JP3773735B2 - Method for predicting springback angle by bending hollow shape material, bending method for hollow shape material, mold design method and recording medium - Google Patents

Description

【００１１】
この式(6) で与えられたスプリングバック角度は、中空形材についての純曲げ理論の解であるが、例えばプレスベンディングなどの張力の作用しない実際の中空形材の曲げ加工では中空形材の曲げ加工は不均等曲げとなり、その場合、スプリングバック角度は加工条件によっては式(6) で表された角度よりも大きくなることが多い。この原因を以下に説明する。
【００１２】
不均等曲げの場合についても、曲げ加工部に生じる曲げモーメントは、おおむね純曲げ理論と同様に断面形状（塑性断面係数、断面二次モーメント）および曲げ半径で定まる。しかし、純曲げ理論の場合と異なり、曲げ加工部と直辺部との境界で曲げ曲げモーメントによる直辺部の変形、そしてこれに伴う曲げ加工部ひずみ量の緩和が生じることになる。これは、見かけ上、曲げ半径が大きくなることと等価であり、実際のスプリングバック角度は式(6) に示す純曲げ理論の解よりも大きくなる。また、この影響は，曲げ角度が小さい条件のように形材全体に占める曲げ加工部と直辺部境界の変形の割合が大きくなるほど大きくなる。従って、より高精度にスプリングバック角度を予測するためには、このような曲げ加工部と直辺部境界近傍の変形の影響をも考慮する必要がある。
【００１３】
本発明者らは、板材の純曲げ理論解を形材に適用した場合の解と、実際のスプリングバック角度の誤差に注目し、これに対する影響因子がほぼ曲げ角度θのみに依存することを知見した。そして、この結果を用いて張力なしの曲げ加工におけるスプリングバック角度を予測する方法を明らかにした。
【００１４】
具体的には、張力なしの曲げ加工における中空形材のスプリングバック角度δθは、次式(7) 、(8) で表されることを見いだした。
【００１５】
【数１８】

【数１９】

【００１６】
なお、曲げ角度θの関数Ｇの具体的な構造（特に、０．００８や１．２２２などの係数）は、上述したものに限られるものではなく、曲げ加工方法、中空形材の材質、形状などに応じて適宜変更されてもよい。例えば、Ｇは下記の式(9) に示すようなものであってもよい。
【００１７】
【数２０】

【００１８】
また、本発明者らは、形材断面に生じる応力がほぼ全面引張になった場合、曲げ加工後のスプリングバック角度もほぼ０になること、および、このときの張力をＴ_L とすれば、張力Ｔが０からＴ_L まで増加するにつれて、無次元スプリングバック角度δθ／θもδθ／θ｜_T=0 から０までほぼ線形に減少すること（図２参照）を明らかにした。これを考慮すると、張力Ｔが加えられたときの曲げ加工による中空形材のスプリングバック角度は、式(7) に基づいて以下の式(1) 、(2) のように記述することができる。（Ｔ：中空形材に加えられる張力，Ｔ_L ：スプリングバック角度がゼロになるときに中空形材に加えられる張力，Ａ：中空形材の断面積）
【数２１】

【数２２】

【００１９】
従って、式(1) および式(2) を解析的に解くことによってスプリングバック角度δθ（＝θ−θ’）を予測することが可能となる。
【００２０】
また、Ｒ、θ、Ｒ’、θ’の間に下の式(4) で示す関係が成り立つことを考慮すると、金属からなる中空形材について所定の曲げ加工形状Ｒ’、θ’を得るための曲げ中心半径Ｒおよび曲げ角度θが式(1) を変形した式(3) および式(2)(4)を満足する解であることが分かる。
【数２３】

【数２４】

【００２１】
さらに、上述の曲げ中心半径Ｒは、曲げ内側半径Ｒ_i と以下の式(5) に示すような関係がある。従って、曲げ加工に使用する金型の形状を式(2) 、(3) 、(5) をともに満たすような曲げ角度θおよび曲げ内側半径Ｒ_i で中空形材に曲げ加工を施すことが可能なものとすれば、スプリングバック角度が生じた後に所望の曲げ角度θ’および曲げ中心半径Ｒ’を得ることができる。
【数２５】

【００２２】
なお、上述の関係式は、中空形材の断面形状が口型であることを仮定して求めたものであるが、目型や田型などの他の断面形状の場合についても同様の関係式が成り立つと考えられる。
【００２３】
また、本発明によるスプリングバック角度の予測方法、中空形材の曲げ加工方法、金型の設計方法をコンピュータに行わせるために、これらの方法に係るプログラムをハードディスク、ＣＤ−ＲＯＭまたはフロッピーディスクなどのコンピュータ読み取り可能な記憶媒体に記憶させておくことができる。
【００２４】
また、本発明によるスプリングバック角度の予測方法は、鉄、鋼、チタンなどのほか広く金属一般に適用することが可能であるが、特に部材の軽量化を実現することが可能で押出によって容易に中空形材を製造可能なＪＩＳ規格による３０００系、５０００系、６０００系、７０００系などのアルミニウム合金に適用することが好ましい。
【００２５】
【実施例】
次に、本発明の実施例について説明する。まず、本発明との比較のためにＦＥＭ解析によって中空形材に生じるスプリングバック角度の調査を行った。ここでは、張力なしの曲げ加工としてプレスベンダー、また張力を加える曲げ加工としてストレッチベンダーでの加工を模擬した。なお、ＦＥＭ解析には、汎用の静的陰解法ソフトＡＢＡＱＵＳ（商品名）を用い、その解析誤差は、無次元スプリングバック角度δθ／θによる評価で最大０．０５程度であることが確認されている。
【００２６】
解析対象である口型の中空形材は、図３に示すように、フランジ幅ｂ＝２０〜８０ｍｍ、高さｈ＝２０〜８０ｍｍ、肉厚ｔ１．５〜５ｍｍとし、曲げ（内側）型半径Ｒ_i ＝２００、５００、１０００ｍｍの３条件、曲げ角度θ＝５°、１０°、１５°、２０°、３０°とした。供試材には、それぞれ代表的な加工硬化特性を持つ押出形材であるアルミニウム合金（６Ｎ０１−Ｔ１材および６Ｎ０１−Ｔ５材）を用いた。
【００２７】
次に、張力なしの曲げ加工におけるスプリングバック角度の予測方法について、その精度を検証した。まず、比較例としての式(6) による予測結果とＦＥＭ解析結果との関係を無次元スプリングバック角度δθ／θによって図４に示す。図４から明らかなように、式(6) のような単純な純曲げ理論による予測結果は、特に曲げ角度が小さい場合に実際よりもスプリングバック角度を小さく見積もることになり、条件によっては非常に誤差が大きくなることが分かる。
【００２８】
これに対して、本発明による式(1) にしたがった予測結果およびＦＥＭ解析結果による無次元スプリングバック角度δθ／θの関係を図５に示す。図５から明らかなように、両者による予測結果はほぼ一致しており、張力なしの場合に本発明によって高精度のスプリングバック角度予測が可能であることが分かる。また、この結果から、式(3) を用いた中空形材の曲げ加工方法および曲げ金型の設計方法の精度も十分なものであることが分かる。
【００２９】
次に、張力のある曲げ加工について、ストレッチベンダーによる曲げ加工解析の精度を検証する。ここでの解析対象は、代表的にフランジ幅４０ｍｍ、高さ４０ｍｍで肉厚２ｍｍの供試材（６Ｎ０１−Ｔ５材）である中空形材とした。そして、加えられた張力Ｔと無次元スプリングバック角度δθ／θとの関係を式(1) およびＦＥＭ解析からそれぞれ求めた。この結果を図６に示す。図６において、式(1) から得られた結果を実線で示し、ＦＥＭ解析結果をプロット点で示している。図６から明らかなように両者は非常によく一致しており、式(1) を用いることで、張力の有無に拘わらず、十分に精度の高いスプリングバック角度予測や金型設計が可能であることが分かる。そのため、本発明によると、アルミニウム合金などの金属からなる中空形材を曲げ加工する際の試作立ち上げにおけるトライアンドエラーを大幅に削減することができるようになる。
【００３０】
【発明の効果】
以上説明したように、本発明によると、予め曲げ加工を行わなくとも高い精度でスプリングバック角度を予測して所望の曲げ加工を行うことが可能となり、金属からなる中空形材の曲げ加工条件を迅速且つ簡易に決定することが可能となる。
【図面の簡単な説明】
【図１】スプリングバック前後における中空形材を示す模式図である。
【図２】中空形材における無次元スプリングバック角度δθ／θと張力Ｔとの関係を示すグラフである。
【図３】本発明の実施例で用いた中空形材の断面形状を示す図である。
【図４】比較例およびＦＥＭ解析での無次元スプリングバック角度δθ／θの予測結果の関係を示すグラフである。
【図５】本発明の実施例（張力＝０）およびＦＥＭ解析での無次元スプリングバック角度δθ／θの予測結果を示すグラフである。
【図６】本発明の実施例およびＦＥＭ解析での無次元スプリングバック角度δθ／θと張力との関係を示すグラフである。
【符号の説明】
１ 中空形材
２ 曲げ中立軸[0001]
BACKGROUND OF THE INVENTION
The present invention relates to a method for predicting a springback angle by bending a hollow shape, which allows a desired bending process with a high accuracy in anticipation of the springback angle, a method for bending a hollow shape, and a mold design. The present invention relates to a method and a recording medium.
[0002]
[Prior art]
In recent years, for structural materials or parts of transportation equipment such as automobiles, ships, trains, etc., or for structural materials or parts of home appliances and building structures, extruded profiles made of aluminum (Al) alloys from the viewpoint of weight reduction. Expected to be used. When such an extruded shape is used, for example, bending such as draw bending, press bending, and multi-bending is often indispensable.
[0003]
[Problems to be solved by the invention]
When the hollow shape is bent, the springback increases the bending radius of the bent portion when the mold restraint is removed due to the difference in flow stress between the inside and outside of the bending neutral shaft. The phenomenon called is generated. When such springback occurs, a predetermined product shape cannot be obtained, and joining with other members becomes difficult. For this reason, when the shape material is to be processed into a predetermined bending radius and bending angle, it is necessary to perform bending with a mold shape and processing conditions set in anticipation of the springback angle. Therefore, in general, the bending process is performed after appropriately selecting the processing conditions such as the bending radius, bending angle, and tension by feeding back the springback angle obtained by performing a bending trial in advance for each material and processing shape. There is a way to do this.
[0004]
However, at present, there is no specific means for predicting the springback angle at the time of bending with high accuracy without actually processing a hollow shape made of metal. Also, with the method of feeding back the result of the bending trial as described above, it is difficult to obtain a product having a desired bending radius although a product having a desired bending angle is obtained. Cannot be obtained. Therefore, when bending a member that is required to have a desired bending angle and bending radius, a complicated operation of repeatedly finding out the shape and processing conditions of the mold used for bending by repeating trial and error. Need to do.
[0005]
As for a plate material, there is a method using a pure bending theory or an uneven bending theory as a method for predicting a springback angle without performing bending work in advance. However, when these methods are applied to the hollow shape member and the springback angle is predicted, there is a problem that the error is large especially when the bending angle is small.
[0006]
Accordingly, an object of the present invention is to provide a hollow shape member capable of obtaining a desired bent shape with a high accuracy of the spring back angle at the time of bending of a hollow shape member made of metal without actually bending. It is intended to provide a method for predicting a springback angle by bending, a method for bending a hollow material, a method for designing a mold, and a recording medium recording a program therefor.
[0007]
[Means for Solving the Problems]
First, with reference to FIG. 1, the relationship of the quantity about the case where a hollow shape material is bent is arranged. FIG. 1A is a schematic view of a hollow member 1 that is constrained to a mold by bending, and FIG. 1B is a diagram showing a state in which the restraint by the mold is removed from the state of FIG. It is a schematic diagram of the hollow shape member 1 after the back is generated. 1A and 1B, θ is a bending angle of the hollow shape member, θ ′ is a bending angle of the hollow shape member after the springback occurs, and R is a bending center radius (bending center) of the hollow shape member 1. , R ′ is the bending center radius of the hollow member 1 after the spring back has occurred.
[0008]
At this time, the springback angle δθ is given by θ−θ ′. H _c is the distance from the bending neutral axis 2 to the innermost bending portion, and R _i is the bending inner radius.
R = R _i + h _c
Holds.
[0009]
Next, the basic concept of the spring back angle prediction method of the present invention will be described. When the general bending theory of a plate material is applied to the bending of a hollow shape, the spring back angle Δθ after bending is given by the following equation (6). In Equation (6), σ _0.2 is the yield strength of the hollow profile, Z _p is the plastic section modulus, E is the elastic modulus of the hollow profile, I is the secondary moment of inertia of the hollow profile, and _Et is the hollow profile. Is the work hardening rate.
[0010]
[Expression 17]

[0011]
The springback angle given by this equation (6) is the solution of the pure bending theory for hollow profiles, but for example, bending of actual hollow profiles without the action of tension, such as press bending, The bending process is non-uniform bending. In this case, the springback angle is often larger than the angle expressed by Equation (6) depending on the processing conditions. The cause of this will be described below.
[0012]
Even in the case of uneven bending, the bending moment generated in the bent portion is determined by the cross-sectional shape (plastic section modulus, cross-sectional secondary moment) and the bending radius, as in the pure bending theory. However, unlike the case of the pure bending theory, the deformation of the straight side portion due to the bending bending moment at the boundary between the bent portion and the straight side portion, and the relaxation of the bending portion distortion accompanying this, occur. This apparently is equivalent to an increase in the bending radius, and the actual springback angle is larger than the solution of the pure bending theory shown in Equation (6). In addition, this effect increases as the ratio of deformation of the boundary between the bent portion and the right side portion occupies the entire shape as in the condition where the bending angle is small. Therefore, in order to predict the springback angle with higher accuracy, it is necessary to consider the influence of deformation near the boundary between the bent portion and the straight side portion.
[0013]
The inventors pay attention to the solution when the pure bending theory solution of the plate material is applied to the profile and the error of the actual springback angle, and find that the influencing factor on this depends only on the bending angle θ. did. Using this result, a method for predicting the springback angle in bending without tension was clarified.
[0014]
Specifically, it has been found that the springback angle Δθ of the hollow shape member in bending without tension is expressed by the following equations (7) and (8).
[0015]
[Formula 18]

[Equation 19]

[0016]
The specific structure of the function G of the bending angle θ (particularly, a coefficient such as 0.008 or 1.222) is not limited to the above-described one, but the bending method, the material and shape of the hollow shape material It may be appropriately changed according to the above. For example, G may be as shown in the following formula (9).
[0017]
[Expression 20]

[0018]
In addition, when the stress generated in the cross section of the profile is almost full-surface tension, the present inventors also have a springback angle after bending of almost zero, and if the tension at this time is T _L , It was clarified that as the tension T increases from 0 to T _L , the dimensionless springback angle δθ / θ also decreases almost linearly from δθ / θ | _{T = 0} to 0 (see FIG. 2). Considering this, the spring back angle of the hollow shape member by bending when the tension T is applied can be described as the following formulas (1) and (2) based on the formula (7). . (T: Tension applied to the hollow profile, T _L : Tension applied to the hollow profile when the springback angle becomes zero, A: Cross-sectional area of the hollow profile)
[Expression 21]

[Expression 22]

[0019]
Therefore, the springback angle δθ (= θ−θ ′) can be predicted by analytically solving the equations (1) and (2).
[0020]
Further, considering that the relationship expressed by the following equation (4) holds among R, θ, R ′, and θ ′, in order to obtain predetermined bending shapes R ′ and θ ′ for the hollow shape made of metal. It can be seen that the bend center radius R and the bend angle θ are the solutions satisfying the expressions (3) and (2) and (4) obtained by modifying the expression (1).
[Expression 23]

[Expression 24]

[0021]
Further, the bending center radius R described above has a relationship as shown in the following formula (5) with the bending inner radius R _i . Therefore, it is possible to bend a hollow shape with a bending angle θ and a bending inner radius R _i that satisfies the equations (2), (3), and (5) for the shape of the mold used for bending. If so, the desired bending angle θ ′ and the bending center radius R ′ can be obtained after the springback angle is generated.
[Expression 25]

[0022]
Note that the above relational expression is obtained on the assumption that the cross-sectional shape of the hollow shape member is a mouth shape, but the same relational expression is applicable to the case of other cross-sectional shapes such as an eye shape and a rice field shape. Is considered to hold.
[0023]
In addition, in order to cause a computer to perform the spring back angle prediction method, the hollow shape bending method, and the mold design method according to the present invention, a program related to these methods is stored in a hard disk, a CD-ROM, a floppy disk, or the like. It can be stored in a computer-readable storage medium.
[0024]
In addition, the method for predicting the springback angle according to the present invention can be applied to a wide range of metals in addition to iron, steel, titanium, etc., but it is particularly possible to reduce the weight of the member and it can be easily hollowed out by extrusion. It is preferable to apply to 3000 series, 5000 series, 6000 series, 7000 series, etc. aluminum alloys according to JIS standards capable of producing a profile.
[0025]
【Example】
Next, examples of the present invention will be described. First, for comparison with the present invention, the springback angle generated in the hollow shape member was investigated by FEM analysis. Here, a press bender was simulated as a bending process without tension, and a stretch bender was simulated as a bending process to apply tension. For general FEM analysis, general-purpose static implicit software ABAQUS (trade name) is used, and it is confirmed that the analysis error is about 0.05 at the maximum by evaluation with the dimensionless springback angle δθ / θ. Yes.
[0026]
As shown in FIG. 3, the mouth-shaped hollow profile to be analyzed has a flange width b = 20 to 80 mm, a height h = 20 to 80 mm, a wall thickness t1.5 to 5 mm, and a bending (inner) mold radius. Three conditions of R _i = 200, 500 and 1000 mm, bending angle θ = 5 °, 10 °, 15 °, 20 ° and 30 ° were set. As the test materials, aluminum alloys (6N01-T1 material and 6N01-T5 material), which are extruded shapes each having typical work hardening characteristics, were used.
[0027]
Next, the accuracy of the method for predicting the springback angle in bending without tension was verified. First, FIG. 4 shows the relationship between the prediction result based on the formula (6) as a comparative example and the FEM analysis result by the dimensionless springback angle δθ / θ. As is clear from FIG. 4, the prediction result based on the simple pure bending theory such as Equation (6) is that the springback angle is estimated to be smaller than the actual one when the bending angle is small. It can be seen that the error increases.
[0028]
In contrast, FIG. 5 shows the relationship between the dimensionless springback angle δθ / θ based on the prediction result according to the equation (1) and the FEM analysis result according to the present invention. As is apparent from FIG. 5, the prediction results of the two are almost the same, and it can be seen that the present invention can predict the springback angle with high accuracy when there is no tension. Also, from this result, it can be seen that the accuracy of the bending method of the hollow shape and the method of designing the bending mold using the equation (3) is sufficient.
[0029]
Next, the accuracy of bending analysis by a stretch bender will be verified for tension bending. The analysis object here was a hollow shape which is a specimen (6N01-T5 material) typically having a flange width of 40 mm, a height of 40 mm and a thickness of 2 mm. Then, the relationship between the applied tension T and the dimensionless springback angle δθ / θ was obtained from Equation (1) and FEM analysis, respectively. The result is shown in FIG. In FIG. 6, the result obtained from the equation (1) is indicated by a solid line, and the FEM analysis result is indicated by a plot point. As is clear from FIG. 6, the two are in good agreement, and by using Equation (1), it is possible to predict the springback angle and mold design with sufficiently high accuracy regardless of the presence or absence of tension. I understand that. Therefore, according to the present invention, it is possible to greatly reduce the trial and error in the start-up of a prototype when bending a hollow shape made of a metal such as an aluminum alloy.
[0030]
【The invention's effect】
As described above, according to the present invention, it is possible to predict a springback angle with high accuracy without performing bending work in advance, and to perform desired bending work. It becomes possible to determine quickly and easily.
[Brief description of the drawings]
FIG. 1 is a schematic view showing a hollow shape before and after a springback.
FIG. 2 is a graph showing a relationship between a dimensionless springback angle δθ / θ and a tension T in a hollow shape member.
FIG. 3 is a view showing a cross-sectional shape of a hollow shape member used in an example of the present invention.
FIG. 4 is a graph showing a relationship between prediction results of dimensionless springback angles δθ / θ in a comparative example and FEM analysis.
FIG. 5 is a graph showing an example of the present invention (tension = 0) and a prediction result of a dimensionless springback angle δθ / θ in FEM analysis.
FIG. 6 is a graph showing a relationship between a dimensionless springback angle δθ / θ and a tension in an example of the present invention and FEM analysis.
[Explanation of symbols]
1 Hollow profile 2 Bending neutral shaft

Claims (7)

As a springback angle generated during bending of a hollow shape made of metal, an angle δθ satisfying the following formulas (1) and (2) (= bending angle θ of the hollow shape material−after a springback occurs) A method for predicting a springback angle, wherein a bending angle θ ′) of a hollow shape member is obtained.

(R: bending center radius of the hollow member (distance from the bending center to the bending neutral axis), T: tension applied to the hollow member, T _L : applied to the hollow member when the springback angle becomes zero Tension, σ _0.2 : Yield strength of hollow profile, Z _p : Plastic section modulus, E: Elastic modulus of hollow profile, I: Secondary moment of inertia of hollow profile, E _t : Work hardening rate of hollow profile, G : Function of bending angle θ, A: Cross section of hollow profile) The method for predicting a springback angle according to claim 1, wherein the metal is an aluminum alloy. In a bending method of a hollow shape made of metal,
Bending angle satisfying the following formulas (3), (2), and (4) so that the bending angle and the bending center radius of the hollow profile after the spring back is generated are θ ′ and R ′. A bending method for a hollow shape, wherein the hollow shape is bent at θ and a bending center radius R.

(T: Tension applied to the hollow profile, T _L : Tension applied to the hollow profile when the springback angle becomes zero, σ _0.2 : Strength of the hollow profile, Z _p : Plastic section modulus of the hollow profile , E: elastic modulus of hollow profile, I: secondary moment of section of hollow profile, E _t : work hardening rate of hollow profile, G: function of bending angle θ, A: cross-sectional area of hollow profile) In the design method of the mold used when bending a hollow shape made of metal,
The shape of the mold is changed to the following equations (3), (2), (5) so that the bending angle and the bending center radius of the hollow shape member after the spring back is generated are θ ′ and R ′. A method for designing a mold, characterized in that the hollow shape member can be bent at a bending angle θ and a bending inner radius R _i that satisfy the following conditions.

(T: Tension applied to the hollow profile, T _L : Tension applied to the hollow profile when the springback angle becomes zero, σ _0.2 : Strength of the hollow profile, Z _p : Plastic section modulus of the hollow profile , E: modulus of elasticity of hollow profile, I: secondary moment of section of hollow profile, E _t : work hardening rate of hollow profile, G: function of bending angle θ, A: cross-sectional area of hollow profile, h _c : distance from the bending neutral axis to the innermost part of the bending) The angle δθ (= θ-θ 'that satisfies the following formulas (1) and (2) as the springback angle that occurs during bending of a hollow shape made of metal, where θ is the bending angle of the hollow shape , Θ ′: a computer-readable recording medium on which a program for causing a computer to perform a process for obtaining a bending angle of the hollow profile after the occurrence of springback is recorded.

(R: bending radius of the hollow member, T: tension applied to the hollow member, T _L : tension applied to the hollow member when the springback angle becomes zero, σ _0.2 : yield strength of the hollow member, Z _p : Plastic section modulus, E: Elastic modulus of hollow profile, I: Secondary moment of section of hollow profile, E _t : Work hardening rate of hollow profile, G: Function of bending angle θ, A: Hollow profile Cross-sectional area of the material) As the bending angle θ and the bending center radius R of the hollow shape material when bending the metal hollow shape material, values satisfying the following expressions (3), (2), and (4) are obtained. A computer-readable recording medium recording a program for causing a computer to perform processing.

(Θ ′: bending angle of the hollow member after the spring back occurs, R ′: bending center radius of the hollow member after the spring back occurs, T: tension applied to the hollow member, T _L : spring Tension applied to the hollow profile when the back angle becomes zero, σ _0.2 : Yield strength of the hollow profile, Z _p : Plastic section modulus of the hollow profile, E: Elastic modulus of the hollow profile, I: Hollow profile Moment of section, E _t : work hardening rate of hollow profile, G: function of bending angle θ, A: cross sectional area of hollow profile) Processing to obtain the bending angle θ and the bending inner radius R _i satisfying the following formulas (3) and (2) and (5) as the shape of the mold used for bending the hollow shape made of metal A computer-readable recording medium on which a program to be executed by a computer is recorded.

(T: Tension applied to the hollow profile, T _L : Tension applied to the hollow profile when the springback angle becomes zero, σ _0.2 : Strength of the hollow profile, Z _p : Plastic section modulus of the hollow profile , E: modulus of elasticity of hollow profile, I: secondary moment of section of hollow profile, E _t : work hardening rate of hollow profile, G: function of bending angle θ, A: cross-sectional area of hollow profile, h _c : distance from the bending neutral axis to the innermost part of the bending)

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