11, 13, 14, 15, 17, 18, 20, 22, 24, and 26 can also be evaluated as beams having seismic performance of the FA type under the current evaluation method.
For example, the beam 10 in Case 4, which was previously evaluated as an FD type H-shaped steel that elastically buckles, can be evaluated as an FA type H-shaped steel that can resist earthquake external forces and fully absorb energy even after plasticization. This makes it possible to bring out the inherent earthquake resistance of the beam 10 in Case 4 far more than the notification.

Next, the results of the calculation and evaluation of beam 10 belonging to the extended region of the FB type are shown in Tables 3 and 4. Table 3 shows the results in the case of standard strength 1, and Table 4 shows the results in the case of standard strength 2.

In this case, like the beams 10 belonging to the expansion area of the FA type, the beams 10 in cases 28, 30, 32, 33, 35, 37, 39, 41, 43, 44, 46, and 48 can be evaluated as beams having seismic performance of the FB type under the current evaluation method.

Next, the results of the calculation and evaluation of beam 10 belonging to the extended region of the FC type are shown in Tables 5 and 6. Table 5 shows the results in the case of standard strength 1, and Table 6 shows the results in the case of standard strength 2.

In the evaluation method of this embodiment, the first buckling stress τ _cr is calculated using equations (61) to (64), and it is evaluated whether or not the ratio of the first buckling stress τ _cr to the yield shear strength F′ satisfies equation (128) and whether or not the width-thickness ratio of the flanges 12, 13 satisfies equation (124).
As shown in Tables 5 and 6, it can be seen that in cases where equation (128) is satisfied and the value of (τ _cr /F') is 1 or more, the beam 10 exerts a maximum strength M _max that is greater than or equal to the full plastic moment M _p (the value of (M _max /M _p ) is 1 or more).
In the cases of Tables 3 to 6, it can be seen that there are cases in which the inherent earthquake resistance performance of the beam 10 is significantly improved beyond what is specified in the notification.

10 Beam 11 Web 12 First flange (flange)
13 Second flange (flange)

Claims (4)

In assessing the seismic performance of beams made of H-shaped steel having a web and a pair of flanges,
Evaluating the type of the beam by constructing an elastic local buckling strength equation considering the coupled buckling of the web and the pair of flanges of the beam;
Evaluating the type of the beam using a first buckling stress τ cr taking into consideration the coupled buckling when a shear force acts on the beam _and the beam buckles, and a second buckling stress σ cr taking into consideration the coupled buckling when a uniform bending moment acts on _the beam and the beam buckles, as indicators;
The first buckling stress intensity τ _cr and the second buckling stress intensity σ _cr are calculated based on an energy method from an out-of-plane displacement of the web and an out-of-plane displacement of the pair of flanges;
The material axis of the beam is defined as the x-axis,
An axis extending in a direction in which the pair of flanges sandwich the web is defined as a y-axis,
The axis extending in the thickness direction of the web is defined as the z-axis,
When a is defined as a half wavelength in the direction along the x-axis of the web, which is displaced alternately in a wave-like manner in the direction along the z-axis toward a first end of the web in the direction along the x-axis,
When a shear force acts on the beam,
The position of the center of the web in the direction along the y-axis is defined as the origin of the y-axis, and the direction from the origin of the y-axis toward one of the pair of flanges is defined as the positive direction of the y-axis.
The first buckling stress intensity τ cr _is calculated using equations (1) to (6) based on the a _n , b _n , λ, and the half wavelength a, which are real numbers that give the first buckling stress intensity τ _cr calculated by equation (7) a minimum positive value ;
When a uniform bending moment acts on the beam,
A center position of one of the pair of flanges in a direction along the y-axis is defined as an origin of the y-axis;
A direction from the origin of the y-axis toward the other of the pair of flanges is defined as a positive direction of the y-axis;
A method for evaluating the seismic performance of a beam, which determines the second buckling stress intensity σ _cr using equations (8) to (14) based on the a _n , b _n and the half wavelength a, which are real numbers that give the second buckling stress intensity σ _cr calculated by equation (15) a minimum positive value .
where N is a natural number of 2 or greater, a0 _, an _, bn _, and λ are undetermined coefficients, E is the Young's modulus of the H-shaped steel, ν is the Poisson's ratio of the H-shaped steel, tw _is the thickness of the web, tf _is the thickness of each of the pair of flanges, bw _is the center-to-center distance in thickness of the pair of flanges, bf _is half the width of each of the pair of flanges, Ww _is the out-of-plane displacement of the web in the direction along the z-axis, and Wf1 _and Wf2 _are the out-of-plane displacement of the pair of flanges in the direction along the y-axis.

The height H of the H-shaped steel constituting the beam and the width W of the pair of flanges are each a predetermined value,
The reference strength of the H-shaped steel is defined as F,
In the H-shaped steel of the type of FA of the beam given by the formulas (16) and (17), the condition of the width-thickness ratio of the pair of flanges and the web in the beam when the equality of the formulas (16) and (17) is established is defined as a reference condition;
The second buckling stress σ _cr of the H-shaped steel under the reference condition is defined as a second reference buckling stress in the type of the FA,
When the ratio of the second reference buckling stress in the type of the FA to the reference strength F is defined as a second reference buckling stress ratio in the type of the FA,
A ratio of the second buckling stress σ _cr to the reference strength F is equal to or greater than the second reference buckling stress ratio in the type of the FA,
The method for evaluating the seismic performance of a beam according to claim 1 , further comprising the step of evaluating the beam that satisfies the formula (16) as the beam in the type of the FA.
where _tf is the thickness of each of the pair of flanges, and _tw is the thickness of the web. In addition, the center-to-center distance in thickness direction of the pair of flanges and the width W of the flanges when calculating the second buckling stress _σcr are equal to the center-to-center distance in thickness direction of the pair of flanges and the width W of the flanges of the H-shaped steel when calculating the second reference buckling stress.

The height H of the H-shaped steel constituting the beam and the width W of the pair of flanges are each a predetermined value,
The reference strength of the H-shaped steel is defined as F,
In the H-shaped steel of the type of FB of the beam given by the formulas (18) and (19), the condition of the width-thickness ratio of the pair of flanges and the web in the beam when the equality of the formulas (18) and (19) is established is defined as the reference condition,
The second buckling stress σ _cr of the H-shaped steel under the reference condition is defined as a second reference buckling stress in the type of the FB,
When the ratio of the second reference buckling stress in the type of the FB to the reference strength F is defined as the second reference buckling stress ratio in the type of the FB,
A ratio of the second buckling stress σ _cr to the reference strength F is equal to or greater than the second reference buckling stress ratio in the type of the FB,
The method for evaluating the seismic performance of a beam according to claim 1 or 2 , wherein the beam that satisfies formula (18) is evaluated as the beam in the type of the FB.
where _tf is the thickness of each of the pair of flanges, and _tw is the thickness of the web. In addition, the center-to-center distance in thickness direction of the pair of flanges and the width W of the flanges when calculating the second buckling stress _σcr are equal to the center-to-center distance in thickness direction of the pair of flanges and the width W of the flanges of the H-shaped steel when calculating the second reference buckling stress.

The yield shear strength of the H-shaped steel is defined as F',
The ratio of the first buckling stress intensity τ _cr to the yield shear strength F′ satisfies formula (20),
The method for evaluating the seismic performance of a beam according to any one of claims 1 to 3, wherein the beam that satisfies formula (21) is evaluated as the beam in the FC type.
Here, F' is F/√3.

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