JP7678326B2 - How to design the support structure - Google Patents

Description

The present invention relates to a method for designing a support structure.

H-shaped cross-section steel materials (H-shaped cross-section members) are used in many building components, such as floor slabs and beams that support roofs, purlins that support roofs, and furring strips that support wall materials. When bending moments occur in components like these due to the weight of the supporting members or external forces, it is rational to use H-shaped cross-sections that have high bending performance around the strong axis relative to the steel weight and good cross-sectional efficiency, so H-shaped cross-section members are commonly used.
On the other hand, although H-shaped cross-section members have excellent cross-sectional performance around the strong axis, they have low cross-sectional performance around the weak axis, which makes lateral buckling (deformation in which the member moves out of the plane while twisting) a problem. When lateral buckling occurs, the strength of the H-shaped cross-section member deteriorates rapidly, and the behavior of the H-shaped cross-section member becomes unstable. For this reason, it is necessary to design the H-shaped cross section so that lateral buckling does not occur.

Specifically, the Ministry of Land, Infrastructure, Transport and Tourism's Notification No. 1024 of 2001, "Determining Special Allowable Stress and Special Material Strength," specifies the allowable stress for buckling of bending members. In this notification, the allowable stress is reduced in cases where the length of a member where lateral buckling may occur is long.
When the H-section member is long or narrow, and in other conditions where lateral buckling is likely to occur, the allowable stress decreases and the H-section member does not exhibit its inherent sectional performance. This requires a larger cross section, which increases the amount of steel, making the H-section member an uneconomical design.

Another method to prevent the reduction in allowable stress due to lateral buckling is to install stiffeners to prevent lateral buckling. With this method, if a sufficient number of stiffeners are installed, the allowable stress will not decrease. However, the processing and installation of the stiffeners requires time and cost, making this method uneconomical.
In response to such a problem, a design method may be adopted in which the restraining effect of the upper flange (first flange) by the floor or roof (plate-like member) is taken into consideration to improve the lateral buckling resistance and stiffening materials for preventing lateral buckling are omitted. In Patent Document 1 and Patent Document 2, an elastic lateral buckling resistance formula is derived for an H-shaped cross-section beam in which the upper flange is fastened to the floor slab by a shear connector, assuming that the lateral movement of the upper flange is completely restrained. This elastic lateral buckling resistance formula is used in the design method and analysis method. In Patent Document 3 and Patent Document 4, an elastic lateral buckling resistance formula is derived for a floor structure having a steel beam and a floor slab joined to the upper surface of the steel beam by a shear connector, assuming that the lateral movement and rotation of the upper flange are completely restrained. This elastic lateral buckling resistance formula is used in the design method.

Patent No. 6340276 Patent No. 6414374 Patent No. 6699639 JP 2020-158953 A

However, although the design methods disclosed in Patent Documents 1 to 4 are useful in enabling efficient design, they can only be expected to be effective under the condition that the lateral movement and rotation of the upper flange are completely restrained. For this reason, they can be applied when the surface material (plate-like member) attached to the upper flange has sufficient rigidity, such as a floor slab. However, they cannot be applied when a surface material that is not rigid enough to expect complete restraint of the upper flange, such as a roofing material or exterior wall material, is attached, or when the shear connector that joins the upper flange and the floor slab does not have sufficient rigidity.

For this reason, when the upper flange is incompletely restrained by roofing materials or exterior wall materials, etc., designs are generally made to install stiffening materials to prevent lateral buckling without considering these effects, even though they do have a certain degree of effect in improving resistance to lateral buckling. In other words, when the upper flange is incompletely restrained, the processing and installation of the stiffening materials incurs effort and cost, making the design uneconomical.

The present invention was made in consideration of these problems, and aims to provide a method for designing a support structure that can set the elastic lateral buckling resistance of an H-shaped cross-section member when the first flange of the H-shaped cross-section member is attached to a plate-like member, taking into account the constraint caused by the plate-like member attached to the first flange.

In order to solve the above problems, the present invention proposes the following means.
The design method of a support structure of the present invention is a design method of a support structure comprising a steel H-shaped cross-section member having a first flange, a second flange, and a web joining the first flange and the second flange to each other, and a plate-shaped member supported by the H-shaped cross-section member by being attached to the first flange, characterized in that the elastic lateral buckling strength _Me of the H-shaped cross-section member is set by equation (1).
where α is an undetermined coefficient that is any real number between 0 and 1, n is the number of half waves of the buckling wave of the H-shaped section member that is any positive integer, E is the Young's modulus of the H-shaped section member, I _fA is the second moment of area of the second flange, l is the length of the H-shaped section member, d _b is the distance between the center of thickness of the first flange and the center of thickness of the second flange, I _fB is the second moment of area of the first flange, G is the shear modulus of the H-shaped section member, J is the Saint-Venant torsional constant of the H-shaped section member, k _{r is the stiffening stiffness coefficient of the rotational movement spring that applies a restraining force to the first flange around an axis along the longitudinal direction of the H-shaped section member, J fA is} the Saint-Venant torsional constant of the second flange, J _fB is the Saint-Venant torsional constant of the first flange, J _w is the Saint-Venant torsional constant of the web, t _w is the thickness of the web, k _h is the shear modulus of the H-shaped section member, : stiffening stiffness coefficient by a parallel moving spring that applies a restraining force in the thickness direction of the web to the first flange, D _w : plate stiffness of the web.

In this invention, it is considered that the first flange is restrained by the plate-like member against the movement in the thickness direction of the web by the parallel movement spring, and against the rotation about the axis along the longitudinal direction of the H-shaped cross-section member by the rotation movement spring. It is assumed that the restraining forces of the parallel movement spring and the rotation movement spring have an arbitrary stiffness. After extensive research, the inventors derived equation (1) as an equation that can be applied when the cross section perpendicular to the longitudinal direction of the H-shaped cross-section member is symmetrical with respect to the first reference plane along the web, that is, when the cross section is symmetrical about one axis.
Therefore, when the cross section of the H-shaped section member is symmetrical about one axis, the elastic lateral buckling strength Me of the H-shaped section member can be set using equation (1) by considering the constraint by the plate-like member attached to the first _flange as a spring element.

In addition, in the support structure design method, the undetermined coefficient α may be set according to equations (3) to (5).

In this invention, the undetermined coefficient α can be precisely set using equations (3) to (5).

Another design method for a support structure of the present invention is a design method for a support structure comprising a steel H-shaped cross-section member having a first flange, a second flange, and a web joining the first flange and the second flange to each other, and a plate-shaped member supported by the H-shaped cross-section member by being attached to the first flange, characterized in that the elastic lateral buckling strength _Me of the H-shaped cross-section member is set by equation (8).
where α is an undetermined coefficient that can take any real number greater than or equal to 0 and less than 1, G is the shear elastic modulus of the H-shaped cross-section member, J is the Saint-Venant torsional constant of the H-shaped cross-section member, d _b is the distance between the thickness centers of the first flange and the second flange, k _r is the stiffening rigidity coefficient of a rotational movement spring that applies a restraining force to the first flange around an axis along the longitudinal direction of the H-shaped cross-section member, J _fA is the Saint-Venant torsional constant of the second flange, J _fB is the Saint-Venant torsional constant of the first flange, J _w is the Saint-Venant torsional constant of the web, t _w is the thickness of the web, _kh is the stiffening rigidity coefficient of a translational movement spring that applies a restraining force to the first flange in the thickness direction of the web, I _fA is the second moment of area of the second flange, I _fB is the second moment of area of the first flange, E is the Young's modulus of the H-shaped cross-section member, and D _w is the plate rigidity of the web.

In this invention, it is considered that the first flange is restrained by the plate-like member by the translation spring against the movement in the thickness direction of the web, and by the rotation spring against the rotation about the axis along the longitudinal direction of the H-shaped cross-section member. It is assumed that the restraining forces of the translation spring and the rotation spring have an arbitrary stiffness. After extensive research, the inventors derived Equation (8) as an equation that can be applied regardless of the number of half waves of the buckling wave of the H-shaped cross-section member and the length of the H-shaped cross-section member when the cross section of the H-shaped cross-section member is symmetrical about one axis.
Therefore, when the cross section of the H-shaped section member is symmetrical about one axis, the elastic lateral buckling strength _Me of the H-shaped section member can be easily set using equation (8) by considering the constraint by the plate-like member attached to the first flange as a spring element.

In addition, in the support structure design method, the undetermined coefficient α may be set according to equations (10) to (16).

In this invention, the undetermined coefficient α can be precisely set using equations (10) to (16).

Another design method for a support structure of the present invention is a design method for a support structure comprising a steel H-shaped cross-section member having a first flange, a second flange, and a web joining the first flange and the second flange to each other, and a plate-shaped member attached to the first flange and thereby supported by the H-shaped cross-section member, characterized in that the elastic lateral buckling strength _Me of the H-shaped cross-section member is set by equation (21).
where α is an undetermined coefficient that takes any real number between 0 and 1, n is the number of half waves of the buckling wave of the H-shaped section member that takes any positive integer, E is the Young's modulus of the H-shaped section member, I is the second moment of area of each of the first flange and the second flange, l is the length of the H-shaped section member, d _b is the distance between the thickness centers of the first flange and the second flange, G is the shear modulus of the H-shaped section member, J is the Saint-Venant torsional constant of the H-shaped section member, k _r is the stiffening stiffness coefficient of a rotational movement spring that applies a restraining force to the first flange around an axis along the longitudinal direction of the H-shaped section member, J _f is the Saint-Venant torsional constant of each of the first flange and the second flange, J _w is the Saint-Venant torsional constant of the web, t _w is the thickness of the web, _kh is the stiffening stiffness coefficient of a translational movement spring that applies a restraining force to the first flange in the thickness direction of the web, and D _w is the plate stiffness of the web.

In this invention, it is considered that the first flange is restrained by the plate-like member with respect to the movement in the thickness direction of the web by the parallel movement spring, and with respect to the rotation around the axis along the longitudinal direction of the H-shaped cross-section member by the rotation movement spring. It is assumed that the restraining forces of the parallel movement spring and the rotation movement spring have an arbitrary rigidity. After extensive research, the inventors derived equation (21) as an equation applicable to the case where the cross section of the H-shaped cross-section member is symmetrical with respect to the first reference plane in addition to the second reference plane along each flange, that is, so-called two-axis symmetry.
Therefore, when the cross section of the H-shaped section member is symmetrical about two axes, the elastic lateral buckling strength Me of the H-shaped section member can be set by equation (21), taking into account the constraint by the plate-like member attached to the first flange as a spring _element .

In addition, in the support structure design method, the undetermined coefficient α may be set according to equations (23) to (25).

In this invention, the undetermined coefficient α can be precisely set using equations (23) to (25).

Another design method for a support structure of the present invention is a design method for a support structure comprising a steel H-shaped cross-section member having a first flange, a second flange, and a web joining the first flange and the second flange to each other, and a plate-shaped member attached to the first flange and supported by the H-shaped cross-section member, characterized in that the elastic lateral buckling strength _Me of the H-shaped cross-section member is set by equation (28).
where α is an undetermined coefficient that can take any real number between 0 and 1, G is the shear elastic modulus of the H-shaped cross-section member, J is the Saint-Venant torsional constant of the H-shaped cross-section member, d _b is the distance between the thickness centers of the first flange and the second flange, k _r is the stiffening stiffness coefficient of a rotational movement spring that applies a restraining force to the first flange around an axis along the longitudinal direction of the H-shaped cross-section member, J _f is the Saint-Venant torsional constant of each of the first flange and the second flange, J _w is the Saint-Venant torsional constant of the web, t _w is the thickness of the web, _kh is the stiffening stiffness coefficient of a parallel movement spring that applies a restraining force to the first flange in the thickness direction of the web, E is the Young's modulus of the H-shaped cross-section member, I is the second moment of area of each of the first flange and the second flange, and D _w is the stiffness of the web.

In this invention, it is considered that the first flange is restrained by the plate-like member with respect to the movement in the thickness direction of the web by the parallel movement spring and the rotational movement spring with respect to the rotation around the axis along the longitudinal direction of the H-shaped cross-section member. It is assumed that the restraining forces of the parallel movement spring and the rotational movement spring have an arbitrary stiffness. After extensive research, the inventors derived Equation (28) as an equation that can be applied regardless of the number of half waves of the buckling wave of the H-shaped cross-section member and the length of the H-shaped cross-section member when the cross section of the H-shaped cross-section member is symmetrical about two axes.
Therefore, when the cross section of the H-shaped section member is symmetrical about two axes, the elastic lateral buckling strength _Me of the H-shaped section member can be easily set using equation (28) by considering the constraint by the plate-like member attached to the first flange as a spring element.

In addition, in the support structure design method, the undetermined coefficient α may be set according to equations (30) to (33).

In this invention, the undetermined coefficient α can be precisely set using equations (30) to (33).

In the design method of the support structure of the present invention, the elastic lateral buckling strength of the H-shaped cross-section member when the first flange of the H-shaped cross-section member is attached to the plate-like member can be set taking into account the constraint by the plate-like member attached to the first flange.

1 is a perspective view showing a cutaway view of a portion of a building in which a support structure to which a support structure design method according to one embodiment of the present invention is applied is used. FIG. 2 is a perspective view of a second H-shaped section member. 2 is a cross-sectional view taken along line A1-A1 in FIG. 1. 11 is a cross-sectional view of the second H-shaped section member before it is displaced. FIG. FIG. 4 is a side view illustrating the bending moment acting on the second H-shaped cross-section member. 13 is a cross-sectional view of the second H-shaped section member after it has been displaced. FIG. 1 is a graph showing the change in elastic lateral buckling strength versus l/H for a given cross-sectional shape of an H-shaped section member. 1 is a graph showing the change in elastic lateral buckling strength versus l/H for a given cross-sectional shape of an H-shaped section member. 1 is a graph showing the change in elastic lateral buckling strength versus l/H for a given cross-sectional shape of an H-shaped section member. 1 is a graph showing the change in elastic lateral buckling strength versus l/H for a given cross-sectional shape of an H-shaped section member. This is an oblique cutaway view of a part of a building in which another support structure is used, to which the support structure design method of one embodiment of the present invention is applied.

Below, a support structure to which a support structure design method according to one embodiment of the present invention (hereinafter, simply referred to as the design method) is applied will be described with reference to Figures 1 to 11.

[1. Structure of buildings using supporting structures]
The support structures 46, 47 are used in the building 1 shown in Fig. 1. The building 1 includes a plurality of columns 10, a plurality of first H-shaped cross-section members (H-shaped cross-section members) 15 which are main beams, a plurality of second H-shaped cross-section members (H-shaped cross-section members) 25 which are minor beams, and a folded plate roof 35.
In addition, the number of columns 10, first H-shaped cross-section members 15, and second H-shaped cross-section members 25 provided in the building 1 may be one each.
The pillars 10 extend in the up-down direction. The pillars 10 are arranged at intervals from one another. The pillars 10 are made of steel, RC (Reinforced Concrete), SRC (Steel Reinforced Concrete), or the like.

The first H-shaped section member 15 and the second H-shaped section member 25 are each made of steel.
The first H-shaped section member 15 includes a first flange 16, a second flange 17, and a web 18. The first flange 16, the second flange 17, and the web 18 are each formed of a steel plate. The first flange 16 and the second flange 17 are each disposed along a horizontal plane and face each other in the up-down direction. The first flange 16 is disposed above the second flange 17.
The web 18 is disposed between the first flange 16 and the second flange 17. The web 18 joins the center of the first flange 16 in the width direction and the center of the second flange 17 in the width direction to each other.

A gusset plate (not shown) is joined by welding or the like to the web 18 of the first H-shaped cross-section member 15, which serves as a main beam.
The first H-shaped section member 15 is disposed between the adjacent columns 10 and extends in a direction along the horizontal plane. Both ends of the first H-shaped section member 15 are joined to the columns 10 by welding or the like.

As shown in Figures 1 to 3, the second H-shaped section member 25 includes a first flange 26, a second flange 27, and a web 28. The first flange 26, the second flange 27, and the web 28 are each formed of a steel plate. The first flange 26 and the second flange 27 are each disposed along a horizontal plane and face each other in the up-down direction. The first flange 26 is disposed above the second flange 27.
The web 28 is disposed between the first flange 26 and the second flange 27. The web 28 joins the center of the first flange 26 in the width direction and the center of the second flange 27 in the width direction to each other.
The cross-sectional shape of the H-shaped cross-section members 15, 25 perpendicular to the longitudinal direction (material axis direction) is each H-shaped.
The orientation of the H-shaped section members 15, 25 is not limited to this. The second H-shaped section member 25 and the first H-shaped section member 15 may be H-shaped steel. More specifically, the second H-shaped section member 25 and the first H-shaped section member 15 may be rolled H-shaped steel or welded H-shaped steel.

Now, the dimensions of the second H-shaped section member 25 are defined as shown in FIG.
The height of the second H-shaped section member 25 is H (mm). The distance between the thickness centers of the first flange 26 and the second flange 27 (the distance between the thickness center of the first flange 26 and the thickness center of the second flange 27) is d _b (mm). The thickness of each of the first flange 26 and the second flange 27 is t _f (mm). The thickness of the web 28 is t _w (mm). The length of the second H-shaped section member 25 is l (mm) (see FIG. 2).
Furthermore, the specifications of the second H-shaped section member 25 are defined as follows.
The Young's modulus of the second H-shaped section member 25 is E (N/mm ² ). The second moment of area of the first flange 26 is I _fB (mm ⁴ ). The second moment of area of the second flange 27 is I _fA (mm ⁴ ). The shear modulus of the second H-shaped section member 25 is G (N/mm ² ). The Saint-Venant torsional constant of the second H-shaped section member 25 is J (mm ⁴ ). The Saint-Venant torsional constant of the first flange 26 is J _fB (mm ⁴ ). The Saint-Venant torsional constant of the second flange 27 is J _fA (mm ⁴ ). The Saint-Venant torsional constant of the web 28 is J _w (mm ⁴ ).
The plate stiffness of the web 28 is _Dw (Nmm). The plate stiffness _Dw can be calculated as _Dw = E· _tw3 /( ¹² (1- ^v2 )) as described later in (56). v represents the Poisson's ratio (-) of the second H-shaped section member 25.

As shown in FIG. 1, the second H-shaped section member 25 is placed between the opposing first H-shaped section members 15 and extends in a direction along the horizontal plane. Both longitudinal ends of the second H-shaped section member 25 are connected to the gusset plates of the first H-shaped section member 15 by high-strength bolts (not shown) or the like.

The folded plate roof 35 is a roofing material. For example, the folded plate roof 35 is formed by folding a metal plate. The folded plate roof 35 is formed by arranging a plurality of corrugated members 36 extending in a first horizontal direction D1 along a horizontal surface in a second horizontal direction D2 along the horizontal surface and perpendicular to the first horizontal direction D1.
The corrugated member 36 has a bottom plate 37, a first inclined plate 38, a top plate 39, and a second inclined plate 40. The bottom plate 37, the first inclined plate 38, the top plate 39, and the second inclined plate 40 are each a plate-shaped member (surface material).

The bottom plate 37 and the top plate 39 are disposed along a horizontal plane. The top plate 39 is disposed higher than the bottom plate 37.
The first inclined plate 38 is inclined gradually upward from the end of the bottom plate 37 on the first side D21 in the second horizontal direction D2 toward the first side D21.
The top plate 39 extends toward the first side D21 from an end of the first side D21 of the first inclined plate 38. The second inclined plate 40 is inclined gradually downward from the end of the first side D21 of the top plate 39 toward the first side D21. The second inclined plate 40 is connected to an end of a second side D22, which is opposite to the first side D21 in the second horizontal direction D2, of the bottom plate 37 of the corrugated member 36 adjacent to the first side D21.

As shown in Fig. 3, the bottom plate 37 of the folded plate roof 35 is disposed on the opposite side (above) of the web 28 of the second H-shaped section member 25, with the first flange 26 sandwiched therebetween. The bottom plates 37 are aligned in the second horizontal direction D2, which is the longitudinal direction of the second H-shaped section member 25.
As shown in Fig. 1, the folded plate roof 35 is supported from below by the first flange 16 of the first H-shaped section member 15 and the first flange 16 of the second H-shaped section member 25. As shown in Fig. 3, in this example, the folded plate roof 35 is attached to the first flange 26 of the second H-shaped section member 25 via a joint member 41 such as a known metal fitting or fastening member.

In this way, the folded plate roof 35 is supported by the second H-shaped section member 25 by being attached to the first flange 26.
1, the second H-shaped section member 25 and the folded plate roof 35 form a support structure 47. Similarly, the first H-shaped section member 15 and the folded plate roof 35 form a support structure 46.
The bottom plate 37, the first inclined plate 38, the top plate 39, and the second inclined plate 40 of the folded plate roof 35 may be directly attached to the first flanges 16, 26 of the H-shaped cross-section members 15, 25, respectively.

Below, we will explain the results of examining the elastic lateral buckling strength of the second H-shaped section member 25, one of the H-shaped section members 15 and 25, as an example.

[2. Consideration of elastic lateral buckling strength formula for H-shaped cross-section members]
2, the z-axis is defined as the longitudinal direction of the second H-section member 25 (hereinafter, simply referred to as the H-section member 25). The y-axis is defined as the direction in which the flanges 26 and 27 face each other, and the x-axis is defined as the thickness direction of the web 28.
The following assumptions (1) to (6) are made for the H-shaped cross-section member 25.
(1) The flange attached to the folded plate roof 35 is defined as the first flange 26. The second flange 27 is not directly restrained by the folded plate roof 35.
(2) The first flange 26 is elastically restrained by the folded plate roof 35 from moving in the x-axis direction and from rotating about the z-axis (the axis along the longitudinal direction of the H-shaped cross-section member 25).
(3) The H-shaped section member 25 is not restrained from warping.
(4) The load condition is a uniform bending moment that compresses the second flange 27. That is, as shown in Fig. 5, the load is a bending moment _Mcr about the x-axis applied to the longitudinal ends 25a and 25b of the H-shaped cross-section member 25.
(5) The intersections of the flanges 26, 27 and the web 28 remain perpendicular even after the H-section member 25 buckles laterally.
(6) The out-of-plane displacement of the flanges 26, 27 is given by an arbitrary function.

As shown in FIG. 6, the out-of-plane displacement (displacement in the x-axis direction) _uA (mm) of the second flange 27 and the out-of-plane displacement _uB (mm) of the first flange 26 are expressed as functions of an arbitrary displacement distribution u (mm) by equations (41) and (42).

Here, α is an undetermined coefficient.
As shown in Fig. 4, it was assumed that the first flange 26 was restrained by a parallel translation spring (horizontal spring) 50 and a rotational translation spring (rotational spring) 51. The parallel translation spring 50 is a spring that simulates the restraint by the folded plate roof 35 to which the first flange 26 is attached, and applies a restraining force to the first flange 26 in the x-axis direction (thickness direction of the web 28).
The rotational movement spring 51 is a spring that simulates the restraint exerted by the folded plate roof 35 to which the first flange 26 is attached, and applies a restraining force to the first flange 26 around an axis along the longitudinal direction of the H-shaped cross-section member 25.
The stiffness of the elastic spring that is attached to the first flange 26 and restrains the horizontal movement and rotation is expressed as a function of the plate stiffness _Dw of the web 28 and the distance _db between the plate thickness centers by equations (43) and (44).

Here, k _h is the stiffness coefficient (horizontal stiffness coefficient) (-) due to the translation spring 50. k _r is the stiffness coefficient (rotational stiffness coefficient) (-) due to the rotational movement spring 51. K _H is the stiffness (N/mm/mm) per unit length in the longitudinal direction of the H-shaped section member 25 due to the translation spring 50. K _R is the stiffness (Nmm/mm) per unit length in the longitudinal direction of the H-shaped section member 25 due to the rotational movement spring 51.
The torsion angle φ _A (rad) of the second flange 27, the torsion angle φ _B (rad) of the first flange 26, and the displacement function w (mm) of the web 28 are expressed as functions of an arbitrary displacement distribution u (mm) by equations (45) to (47).

However, the intersection point between the first flange 26 and the web 28 is defined as the origin of the y-axis.
At this time, the translation of the first flange 26 in the x-axis direction and the rotation around the z-axis are restrained by the elastic spring, and a uniform bending external force that compresses the second flange 27 acts on both longitudinal ends of the H-shaped cross-section member 25. The total potential energy Π (Nmm) in this case is expressed by equation (48).

where ν is the Poisson's ratio (-) of the H-shaped section member 25. _{M cr} is the elastic lateral buckling strength (elastic lateral buckling moment) (Nmm) of the H-shaped section member 25. _{u A} ^' and u _B ^' are functions obtained by first-order differentiation of u _A and u _B with respect to z. _{u A} ^'' and u _B ^'' are functions obtained by second-order differentiation of u _A and u _B with respect to z. _{φ A} ^' and φ _B ^' are functions obtained by first-order differentiation of φ _A and φ _B with respect to z.
By substituting equations (41) to (47) into equation (48) and rearranging, the total potential energy Π is expressed by equation (49).

By rearranging equation (49) using the relationship between equations (55) and (58) below, the total potential energy Π is expressed by equation (59).

where _Jw is the Saint-Venant torsional constant of the web 28 (mm ⁴ ).
Next, the equation is rearranged using equation (60) where the displacement distribution u is a sine wave: Here, n represents the number of half waves of the sine wave (the buckling wave of the H-shaped cross-section member 25) and can be any positive integer.

The equations obtained by first-order or second-order differentiation of the displacement distribution u are expressed as equations (61) and (62).

Using equations (60) to (62), each definite integral is expressed by equations (63) to (65).

Substituting equations (63) to (65) into equation (59) and rearranging, the total potential energy Π is expressed by equation (71).

According to the principle of minimum potential energy (Π=0), equation (71) can be rearranged for the elastic lateral buckling strength M _cr to obtain equation (72).

Here, α is an undetermined coefficient related to the displacement of the first flange 26. The elastic lateral buckling strength M cr is given by taking any real number in the range of 0 or more and less than 1 according to the assumed displacement distribution of any first flange 26. Note that it is preferable to use a _value for the undetermined coefficient α that minimizes the elastic lateral buckling strength M _cr . Therefore, an equation obtained by partially differentiating equation (72) with respect to the undetermined coefficient α is expanded. The undetermined coefficient α is any real number between 0 and 1. Taking this into consideration, the undetermined coefficient α that provides the minimum solution for the elastic lateral buckling strength M _cr according to equation (72) is expressed by equation (73).
Incidentally, the undetermined coefficient α may be determined not by equation (73) but by using a convergence calculation, a solver in a spreadsheet software, or the like to find a value that minimizes the elastic lateral buckling strength M _cr .

Here, X and Y are expressed by equations (74) and (75).

The elastic lateral buckling strength M _cr according to equation (72) depends on the number n of half waves of the sine wave. However, in the elastic lateral buckling strength M _cr , the minimum value of the buckling load obtained by substituting any positive integer for n is a constant value regardless of n. The minimum value M _cr,min of the elastic lateral buckling strength in this case is given by equation (76).

As in the case of the elastic lateral buckling strength M _cr , α is an undetermined coefficient related to the displacement of the first flange 26, and the minimum value M _cr,min of the elastic lateral buckling strength is given by taking any real number in the range of 0 or more and less than 1 according to the assumed displacement distribution of any first flange 26. Note that it is preferable to use a value for the undetermined coefficient α that minimizes the minimum value M _cr,min of the elastic lateral buckling strength. Therefore, an equation obtained by partially differentiating equation (76) with respect to the undetermined coefficient α is developed, and higher-order terms are ignored as being minute and rearranged. Considering that the undetermined coefficient α is a value between 0 and less than 1, the undetermined coefficient α that gives the minimum solution for the minimum value M _cr,min of the elastic lateral buckling strength according to equation (76) is expressed by equation (77).
Since equation (77) is an approximate solution, under special conditions such as when the horizontal stiffening stiffness coefficient k _h and the rotational stiffening stiffness coefficient k _r are extremely small, the square root of equation (77) may become a negative value, and α may not be obtained. In such cases, α can be obtained by treating the square root of equation (77) as 0, and the minimum value M _cr,min of the elastic lateral buckling strength according to equation (76) can be obtained. In addition, the undetermined coefficient α may be obtained by using a convergence calculation or a solver in a spreadsheet software to obtain a value that minimizes the elastic lateral buckling strength M _cr , without using equation (77).

Here, U, V, W, X, Y, and Z are expressed by equations (78) to (83).

The above formulas have been derived for a uniaxially symmetrical cross section in which the cross-sectional dimensions of the flanges 26 and 27 are different from each other. As shown in Fig. 3, in the uniaxially symmetrical cross section, the cross section perpendicular to the longitudinal direction of the H-shaped section member 25 is symmetrical with respect to the first reference plane S1 along the web 28.
On the other hand, in the case where the cross section of the H-shaped cross-section member 25 is symmetrical with respect to the first reference plane S1 in addition to the second reference plane S2 along each flange 26, 27, that is, is so-called two-axis symmetrical, the equations can be rearranged using the relationships shown in equations (87) and (88).

where I is the second moment of area (mm ⁴ ) of each of the flanges 26, 27. _Jf is the Saint-Venant torsional constant (mm ⁴ ) of each of the flanges 26, 27.
In this case, the elastic lateral buckling strength M _cr is expressed by equations (89) to (92).

The minimum value M _cr,min of the elastic lateral buckling strength is expressed by equations (93) to (97).

3. Support structure design method
As described above, in order to precisely set the elastic lateral buckling strength M _e when the cross section of the H-shaped section member 25 is symmetrical about one axis, in the design method of the support structure of this embodiment, it is set by the elastic lateral buckling strength M _cr of the H-shaped section member 25 shown in formula (72). At this time, it is preferable to set the undetermined coefficient α by formulas (73) to (75).
In order to easily set the elastic lateral buckling strength M _e when the cross section of the H-shaped section member 25 is symmetrical about one axis, in the design method of the support structure of this embodiment, it is set by the elastic lateral buckling strength M _cr,min of the H-shaped section member 25 shown in formula (76). Formula (76) is an equation that does not depend on the number n of half waves of the buckling wave of the H-shaped section member 25 and the length l of the H-shaped section member 25.
At this time, it is preferable to set the undetermined coefficient α according to equations (77) to (83).

In order to precisely set the elastic lateral buckling strength M _e when the cross section of the H-shaped section member 25 is biaxially symmetrical, in the design method of the support structure of this embodiment, the elastic lateral buckling strength M _cr of the H-shaped section member 25 is set as shown in formula (89). At this time, it is preferable to set the undetermined coefficient α from formulas (90) to (92).
In order to easily set the elastic lateral buckling strength M _e when the cross section of the H-shaped section member 25 is biaxially symmetric, in the design method of the support structure of this embodiment, it is set by the elastic lateral buckling strength M _cr,min of the H-shaped section member 25 shown in formula (93). Formula (93) is an equation that does not depend on the number n of half waves of the buckling wave of the H-shaped section member 25 and the length l of the H-shaped section member 25.
At this time, it is preferable to set the undetermined coefficient α according to equations (94) to (97).

[4. Consideration of elastic lateral buckling strength of H-shaped cross-section members]
In order to confirm the accuracy of the evaluation formula for the elastic lateral buckling strength of the H-shaped section member 25, a comparison was made with the results of an elastic buckling analysis using the Finite Element Method (FEM). In the analysis model shown in Figure 2, the H-shaped section member 25 is configured using four-node shell elements.

As a restraining effect of the folded plate roof 35 attached to the first flange 26, the node at the center of the cross section of the first flange 26 in FIG. 4 is restrained as follows. That is, for this node, movement in the x-axis direction is restrained by a horizontal spring 50, and rotation around the z-axis is restrained by a rotational spring 51.

In addition, both ends of the H-shaped cross-section member 25 in the longitudinal direction are fixed ends with respect to the twisting of the H-shaped cross-section member 25, and are free ends with respect to the warping of the flanges 26, 27. That is, as shown in FIG. 2, at the first end 25a in the longitudinal direction of the H-shaped cross-section member 25, dx = 0, dy = 0, dz = 0, and rotz (rotation around the z-axis) = 0. At the second end 25b opposite the first end 25a in the longitudinal direction of the H-shaped cross-section member 25, dy = 0, dx = 0, and rotz = 0.

For this analysis model, the dimensions of the H-shaped section member 25, the horizontal stiffening stiffness coefficient k _h of the horizontal spring 50, and the rotational stiffening stiffness coefficient k _r of the rotational spring 51 were set as variables. Specifically, four types of H-shaped section members 25 with different longitudinal cross-sectional dimensions were used. Two of the four types had cross sections symmetrical about one axis with different widths for the two flanges 26, 27. In the cross sections symmetrical about one axis, the cross section perpendicular to the longitudinal direction of the H-shaped section member 25 is symmetrical with respect to the first reference plane S1.
The stiffening stiffness of the springs was set in five stages, ranging from 0 (equivalent to a state in which the horizontal spring 50 and the rotational spring 51 are not present) to a sufficiently large range for the stiffening stiffness coefficients k _h and k _r . For a total of 20 samples shown in Table 1, which are combinations of these, analysis was performed by changing the length of the H-shaped cross-section member 25 so that the ratio (l/H) of the length l of the H-shaped cross-section member 25 to the height H of the H-shaped cross-section member 25 was in the range of 6 to 50.

For example, in sample No. 1-1, the height H of the H-section member 25 is 400 mm. The width _B1 of the first flange 26 is 100 mm, and the width _B2 of the second flange 27 is 150 mm. The thickness _tw of the web 28 is 6 mm, and the thickness _tf of each of the flanges 26 and 27 is 9 mm.
As pairs of (horizontal stiffening stiffness coefficient k _h , rotational stiffening stiffness coefficient k _r ), (0,0), (1,1), (6,2), (10,20), and (1000,1000) were used.

Figures 7 to 10 show the results of elastic buckling analysis for each cross-sectional shape of the H-shaped section member 25, comparing the elastic lateral buckling strength M _e according to the present invention and the minimum value of the elastic lateral buckling strength M _e . The cross sections of the H-shaped section member 25 corresponding to Figures 7 and 9 are symmetrical about one axis, with the two flanges 26, 27 having different widths. The curves in the figures represent the elastic lateral buckling strength M _e calculated using equation (72). The straight lines in the figures represent the elastic lateral buckling strength M _e calculated using equation (76).
8 and 10 is a biaxially symmetrical cross section in which the cross-sectional dimensions of the two flanges 26, 27 are equal. The curves in the figures represent the elastic lateral buckling strength _Me calculated from equation (89). The straight lines in the figures represent the elastic lateral buckling strength _Me calculated from equation (93).
In addition, in Figures 7 to 10, for cases where the horizontal stiffening stiffness coefficient k _h and the rotational stiffening stiffness coefficient k _r are 0, the minimum value of the elastic lateral buckling strength according to equation (76) or equation (93) is 0, so they are omitted.
7 to 10 also show the elastic lateral buckling strength M _FEM , which is the result of elastic buckling analysis by FEM.

Regardless of the dimensions of the H-shaped section member 25 or the stiffening coefficients k _h , k _r of the springs, the elastic lateral buckling strength M _e from equations (72) and (89) shows a good correspondence with the elastic lateral buckling strength M _FEM . Moreover, the minimum value of the elastic lateral buckling strength M _e from equations (76) and (93) accurately captures the lower limit of the elastic lateral buckling strength M _FEM , regardless of the dimensions of the H-shaped section member 25 or the stiffening coefficients k _h , k _r .

5. Effects of this embodiment
As described above, in the design method of this embodiment, when the cross section of the H-shaped section member 25 is symmetrical about one axis, it is sometimes necessary to precisely set the elastic lateral buckling strength M _e . In this design method, it is considered that the first flange 26 is restrained by the folded plate roof 35, and since the restraining force depends on the rigidity of the folded plate roof, it is assumed that the first flange 26 is restrained by the translation spring 50 and the rotation spring 51 having arbitrary rigidity. The translation spring 50 is a spring element that applies a restraining force in the thickness direction of the web 28. The rotation spring 51 is a spring element that applies a restraining force around an axis along the longitudinal direction of the H-shaped section member 25. As a result of intensive study, the inventors have derived the formula (72) as a formula that can be applied when the cross section perpendicular to the longitudinal direction of the H-shaped section member 25 is symmetrical about one axis.
Therefore, when the cross section of the H-shaped section member 25 is symmetrical about one axis, the elastic lateral buckling strength _Me of the H-shaped section member 25 can be set by equation (72), taking into account the constraint of the folded plate roof 35 attached to the first flange 26 as a spring element.

By using the design method according to this embodiment, it is possible to appropriately evaluate the elastic lateral buckling strength Me taking into account the restraining effect of the H-shaped cross-section member 25 having horizontal stiffening stiffness due to the translation spring 50 and rotational stiffening stiffness due to the rotational spring 51 in the restraining portion of the first _flange 26. This makes it possible to estimate a higher elastic lateral buckling strength than when the restraining effect is not taken into account, resulting in the omission of stiffening materials and reduction in processing and construction costs, enabling an economical design.

The undetermined coefficient α is set using equations (73) to (75). Therefore, the undetermined coefficient α can be precisely set using equations (73) to (75).

In addition, in the design method of this embodiment, when the cross section of the H-shaped section member 25 is symmetrical about one axis, it may be necessary to simply set the elastic lateral buckling strength M _e . In the design method in this case, it is assumed that the first flange 26 is restrained by the folded plate roof 35, and the restraining force depends on the rigidity of the folded plate roof, so it is assumed that it is restrained by the translation spring 50 and the rotation spring 51 having arbitrary rigidity. After extensive research, the inventors derived equation (76) as an equation that can be applied when the cross section of the H-shaped section member 25 is symmetrical about one axis, regardless of the number n of half waves of the buckling wave of the H-shaped section member 25 and the length l of the H-shaped section member 25.
Therefore, when the cross section of the H-shaped section member 25 is symmetrical about one axis, the elastic lateral buckling strength M _e of the H-shaped section member 25 can be easily set by equation (76), taking into account the constraint of the folded plate roof 35 attached to the first flange 26 as a spring element.

The undetermined coefficient α is set using equations (77) to (83). This allows the undetermined coefficient α to be precisely set using equations (77) to (83).

In addition, in the design method of this embodiment, when the cross section of the H-shaped section member 25 is biaxially symmetrical, it may be necessary to precisely set the elastic lateral buckling strength _Me . In the design method in this case, it is considered that the first flange 26 is restrained by the folded plate roof 35, and since the restraining force depends on the rigidity of the folded plate roof 35, it is assumed that the first flange 26 is restrained by the translation spring 50 and the rotation spring 51 having arbitrary rigidity. After extensive investigation, the inventors have derived the formula (89) as a formula applicable when the cross section of the H-shaped section member 25 is biaxially symmetrical.
Therefore, when the cross section of the H-shaped section member 25 is symmetrical about two axes, the elastic lateral buckling strength _Me of the H-shaped section member 25 can be set by equation (89), taking into account the constraint of the folded plate roof 35 attached to the first flange 26 as a spring element.

The undetermined coefficient α is set using equations (90) to (92). Therefore, the undetermined coefficient α can be precisely set using equations (90) to (92).

In addition, in the design method of this embodiment, when the cross section of the H-shaped section member 25 is biaxially symmetrical, it may be desired to simply set the elastic lateral buckling strength M _e . In the design method for this case, it is considered that the first flange 26 is restrained by the folded plate roof 35, and since the restraining force depends on the rigidity of the folded plate roof 35, it is assumed that the first flange 26 is restrained by the translation spring 50 and the rotation spring 51 having arbitrary rigidity. After extensive research, the inventors derived equation (93) as an equation that can be applied when the cross section of the H-shaped section member 25 is biaxially symmetrical, regardless of the number n of half waves of the buckling wave of the H-shaped section member 25 and the length l of the H-shaped section member 25.
Therefore, when the cross section of the H-shaped section member 25 is biaxially symmetrical, the elastic lateral buckling strength M e of the H-shaped section member 25 can be easily set by equation (93), taking into account the constraint of the folded plate roof 35 attached to the first flange ₂₆ as a spring element.

The undetermined coefficient α is set using equations (94) to (97). This allows the undetermined coefficient α to be precisely set using equations (94) to (97).

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, as shown in FIG. 11, in a building 1A, a folded plate roof 35 is attached to a first H-shaped cross-section member 15, which is a main girder, via a second H-shaped cross-section member 25, which is a purlin.
Furthermore, a second H-shaped section member 25, which is a furring strip, is fixed to the column 10, and an exterior wall material 55 is attached to this second H-shaped section member 25 via a joint member (not shown). The second H-shaped section member 25 and the exterior wall material 55 form a support structure 57.
For example, the exterior wall material 55 has a plurality of siding boards (plate-shaped members) 56. Each siding board 56 is made of steel, cement, or the like, and extends in the vertical direction. The plurality of siding boards 56 are arranged in the longitudinal direction of the second H-shaped cross-section member 25. The plurality of siding boards 56 are disposed on opposite sides of the web 28 with the first flange 26 sandwiched therebetween, and are each attached to the first flange 26 via a joint member (not shown).

In this design method, when the cross section of the H-shaped section member 25 is biaxially symmetrical, the elastic lateral buckling strength _Me of the H-shaped section member 25 may be set by equation (72), or the elastic lateral buckling strength _Me of the H-shaped section member 25 may be set by equation (76).
The first flange 26 of the H-shaped cross-section member 25 may be attached directly to the bottom plate 37 of the folded plate roof 35, or may be attached via a joint member.

15 First H-shaped cross-sectional member (H-shaped cross-sectional member)
16, 26 First flange 17, 27 Second flange 18, 28 Web 25 Second H-shaped section member (H-shaped section member)
35 Folded plate roof (plate-shaped member)
37 Bottom plate (plate-shaped member)
38 First inclined plate (plate-shaped member)
39 Top plate (plate-shaped member)
40 Second inclined plate (plate-shaped member)
46, 47, 57 Support structure 50 Translation spring 51 Rotation spring 56 Siding board (plate-shaped member)

Claims (8)

A method for designing a support structure comprising a steel H-shaped section member having a first flange, a second flange, and a web joining the first flange and the second flange to each other, and a plate-shaped member attached to the first flange and supported by the H-shaped section member, wherein the elastic lateral buckling strength _Me of the H-shaped section member is set by equation (1).
where α is an undetermined coefficient that is any real number between 0 and 1, n is the number of half waves of the buckling wave of the H-shaped section member that is any positive integer, E is the Young's modulus of the H-shaped section member, I _fA is the second moment of area of the second flange, l is the length of the H-shaped section member, d _b is the distance between the center of thickness of the first flange and the center of thickness of the second flange, I _fB is the second moment of area of the first flange, G is the shear modulus of the H-shaped section member, J is the Saint-Venant torsional constant of the H-shaped section member, k _{r is the stiffening stiffness coefficient of the rotational movement spring that applies a restraining force to the first flange around an axis along the longitudinal direction of the H-shaped section member, J fA is} the Saint-Venant torsional constant of the second flange, J _fB is the Saint-Venant torsional constant of the first flange, J _w is the Saint-Venant torsional constant of the web, t _w is the thickness of the web, k _h is the shear modulus of the H-shaped section member, : stiffening stiffness coefficient by a parallel moving spring that applies a restraining force in the thickness direction of the web to the first flange, D _w : plate stiffness of the web.

2. The method for designing a support structure according to claim 1, wherein the undetermined coefficient α is set by any one of equations (3) to (5).

A method for designing a support structure comprising a steel H-shaped section member having a first flange, a second flange, and a web joining the first flange and the second flange to each other, and a plate-shaped member attached to the first flange and supported by the H-shaped section member, wherein the elastic lateral buckling strength _Me of the H-shaped section member is set by equation (8).
where α is an undetermined coefficient that can take any real number greater than or equal to 0 and less than 1, G is the shear elastic modulus of the H-shaped cross-section member, J is the Saint-Venant torsional constant of the H-shaped cross-section member, d _b is the distance between the thickness centers of the first flange and the second flange, k _r is the stiffening rigidity coefficient of a rotational movement spring that applies a restraining force to the first flange around an axis along the longitudinal direction of the H-shaped cross-section member, J _fA is the Saint-Venant torsional constant of the second flange, J _fB is the Saint-Venant torsional constant of the first flange, J _w is the Saint-Venant torsional constant of the web, t _w is the thickness of the web, _kh is the stiffening rigidity coefficient of a translational movement spring that applies a restraining force to the first flange in the thickness direction of the web, I _fA is the second moment of area of the second flange, I _fB is the second moment of area of the first flange, E is the Young's modulus of the H-shaped cross-section member, and D _w is the plate rigidity of the web.

4. The method for designing a support structure according to claim 3, wherein the undetermined coefficient α is set by any one of equations (10) to (16).

A method for designing a support structure comprising a steel H-shaped section member having a first flange, a second flange, and a web joining the first flange and the second flange to each other, and a plate-like member attached to the first flange and supported by the H-shaped section member, wherein the elastic lateral buckling strength _Me of the H-shaped section member is set by equation (21).
where α is an undetermined coefficient that takes any real number between 0 and 1, n is the number of half waves of the buckling wave of the H-shaped section member that takes any positive integer, E is the Young's modulus of the H-shaped section member, I is the second moment of area of each of the first flange and the second flange, l is the length of the H-shaped section member, d _b is the distance between the thickness centers of the first flange and the second flange, G is the shear modulus of the H-shaped section member, J is the Saint-Venant torsional constant of the H-shaped section member, k _r is the stiffening stiffness coefficient of a rotational movement spring that applies a restraining force to the first flange around an axis along the longitudinal direction of the H-shaped section member, J _f is the Saint-Venant torsional constant of each of the first flange and the second flange, J _w is the Saint-Venant torsional constant of the web, t _w is the thickness of the web, _kh is the stiffening stiffness coefficient of a translational movement spring that applies a restraining force to the first flange in the thickness direction of the web, and D _w is the plate stiffness of the web.

6. The method for designing a support structure according to claim 5, wherein the undetermined coefficient α is set by any one of equations (23) to (25).

A method for designing a support structure comprising a steel H-shaped section member having a first flange, a second flange, and a web joining the first flange and the second flange to each other, and a plate-like member attached to the first flange and supported by the H-shaped section member, wherein the elastic lateral buckling strength _Me of the H-shaped section member is set by equation (28).
where α is an undetermined coefficient that can take any real number between 0 and 1, G is the shear elastic modulus of the H-shaped cross-section member, J is the Saint-Venant torsional constant of the H-shaped cross-section member, d _b is the distance between the thickness centers of the first flange and the second flange, k _r is the stiffening stiffness coefficient of a rotational movement spring that applies a restraining force to the first flange around an axis along the longitudinal direction of the H-shaped cross-section member, J _f is the Saint-Venant torsional constant of each of the first flange and the second flange, J _w is the Saint-Venant torsional constant of the web, t _w is the thickness of the web, _kh is the stiffening stiffness coefficient of a parallel movement spring that applies a restraining force to the first flange in the thickness direction of the web, E is the Young's modulus of the H-shaped cross-section member, I is the second moment of area of each of the first flange and the second flange, and D _w is the stiffness of the web.

8. The method for designing a support structure according to claim 7, wherein the undetermined coefficient α is set by equations (30) to (33).

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