JP2021006786A - Apparatus, method and program for estimating buckling stress intensity - Google Patents

Abstract

To provide an apparatus for estimating buckling stress intensity which allows for simpler calculation necessary for estimating buckling stress intensity compared with a case using Fourier series, and can formulate estimated results so as to be physically understood.SOLUTION: A buckling stress intensity estimation apparatus 50 for estimating buckling stress intensity of a plate element which buckles with bending moment applied around a z-axis extending in a plate thickness direction and spreads along an x-axis extending in a material axis direction and a y-axis extending in a plate width direction includes: a displacement estimation unit 52 which estimates out-of-plane displacement of the plate element facing a direction along the z-axis when a half wavelength in a direction along the x-axis of the plate element which alternately displaces in a wavelike fashion in the direction along the z-axis as it goes toward a first end of the plate element in the direction along the x-axis is defined as (a); and a stress intensity calculation unit 53 which determines buckling stress intensity of the plate element which generates out-of-plane displacement based on energy approach.SELECTED DRAWING: Figure 2

Description

The present invention relates to a buckling stress estimation device, a buckling stress estimation method, and a buckling stress estimation program.

Conventionally, the buckling stress degree (local buckling stress degree) of a plate element that is bent in a plane by an action of a bending moment has been estimated. In this buckling stress estimation method, it is common to use the Fourier series expressed by Eq. (1). When estimating the out-of-plane displacement W _w of the plate element, Eq. (1) is used as (1). -1) The buckling stress degree is estimated after changing to the expression of Eq. 1) (see, for example, Non-Patent Document 1).
However, b _w is the length in the direction orthogonal to the thickness direction of the plate element.

Then, the buckling stress degree is estimated by applying the energy method to this Fourier series. Since the basis of the Fourier series has orthogonality, more curved surfaces can be accurately approximated by the Fourier series.

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

However, the Fourier series has a problem that the convergence of the approximation is slow. For example, when trying to approximate the buckling deformation that occurs when a bending moment acts on a member of a building or a plate element that constitutes the member, the 25th partial sum of the Fourier series, that is, the equation (1-1) It is necessary to set N to about 25. In this case, the approximation function 50 includes a term by unknown coefficients a ₀ non-trigonometric functions.
As described above, since the Fourier class requires a large number of terms for the convergence of the approximation, the analysis by the Fourier class cannot be performed by manual calculation, and numerical calculation using a computer is required. Further, the conventional buckling stress degree estimation method has a technical problem that the estimation result cannot be easily understood and formulated.

The present invention has been made in view of such a problem, and the calculation necessary for estimating the buckling stress degree of the plate element on which the bending moment acts is easier than when the Fourier class is used. It is an object of the present invention to provide a buckling stress estimation device, a buckling stress estimation method, and a buckling stress estimation program that can formulate the estimation results in an easy-to-understand manner.

In order to solve the above problems, the present invention proposes the following means.
The buckling stress estimation device of the present invention spreads along the x-axis extending in the material axis direction and the y-axis extending in the plate width direction, respectively, and a bending moment acts on the z-axis extending in the plate thickness direction to buckle. A buckling stress estimation device for estimating the buckling stress degree of a plate element, in which the z-axis is positively oriented and the z-axis is oriented toward the first end of the plate element in a direction along the x-axis. The out-of-plane displacement W _{w of the} plate element in the direction along the z-axis, where a is the half wavelength in the direction along the x-axis of the plate element that is alternately wavy in the negative direction of A displacement estimation unit that estimates the above equation (2), and a stress degree calculation unit that obtains the buckling stress degree of the plate element that produces the out-of-plane displacement W _w based on the equation (2) based on the energy method. It is characterized by having.
However, N is a natural number of 2 or _{more, a 0, a n, b} n are unknown coefficients, b _w is the length of the plate elements in the direction along the y-axis.

Further, in the method for estimating the buckling stress degree of the present invention, the bending moment acts around the x-axis extending in the material axis direction and the y-axis extending in the plate width direction, respectively, and around the z-axis extending in the plate thickness direction. A method for estimating the degree of buckling stress of a plate element that buckles, in which the z-axis is positively oriented and the z-axis is oriented toward the first end side of the plate element in the direction along the x-axis. When the half wavelength of the plate element that is alternately wavy in the negative direction of the z-axis in the direction along the x-axis is a, the plate element is out of the plane in the direction along the z-axis. The displacement estimation process for estimating the displacement W _w by the equation (3) and the buckling stress degree of the plate element that produces the out-of-plane displacement W _w based on the equation (3) are obtained based on the energy method. It is characterized by performing a calculation process.
However, N is a natural number of 2 or _{more, a 0, a n, b} n are unknown coefficients, b _w is the length of the plate elements in the direction along the y-axis.

Further, the buckling stress estimation program of the present invention spreads along the x-axis extending in the material axis direction and the y-axis extending in the plate width direction, respectively, and a bending moment acts around the z-axis extending in the plate thickness direction. A buckling stress estimation program for an estimation device that estimates the buckling stress of a buckling plate element, in the z-axis direction toward the first end of the plate element in the direction along the x-axis. When the half wavelength of the plate element that is alternately wavy in the positive direction and the negative direction in the z-axis direction in the direction along the x-axis is a, the estimation device is set along the z-axis. A displacement estimation unit that estimates the out-of-plane displacement W _w of the plate element in the direction by equation (4), and the buckling stress degree of the plate element that produces the out-of-plane displacement W _w based on equation (4). Is characterized by functioning as a stress degree calculation unit obtained based on the energy method.
However, N is a natural number of 2 or _{more, a 0, a n, b} n are unknown coefficients, b _w is the length of the plate elements in the direction along the y-axis.

The inventors have investigated multiple functions that can estimate the out-of-plane displacement of the plate element on which the bending moment acts, with a number of terms smaller than the Fourier series, while using trigonometric functions. As a result, it was found that the out-of-plane displacement W _w of the plate element can be estimated with a number of terms smaller than the Fourier series using the equation (2). Equations (3) and (4) are the same as equations (2).
Since the equation (2) can estimate the out-of-plane displacement W _w of the plate element with a number of terms smaller than the Fourier series, the calculation required for the estimation can be performed more easily than when the Fourier series is used. Further, when the number of terms required to estimate the out-of-plane displacement W _w with a certain accuracy is estimated by using the equation (2) rather than by using the Fourier series of the equation (1-1). Therefore, the estimation result can be physically understood and formulated as compared with the case of using the Fourier series.

Further, in the buckling stress estimation device, the buckling stress estimation method, and the buckling stress estimation program, N may be 2.
According to these inventions, in equation (2), the undetermined coefficients a ₁ , a ₂ , b ₁ and b ₂ are first obtained in the state where the half wavelength a is treated as a constant, and then the half wavelength a is treated as a variable. , The half wavelength a in the equation (2) is obtained. (2) when determining the unknown coefficients a _1, etc. In the equation, since the number of undetermined coefficients asking have four or less time, pending with the fourth order following equation formulas known solutions coefficients a ₁ , A ₂ , b ₁ , and b _2, etc. can be easily obtained.

Further, in the buckling stress estimation device, the stress calculation unit uses equations (5) and (6) to obtain the minimum positive value for the buckling stress σ _{cr according} to equation (7). wherein a _n is a real number giving the _value, based on b _n and the half-wave a, may be determined the buckling stress of sigma _cr.
However, E is the Young's modulus of the plate element, ν is the Poisson's ratio of the plate element, and t _w is the thickness of the plate element.

Further, in the method for estimating the buckling stress degree, in the stress degree calculation step, the equations (8) and (9) are used, and the buckling stress degree σ _{cr according} to the equation (10) is the smallest positive. wherein a _n is a real number giving the _value, based on b _n and the half-wave a, may be determined the buckling stress of sigma _cr.
However, E is the Young's modulus of the plate element, ν is the Poisson's ratio of the plate element, and t _w is the thickness of the plate element.

Further, in the buckling stress estimation program, the stress calculation unit uses equations (11) and (12) to obtain the minimum positive value for the buckling stress σ _{cr according} to equation (13). wherein a _n is a real number giving the _value, based on b _n and the half-wave a, may be determined the buckling stress of sigma _cr.
However, E is the Young's modulus of the plate element, ν is the Poisson's ratio of the plate element, and t _w is the thickness of the plate element.

According to these aspects of the invention, when determining the buckling stress of sigma _cr based on the energy method, it is possible to accurately determine the buckling stress of sigma _cr using Equation.

According to the buckling stress estimation device, the buckling stress estimation method, and the buckling stress estimation program of the present invention, the buckling stress of the plate element on which the bending moment acts as compared with the case of using the Fourier class. The calculation required for estimating the degree can be easily performed, and the estimation result can be formulated in an easy-to-understand manner.

It is a perspective view of the building provided with the web to which the buckling stress degree estimation device of one Embodiment of this invention is applied. It is a figure which shows the outline of the device for estimating the buckling stress degree. It is a perspective view which shows typically the state in which the web on which a bending moment acted is buckling when the web is sufficiently long in the direction along the x-axis. FIG. 3 is an enlarged perspective view of a half wavelength in the direction along the x-axis in the web of FIG. It is a figure which shows an example of the estimation result of the out-of-plane displacement of the web by the coordinates of the y-axis when the bending moment acts on the web. It is a figure which shows an example of the change of the buckling coefficient with respect to N when a bending moment acts on a web.

Hereinafter, an embodiment of the buckling stress estimation device according to the present invention will be described with reference to FIGS. 1 to 6.
This buckling stress estimation device (estimator, hereinafter simply referred to as an estimation device) is a seat of a web (plate element) 13 of H-shaped steel 10 used as a steel beam in a building 1 shown in FIG. 1, for example. It is used to estimate the degree of buckling stress. The H-section steel 10 includes an upper flange (first flange) 11, a lower flange (second flange) 12, and a web 13 that connects the upper flange 11 and the lower flange 12 to each other. In FIG. 1, the floor slab 20 described later is shown by a chain double-dashed line.
The upper flange 11 and the lower flange 12 of the H-shaped steel 10 are also plate elements. The plate element is an elastic element. The elastic element is an element that does not consider the material non-linearity.

The H-section steel 10 extends in a direction along a horizontal plane, for example. The upper flange 11 is formed in a flat plate shape, and is arranged so that the thickness direction of the upper flange 11 is along the vertical direction. The lower flange 12 is formed in a flat plate shape and is arranged below the upper flange 11. The lower flange 12 is arranged so that the thickness direction of the lower flange 12 is along the vertical direction.
The web 13 is formed in a flat plate shape having a rectangular shape when viewed in the thickness direction of the web 13. The web 13 is arranged so that the thickness direction of the web 13 is along the horizontal plane. The web 13 connects the center in the width direction on the lower surface of the upper flange 11 and the center in the width direction on the upper surface of the lower flange 12.

The end portion of the H-shaped steel 10 in the longitudinal direction is fixed to a column 15 or the like. The H-section steel 10 supports the floor slab 20 from below the floor slab 20. The upper flange 11 of the H-shaped steel 10 is provided with a shear connector 21 such as a headed stud. The shear connector 21 is embedded in the floor slab 20.
The building 1 is used by installing equipment (not shown) on the floor slab 20.

FIG. 2 shows the estimation device 50 of this embodiment. The estimation device 50 is a computer, and includes a CPU (Central Processing Unit) 51, a main storage device 55, an auxiliary storage device 60, an input / output interface (IO / I / F) 65, and a recording / playback device 70. I have. The CPU 51, the main storage 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 storage device 55 is a RAM (Random Access Memory) or the like that serves as a work area or the like 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 / reproducing device 70 records and reproduces data on a recording medium 71 such as a USB (Universal Serial Bus) memory.

The auxiliary storage device 60 is a hard disk drive device or the like that stores various data, programs, and the like. The auxiliary storage device 60 stores a buckling stress estimation program (hereinafter, simply referred to as an estimation program) 61 for causing the computer to function as the estimation device 50, various programs such as an OS program, and the like. Various programs including the estimation program 61 are taken into the auxiliary storage device 60 from the recording medium 71 via the recording / playback device 70. The estimation program 61 and the like are stored in the recording medium 71.
These programs may be taken into the auxiliary storage device 60 from an external device via a disc-type recording medium such as a CD or DVD or a communication device (not shown).

The CPU 51 executes various arithmetic processes. The CPU 51 functionally comprises a displacement estimation unit 52 that estimates the out-of-plane displacement of the web 13, and a stress degree calculation unit 53 that obtains the buckling stress degree of the web 13 that causes the out-of-plane displacement based on the energy method. I have. The displacement estimation unit 52 and the stress degree calculation unit 53, which are functional components of the CPU 51, function when the CPU 51 executes an estimation program 61 or the like stored in the auxiliary storage device 60. The estimation program 61 and the like are programs for the estimation device 50. The estimation program 61 causes the estimation device 50 to function as the displacement estimation unit 52 and the stress degree calculation unit 53.

In the displacement estimation unit 52 and the stress degree calculation unit 53 of the estimation device 50 of the present embodiment, as shown in FIG. 3, the position coordinates of the web 13 are orthogonal to the right-handed system composed of the x-axis, the y-axis, and the z-axis. Recognize based on the coordinate system. In addition, in FIG. 3 and FIG. 5 described later, the out-of-plane displacement of the web 13 is shown larger than the estimated value.
It is assumed that the web 13 spreads along the x-axis extending in the material axis direction of the web 13 and the y-axis extending in the plate width direction of the web 13. Of the x-axis and the y-axis, for example, the x-axis extends along the horizontal plane and the y-axis extends along the vertical direction. The axis extending in the plate thickness direction of the web 13 is defined as the z-axis. The x-axis, y-axis, and z-axis are orthogonal to each other. When viewed in the direction along the z-axis (hereinafter referred to as the z-axis direction), the web 13 has a side extending in the direction along the x-axis (hereinafter referred to as the x-axis direction) and a direction along the y-axis (hereinafter referred to as y). Each has a side extending in the axial direction). It is assumed that the web 13 is sufficiently long in the x-axis direction. The fact that the web 13 is sufficiently long in the x-axis direction here means that the boundary condition of the surface 13a arranged at each end of the web 13 in the x-axis direction and extending in the y-axis direction (hereinafter referred to as the end face in the x-axis direction) This means that the web 13 has a length that can ignore the influence on the buckling deformation.
The out-of-plane displacement of the web 13 is a displacement of the web 13 in the z-axis direction.

When the bending moment F1 around the z-axis acts on the end faces 13a of the web 13 in the x-axis direction, the web 13 may buckle (local buckling). The bending moment F1 is transmitted over the entire length of the web 13 in the x-axis direction. The transmitted bending moment F1 acts around an axis parallel to the z-axis (hereinafter, referred to as a rotation axis of the bending moment F1) through the neutral axis of the web 13. The neutral axis is generated at a position in the y-axis direction of the web 13 in which the tensile force and the compressive force acting on the web 13 due to the bending moment F1 are balanced.
The bending moments F1 acting on each end surface 13a of the web 13 in the x-axis direction are external forces of equal magnitude. In this example, a tensile stress acts on the lower end of the web 13 and a compressive stress acts on the upper end of the web 13, and a bending moment F1 acts on the web 13 so that the web 13 bends downward.

In this case, the web 13 is displaced alternately in the positive direction of the z-axis and the negative direction of the z-axis toward the first end (one of the end faces 13a in the x-axis direction) of the web 13. As a whole, it may be displaced in a wavy shape for a plurality of wavelengths (hereinafter, referred to as a wavy shape in the x-axis direction).
For one wavelength of the web 13 displaced along the x-axis, the second end opposite to the first end in the x-axis direction is set as the origin of the x-axis, and the second end toward the first end in the x-axis direction. Let the orientation be the positive orientation of the x-axis.
The origin of the y-axis is the lower end (first end) of the web 13 in the y-axis direction. The positive direction of the y-axis is upward (direction from the first end to the second end).
The origin of the z-axis is the center of the web 13 in the z-axis direction (center in the thickness direction). The positive direction of the z-axis is the direction that constitutes the Cartesian coordinate system of the right-handed system with respect to the positive direction of the x-axis and the positive direction of the y-axis.

As the unit such as the length described below, an SI unit such as "m" is preferably used for the length. Let b _w be the length of the web 13 in the y-axis direction, and let t _w be the thickness of the web 13 (the length in the z-axis direction). Let E be the Young's modulus of Web 13 and ν be the Poisson's ratio of Web 13.
The estimation device 50 estimates the degree of buckling stress of the web 13 when the bending moment F1 acts on the web 13 to which the flanges 11 and 12 are not connected to buckle. The estimation device 50 is used when the thickness of the flanges 11 and 12 is negligibly thin and the rigidity of the flanges 11 and 12 is negligibly low in the H-shaped steel 10. The degree of bending stress may be estimated as the degree of buckling stress of the web 13.

The displacement estimation unit 52 and the stress degree calculation unit 53 of the estimation device 50 of the present embodiment make the following assumptions 1 to 6 when estimating the buckling stress degree of the web 13.
1. 1. The thickness of the web 13 is thin, and the thickness of the web 13 is shorter than the length in the x-axis direction and the length in the y-axis direction of the web 13.
2. 2. The deflection (out-of-plane displacement due to buckling) of the web 13 is small, which is smaller than the thickness of the web 13.
3. 3. The central surface of the web 13 in the thickness direction maintains a neutral surface without expanding or contracting due to bending of the web 13.
4. In the cross section of the web 13, the assumption of plane holding is established for bending.
5. The material of the web 13 is homogeneous and isotropic.
6. The displacement when an external force acts on the web 13 follows Hooke's law.

As shown in FIG. 3, when the bending moment F1 acts on the web 13, the web 13 may be displaced in a wavy shape in the x-axis direction. Assuming that the out-of-plane displacement in the x-axis direction of the web 13 wavy in the x-axis direction is expressed by the equation of sin (πx / a), the x of the web 13 wavy in the x-axis direction of the web 13 is assumed. The axial wavelength is 2a. The half wavelength in the x-axis direction (half the length of the wavelength) is a.
FIG. 4 is a diagram showing the out-of-plane displacement of the portion of the web 13 in the central portion in the x-axis direction in which the length in the x-axis direction is half a wavelength a.

When the bending moment F1 acts on the web 13, conventionally, the Fourier series expressed by Eq. (1-1) is used to direct the web 13 in the z-axis direction at a certain coordinate on the x-axis and a certain coordinate on the y-axis. It was estimated out-of-plane displacement _w w of the web 13.
The inventors have used trigonometric functions, but with a number of terms smaller than the Fourier series, the out-of-plane displacement of the web 13 in the z-axis direction at a coordinate on the x-axis and a coordinate on the y-axis (first). a function that can be estimated out-of-plane displacement) _w w was more study. The out-of-plane displacement w _w is a function of the y-axis coordinate, not a function of the x-axis coordinate (it is a function at a certain coordinate on the x-axis).
Consequently, if the moment F1 bending the web 13 is applied, out-of-plane displacement w _w web 13 toward the z-axis direction in a top coordinate and y-axis is on the x-axis coordinates by equation (16), Fourier We found that it is estimated with fewer terms than the series. However, N is the 2 or more is a natural _{number, a 0, a n, b} n are undetermined coefficients.
Equation (16) includes a term by trigonometric function using a power function of y-axis coordinates. cos (2πy ⁿ / b _w ⁿ ) and sin (π y ⁿ / b _w ⁿ ) are the basis. Equation (16) can be preferably used for estimating the out-of-plane deformation of a plate element such as a web 13.

On the other hand, the out-of-plane displacement (second out-of-plane displacement) W _w of the web 13 in the z-axis direction at an arbitrary coordinate on the x-axis and a certain coordinate on the y-axis is estimated by the equation (18). Equation (18) is based on the assumption that, as described above, the out-of-plane displacement of the web 13 at a certain coordinate on the x-axis in the z-axis direction is represented by the equation of sin (πx / a). The out-of-plane displacement W _w is a function of each of the y-axis coordinate and the x-axis coordinate, and is used when estimating the buckling stress degree of the web 13. Incidentally, out-of-plane displacement W _w of FIG. 3 is a repeat of plane displacement W _w in FIG. 4 in the x-axis direction. Since estimating the buckling stress degree of the web 13 of FIG. 3 and estimating the buckling stress degree of the web 13 of FIG. 4 are synonymous, the above equation (18) is the out-of-plane displacement W _w of FIG. Is estimated.
By substituting the equation (16) into the equation (18), the equation (19) is obtained. When N is 2 in the equation (19), it can be transformed as in the equation (20).

For example, before the bending moment F1 acts on the web 13 of FIG. 4, coordinates the coordinates of the x-axis is x _0, y-axis in the web 13 is part of y ₀ of the z-axis (hereinafter, referred to as the estimated target portion) The coordinates are 0. After the bending moment F1 acts on the web 13, the out-of-plane displacement W _w in this estimation target portion is estimated by the equation (19-1), and the z-axis coordinate at this time becomes 0 (19-1). It is the value obtained by adding the out-of-plane displacement W _w estimated by the formula. That is, after the bending moment F1 acts, it is estimated that the estimation target portion is arranged at a position where the x-axis coordinate is x ₀ , the y-axis coordinate is y ₀ , and the z-axis coordinate is W _w. To.

The displacement estimation unit 52 estimates the out-of-plane displacement W _w of the web 13 by the equation (19).

The web 13 is arranged in a closed section where the coordinates of the x-axis are 0 or more and a or less, and the coordinates of the y-axis are 0 or more and b _w or less. At this time, the strain energy U generated in the web 13 due to the buckling deformation is expressed by the equation (21), and the external force potential V given by the bending moment F1 is expressed by the equation (22).
The total potential energy Π is expressed by Eq. (23) as the sum of the strain energy U and the external force potential V.

Here, D _w is the plate rigidity of the web 13, and is expressed by the equation (25). σ _w is a stress function of the web 13 on which the bending moment F1 acts, and is expressed by Eq. (26). However, σ _cr is the degree of buckling stress of the web 13, and the compression is positive. The function δ _w is a function expressing the deformation of the web 13 in the x-axis direction when buckling occurs, and is expressed by Eq. (27). However, δ is the displacement in the x-axis direction at the second end (upper end) in the y-axis direction on the second end in the x-axis direction of the web 13.
For _{^{example, (∂ 2 W w / ∂x}} 2) means a function that the _{W w} and second order partial derivative by _{^{x, (∂ 2 W w /}} (∂x∂y)) respectively the _{W w} x, a y one It means a function that is partially differentiated.

The stress degree calculation unit 53 obtains the buckling stress degree σ _cr of the web 13 that produces the out-of-plane displacement W _w based on the equation (19) based on the energy method. The stress degree calculation unit 53 gives the buckling stress degree σ _cr of the web 13 a minimum positive value to the buckling stress degree σ _{cr according} to the equation (32) by using the equations (30) and (31). The buckling stress degree σ _cr is obtained based on the real numbers an _n , b _n and the half wavelength a.
However, the buckling coefficient k is obtained by the equation (33).

Specifically, in a state where the half wavelength a is treated as a constant, the equations indicating that the function obtained by partially differentiating the total potential energy Π with the undetermined coefficients an _n and b _n is equal to 0 are combined to form a real number. Find an _n and b _n . A _n which is a solution of the simultaneous _equations, when the set of b _n have more than one of the plurality of sets of a _n, b _n, a minimum positive value (32) buckling stress degree of expression sigma _cr Request _a n, based on a set of _{b n (a} n, a set of _{b n} (32) are substituted into equation) buckling stress of _{sigma cr} give. Then, treat the half-wave a as a variable, the a _n, the total potential energy Π a set has been obtained in b _n from the equation representing the function obtained by partially differentiating a half wave a is equal to 0, a half-wave a Ask. Wherein a _n obtained as described _above, b _n set and the half-wave a buckling stress of sigma _cr determined based on becomes the buckling stress of sigma _cr seeking.
When there is only one set of undecided coefficients an _n and b _n that is the solution of the simultaneous equations, the set of an _n and b _n gives the minimum positive value to the buckling stress degree σ _{cr according} to equation (32). in _granting, seeking a n, _{b n} buckling stress of _{sigma cr} based on a set of. Next, the half wavelength a is treated as a variable, and the buckling stress degree σ _cr is obtained as described above.

Incidentally, the method of estimating the buckling stress of the present embodiment (hereinafter, simply referred to as estimation method), the displacement estimation step of estimating the out-of-plane displacement W _w of the web 13 (19), out-of-plane displacement W _w The buckling stress degree σ _cr of the web 13 that causes the above is obtained by performing a stress degree calculation step based on the energy method.
The bending moment F1 acting on the web 13 when the buckling stress degree σ _cr occurs on the web 13 can be obtained from a known equation regarding the plate element. By making the bending moment F1 acting on the web 13 smaller than this bending moment F1, it is possible to suppress the web 13 from buckling.

[Evaluation of estimation accuracy of out-of-plane displacement and buckling stress]
(Comparison by out-of-plane displacement)
FIG. 5 shows the estimation result of the out-of-plane displacement W _w of the web 13 based on the y-axis coordinates when the bending moment F1 acts on the web 13. It is assumed that the surfaces 13b arranged at each end of the web 13 in the y-axis direction and extending in the x-axis direction (hereinafter referred to as end faces in the y-axis direction) 13b are fixedly supported (non-rotatably supported and rigidly joined). This out-of-plane displacement W _w is an out-of-plane displacement in the cross section of the cutting line AA in FIG.
In FIG. 5, the horizontal axis represents the coordinates of the y-axis with respect to the length b _w of the web 13, and the vertical axis represents the estimated out-of-plane displacement W _w of the web 13. 0 on the horizontal axis represents the lower end of the web 13, and 1 on the horizontal axis represents the upper end of the web 13. In the vertical axis, the value of an out-of-plane displacement W _w is dimensionless by the maximum value of an out-of-plane displacement W _w.

In FIG. 5, the solid line L1 represents the analysis result by the FEM (Finite Element Method). The dotted line L2 represents the result estimated using the equation (19) of the present embodiment (where N is 2), and the alternate long and short dash line L3 represents the result estimated using the equation (1-1) (where N is). 15) is represented. Since the line L2 and the line L3 substantially coincide with the line L1, the line L2 and the line L3 overlap the line L1.
N in Eq. (19) of the present embodiment, which is necessary to approximate the analysis result by FEM with the same accuracy as in the case where N is 15 in the Fourier series of Eq. (1-1), is 2. I understood.

(Comparison by buckling stress)
The buckling stress degree σ _cr of the web 13 was obtained by the formula (1-1) and the formula (19) of the present embodiment, and the results of comparison by the buckling coefficient are shown in Table 1 and FIG.

When the equation (19) of this embodiment was used, N was set to 2. On the other hand, when the Fourier series of Eq. (1-1), which is a comparative example, was used, N was changed to 4, 5, 7, ..., 25.
In Table 1, the buckling coefficient when the end surface 13b in the y-axis direction of the web 13 is simply supported converges to 23.88 as the estimation accuracy of the out-of-plane displacement W _w and the buckling stress degree σ _cr increases. (See line L11 in FIG. 6). Seat屈係number of cases in the y-axis direction of the end face 13b is fixedly supported web 13, in accordance with the out-of-plane displacement W _w and buckling stress of sigma _cr estimation accuracy is high, converges to 39.6 (Fig. 6 See line L12 inside).

In FIG. 6, the horizontal axis represents N and the vertical axis represents the buckling coefficient k. The white circles represent the estimation results in the case of simple support using the equation (19) of the present embodiment. The black circles represent the estimation results in the case of simple support using the Fourier series of Eq. (1-1). The white square marks represent the estimation results in the case of fixed support using the equation (19) of the present embodiment. The black square marks represent the estimation results in the case of fixed support using the Fourier series of Eq. (1-1).
The accuracy when the buckling coefficient k is evaluated using the equation (19) of the present embodiment is the accuracy when N is 25 in the Fourier class of the equation (1-1) in both the simple support and the fixed support. It turned out to be about the same as.

As described above, according to the estimation device 50, the estimation method, and the estimation program 61 of the present embodiment, the inventors use a trigonometric function but have fewer terms than the Fourier series of equation (1-1). We examined a plurality of functions that can estimate the out-of-plane displacement of the web 13 on which the bending moment F1 acts. As a result, it was found that the out-of-plane displacement W _w of the web 13 can be estimated with a number of terms smaller than the Fourier series by using the equation (19).
Since equation (19) can estimate the out-of-plane displacement W _w of the web 13 with a smaller number of terms than the Fourier series of equation (1-1), the calculation required for estimation is easier than when the Fourier series is used. It can be carried out. Further, when the number of terms required to estimate the out-of-plane displacement W _w with a certain accuracy is estimated using the equation (19) rather than using the Fourier series of the equation (1-1). Therefore, the estimation result can be physically understood and formulated as compared with the case of using the Fourier series.

By using the equation (19), the out-of-plane displacement W _w of the web 13 can be quickly approximated. That is, it is possible to quickly approximate the buckling waveform when a member of a building undergoing in-plane bending or a plate element constituting the member causes local buckling.

Also, N is 2. First, in equation (19), the undetermined coefficients a ₁ , a ₂ , b ₁ , and b ₂ are obtained with the half wavelength a treated as a constant, then the half wavelength a is treated as a variable, and the half in equation (19). Find the wavelength a. (19) when determining the unknown coefficients a _1, etc. In the equation, the number of undetermined coefficients is four or less for obtaining a time. Therefore, the undetermined coefficients a ₁ , a ₂ , b ₁ , and b ₂ can be easily obtained by using the fourth-order or lower equation for which the solution formula is known.
In addition, the stress degree calculation unit 53 (in the stress degree calculation step) uses equations (30) and (31) to give the minimum positive value to the buckling stress degree σ _{cr according} to equation (32). The buckling stress degree σ _cr is obtained based on a certain an _n , b _n and a half wavelength a. When obtaining the buckling stress of sigma _cr based on the energy method, it is possible to accurately determine the buckling stress of sigma _cr using Equation (30) expression (32) below.

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 the configuration is changed, combined, or deleted without departing from the gist of the present invention. Etc. are also included.
For example, in the above embodiment, N in the equation (19) may be 3 or more.
When the buckling stress degree σ _cr is obtained based on the energy method, it is not necessary to use equations (30) to (32).
In the present embodiment, the estimation device 50 estimates the buckling stress degree of the web 13 which is a plate element, but the plate element is not limited to the web 13, and may be an upper flange 11 or a lower flange 12.

13 Web (board element)
50 Suppression stress estimation device 52 Displacement estimation unit 53 Stress calculation unit 61 Swivel stress estimation program F1 Bending moment

Claims (9)

The degree of buckling stress of a plate element that spreads along the x-axis extending in the material axis direction and the y-axis extending in the plate width direction and buckling due to a bending moment acting around the z-axis extending in the plate thickness direction is estimated. It is a stress degree estimation device
Along the x-axis of the plate element, the plate element is alternately wavy in the positive direction of the z-axis and the negative direction of the z-axis toward the first end in the direction along the x-axis of the plate element. When the half wavelength in the direction is a,
A displacement estimation unit that estimates the out-of-plane displacement W _w of the plate element in the direction along the z-axis by equation (1), and
A stress degree calculation unit that obtains the buckling stress degree of the plate element that causes the out-of-plane displacement W _w based on the equation (1) based on the energy method.
A device for estimating the degree of buckling stress.
However, N is a natural number of 2 or _{more, a 0, a n, b} n are unknown coefficients, b _w is the length of the plate elements in the direction along the y-axis.

The buckling stress estimation device according to claim 1, wherein N is 2. Said stress calculation unit, (2) using the formula and (3), (4) the a _n, b _n and is a real number that gives the smallest positive value to the buckling stress of sigma _cr by formula The device for estimating a buckling stress degree according to claim 1 or 2, wherein the buckling stress degree σ _cr is obtained based on the half wavelength a.
However, E is the Young's modulus of the plate element, ν is the Poisson's ratio of the plate element, and t _w is the thickness of the plate element.

The degree of buckling stress of a plate element that spreads along the x-axis extending in the material axis direction and the y-axis extending in the plate width direction and buckling due to a bending moment acting around the z-axis extending in the plate thickness direction is estimated. It is a method of estimating the degree of stress.
On the x-axis of the plate element, the plate element is alternately wavy in the positive direction of the z-axis and the negative direction of the z-axis toward the first end side in the direction along the x-axis of the plate element. When the half wavelength in the along direction is a,
A displacement estimation step of estimating the out-of-plane displacement W _w of the plate element in the direction along the z-axis by equation (5), and
A stress degree calculation step of obtaining the buckling stress degree of the plate element that causes the out-of-plane displacement W _w based on the equation (5) based on the energy method.
How to estimate the degree of buckling stress.
However, N is a natural number of 2 or _{more, a 0, a n, b} n are unknown coefficients, b _w is the length of the plate elements in the direction along the y-axis.

The method for estimating a buckling stress degree according to claim 4, wherein N is 2. In the stress calculation step, (6) using the formula and (7), wherein a _n, b _n and is a real number that gives the smallest positive value to the buckling stress of sigma _cr by (8) The method for estimating a buckling stress degree according to claim 4 or 5, wherein the buckling stress degree σ _cr is obtained based on the half wavelength a.
However, E is the Young's modulus of the plate element, ν is the Poisson's ratio of the plate element, and t _w is the thickness of the plate element.

An estimation device that estimates the degree of buckling stress of a plate element that spreads along the x-axis extending in the material axis direction and the y-axis extending in the plate width direction, and buckles due to a bending moment acting around the z-axis extending in the plate thickness direction. Buckling stress estimation program for
Along the x-axis of the plate element, the plate element is alternately wavy in the positive direction of the z-axis and the negative direction of the z-axis toward the first end in the direction along the x-axis of the plate element. When the half wavelength in the direction is a,
The estimation device
A displacement estimation unit that estimates the out-of-plane displacement W _w of the plate element in the direction along the z-axis by equation (9), and
A stress degree calculation unit that obtains the buckling stress degree of the plate element that causes the out-of-plane displacement W _w based on the equation (9) based on the energy method.
A buckling stress estimation program that works.
However, N is a natural number of 2 or _{more, a 0, a n, b} n are unknown coefficients, b _w is the length of the plate elements in the direction along the y-axis.

The buckling stress estimation program according to claim 7, wherein N is 2. It said stress calculation unit (10) using the formula and (11), (12) the a _n, b _n and the buckling stress of sigma _cr is a real number that gives the minimum positive value by an equation The buckling stress estimation program according to claim 7 or 8, wherein the buckling stress degree σ _cr is obtained based on the half wavelength a.
However, E is the Young's modulus of the plate element, ν is the Poisson's ratio of the plate element, and t _w is the thickness of the plate element.

Images