Buckling strength estimation method, steel member design method and buckling strength estimation program

Abstract

To provide a buckling strength estimation method capable of accurately and easily estimating buckling strength of an elastic element in a case where a cross-sectional shape or material is not uniform along the entire length.SOLUTION: A buckling strength estimation method S5 for estimating buckling strength of a rod-shaped or flat-shaped elastic element when an external force acts on the elastic element and the elastic element buckles, comprises: a displacement estimation step S7 for estimating the out-of-plane displacement of the elastic element due to buckling using a partial sum of a Fourier series using predetermined Fourier coefficients; and a strength calculation step S8 for acquiring the buckling strength of the elastic element that generates the out-of-plane displacement based on the energy method.SELECTED DRAWING: Figure 21

Description

Technical Field

[0001] The present invention relates to a method for estimating buckling resistance, a method for designing a steel member, and a program for estimating buckling resistance.

Background Art

[0002] Conventionally, square steel pipes, H-shaped steels, etc. have been used for columns and beams of steel-frame buildings. These steel members often have a relatively thin plate thickness compared to their outer dimensions, and when a load acts, buckling phenomena such as local buckling in which the plate elements constituting the member deform out of the plane and lateral buckling in which the entire member deforms out of the plane may occur. Therefore, in the design of steel members, various buckling resistances are calculated, and a method is adopted in which the design is made such that the various buckling resistances exceed the forces generated in the members by the assumed external forces. A steel member may be configured by joining one or a plurality of elastic elements formed in a rod shape or a flat plate shape, for example.

[0003] Here, since the correct solutions of the conventional basic equations of buckling (differential equations of deflection) are often unknown or are too complex in many cases, the energy method is often used for calculating buckling resistance. This energy method assumes the deformation (displacement function) u associated with the buckling of the member, obtains the strain energy ΔU stored in the member due to this deformation and the potential energy ΔV of the external force, and obtains the buckling resistance based on the fact that both are in a neutral equilibrium state (ΔU = ΔV) under the buckling resistance. When estimating the buckling resistance using this energy method, it is necessary to assume the deformation (displacement function) u associated with the buckling of the member. Generally, a so-called Fourier series (Fourier sine series, Fourier cosine series) expressed by equations (1) to (2) is used. The Fourier series has an infinite number of terms.

[0004]

Equations

[0005] For example, in Patent Documents 1 and 2, something corresponding to this Fourier series (with a slightly different form of the formula) is used to estimate the lateral buckling resistance. In the following formulas, the total length of the range where buckling deformation is assumed is represented as l, and the coordinate in the material length direction is represented as z. a0, a1 to a N are undetermined coefficients, and n is a natural number.

[0006] Since the basis of the Fourier series has orthogonality, the deformation due to buckling can be accurately approximated by the Fourier series. As a result, the buckling resistance obtained by the energy method has high evaluation accuracy. However, the Fourier series has a problem of slow convergence of approximation. When trying to approximate the buckling deformation occurring in the members of a building or the plate elements constituting the members, depending on the type of buckling and the acting external force that occur, it is necessary to use a partial sum from the 10th to the 25th term of the Fourier series. That is, it is necessary to set N to about 10 to 25 in formula (1) or (2). In this case, the deformation u includes a total of about 10 to 25 undetermined coefficients of a0 to a N and becomes a high-order eigenvalue problem in the calculation of the buckling resistance by the energy method. Therefore, it cannot be calculated in an explicit form, and it becomes necessary to perform numerical calculations using a computer. For this reason, in the prior art Patent Documents 1 and 2, the displacement function is approximately given by performing a regression analysis on the FEM analysis results so that it can be evaluated in an explicit form, but the evaluation accuracy is low. In response to such problems, the inventors are examining a method of approximating buckling deformation using trigonometric functions with exponents of the phase as represented by formulas (3) to (4).

[0007]

Number

[0008] n, which is the exponent in formulas (3) to (4), is a natural number. By using these functions, the out-of-plane displacement due to the buckling of the elastic element can be estimated with fewer terms than a Fourier series, and the number of undetermined coefficients required to evaluate the buckling strength with sufficient accuracy can be suppressed to four or less. Therefore, the calculation of the buckling strength only needs to use an equation of degree four or less whose solution formula is known, and the calculation can be easily and accurately performed in an explicit form.

Prior Art Documents

Patent Documents

[0009]

Patent Document 1

Patent Document 2

Summary of the Invention

Problems to be Solved by the Invention

[0010] However, when using these trigonometric functions with the phases raised to powers, since this function cannot be integrated by algebraic operations, it is necessary to perform numerical integration by determining the integration interval in advance. Therefore, on the premise that the cross-sectional shape of the member for evaluating the buckling strength is uniform over the entire length, the entire length of the member is numerically integrated in advance as the integration interval, and the integration terms are treated as constant coefficients. On the other hand, steel members of steel frame structures often do not have a uniform cross-sectional shape over the entire length, such as when the cross-section changes in the middle of the member, when there are defects in a part of the member, or when a reinforcing member is attached in the middle of the member. Also, during earthquakes or fires, due to plasticization or temperature changes of the member, the material of the member may not be uniform over the entire length.

[0011] In order to evaluate the buckling resistance in such a case, it is necessary to change the integration interval according to the assumptions of the cross-sectional shape and material changes and perform the calculation. However, in the case of a trigonometric function raised to a power with a phase where algebraic integration is not possible, numerical integration is required each time. Numerical integration has a high computational load and takes time, so it is not practical for calculating the buckling resistance. That is, with the prior art methods, when the cross-sectional shape and material of the steel member are not uniform, it is not possible to easily and accurately calculate the buckling resistance.

[0012] The present invention has been made in view of such problems, and an object thereof is to provide a buckling resistance estimation method capable of accurately and easily estimating the buckling resistance of an elastic element when the cross-sectional shape and material of the elastic element are not uniform over the entire length, a buckling resistance estimation program, and a design method for a steel member using this buckling resistance estimation method.

Means for Solving the Problem

[0013] In order to solve the above problems, the present invention proposes the following means. (1) Aspect 1 of the present invention is a method for estimating the buckling resistance of an elastic element formed in a rod shape or a flat plate shape when an external force acts on the elastic element and the elastic element buckles. The method includes a displacement estimation step of estimating the out-of-plane displacement due to the buckling of the elastic element by a partial sum of a Fourier series using predetermined Fourier coefficients, and a strength calculation step of obtaining the buckling resistance of the elastic element that causes the out-of-plane displacement based on the energy method. The elastic element referred to here means an element that does not consider material non-linearity.

[0014] In the present invention, a Fourier series is used to estimate the out-of-plane displacement due to buckling of an elastic element formed in a rod shape or a flat plate shape. Therefore, in the ultimate strength calculation step, it can be divided into arbitrary intervals and calculated algebraically by integration, and the buckling strength can be accurately obtained even when the cross-sectional shape and material of the elastic element are not uniform over the entire length. Further, the Fourier series used to estimate the out-of-plane displacement is a partial sum of a Fourier series in which the Fourier coefficients are determined in advance. Therefore, the number of undetermined coefficients is significantly smaller than when obtaining the Fourier coefficients when obtaining the buckling strength, and the out-of-plane displacement can be easily estimated in the displacement estimation step, and the buckling strength can be easily obtained in the ultimate strength calculation step.

[0015] (2)Aspect 2 of the present invention may be the method for estimating buckling strength according to (1), which includes a coefficient determination step of obtaining the Fourier coefficients by Fourier-transforming a function in which the phase is raised to a power before the displacement estimation step. The power mentioned here means an exponent that is a real number. In the present invention, for example, compared with the case of Fourier-transforming a function in which the phase is raised to a power by an exponent that is a natural number, the number of terms of the Fourier series required to accurately estimate the out-of-plane displacement can be reduced in the displacement estimation step.

[0016] (3)Aspect 3 of the present invention may be the method for estimating buckling strength according to (1) or (2), in which the number of undetermined coefficients of the partial sum of the Fourier series is 4 or less. Generally, an equation of degree 4 or less can be solved by a formula for solutions. In the present invention, the undetermined coefficients of 4 or less of the partial sum of the Fourier series can be easily calculated by the formula for solutions in an explicit form without using a convergence calculation.

[0017] (4)Aspect 4 of the present invention is a method for designing a steel frame member, which includes a cross-section determination step of determining the cross-sectional dimensions of the steel frame member having the elastic element based on the buckling strength obtained by the method for estimating buckling strength according to any one of (1) to (3).

[0018] In the present invention, in the cross-section determination step, the cross-sectional dimensions of the steel frame member having an elastic element can be determined so that the steel frame member does not buckle.

[0019] (5) Aspect 5 of the present invention is a buckling strength estimation program for an estimation device that estimates the buckling strength when an external force acts on an elastic element formed in a rod shape or a flat plate shape and the elastic element buckles. The estimation device includes a displacement estimation unit that estimates the out-of-plane displacement due to the buckling of the elastic element by a partial sum of a Fourier series using predetermined Fourier coefficients, and a strength calculation unit that obtains the buckling strength of the elastic element that causes the out-of-plane displacement based on the energy method.

[0020] In the present invention, a Fourier series is used to estimate the out-of-plane displacement due to the buckling of an elastic element formed in a rod shape or a flat plate shape. Therefore, it can be algebraically integrated by being divided into arbitrary sections by the strength calculation unit, and the buckling strength can be accurately obtained even when the cross-sectional shape and material of the elastic element are not uniform over the entire length. Further, the Fourier series used to estimate the out-of-plane displacement is a partial sum of a Fourier series in which the Fourier coefficients are determined in advance. Therefore, the number of undetermined coefficients is significantly smaller than when obtaining the Fourier coefficients when obtaining the buckling strength, and the out-of-plane displacement can be easily estimated by the displacement estimation unit, and the buckling strength can be easily obtained by the strength calculation unit.

Advantages of the Invention

[0021] In the buckling strength estimation method, the steel frame member design method, and the buckling strength estimation program of the present invention, the buckling strength of the elastic element can be accurately and easily estimated even when the cross-sectional shape and material of the elastic element are not uniform over the entire length.

Brief Description of the Drawings

[0022]

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

Figure 7

Figure 8

Figure 9

Figure 10

Figure 11

Figure 12

Figure 13

Figure 14

Figure 15

Figure 16

Figure 17

Figure 18

Figure 19

Figure 20

Figure 21

Mode for Carrying Out the Invention

[0023] Hereinafter, an embodiment of a method for estimating buckling strength, a program for estimating buckling strength, and a design method for a steel member according to the present invention will be described with reference to FIGS. 1 to 21.

[0024] 〔1. Configuration of elastic elements〕 Hereinafter, as an elastic element aggregate having a plurality of elastic elements, the H-shaped cross-sectional member 10 shown in FIGS. 1 to 3 will be described as an example. Note that the H-shaped cross-sectional member may be an H-shaped steel. As shown in FIGS. 1 to 3, the H-shaped cross-sectional member 10 has an upper flange 11, a lower flange 12, and a web 13. The upper flange 11, the lower flange 12, and the web 13 are elastic elements each formed in a flat plate shape by a steel plate or the like. The upper flange 11 is disposed above the lower flange 12. The web 13 is disposed between the upper flange 11 and the lower flange 12 and is connected to the center in the width direction of the upper flange 11 and the center in the width direction of the lower flange 12, respectively.

[0025] 〔2. Consideration of method for estimating buckling strength〕 〔2.1. Case where cross-sectional shape is uniform over the entire length〕 Hereinafter, first, the case where the cross-sectional shape of the H-shaped cross-sectional member 10 in the material length direction (longitudinal direction) is uniform over the entire length will be described. The lateral buckling of the H-shaped cross-sectional member 10 in which the deformation of the upper flange 11 is restricted by a floor slab 40 or the like (see FIG. 3) will be considered. Here, coordinates are defined for the H-shaped cross-sectional member 10 as follows. As shown in FIGS. 1 and 2, the center of the upper flange 11 at the end in the material length direction of the H-shaped cross-sectional member 10 is defined as the origin O. The material length direction of the H-shaped cross-sectional member 10 is defined as the z-axis, the height direction (the direction in which the flanges 11 and 12 face each other) is defined as the y-axis, and the out-of-plane direction (the thickness direction of the web 13) is defined as the x-axis.

[0026] For quantities such as length defined below, SI units such as "m (meter)" are preferably used. Let the length of the H-shaped cross-sectional member 10 be l, and the distance between the center of the upper flange 11 and the center of the lower flange 12 in the y-axis direction (opposing direction) be d. b It is defined as such. Assuming the case where the deformation of the upper flange 11 is restricted, the lateral movement (movement in the x-axis direction) and rotation (rotation about the z-axis) at the center position (x = 0, y = 0) of the upper flange 11 are assumed to be completely restricted over the entire length in the z-axis direction. Bending moments about the x-axis act as external forces at both ends of the H-shaped cross-sectional member 10 in the z-axis direction. Let the bending moment at the position z = 0 be αM cr and the bending moment at the position z = l be βM. cr It is defined as such.

[0027] Since the lateral movement and rotation of the upper flange 11 of the H-shaped cross-sectional member 10 are restricted over the entire length of the H-shaped cross-sectional member 10, buckling deformation occurs on the lower flange 12 side. Therefore, in order to estimate the buckling strength, it is necessary to estimate the out-of-plane displacement u of the lower flange 12 due to buckling (the out-of-plane displacement in the thickness direction of the web 13 in the lower flange 12. See Figure 3). For the approximation of the out-of-plane displacement u, a partial sum (linear sum) of a Fourier series using a plurality of predetermined Fourier coefficients shown in equations (9) and (10) is used. In other words, equations (9) and (10) use a partial sum of a Fourier series with predetermined Fourier coefficients.

[0028]

Equation

[0029] Here, a1 to a M (M is a natural number) are undetermined coefficients, and b m-n (m = 1, …, M, n = 1, …, N. N is a natural number) are Fourier coefficients. That is, for example, each of equations (9-1), …, (9-2) included in equation (9) is a partial sum of a Fourier series up to n = 1, …, N using predetermined Fourier coefficients.

[0030]

Equation

[0031] Therefore, equation (9) is the partial sum of a Fourier series using two or more (a plurality of) predetermined Fourier coefficients. The same applies to equation (10). Note that the out-of-plane displacement u may be defined by one partial sum of a Fourier series using predetermined Fourier coefficients. When the ends of the H-shaped cross-sectional member 10 in the z-axis direction are constrained against rotation about the y-axis, the Fourier cosine series shown in equation (9) is used. When the ends of the H-shaped cross-sectional member 10 in the z-axis direction are not constrained against rotation about the y-axis, the Fourier sine series shown in equation (10) is used.

[0032] The constraint against rotation about the y-axis mentioned here will be described with reference to FIG. 4. In FIG. 4, the state when no external force is acting on the H-shaped cross-sectional member 10 is shown by a solid line. The central axis O1 is the central axis when no external force is acting on the H-shaped cross-sectional member 10. For example, the case where it is not constrained against rotation about the y-axis means that when an external force acts on the H-shaped cross-sectional member 10, as shown by the one-dot chain line in FIG. 4, in plan view, the extension line of the end face 12a of the lower flange 12 and the central axis O1 are not orthogonal. On the other hand, the case where it is constrained against rotation about the y-axis means that when an external force acts on the H-shaped cross-sectional member 10, as shown by the broken line in FIG. 4, in plan view, the extension line of the end face 11a of the upper flange 11 and the central axis O1 are orthogonal.

[0033] Note that in this embodiment, since the upper flange 11 is constrained by a floor slab or the like, regardless of the presence or absence of the constraint against rotation about the y-axis, when an external force acts on the H-shaped cross-sectional member 10, in plan view, the extension line of the end face 11a of the upper flange 11 and the central axis O1 are orthogonal.

[0034] Individual Fourier coefficient b m-nis obtained by performing a Fourier transform (Fourier sine transform or Fourier cosine transform) on a trigonometric function (function) with a powered phase that converges quickly when approximating buckling deformation but cannot be integrated, as in equations (11) to (14).

[0035]

Number

[0036] In equations (11) to (14), "z / l" and "1 - z / l" are values obtained by non-dimensionalizing the coordinate z in the z-axis direction of the H-shaped cross-sectional member 10 by the length l of the H-shaped cross-sectional member 10, and represent the phase in the trigonometric function used for the displacement function. Note that p in the trigonometric function with a powered phase takes any positive real number, and κ takes any real number greater than or equal to -p. For this reason, f m becomes a real number. For example, the right side of equations (11) to (14) becomes a "trigonometric function with a powered phase". When using such a trigonometric function with a powered phase for the displacement function, the degree of freedom in setting the shape of the displacement function increases compared to the case of using a function with a phase powered by a natural number exponent, and as a result, it becomes easier to accurately estimate the buckling deformation. Also, the Fourier coefficient b m-n obtained by the Fourier transform can evaluate the buckling strength with sufficient accuracy by considering up to about the 20th term.

[0037] Here, the case of selectively using equations (11) to (14) will be described with reference to FIG. 5. In FIG. 5, the central axis O1 indicated by the dashed line is the central axis when no external force is acting on the H-shaped cross-sectional member 10. For example, when rotation about the y-axis is not restricted at the end of the H-shaped cross-sectional member 10 in the z-axis direction and an external force acts, and the H-shaped cross-sectional member 10 deforms like the central axis O2, equations (11) and (12) using a sine wave are used. On the other hand, at the ends of the H-shaped cross-sectional member 10 in the z-axis direction, the rotation about the y-axis is restricted. When an external force acts and the H-shaped cross-sectional member 10 deforms like the central axis O3, the cosine wave is used, and equations (13) and (14) are used.

[0038] Here, for example, the shape of the out-of-plane displacement u represented by equations (13) and (14) will be described with reference to FIG. 6. Equation (14) is used when the out-of-plane displacement u becomes maximum between the position of z = 0 and the position of z = (l / 2), like the central axis O6 shown by the dashed line. Equation (13) is used when the out-of-plane displacement u becomes maximum between the position of z = (l / 2) and the position of z = l, like the central axis O7 shown by the one-dot chain line.

[0039] Next, a method for estimating the buckling resistance by the energy method will be shown. Here, first, a method for estimating the buckling resistance will be shown in the case where the cross-sectional shape of the H-shaped cross-sectional member 10, which is the object of estimating the buckling resistance, can be represented in a simpler notation and is uniform over the entire length in the z-axis direction (member length direction). The method for estimating the buckling resistance when the cross-sectional shape of the H-shaped cross-sectional member 10 is not uniform over the entire length in the z-axis direction will be described later. Due to the deformation accompanying the buckling of the H-shaped cross-sectional member 10, the strain energy ΔU stored inside the H-shaped cross-sectional member 10 and the potential energy ΔV of the external force are represented by equations (15) and (16), respectively.

[0040]

Equation

[0041] Note that the deformation of the flanges 11 and 12 is treated as a bar-like problem. The first term on the right side of equation (15) representing the strain energy ΔU is represented as the bending of the lower flange 12, and the second term is represented as the energy due to the torsion of the lower flange 12. On the other hand, the deformation of the web 13 is treated as a flat plate-like problem, and the third term on the right side of equation (15) represents the two-way bending and torsion of the web 13.

[0042] Here, u represents the out-of-plane displacement of the lower flange 12 and is given by the method for estimating the out-of-plane deformation due to buckling described above. Also, φ represents the twist angle of the lower flange 12 (see Fig. 3), w represents the out-of-plane displacement of the web 13, and they are expressed by equations (17) and (18) as functions of the out-of-plane displacement of the lower flange 12, respectively.

[0043]

Number

[0044] Other symbols mean d b : Distance between the plate thickness centers of the flanges 11 and 12, I f : Second moment of area of the flanges 11 and 12, J f : Saint-Venant torsion constant of the flanges 11 and 12, D w : Bending stiffness of the web 13, E: Young's modulus, G: Shear modulus, ν: Poisson's ratio, respectively. u’ represents the value obtained by differentiating u with respect to z once, and u’’ represents the value obtained by differentiating u with respect to z twice. φ’ represents the value obtained by differentiating φ with respect to z once. When lateral buckling occurs in the H-shaped cross-sectional member 10, the total potential energy Π of the H-shaped cross-sectional member 10 is given by equation (19).

[0045]

Number

[0046] Under the buckling resistance, since the strain energy and the potential energy of the external force are in a neutral equilibrium state, when the equation is rearranged with "Π = 0", equation (20) is obtained. Here, a part of the third term on the right side of equation (15) representing the strain energy ΔU, which is a small term, is set to 0 and expressed in a simplified form.

[0047]

Number

[0048] Here, each definite integral is defined as in equations (21) to (25).

[0049]

number

[0050] The constants A to K include the out-of-plane displacement u of the bottom flange 12, and the torsion angle φ of the bottom flange 12 and the out-of-plane displacement w of the web 13, which are calculated using the out-of-plane displacement u. For this reason, the constants A to K each include the undetermined coefficients a0 to a N is contained within.

[0051] By rearranging the formula using constants A to K, the buckling strength M cr can be expressed by equation (26).

[0052]

number

[0053] In the calculation process of the buckling strength, constants A to K are used to calculate the buckling strength M cr The undetermined coefficients a0 to a are real numbers that give the smallest positive value to N Based on this, the buckling strength M cr Ask for. Specifically, the total potential energy is calculated by dividing the undetermined coefficient a n By solving a simultaneous equation that shows that the function partially differentiated by is equal to 0, we can n Here, if the total number of undetermined coefficients is 4 or less, the equation will be of degree 4 or less, so by using the solution formula, the undetermined coefficient a n can be obtained. In addition, the undetermined coefficient a n If there are multiple pairs of buckling strength M cr The undetermined coefficient a that gives the smallest positive value to n Based on the set of cr Ask for. By the above method, the buckling strength M of the H-shaped section member 10 when the cross-sectional shape is uniform over the entire length can be calculated. cr can be estimated.

[0054] 〔2.2. When the cross-sectional shape and material are not uniform over the entire length (the width of the flange changes)〕 When the cross-sectional shape and material of the H-shaped cross-sectional member are not uniform over the entire length, the calculation method of strain energy is different. The H-shaped cross-sectional member 15 of the first modified example shown in FIGS. 7 and 8 has upper flanges 16 and lower flanges 17 instead of the upper flange 11 and the lower flange 12 in each configuration of the H-shaped cross-sectional member 10. The widths (lengths in the x-axis direction) of the flanges 16 and 17 are wide between z = 0 and l1 (hereinafter referred to as the base end side) and narrow between z = l1 and l (hereinafter referred to as the tip end side). The calculation method of strain energy in this case will be described.

[0055] Strain energy ΔU of the flange f is represented by the formula (27) when the widths of the flanges 11 and 12 are uniform, whereas it is represented by the formula (28) when the widths of the flanges 16 and 17 change.

[0056]

Equation

[0057] Here, I f1 and I f2 are the second moments of area of the flanges 16 and 17 on the base end side and the tip end side, respectively, J f1 and J f2 are the shear torsional constants of the flanges on the base end side and the tip end side, respectively, E1 and E2 are the Young's moduli on the base end side and the tip end side, respectively, and G1 and G2 are the shear elastic moduli on the base end side and the tip end side, respectively.

[0058] 〔2.3. When the cross-sectional shape and material are not uniform over the entire length (web defect)〕 The H-shaped cross-sectional member 20 of the second modified example shown in FIGS. 9 and 10 has a defect (through hole) 13a formed in the web 13 in each configuration of the H-shaped cross-sectional member 10. The defect 13a has a length (d at the center of the web 13 in the y-axis direction bIt is formed from the position of z = l1 to the position of z = l2 within the range of ( / 2). The calculation method of the strain energy in this case will be described.

[0059] The strain energy ΔU of the web w is represented by the formula (29) when there is no defect 13a in the web 13, while it is represented by the formula (30) when there is a defect 13a in the web 13.

[0060]

Equation

[0061] Thus, when the cross-sectional shape and material of the H-shaped cross-sectional member are not uniform over the entire length, the strain energy ΔU of these flanges 11, 12 (16, 17) f and the strain energy ΔU of the web 13 w are used to obtain the strain energy ΔU stored inside the H-shaped cross-sectional member from the formula (31).

[0062]

Equation

[0063] And by calculating using the above-described energy method, the buckling strength M cr can be estimated.

[0064] As described above, by dividing and handling the integration interval in the calculation of the strain energy, even when the cross-sectional shape of the H-shaped cross-sectional member whose buckling strength is to be evaluated is not uniform over the entire length, such as when the cross-sectional shape or material changes in the middle of the H-shaped cross-sectional member, or when there is a defect 13a in the flanges 11, 12 or the web 13, it is possible to estimate the buckling strength. Such a calculation is possible because the displacement function approximating the deformation due to buckling is a function that can be expressed in the form of an indefinite integral so that it can be integrated for any interval. In [2], the flanges 11, 12 (16, 17) were treated as rod-shaped elastic elements, and the web 13 was treated as a flat-plate-shaped elastic element. That is, the displacement estimation method according to the present invention can be applied to both rod-shaped elastic elements and flat-plate-shaped elastic elements.

[0065] 〔3. Verification of evaluation accuracy〕 〔3.1. Verification 1 of evaluation accuracy of the present embodiment〕 Next, verification is performed on the evaluation accuracy of the buckling resistance when the cross-sectional shape of the H-shaped cross-sectional member is not uniform over the entire length. The verification of the evaluation accuracy is carried out by comparing the eigenvalue analysis results using the finite element method and the buckling resistance estimation results by the above-described energy method.

[0066] The object to be verified is the H-shaped cross-sectional member shown in FIG. 11, and the basic cross-sectional dimensions are H-700x200x9x19. As boundary conditions, the rotations about the y-axis at both ends of the H-shaped cross-sectional member in the z-axis direction were restrained, and the transverse movement and rotation of the upper flange were restrained over the entire length. As load conditions, antisymmetric bending moments M were applied to both ends of the H-shaped cross-sectional member in the z-axis direction. That is, at the first end 10a of the H-shaped cross-sectional member, the displacements in the x-axis direction, y-axis direction, and z-axis direction were restrained (Ux = Uy = Uz = 0), and the rotations about the y-axis and z-axis were restrained (Ry = Rz = 0). At the second end 10b of the H-shaped cross-sectional member, the displacements in the x-axis direction and y-axis direction were restrained (Ux = Uy = 0), and the rotations about the y-axis and z-axis were restrained (Ry = Rz = 0). At the upper flange of the H-shaped cross-sectional member, the displacement in the x-axis direction was restrained (Ux = 0), and the rotation about the z-axis was restrained (Rz = 0).

[0067] Here, three types of analysis cases were examined. CASE1 is a basic H-shaped cross-sectional member that is uniformly used over the entire length of the cross-section. CASE2 is an H-shaped cross-sectional member in which the flange width is 250 mm in the range of 0.15l at both ends of the H-shaped cross-sectional member (the flange width is 200 mm outside this part). CASE3 is an H-shaped cross-sectional member with rectangular holes in the web over the entire length. Regarding the rectangular holes in CASE3, both the height and width are 350 mm, and they are continuously arranged in the z-axis direction such that the center-to-center distance of the rectangular holes is 1050 mm.

[0068] When estimating the buckling resistance by the energy method, the buckling deformation is estimated using Equation (32). Since it is assumed that the rotation about the y-axis at both ends of the H-shaped cross-sectional member is restricted, Equation (32) is the partial sum of four Fourier cosine series using the Fourier cosine series shown in Equation (9). The Fourier cosine coefficients in the four Fourier cosine series use the values up to the 20th term obtained by performing a Fourier cosine transform (Fourier transform) on the cosine function with the phase shown in Equation (33) raised to a power.

[0069]

Equation

[0070] Note that in Equation (33), p is a positive integer from 1 to 4, and κ is set to 1.5 regardless of the value of p.

[0071] Here, taking the case of l / H = 18 for CASE1 as an example, the estimation of the buckling deformation is shown. Note that H is the beam depth of the H-shaped cross-sectional member. Figure 12 shows the out-of-plane deformation amount of the lower flange that occurs as lateral buckling deformation in the case of l / H = 18 for CASE1. In Figure 12, the vertical axis shows the value obtained by non-dimensionalizing the out-of-plane deformation amount u by the maximum value u of the out-of-plane deformation amount, and the horizontal axis shows the value obtained by non-dimensionalizing the coordinate z in the material length direction by the length l of the H-shaped cross-sectional member. max The value is shown, and the horizontal axis shows the value obtained by non-dimensionalizing the coordinate z in the material length direction by the length l of the H-shaped cross-sectional member.

[0072] Regarding this buckling deformation, the results of estimating the buckling deformation using a cosine function with the phase shown in Equation (33) raised to a power are shown in FIGS. 13 and 14. FIG. 13 shows each displacement function used for estimating the buckling deformation, and FIG. 14 is a diagram showing the superposition of the buckling deformation of the lower flange obtained from the linear sum of the displacement functions and the analysis result of FEM. From FIGS. 13 and 14, it can be seen that by using a cosine function with the phase raised to a power as the displacement function, the buckling deformation can be accurately estimated by the linear sum of four functions.

[0073] Since these functions are difficult to integrate, they are converted into the form of a linear sum of Fourier cosine series shown in Equation (32). The Fourier cosine coefficients (Fourier coefficients) b m-n obtained by Fourier-transforming each displacement function are shown in Table 1.

[0074]

Table 1

[0075] In Table 1, for example, when m = 1, the Fourier coefficient b 1-1 when n = 1 is 0.35048, and the Fourier coefficient b 1-2 when n = 2 is 0.24562. By substituting the Fourier cosine coefficients shown in Table 1 into Equation (32) and calculating, the buckling deformation can be estimated in the form with four undetermined coefficients up to a1 to a4. As shown in Table 1, it can be seen that when representing a trigonometric function with the phase raised to a power by a Fourier series, sufficient accuracy can be obtained by considering the Fourier coefficients up to about the 20th term.

[0076] 〔3.2. Verification of Conventional Evaluation Accuracy〕 On the other hand, the results of using the generally used Fourier cosine series shown in Equation (34) for estimating the buckling deformation are shown in FIGS. 15 and 16.

[0077]

Equation

[0078] As can be seen from FIGS. 15 and 16, when using the Fourier cosine series for the displacement function, in order to achieve the same estimation accuracy as when using a cosine function with the phase raised to a power in the displacement function, it is necessary to use a linear sum of seven functions. Note that in this example, even when using the Fourier cosine function, the buckling deformation can be estimated with a relatively small number of terms, but the number of required displacement functions exceeds 4. Therefore, when estimating the buckling strength using the energy method, it cannot be calculated in an explicit form, and numerical calculations using a computer are required, which poses a problem.

[0079] 〔3.3. Verification of Evaluation Accuracy of this Embodiment 2〕 The estimated results of the buckling strength are shown in FIGS. 17 to 19. In FIGS. 17 to 19, the vertical axis represents the elastic lateral buckling strength (buckling strength) M cr , and the horizontal axis represents the value obtained by dividing the beam length by the beam section (= l / H). The results of the eigenvalue analysis (FEM) are plotted (open circles), and the estimated results of the buckling strength by the energy method are shown by solid lines, respectively. In each analysis result, it can be seen that for the region of l / H where the buckling strength is determined by lateral buckling and is large, the estimated strength using the energy method according to the present invention can accurately estimate the strength. Note that in the region where l / H is small, the buckling strength is determined by shear buckling, which is a final behavior different from lateral buckling, so it is difficult for the results of the eigenvalue analysis and the estimated results of the buckling strength by the energy method to match. Also, for the H-shaped cross-section member of CASE2 with wide flange widths at both ends or the H-shaped cross-section member of CASE3 with rectangular holes in the web over the entire length, the buckling strength can be accurately estimated.

[0080] So far, the object to be estimated for the buckling strength has been an H-shaped cross-section member, which is an elastic element assembly having a plurality of elastic elements formed in a flat plate shape. However, the elastic elements may be formed in a rod shape, and the object to be estimated for the buckling strength may be composed of one elastic element.

[0081] 〔4. Steel Structure Member Design Device〕 Fig. 20 shows a steel member design device 50 that can preferably perform the steel member design method of the present embodiment. The steel member design device 50 is a device that estimates the buckling strength when an external force acts on an H-shaped cross-section member and the H-shaped cross-section member buckles. The steel member design device 50 is a computer and includes a CPU (Central Processing Unit) 51, a main memory device 55, an auxiliary storage device 60, an input / output interface (IO·I / F) 65, and a recording / playback device 70. The CPU 51, the main memory device 55, the auxiliary storage device 60, the input / output interface 65, and the recording / playback device 70 are connected to each other by a bus 75. The main memory device 55 is a RAM (Random Access Memory) or the like that serves as a work area of the CPU 51. The input / output interface 65 is connected to an input device 66 such as a keyboard and a mouse, and a display device 67. The recording / playback device 70 records and plays back data with respect to a recording medium 71 such as a USB (Universal Serial Bus) memory.

[0082] The auxiliary storage device 60 is a hard disk drive device or the like that stores various data and programs. The auxiliary storage device 60 stores a steel member design program 61 for causing the computer to function as the steel member design device 50 and various programs such as an OS program. Various programs including the steel member design program 61 are taken into the auxiliary storage device 60 from the recording medium 71 via the recording / playback device 70. The steel member design program 61 and the like are stored in the recording medium 71. Note that these programs may be taken into the auxiliary storage device 60 from an external device via a disk-type recording medium such as a CD or a DVD or a communication device (not shown).

[0083] The CPU 51 executes various arithmetic processes. Functionally, the CPU 51 has a coefficient determination unit 51a, a displacement estimation unit 51b, a strength calculation unit 51c, and a cross-section determination unit 51d. The coefficient determination unit 51a obtains the Fourier coefficients b m-n (m = 1, …, M, n = 1, …, N), for example, by performing a Fourier transform on a function in which the phase from the expressions (11) to (14) is raised to a power. Here, f in the expressions (11) to (14) is manually or automatically adjusted according to the boundary conditions, load conditions, cross-sectional shape, and material changes of the H-shaped cross-sectional member. m-n (m = 1, …, M, n = 1, …, N). Here, f in the expressions (11) to (14) m is manually or automatically adjusted according to the boundary conditions, load conditions, cross-sectional shape, and material changes of the H-shaped cross-sectional member. The displacement estimation unit 51b estimates the out-of-plane displacement u due to buckling of the H-shaped cross-sectional member using the expression (9) or (10) (partial sum of Fourier series) with the Fourier coefficients b m-n obtained by the coefficient determination unit 51a. m-n using the expression (9) or (10) (partial sum of Fourier series) with the Fourier coefficients b m-n obtained by the coefficient determination unit 51a.

[0084] The buckling strength calculation unit 51c obtains the buckling strength of the H-shaped cross-sectional member that causes an out-of-plane displacement by using an expression such as (26) based on the energy method. Based on the buckling strength obtained by the buckling strength calculation unit 51c, the cross-section determination unit 51d determines the cross-sectional dimensions of the steel frame member having elastic elements so that buckling does not occur in the steel frame member, for example. Examples of the steel frame member include steel columns and beams.

[0085] Note that the coefficient determination unit 51a, the displacement estimation unit 51b, and the buckling strength calculation unit 51c constitute a buckling strength estimation device (estimation device) 50a. The coefficient determination unit 51a, the displacement estimation unit 51b, the buckling strength calculation unit 51c, and the cross-section determination unit 51d, which are functional components of the CPU 51, function when the CPU 51 executes a design program 61 of the steel frame member stored in the auxiliary storage device 60. The design program 61 of the steel frame member is a program for the steel frame member design device 50. The design program 61 of the steel frame member causes the steel frame member design device 50 to function as the coefficient determination unit 51a, the displacement estimation unit 51b, the buckling strength calculation unit 51c, and the cross-section determination unit 51d.

[0086] The design program 61 of the steel frame member has a buckling strength estimation program 61a. The buckling strength estimation program 61a is a program for the buckling strength estimation device 50a that estimates the buckling strength when an external force acts on the H-shaped cross-section member and the H-shaped cross-section member buckles. The buckling strength estimation program 61a causes the buckling strength estimation device 50a to function as a coefficient determination unit 51a, a displacement estimation unit 51b, and a strength calculation unit 51c.

[0087] [5. Design method of steel frame member] Next, the design method of the steel frame member of the present embodiment will be described. FIG. 21 is a flowchart showing the design method S1 of the steel frame member. First, the buckling strength estimation method S5 is performed. The buckling strength estimation method S5 is a method for estimating the buckling strength when an external force acts on the H-shaped cross-section member and the H-shaped cross-section member buckles. In the buckling strength estimation method S5, first, the process proceeds to step S6. In the coefficient determination step S6, for example, by performing a Fourier transform on a function in which the phases of equations (11) to (14) are raised to a power, the Fourier coefficient b m-n is obtained. The coefficient determination step S6 is a step performed before the displacement estimation step S7 described later. When the coefficient determination step S6 ends, the process proceeds to step S7.

[0088] Next, in the displacement estimation step S7, the out-of-plane displacement u due to the buckling of the H-shaped cross-section member is estimated by equation (9) or (10) using the Fourier coefficient b m-n obtained in the coefficient determination step S6. When the displacement estimation step S7 ends, the process proceeds to step S8. Next, in the strength calculation step S8, the buckling strength of the H-shaped cross-section member that causes an out-of-plane displacement is obtained by equation (26) or the like based on the energy method. The number of undetermined coefficients a1 to a M in equation (9) or (10) (partial sum of Fourier series) is preferably 4 or less. In other words, the natural number M is preferably 4 or less. When the strength calculation step S8 ends, all steps of the buckling strength estimation method S5 end, and the buckling strength of the H-shaped cross-section member is estimated. When the buckling strength estimation method S5 ends, the process proceeds to step S10.

[0089] Next, in the cross-section determination step S10, based on the buckling strength obtained by the buckling strength estimation method S5, the cross-sectional dimensions of the steel frame member having elastic elements are determined. When the cross-section determination step S10 is completed, all steps of the steel frame member design method S1 are completed, and a steel frame member having a desired cross-sectional dimension is designed. Note that the steel frame member design method S1 may be performed by a human.

[0090] 〔6. Effects of this Embodiment〕 As described above, in the buckling strength estimation method S5 of this embodiment, a Fourier series is used to estimate the out-of-plane displacement due to the buckling of the H-shaped cross-section member. Therefore, in the displacement estimation step S7, the out-of-plane displacement can be accurately estimated, and in the strength calculation step S8, the buckling strength can be accurately obtained even when the cross-sectional shape and material of the H-shaped cross-section member are not uniform over the entire length. Further, the Fourier series used to estimate the out-of-plane displacement is a partial sum of a Fourier series whose Fourier coefficients are determined in advance. Therefore, the number of undetermined coefficients is significantly smaller than when obtaining Fourier coefficients when obtaining the buckling strength, the out-of-plane displacement can be easily estimated in the displacement estimation step S7, and the buckling strength can be easily obtained in the strength calculation step S8.

[0091] The buckling strength estimation method S5 includes a coefficient determination step S6 before the displacement estimation step S7. For example, compared with the case of performing a Fourier transform on a function in which the phase is raised to a power with a natural number as an exponent, in the displacement estimation step S7, the number of terms of the Fourier series required to accurately estimate the out-of-plane displacement can be reduced. The number of undetermined coefficients of the formula (9) or (10) is 4 or less. In this case, the 4 or less undetermined coefficients of the formula (9) or (10) can be easily calculated by the solution formula in a positive form without using a convergence calculation.

[0092] The design method S1 of the steel frame member includes a cross-section determination step S10. Therefore, in the cross-section determination step S10, the cross-sectional dimensions of the steel frame member can be determined so that the steel frame member having elastic elements does not buckle.

[0093] Also, in the buckling strength estimation program 61a of the present embodiment, a Fourier series is used to estimate the out-of-plane displacement due to the buckling of the H-shaped cross-section member. Therefore, the out-of-plane displacement can be accurately estimated by the displacement estimation unit 51b, and the buckling strength can be accurately obtained by the strength calculation unit 51c even when the cross-sectional shape and material of the H-shaped cross-section member are not uniform over the entire length. Further, the Fourier series used to estimate the out-of-plane displacement is a partial sum of a Fourier series whose Fourier coefficients are determined in advance. Therefore, the number of undetermined coefficients is significantly smaller compared to the case where Fourier coefficients are obtained when obtaining the buckling strength, and the out-of-plane displacement can be easily estimated by the displacement estimation unit 51b, and the buckling strength can be easily obtained by the strength calculation unit 51c.

[0094] As described above, although one embodiment of the present invention has been described in detail with reference to the drawings, the specific configuration is not limited to this embodiment, and modifications, combinations, deletions, etc. of the configuration within the scope not departing from the gist of the present invention are also included. Needless to say, the configurations shown in each embodiment can be used in appropriate combinations. For example, in the above embodiment, in the buckling strength estimation method S5, the coefficient determination step S6 may not be performed. In this case, the CPU 51 of the steel frame member design device 50 does not have the coefficient determination unit 51a, and the steel frame member design program 61 does not cause the steel frame member design device 50 to function as the coefficient determination unit 51a.

[0095] Although the effects of the present invention have been shown by taking the lateral buckling of the H-shaped cross-section member as an example, the present invention can be similarly applied to the local buckling of the flange and web plate elements constituting the H-shaped cross-section member and the buckling problems of members having different cross-sectional shapes such as square steel pipes.

Description of Reference Numerals

[0096] 50a Buckling strength estimation device (estimation device) 51b Displacement estimation unit 51c Stiffness calculation unit 61a Buckling strength estimation program S1 Steel member design method S5 Buckling strength estimation method S6 Coefficient determination process S7 Displacement estimation process S8 Stiffness calculation process S10 Cross-section determination process

Claims

1. A buckling strength estimation method for estimating the buckling strength when an external force acts on an elastic element formed in a rod shape or a flat plate shape and the elastic element buckles, comprising: a displacement estimation step of estimating an out-of-plane displacement due to buckling of the elastic element by a partial sum of a Fourier series using predetermined Fourier coefficients; a strength calculation step of obtaining the buckling strength of the elastic element that causes the out-of-plane displacement based on the energy method; A buckling strength estimation method comprising the above steps.

2. The buckling strength estimation method according to claim 1, further comprising a coefficient determination step of obtaining the Fourier coefficients by Fourier-transforming a function in which the phase is raised to a power before the displacement estimation step.

3. The buckling strength estimation method according to claim 1 or 2, wherein the number of undetermined coefficients of the partial sum of the Fourier series is 4 or less.

4. A steel frame member design method comprising a cross-section determination step of determining the cross-sectional dimensions of a steel frame member having the elastic element based on the buckling strength obtained by the buckling strength estimation method according to claim 1 or 2.

5. A buckling strength estimation program for an estimation device that estimates the buckling strength when an external force acts on an elastic element formed in a rod shape or a flat plate shape and the elastic element buckles, causing the estimation device to function as a displacement estimation unit that estimates an out-of-plane displacement due to buckling of the elastic element by a partial sum of a Fourier series using predetermined Fourier coefficients; a strength calculation unit that obtains the buckling strength of the elastic element that causes the out-of-plane displacement based on the energy method; A buckling strength estimation program.

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