JP2021006791A - Apparatus, method and program for estimating buckling stress intensity - Google Patents

Abstract

To provide an apparatus for estimating buckling stress intensity which estimates buckling stress intensity in consideration of coupled buckling generated by a web plate element and a flange plate element.SOLUTION: A buckling stress intensity estimation apparatus 50 for estimating buckling stress intensity of an H-steel having a web plate element spreading along an x-axis extending in a material axis direction and a y-axis extending in a plate width direction and first and second flange plate elements arranged to hold the web plate element in a direction along the y-axis, and buckling with bending moment applied around a z-axis extending in a plate thickness direction of the web plate element, includes: a displacement estimation unit 52 which estimates out-of-plane displacement Ww in the web plate element facing a direction along the z-axis using a mathematical formula, and estimates out-of-plane displacement Wf1 and Wf2 in the first and second flange plate elements facing the direction along the y-axis using mathematical formulae, respectively; and a stress intensity calculation unit 53 which determines buckling stress intensity of the H-steel based on the out-of-plane displacement Ww, the out-of-plane displacement Wf1 and Wf2, and 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, in the width-thickness ratio classification of H-section steel to bending moment, coupled buckling between plate elements such as web (web plate element) and flange (flange plate element) is not considered, and under simple support conditions. The width-thickness ratio classification of H-section steel is constructed based on the strength formula elucidated for a certain plate element.
However, in the local buckling of the H-shaped steel, the web and the flange interact with each other, so that the buckling stress degree (local buckling stress degree) of the H-shaped steel cannot be accurately estimated by the conventional width-thickness ratio classification. Since the shape of the local buckling when the bending moment acts on the H-section steel shows complicated periodic behavior on both the web and the flange, the Fourier series expressed by Eq. (1) is changed to Eq. (1-1). it is possible to estimate the out-of-plane displacement W _w of the web such as a plate element of H-section steel be used to change the representation (e.g., see non-Patent Document 1).
However, b _w is the distance between the centers in the thickness direction of both flanges of the H-section steel.

Then, the behavior of the H-shaped steel can be reproduced, such as estimating the buckling stress degree of the H-shaped steel based on the estimated out-of-plane displacement W _w .

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, in order to reproduce the behavior of H-section steel, it is necessary to set N to 30 or more in the Fourier series of Eq. (1-1). For this reason, the calculation for estimating the buckling stress degree becomes extremely complicated, and there arises a problem that it is difficult to mathematically express the estimated out-of-plane displacement W _w . Therefore, the conventional width-thickness ratio classification is constructed under the condition of simple support without considering the coupled buckling of the web and the flange (buckling considering the interaction).

The present invention has been made in view of such a problem, and in the H-shaped steel on which a bending moment acts, the buckling stress degree is taken into consideration in consideration of the coupled buckling of the web plate element and the flange plate element. It is an object of the present invention to provide a buckling stress estimation device for estimating a buckling stress, a buckling stress estimation method, and a buckling stress estimation program.

In order to solve the above problems, the present invention proposes the following means.
The buckling stress estimation device of the present invention sandwiches a web plate element extending along the x-axis extending in the material axis direction and the y-axis extending in the plate width direction, and the web plate element in the direction along the y-axis. The buckling stress of an H-shaped steel provided with a first flange plate element and a second flange plate element arranged in such a manner, and a bending moment acts on the z-axis extending in the plate thickness direction of the web plate element to buckle. A device for estimating the degree of buckling stress, in which the first flange plate element receives a tensile force due to the bending moment, and the position of the center of the first flange plate element in the direction along the y-axis. Is the origin of the y-axis, the direction from the first flange plate element to the second flange plate element is the positive direction of the y-axis, and the direction of the first flange plate element in the direction along the y-axis. centering the distance between the center of the second flange plate element and b _w, toward the first end of the direction along the x-axis of said web plate element, the negative of the positive direction and the z-axis of the z-axis When the half wavelength of the web plate element that is alternately wavy in the direction along the x-axis is a, the out-of-plane displacement W _w of the web plate element in the direction along the z-axis is defined as a. The out-of-plane displacements W _f1 and W _f2 of the first flange plate element and the second flange plate element in the direction along the y-axis, which are estimated by the equation (2), are the equations (3) and (4). The displacement estimation unit estimated by the equation and the stress degree calculation unit for obtaining the buckling stress degree of the H-shaped steel based on the out-of-plane displacement W _w , the out-of-plane displacement W _f1 , W _f2 , and the energy method. It is characterized by having.
However, N is the 2 or more is a natural _{number, a 0, a n, b} n are undetermined coefficients.

Further, in the method for estimating the buckling stress degree of the present invention, a web plate element extending along the x-axis extending in the material axis direction and the y-axis extending in the plate width direction, respectively, and the web plate element in the direction along the y-axis. A seat of an H-shaped steel provided with a first flange plate element and a second flange plate element arranged so as to be sandwiched between the web plate elements, and a bending moment acts on the z-axis extending in the plate thickness direction of the web plate element to buckle. It is a method of estimating the degree of buckling stress for estimating the degree of bending stress, and it is assumed that the first flange plate element receives a tensile force due to the bending moment, and the center of the first flange plate element in the direction along the y-axis. Is the origin of the y-axis, the direction from the first flange plate element to the second flange plate element is the positive direction of the y-axis, and the first flange plate in the direction along the y-axis. the distance between the center of the element and the center of the second flange plate element and b _w, toward the first end of the direction along the x-axis of said web plate element, the positive direction and the z-axis of the z-axis When the half wavelength of the web plate element that is alternately wavy in the negative direction in the direction along the x-axis is a, the out-of-plane displacement W of the web plate element in the direction along the z-axis _w is estimated by the equation (5), and the out-of-plane displacements W _f1 and W _f2 of the first flange plate element and the second flange plate element in the direction along the y-axis are _defined by the equation (6), (6). The displacement estimation process estimated by Eq. 7) and the buckling stress degree of the H-shaped steel are obtained based on the out-of-plane displacement W _w , the out-of-plane displacement W _f1 , W _f2 , and the energy method. It is characterized by performing the calculation process.
However, N is the 2 or more is a natural _{number, a 0, a n, b} n are undetermined coefficients.

Further, in the buckling stress estimation program of the present invention, the web plate element extending along the x-axis extending in the material axis direction and the y-axis extending in the plate width direction, respectively, and the web plate element in the direction along the y-axis. A seat of an H-shaped steel provided with a first flange plate element and a second flange plate element arranged so as to be sandwiched between the web plate elements, and a bending moment acts on the z-axis extending in the plate thickness direction of the web plate element to buckle. It is an estimation program of buckling stress degree for an estimation device for estimating a bending stress degree, and it is assumed that the first flange plate element receives a tensile force by the bending moment, and the y-axis of the first flange plate element The position of the center in the along direction is defined as the origin of the y-axis, the direction from the first flange plate element toward the second flange plate element is defined as the positive direction of the y-axis, and the direction along the y-axis is described. according the distance between a center of said second flange plate elements of the first flange plate element and b _w, toward the first end of the direction along the x-axis of said web plate element, the positive direction of the z-axis And when the half wavelength of the web plate element that is alternately wavy in the negative direction of the z-axis in the direction along the x-axis is a, the estimation device is set to the z-axis of the web plate element. The out-of-plane displacement W _w in the direction along the line is estimated by the equation (8), and the out-of-plane displacement W _{f1 of} the first flange plate element and the second flange plate element in the direction along the y-axis, The displacement estimation unit that estimates W _f2 by equations (9) and (10), and the buckling stress degree of the H-shaped steel are determined by the out-of-plane displacement W _w , the out-of-plane displacement W _f1 , W _f2 , It is also characterized by functioning as a stress degree calculation unit obtained based on the energy method.
However, N is the 2 or more is a natural _{number, a 0, a n, b} n are undetermined coefficients.

The equations (5) and (8) are the same as the equation (2), the equations (6) and (9) are the same as the equation (3), and the equations (7) and (10) are the same. It is the same as the equation (4).
According to these aspects of the invention, with respect to H-section steel bending moment acts, (2) by equation estimates the plane displacement W _w of the web plate elements in a predetermined coordinate of x-axis and y-axis. As shown in equation (3), the out-of-plane displacement W _w of the web plate element is out-of-plane of the first flange plate element corresponding to the function in which 0 is substituted for the y-axis coordinate of the function obtained by first-order partial differential equation with y. The displacement W _f1 is determined. Then, as shown in Eq. (4), the second flange plate element corresponds to the function in which b _w is substituted for the y-axis coordinate of the function obtained by first-order partial differential equation of the out-of-plane displacement W _w of the web plate element by y. The out-of-plane displacement W _{f2 of} is determined.

The degree of mutual influence between the web plate element and each flange plate element changes according to the cross-sectional quantities and material properties of the web plate element and each flange plate element, and the out-of-plane displacement W _w of the web plate element and each The out-of-plane displacements W _f1 and W _{f2 of the} flange plate element are determined. As described above, by obtaining the degree of buckling stress of the H-shaped steel based on the out-of-plane displacement W _w , the out-of-plane displacement W _f1 , W _f2 , and the energy method, the web plate element and the first and second flange plates The degree of buckling stress can be estimated in consideration of coupled buckling with the element.

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. Since the number of undecided coefficients to be obtained at one time is 4 or less when finding the undecided coefficient a ₁ etc. in the equation (2), a ₁ , a by using an equation of degree 4 or less for which the formula of the solution is known. The undecided coefficients of ₂ , b ₁ , and b ₂ can be easily obtained.

Further, in the buckling stress estimation device, the stress calculation unit uses equations (11) to (14) to obtain the minimum positive value for the buckling stress σ _{cr according} to equation (15). 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 H-shaped steel, ν is the Poisson ratio of the H-shaped steel, t _w is the thickness of the web plate element, and t _f is the first flange plate element and the said. It is the thickness of each of the second flange plate elements, and b _f is a value of half the width of each of the first flange plate element and the second flange plate element.

Further, in the method for estimating the buckling stress degree, in the stress degree calculation step, the equations (16) to (19) are used, and the buckling stress degree σ _{cr according} to the equation (20) 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 H-section steel, ν is the Poisson's ratio of the H-section steel, t _w is the thickness of the web plate element, and t _f is the first flange plate element and the said. It is the thickness of each of the second flange plate elements, and b _f is a value of half the width of each of the first flange plate element and the second flange plate element.

Further, in the buckling stress estimation program, the stress calculation unit uses equations (21) to (24) to obtain the minimum positive value for the buckling stress σ _{cr according} to equation (25). wherein a _n is a real number giving the _value, based on b _n and the half-wave a, may be the buckling stress of sigma _cr determined.
However, E is the Young's modulus of the H-section steel, ν is the Poisson's ratio of the H-section steel, t _w is the thickness of the web plate element, and t _f is the first flange plate element and the said. It is the thickness of each of the second flange plate elements, and b _f is a value of half the width of each of the first flange plate element and the second flange plate element.

According to these inventions, when calculating the buckling stress degree based on the out-of-plane displacement W _w , the out-of-plane displacement W _f1 , W _f2 , and the energy method, the buckling stress degree is accurately obtained by using a mathematical formula. Can be done.

According to the buckling stress estimation device, the buckling stress estimation method, and the buckling stress estimation program of the present invention, in the H-shaped steel on which the bending moment acts, the web plate element and the flange plate element The degree of buckling stress can be estimated in consideration of coupled buckling.

It is a perspective view of the building provided with H-section steel 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 H-section steel on which a bending moment acts is buckled when the H-section steel is sufficiently long in the direction along the x-axis. It is sectional drawing which is orthogonal to the longitudinal direction of the H-section steel. It is an enlarged perspective view of the H-section steel of FIG. It is sectional drawing of the cutting line AA in FIG. It is a figure which shows the comparison result of the buckling stress degree estimated by the same buckling stress degree estimation apparatus, and the eigenvalue analysis estimated by FEM.

Hereinafter, an embodiment of the buckling stress estimation device according to the present invention will be described with reference to FIGS. 1 to 7.
This buckling stress estimation device (estimator, hereinafter simply referred to as an estimation device) is used to estimate the buckling stress of the H-shaped steel 10 used as a steel beam in the building 1 shown in FIG. 1, for example. Used. The H-section steel 10 includes a first flange plate element 11, a second flange plate element 12, and a web plate element 13 that connects the first flange plate element 11 and the second flange plate element 12 to each other. In FIG. 1, the floor slab 20 described later is shown by a chain double-dashed line.
The first flange plate element 11, the second flange plate element 12, and the web plate element 13 included in the H-shaped steel 10 are formed of a steel plate which 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 first flange plate element 11 is formed in a flat plate shape, and is arranged so that the thickness direction of the first flange plate element 11 is along the vertical direction. The second flange plate element 12 is formed in a flat plate shape and is arranged above the first flange plate element 11. The second flange plate element 12 is arranged so that the thickness direction of the second flange plate element 12 is along the vertical direction.
The web board element 13 is formed in a flat plate shape having a rectangular shape when viewed in the thickness direction of the web board element 13. The web board element 13 is arranged so that the thickness direction of the web board element 13 is along the horizontal plane. The web plate element 13 connects the center in the width direction on the upper surface of the first flange plate element 11 and the center in the width direction on the lower surface of the second flange plate element 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 second flange plate element 12 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 has a displacement estimation unit 52 that estimates the out-of-plane displacement of the web plate element 13, and the out-of-plane displacement of each of the first flange plate element 11 and the second flange plate element 12, and the seat of the H-shaped steel 10. The buckling stress degree is provided with the out-of-plane displacement of the web plate element 13, the out-of-plane displacement of each of the first flange plate element 11 and the second flange plate element 12, and the stress degree calculation unit 53 for obtaining the bending stress degree based on the energy method. There is. 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 right-handed system in which the position coordinates of the H-shaped steel 10 are composed of the x-axis, the y-axis, and the z-axis. Recognize based on the Cartesian coordinate system of. In addition, in FIG. 3 and FIG. 5 described later, the out-of-plane displacement of the web plate element 13 and the like is shown larger than the estimated value. FIG. 3 shows a state in which the H-section steel 10 is buckled.
It is assumed that the web board element 13 extends along the x-axis extending in the material axis direction of the web board element 13 and the y-axis extending in the plate width direction of the web board element 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 plate element 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 plate element 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 the y-axis direction). , Y-axis direction). The out-of-plane displacement of the web plate element 13 is a displacement of the web plate element 13 in the z-axis direction.

The first and second flange plate elements 11 and 12 are arranged so as to sandwich the web plate element 13 in the y-axis direction. The out-of-plane displacement of the first flange plate element 11 and the second flange plate element 12 is a displacement in the y-axis direction.
It is assumed that the H-section steel 10 is sufficiently long in the x-axis direction. The term that the H-shaped steel 10 is sufficiently long in the x-axis direction here means that the surface (hereinafter referred to as the end face in the x-axis direction) 10a arranged at each end of the H-shaped steel 10 in the x-axis direction and extending in the y-axis direction. It means that the H-section steel 10 has a length such that the influence of the boundary condition on the buckling deformation can be ignored.

When the bending moment F1 around the z-axis acts on the end faces 10a of the H-shaped steel 10 in the x-axis direction, the H-shaped steel 10 may buckle. The bending moment F1 is transmitted over the entire length of the H-shaped steel 10 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 H-shaped steel 10.
The bending moments F1 acting on each end face 10a of the H-shaped steel 10 in the x-axis direction are external forces of equal magnitude. In this example, the first flange plate element 11 receives a tensile force due to the bending moment F1, the second flange plate element 12 receives a compressive force due to the bending moment F1, and the H-shaped steel 10 bends as a convex downward. As described above, the bending moment F1 acts on the H-shaped steel 10.

In this case, the web plate element 13 alternates between the positive direction of the z-axis and the negative direction of the z-axis toward the first end of the web plate element 13 in the x-axis direction (one of the end faces 10a in the x-axis direction). As a result of the displacement, the web plate element 13 is displaced in a wavy shape (hereinafter, referred to as a wavy shape in the x-axis direction) for a plurality of wavelengths as a whole. The second flange plate element 12 that receives the compressive force is displaced in a wavy shape corresponding to the web plate element 13, but the first flange plate element 11 that receives the tensile force is hardly displaced.
For one wavelength of the web plate element 13 displaced along the x-axis, the second end opposite to the first end in the x-axis direction is the origin of the x-axis, and the first end in the x-axis direction from this second end. Let the direction toward the x-axis be the positive direction.

As shown in FIGS. 3 and 4, the position of the center of the first flange plate element 11 in the y-axis direction is set as the origin of the y-axis. The direction from the first flange plate element 11 to the second flange plate element 12 is defined as the positive direction of the y-axis.
The origin of the z-axis is the center of the web plate element 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.

Here, as shown in FIG. 4, the dimensions of the H-section steel 10 in the cross section orthogonal to the longitudinal direction are defined. As the unit such as the length described below, an SI unit such as "m" is preferably used for the length.
The thickness of the web plate element 13 (length in the z-axis direction), and t _w. The distance between the center of the first flange plate element 11 and the center of the second flange plate element 12 in the y-axis direction is defined as b _w .
The width of the first flange plate element 11 (length in the z-axis direction) and the width of the second flange plate element 12 are equal to each other, and are half the width of each of the first flange plate element 11 and the second flange plate element 12. _Let b _f . The width of each of the first flange plate element 11 and the second flange plate element 12 is defined as B _f .
The thickness of the first flange plate element 11 and the thickness of the second flange plate element 12 are equal to each other, and the thickness of each of the first flange plate element 11 and the second flange plate element 12 is t _f .
Let E be the Young's modulus of the H-shaped steel 10 (web plate elements 13, 1st and 2nd flange plate elements 11 and 12), and let ν be the Poisson ratio of the H-shaped steel 10.

The estimation device 50 of the present embodiment estimates the degree of buckling stress of the H-shaped steel 10 when the bending moment F1 acts on the H-shaped steel 10 to buckle. The displacement estimation unit 52 and the stress degree calculation unit 53 of the estimation device 50 make the following assumptions 1 to 6 when estimating the buckling stress degree of the H-shaped steel 10.
1. 1. The thickness of the web board element 13 is thin, and the thickness of the web board element 13 is shorter than the length of the web board element 13 in the x-axis direction and the length in the y-axis direction.
2. 2. The deflection (out-of-plane displacement due to buckling) of the web plate element 13 is small, which is smaller than the thickness of the web plate element 13.
3. 3. The central surface of the web plate element 13 in the thickness direction maintains a neutral surface without expanding or contracting due to bending of the web plate element 13.
4. In the cross section of the H-section steel 10, the assumption of flattening against bending holds.
5. The material of the H-section steel 10 is homogeneous and isotropic.
6. The displacement when an external force acts on the H-section steel 10 follows Hooke's law.

As shown in FIG. 3, when the bending moment F1 acts on the H-shaped steel 10, the web plate element 13 and the like of the H-shaped steel 10 may be displaced in a wavy shape in the x-axis direction. When the y-axis coordinate of the web board element 13 wavy in the x-axis direction is a certain value, the out-of-plane displacement of the web board element 13 in the z-axis direction at the x-axis coordinate is sin (πx). It is assumed that it is expressed by the equation / a). At this time, the wavelength in the x-axis direction of the web plate element 13 displaced in a wavy manner in the x-axis direction of the web plate element 13 becomes 2a. The half wavelength (half the length of the wavelength) in the x-axis direction of the web board element 13 is a.
FIG. 5 is a diagram showing an out-of-plane displacement W _w in the entire region of the H-shaped steel 10 in a portion where the length of the H-shaped steel 10 in the x-axis direction is a half wavelength a.

FIG. 6 shows the state when the bending moment F1 acts on the H-shaped steel 10 with a solid line. The dotted line in FIG. 6 shows the state when the bending moment F1 does not act on the H-shaped steel 10.

When the bending moment F1 acts on the web plate element 13, the out-of-plane displacement of the web plate element 13 has been estimated by using the Fourier series represented by the equation (1-1).
While using trigonometric functions, the inventors have found that when the x-axis coordinate is a certain value with a number of terms smaller than the Fourier series, the web board element 13 oriented in the z-axis direction at the y-axis coordinate. the functions that can estimate the out-of-plane displacement (first out-of-plane displacement) w _w a plurality studied. 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).
As a result, when the bending moment F1 acts on the H-shaped steel 10, when the x-axis coordinate is a certain value, the out-of-plane displacement w _w of the web plate element 13 in the z-axis direction at the y-axis coordinate. Found that it is estimated by the equation (30) with a number of terms smaller than the Fourier class. However, N is the 2 or more is a natural _{number, a 0, a n, b} n are undetermined coefficients.
Equation (30) 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 (30) can be preferably used for estimating the out-of-plane displacement of a plate element such as the web plate element 13.

On the other hand, when the x-axis coordinate is an arbitrary value, the out-of-plane displacement (second out-of-plane displacement) W _w of the web plate element 13 in the z-axis direction at a certain coordinate on the y-axis is (31). Estimated by the formula. In equation (31), as described above, when the y-axis coordinate of the web board element 13 is a certain value, the out-of-plane displacement of the web board element 13 in the z-axis direction at the x-axis coordinate is It is based on the assumption that it is expressed by the equation of sin (πx / a).
Incidentally, out-of-plane displacement W _w of FIG. 3 is a repeat of plane displacement W _w in FIG. 5 in the x-axis direction. Since estimating the out-of-plane displacement W _w of the web plate elements 13 that and 5 for estimating the out-of-plane displacement W _w of the web plate element 13 of FIG. 3 are as defined, the equation (31) is 5 The out-of-plane displacement W _{w of} is estimated. The out-of-plane displacement W _w is a function of the y-axis coordinate and the x-axis coordinate, and is used when estimating the buckling stress degree of the web plate element 13.
By substituting the equation (30) into the equation (31), the equation (32) is obtained. When N is 2 in the equation (32), it can be transformed as in the equation (33).

Further, the out-of-plane displacement W _f1 of the first flange plate element 11 at a predetermined coordinate of the z-axis is estimated by the equation (34), and the out-of-plane displacement W _f2 of the second flange plate element 12 at a predetermined coordinate of the z-axis is estimated. Estimated by equation (35).
Equations (34) and (35) include terms by trigonometric functions using power functions of y-axis coordinates.

For example, before acting moment F1 bent H-beam 10 in FIG. 5, the coordinates of the x-axis in the web plate element 13 is x _0, coordinate the portion of y ₀ of the y-axis (hereinafter, a first estimated target portion The z-axis coordinates of) are 0. The out-of-plane displacement W _w in the first estimation target portion is estimated by the equation (32-1). The z-axis coordinates of the first estimation target portion after the bending moment F1 acts on the H-shaped steel 10 in FIG. 5 are the values obtained by adding the out-of-plane displacement W _w estimated by Eq. (32-1) to 0. .. That is, after the bending moment F1 acts, the first 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 . Presumed.
For example, before acting moment F1 bent H-beam 10 in FIG. 5, the coordinates of the x-axis in the second flange plate element 12 is x _0, coordinate the portion of z ₀ of the z-axis (hereinafter, the second estimation target The coordinates of the y-axis of (referred to as a part) are b _w . The out-of-plane displacement W _f2 in the second estimation target portion is estimated by the equation (35-1). The y-axis coordinates of the second estimation target portion after the bending moment F1 acts on the H-shaped steel 10 in FIG. 5 are the value obtained by adding the out-of-plane displacement W _f2 estimated by Eq. (35-1) to b _w. Become. That is, after the bending moment F1 acts, the second estimation target portion is arranged at a position where the x-axis coordinate is x ₀ , the y-axis coordinate is (b _w + W _f2 ), and the z-axis coordinate is z _0. It is presumed that it has been done.

The displacement estimation unit 52 estimates the out-of-plane displacement W _w of the web plate element 13 by the equation (32), estimates the out-of-plane displacement W _f1 of the first flange plate element 11 by the equation (34), and estimates the second flange plate. The out-of-plane displacement W _f2 of the element 12 is estimated by Eq. (35).
Here, based on the energy method, the strain energy U _w generated in the web plate element 13 due to buckling deformation is expressed as in Eq. (37), and the strain energy U _{f of the} first and second flange plate elements 11 and 12 is expressed. Is expressed as in Eq. (38).

However, the plate rigidity D _w of the web plate element 13 is expressed as in the equation (39). The plate rigidity D _f of the first and second flange plate elements 11 and 12 is expressed by the equation (40). The stress function σ _w of the web plate element 13 on which the bending moment F1 acts is expressed by Eq. (41) with the buckling stress degree of the H-section steel 10 as σ _cr . However, the stress function σ _w makes compression positive.
The function δ _w is the displacement in the x-axis direction at a certain coordinate of the y-axis of the web plate element 13 when buckling occurs, and the displacement in the x-axis direction generated at the center of the second flange plate element 12 is δ (42). ) Is expressed as an equation.

The stress functions σ _f1 and σ _f2 are stress functions of the first and second flange plate elements 11 and 12 on which the bending moment F1 acts. The stress function σ _f1 is equal to −σ _cr and the stress function σ _f2 is equal to σ _cr . However, the stress functions σ _f1 and σ _{f2 make} compression positive.
The functions δ _f1 and δ _f2 are functions expressing the displacements of the first and second flange plate elements 11 and 12 in the x-axis direction when buckling occurs. The function δ _f1 is equal to −δ and the function δ _f2 is equal to δ.

Further, the external force potential energy V _w of the web plate element 13 is expressed by the equation (48), and the external force potential energy V _f of the first and second flange plate elements 11 and 12 is expressed by the equation (49). ..

The total potential energy Π of the H-beam 10 is expressed as the sum of the strain energy and the external potential energy as in Eq. (50).

The stress degree calculation unit 53 sets the buckling stress degree σ _cr of the H-shaped steel 10 as the out-of-plane displacement W _w of the web plate element 13 and the out-of-plane displacement W _f1 and W _{f2 of the} first and second flange plate elements 11 and 12. , And based on the energy method. That is, the stress calculating unit 53 (51) from equation using expression (54), (55) _a n, _{b n} and buckling stress of _{sigma cr} is a real number that gives the minimum positive value by an equation The buckling stress degree σ _cr is obtained based on the half wavelength a.

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 buckling stress of sigma _cr by equation (55) Request _a n, based on a set of _{b n (a} n, a set of _{b n} (55) 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 pair of undetermined coefficients an _n and b _n that is the solution of the simultaneous equations, the pair of an _n and b _n gives the minimum positive value to the buckling stress degree σ _{cr according} to equation (55). 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.

The buckling coefficient k _w of the web plate element 13 is expressed by the equation (58), and the buckling coefficient k _f of the first and second flange plate elements 11 and 12 is expressed by the equation (59), respectively.

In the buckling stress degree estimation method of the present embodiment (hereinafter, simply referred to as an estimation method), a displacement estimation step and a stress degree calculation step are performed. In the displacement estimation step, the out-of-plane displacement W _w of the web plate element 13 is estimated by the equation (32), and the out-of-plane displacements W _f1 and W _f2 of the first and second flange plate elements 11 and 12 are estimated by the equation (34). , (35), respectively. In the stress degree calculation step, the buckling stress degree σ _cr is determined based on the out-of-plane displacement W _w of the web plate element 13, the out-of-plane displacement W _f1 and W _{f2 of} the first and second flange plate elements 11 and 12, and the energy method. Ask.

The bending moment F1 acting on the H-shaped steel 10 when the buckling stress degree σ _cr is generated on the H-shaped steel 10 can be obtained from a known equation regarding a beam. By making the bending moment F1 acting on the H-shaped steel 10 smaller than this bending moment F1, it is possible to prevent the H-shaped steel 10 from buckling.
Further, the applicable range of dimensions such as the thickness t _f of the first and second flange plate elements 11 and 12 in the estimation device 50, the estimation method, and the estimation program 61 of the present embodiment is preferably as follows.
-Width-thickness ratio of the first and second flange plate elements 11 and 12 (b _f / t _f ): 20 or less-Width-thickness ratio of the web plate element 13 (b _w / t _w ): 300 or less-Aspect ratio (b _w) / B _f ): 0.5 or more and 10 or less ・ Plate thickness ratio (t _f / t _w ): 0.5 or more and 5.0 or less

[Evaluation of estimation accuracy of buckling stress]
FIG. 7 shows the results of comparing the buckling stress degree estimated by the estimation method (estimating device 50, estimation program 61) of the present embodiment and the buckling stress degree estimated by the FEM (Finite Element Method). Shown. M _p represents the total plastic moment of the H-shaped steel 10 (the bending moment when the H-shaped steel 10 is completely plasticized). M _{cr_pre} represents the bending moment at the time of buckling of the H-shaped steel 10 estimated by the estimation method of the present embodiment. M _{cr_FEM} represents the bending moment at buckling of the H-section steel 10 estimated by FEM. In FIG. 7, the horizontal axis represents the value of √ (M _p / M _{cr_pre} ), and the vertical axis represents the value of (M _{cr_pre} / M _p ) or (M _{cr_FEM} / M _p ). By FEM and the estimation method of the present embodiment, the buckling stress degree of H-shaped steel 10 of various specifications was estimated.
In FIG. 7, ◯ indicates the result estimated by FEM, that is, the result with the vertical axis as the value of (M _{cr_FEM} / M _p ). On the horizontal axis marked with a _circle , the value of √ (M _p / M _{cr_pre} ) estimated by the estimation method of this embodiment was used for the H-shaped steel 10 having the same specifications as the H-shaped steel 10 analyzed by FEM. The curve represents the result estimated by the estimation method of the present embodiment, that is, the result in which the vertical axis is the value of (M _{cr_pre} / M _p ).

When the buckling stress degree σ _cr is multiplied by the cross-sectional coefficient Z of the H-shaped steel 10, the bending moment M _cr at the time of buckling is obtained.
All plastic moment M _p does not depend on the estimation method of the FEM and the present embodiment, a constant value corresponding to the material strength. By comparing the bending moment M _{cr_pre} at the time of buckling obtained from the estimation method of the present embodiment with the degree of buckling stress estimated by FEM, the estimation accuracy of the present embodiment can be evaluated. For the H-shaped steel 10 having the same specifications, the results estimated by FEM and the results estimated by the estimation method of the present embodiment show that the values on the horizontal axis are the same as each other and the values on the vertical axis are errors in FIG. It shifts accordingly.

From Figure 7, the bending moment _{M Cr_FEM} of buckling estimated in FEM, and bending moment _{M Cr_pre} of buckling estimated by the estimation method of the present embodiment is well supported. Since there is a certain relationship between the bending moment M _{cr_pre} at the time of buckling estimated from the estimation method of the present embodiment and the buckling stress degree estimated by FEM, the buckling stress is estimated by the estimation method of the present embodiment. Even if the degree σ _cr is estimated, the buckling stress degree σ _cr can be estimated with practically sufficient accuracy.

When trying to estimate the buckling stress of an H-section steel on which a bending moment acts from the Fourier series of equation (1-1), it is necessary to set N to 30 or more, and the buckling stress is positive. Cannot be shown in the solution. On the other hand, in the estimation method of the present embodiment, the out-of-plane displacement W _w of the web plate element 13 is set to the equation (32), and the out-of-plane displacements W _f1 and W _f2 of the first and second flange plate elements 11 and 12 are set to (34). ) And (35), the buckling stress degree σ _cr can be calculated by an explicit solution as in Eq. (55).

As described above, according to the estimation device 50, the estimation method, and the estimation program 61 of the present embodiment, the surface of the web plate element 13 is subjected to the equation (32) with respect to the H-shaped steel 10 on which the bending moment F1 acts. The outer displacement W _w is estimated. As shown in equation (34), the first flange plate element 11 corresponds to a function in which 0 is substituted for the y-axis coordinate of the function obtained by first-order partial differential equation of the out-of-plane displacement W _w of the web plate element 13 with y. The out-of-plane displacement W _f1 is determined. Then, as shown in Eq. (35), the second flange plate corresponds to the function in which b _w is substituted for the y-axis coordinate of the function obtained by first-order partial differential equation of the out-of-plane displacement W _w of the web plate element 13 with y. The out-of-plane displacement W _f2 of the element 12 is determined.

The degree of mutual influence between the web plate element 13 and the first and second flange plate elements 11 and 12 depends on the cross-sectional displacement and material properties of the web plate element 13 and the first and second flange plate elements 11 and 12. As a result, the out-of-plane displacement W _w of the web plate element 13 and the out-of-plane displacements W _f1 and W _{f2 of the} first and second flange plate elements 11 and 12 are determined. As described above, by obtaining the buckling stress degree σ _cr of the H-section steel 10 based on the out-of-plane displacement W _w , the out-of-plane displacement W _f1 , W _f2 , and the energy method, the web plate element 13 and the first , 2 The buckling stress degree σ _cr can be estimated in consideration of the coupled buckling with the flange plate elements 11 and 12.

Even while using a trigonometric function, in the section number is less than the Fourier series, can estimate plane displacement W _w such web plate elements 13, calculations can be easily performed necessary for the estimation of such out-of-plane displacement W _w ..
When the number of terms required to estimate the out-of-plane displacement W _{w and the} like with a certain accuracy is estimated using the equation (32) rather than using the Fourier series of the equation (1-1). Since there are fewer of them, it is possible to formulate the estimation results such as the out-of-plane displacement W _w in an easy-to-understand physical manner.

Furthermore, the estimation device 50, the estimation method, and the estimation program 61 of the present embodiment have the following features 1 to 8.
1. 1. By considering the resistance element of the web plate element 13 to the first and second flange plate elements 11 and 12, the estimated buckling stress degree to the first and second flange plate elements 11 and 12 increases, and the safety side It is possible to estimate the buckling stress degree of the first and second flange plate elements 11 and 12 more accurately than the conventional estimation method evaluated in 1.
2. 2. By considering the resistance element of the first and second flange plate elements 11 and 12 to the web plate element 13, the estimated buckling stress degree with respect to the web plate element 13 increases, and the conventional evaluation has been made on the safe side. The buckling stress degree of the web plate element 13 can be estimated more accurately than the estimation method of.
3. 3. The above 1. And 2. By considering the resistance factor against the local buckling of the above, the estimated buckling stress degree is increased as compared with the buckling stress degree of the entire H-section steel 10 derived under the simple support condition.
4. Since the behavior of the local buckling of the H-shaped steel 10 is complicated, the formula of the local buckling stress degree of the H-shaped steel 10 has not been derived so far. However, by using the power function, the equation of the local buckling stress degree of the H-section steel 10 can be derived as a simple equation having a relatively small number of terms.

5. By using the equation of the local buckling stress degree of the H-shaped steel 10 based on the power function, the local buckling stress degree of the H-shaped steel 10 receiving the bending moment F1 around the z-axis is estimated without using numerical simulation. be able to.
6. From the relationship between the equation of the local buckling stress degree of the H-section steel 10 receiving the bending moment F1 and the ratio of the buckling stress degree to the total plastic state, the plate thickness considering the interaction between the plate elements 11, 12 and 13. The ratio classification can be set.
7. In the width-thickness ratio classification considering the interaction between the board elements 11, 12, and 13, the width-thickness ratio of the web board element 13 is larger than that of the conventional width-thickness ratio classification (to the thickness of the web board element 13). On the other hand, a range (where the width of the web plate element 13 is wide) can also be treated as a seismic member.
8. In the width-thickness ratio classification considering the interaction between the plate elements 11, 12, and 13, it is possible to replace the H-section steel 10 with a member having a smaller mass (steel weight) per unit length as compared with the conventional estimation method. It will be possible and cheaper buildings can be designed.

Further, N is 2 in the equation (32). First, in equation (32), 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 (32). Find the wavelength a. (32) 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 of a ₁ , a ₂ , b ₁ , and b ₂ can be easily obtained by using equations of degree 4 or less for which the formula of the solution is known.
Further, the stress degree calculation unit 53 (in the stress degree calculation step) uses equations (51) to (54) to give the minimum positive value to the buckling stress degree σ _{cr according} to equation (55). there _a n, based on _{b n,} determine the buckling stress of _{sigma cr.} Therefore, when obtaining the buckling stress degree σ _cr based on the out-of-plane displacement W _w , the out-of-plane displacement W _f1 , W _f2 , and the energy method, it is possible to accurately obtain the buckling stress degree σ _cr using a mathematical formula. it can.

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 (32) 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 (51) to (55).

10 H-shaped steel 11 1st flange plate element 12 2nd flange plate element 13 Web plate element 50 Suction stress estimation device (estimation device)
52 Displacement estimation unit 53 Stress calculation unit 61 Buckling stress estimation program (estimation program)
F1 bending moment

Claims (9)

A web plate element extending along the x-axis extending in the material axis direction and a y-axis extending in the plate width direction, and a first flange plate element and a first flange plate element arranged so as to sandwich the web plate element in a direction along the y-axis. A device for estimating the degree of buckling stress, which comprises two flange plate elements and estimates the degree of buckling stress of an H-shaped steel that buckles due to a bending moment acting around the z-axis extending in the plate thickness direction of the web plate element. There,
It is assumed that the first flange plate element receives a tensile force due to the bending moment.
The position of the center of the first flange plate element in the direction along the y-axis is defined as the origin of the y-axis.
The direction from the first flange plate element to the second flange plate element is defined as the positive direction of the y-axis.
The distance between a center of said second flange plate element of said first flange plate element in a direction along the y-axis and b _w,
On the x-axis of the web board element, the web board 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 web board element. When the half wavelength in the along direction is a,
The out-of-plane displacement W _w of the web plate element in the direction along the z-axis is estimated by the equation (1), and the direction of the first flange plate element and the second flange plate element along the y-axis. Displacement estimation unit that estimates the out-of-plane displacements W _f1 and W _f2 toward the above by equations (2) and (3), respectively.
The stress degree calculation unit for obtaining the buckling stress degree of the H-section steel based on the out-of-plane displacement W _w , the out-of-plane displacement W _f1 , W _f2 , and the energy method.
A device for estimating the degree of buckling stress.
However, N is the 2 or more is a natural _{number, a 0, a n, b} n are undetermined coefficients.

The buckling stress estimation device according to claim 1, wherein N is 2. It said stress calculation unit, (4) using the equation (7) from the equation, the 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 device for estimating the degree of buckling stress according to claim 1 or 2, wherein the degree of buckling stress σ _cr is obtained based on the half wavelength a.
However, E is the Young's modulus of the H-section steel, ν is the Poisson's ratio of the H-section steel, t _w is the thickness of the web plate element, and t _f is the first flange plate element and the said. It is the thickness of each of the second flange plate elements, and b _f is a value of half the width of each of the first flange plate element and the second flange plate element.

A web plate element extending along the x-axis extending in the material axis direction and a y-axis extending in the plate width direction, and a first flange plate element and a first flange plate element arranged so as to sandwich the web plate element in a direction along the y-axis. A method for estimating the degree of buckling stress, which comprises two flange plate elements and estimates the degree of buckling stress of an H-shaped steel that buckles due to a bending moment acting around the z-axis extending in the plate thickness direction of the web plate element. There,
It is assumed that the first flange plate element receives a tensile force due to the bending moment.
The position of the center of the first flange plate element in the direction along the y-axis is defined as the origin of the y-axis.
The direction from the first flange plate element to the second flange plate element is defined as the positive direction of the y-axis.
The distance between a center of said second flange plate element of said first flange plate element in a direction along the y-axis and b _w,
On the x-axis of the web board element, the web board 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 web board element. When the half wavelength in the along direction is a,
The out-of-plane displacement W _w of the web plate element in the direction along the z-axis is estimated by the equation (9), and the direction of the first flange plate element and the second flange plate element along the y-axis. Displacement estimation process for estimating out-of-plane displacement W _f1 and W _f2 toward Eq. (10) and (11), respectively.
The stress degree calculation step of obtaining the buckling stress degree of the H-section steel based on the out-of-plane displacement W _w , the out-of-plane displacement W _f1 , W _f2 , and the energy method.
How to estimate the degree of buckling stress.
However, N is the 2 or more is a natural _{number, a 0, a n, b} n are undetermined coefficients.

The method for estimating a buckling stress degree according to claim 4, wherein N is 2. In the stress calculation step, (12) using (15) from the equation, the a _n, b _n and is a real number that gives the smallest positive value to the buckling stress of sigma _cr by (16) 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 H-section steel, ν is the Poisson's ratio of the H-section steel, t _w is the thickness of the web plate element, and t _f is the first flange plate element and the said. It is the thickness of each of the second flange plate elements, and b _f is a value of half the width of each of the first flange plate element and the second flange plate element.

A web plate element extending along the x-axis extending in the material axis direction and a y-axis extending in the plate width direction, and a first flange plate element and a first flange plate element arranged so as to sandwich the web plate element in a direction along the y-axis. A buckling stress degree for an estimation device that includes two flange plate elements and estimates the buckling stress degree of an H-shaped steel that buckles due to a bending moment acting on the z-axis extending in the plate thickness direction of the web plate element. It is an estimation program of
It is assumed that the first flange plate element receives a tensile force due to the bending moment.
The position of the center of the first flange plate element in the direction along the y-axis is defined as the origin of the y-axis.
The direction from the first flange plate element to the second flange plate element is defined as the positive direction of the y-axis.
The distance between a center of said second flange plate element of said first flange plate element in a direction along the y-axis and b _w,
On the x-axis of the web board element, the web board 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 web board element. When the half wavelength in the along direction is a,
The estimation device
The out-of-plane displacement W _w of the web plate element in the direction along the z-axis is estimated by the equation (17), and the direction of the first flange plate element and the second flange plate element along the y-axis. The displacement estimation unit that estimates the out-of-plane displacements W _f1 and W _f2 toward the above by equations (18) and (19), respectively.
The stress degree calculation unit for obtaining the buckling stress degree of the H-section steel based on the out-of-plane displacement W _w , the out-of-plane displacement W _f1 , W _f2 , and the energy method.
A buckling stress estimation program that works.
However, N is the 2 or more is a natural _{number, a 0, a n, b} n are undetermined coefficients.

The buckling stress estimation program according to claim 7, wherein N is 2. It said stress calculation unit (20) using (23) from the equation, the a _n, b _n and is a real number that gives the smallest positive value to the buckling stress of sigma _cr by (24) 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 H-shaped steel, ν is the Poisson ratio of the H-shaped steel, t _w is the thickness of the web plate element, and t _f is the first flange plate element and the said. It is the thickness of each of the second flange plate elements, and b _f is a value of half the width of each of the first flange plate element and the second flange plate element.

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