JPH11352027A - Method for analyzing reinforced concrete member and its recording medium - Google Patents
Method for analyzing reinforced concrete member and its recording mediumInfo
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- JPH11352027A JPH11352027A JP10165145A JP16514598A JPH11352027A JP H11352027 A JPH11352027 A JP H11352027A JP 10165145 A JP10165145 A JP 10165145A JP 16514598 A JP16514598 A JP 16514598A JP H11352027 A JPH11352027 A JP H11352027A
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Abstract
Description
【０００１】[0001]
【発明の属する技術分野】この発明は、たとえば鉄筋コ
ンクリート部材を有限要素法により解析する方法および
その記録媒体に関する。BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a method for analyzing a reinforced concrete member by a finite element method and a recording medium for the method.
【０００２】[0002]
【従来の技術】鉄筋コンクリート部材の強度や変形を予
測する手法として有限要素法が用いられている。鉄筋コ
ンクリート部材として例えば図２（ａ），（ｂ）に示さ
れるような鉄筋コンクリート柱を有限要素法により解析
する場合、図３に示されるように平面応力要素でモデル
化して解析する二次元解析と、図４に示されるように鉄
筋コンクリート部材を六面体要素にモデル化して解析す
る三次元解析とが知られている。2. Description of the Related Art A finite element method is used as a method for predicting the strength and deformation of a reinforced concrete member. When a reinforced concrete column as shown in FIGS. 2 (a) and 2 (b) is analyzed by a finite element method as a reinforced concrete member, for example, as shown in FIG. As shown in FIG. 4, a three-dimensional analysis in which a reinforced concrete member is modeled into a hexahedral element and analyzed is known.
【０００３】[0003]
【発明が解決しようとする課題】しかしながら、柱など
では帯筋の拘束効果により、コンクリートが三次元応力
状態になり、三方向から拘束されることで強度や変形能
が高まるという特徴があるところ、二次元解析は、モデ
ル作成は比較的容易であるが、本来三次元である部材を
二次元で解析するものであるから、正しい解析ができな
いという課題がある。具体的には、図５に示されるよう
な、平面要素に直交する部材の厚さ方向の鉄筋（ハッチ
部分）の拘束効果を考慮することができないため、部材
の厚さ方向の応力が０と仮定され、コンクリートの強度
が過小評価されてしまい、結果として、必要以上に高強
度の部材を使用せざるを得ないという欠点があった。However, in columns and the like, concrete has a three-dimensional stress state due to the restraining effect of the stirrups, and there is a feature that strength and deformability are increased by being restrained from three directions. In the two-dimensional analysis, although model creation is relatively easy, there is a problem that correct analysis cannot be performed because members that are originally three-dimensional are analyzed in two dimensions. Specifically, as shown in FIG. 5, it is not possible to consider the restraining effect of the reinforcing bar (hatched portion) in the thickness direction of the member orthogonal to the plane element, so that the stress in the thickness direction of the member is 0. It is assumed that the strength of the concrete is underestimated, and as a result, there is a disadvantage that a member having a higher strength than necessary must be used.
【０００４】これに対し、二次元解析として平面応力要
素ではなく平面ひずみ要素を用いる方法もあるが、その
場合は部材の厚さ方向の変形が完全に拘束されてしまう
ため、逆に強度が過大評価されてしまう。平面ひずみ要
素で厚さ方向の変形を許容した一般化平面ひずみ要素も
開発されているが、この場合は厚さ方向の鉄筋の効果を
考慮することができない上に、モデル全域で厚さ方向の
変形が同一となってしまうため、実際の強度を精度よく
解析することができないという欠点があった。On the other hand, there is a method of using a plane strain element instead of a plane stress element as a two-dimensional analysis. In this case, however, the deformation in the thickness direction of the member is completely restrained, and conversely, the strength is excessively large. Will be evaluated. A generalized plane strain element that allows deformation in the thickness direction with a plane strain element has also been developed, but in this case, the effect of the reinforcing bar in the thickness direction cannot be considered, and in addition, Since the deformation is the same, there is a disadvantage that the actual strength cannot be analyzed with high accuracy.
【０００５】一方、三次元解析にあっては、モデル作成
の手間がかかるほか、計算時間が多くかかるという欠点
があった。[0005] On the other hand, the three-dimensional analysis has the drawbacks that it takes time and effort to create a model and that it takes a lot of calculation time.
【０００６】そこで、本発明は、モデル作成および計算
の手間がかかることなく、鉄筋の拘束効果を考慮した有
限要素法による解析を行うことができる鉄筋コンクリー
ト部材の解析方法およびその記録媒体を提供することを
目的とする。It is an object of the present invention to provide a method for analyzing a reinforced concrete member and an analysis method for a reinforced concrete member capable of performing an analysis by a finite element method in consideration of the restraining effect of a reinforcing bar without taking time and effort for model creation and calculation, and a recording medium therefor. With the goal.
【０００７】[0007]
【課題を解決するための手段】この発明は、上記目的を
達成するためになされたもので、鉄筋コンクリート部材
を平面要素にモデル化して解析する二次元解析にあた
り、平面要素に直交する方向の鉄筋とコンクリートとの
応力の釣り合い式から厚さ方向の等価剛性Means for Solving the Problems The present invention has been made to achieve the above-mentioned object. In a two-dimensional analysis in which a reinforced concrete member is modeled and analyzed as a plane element, the present invention relates to a method in which a reinforcing bar in a direction perpendicular to the plane element is used. Equivalent stiffness in the thickness direction from the balance equation of stress with concrete
【数２】 を定義し、これを用いて要素剛性マトリクスを作成し、
荷重および変位を計算し、要素ひずみおよび要素応力の
計算にあたっても、厚さ方向の剛性として前記等価剛性
を用いて計算することを特徴とする。(Equation 2) Is defined, and an element rigidity matrix is created using the
The load and the displacement are calculated, and the element strain and the element stress are also calculated by using the equivalent rigidity as the rigidity in the thickness direction.
【０００８】また、本発明は、上記鉄筋コンクリート部
材の解析をコンピュータに実行させるプログラムを記録
したコンピュータ読み取り可能な記録媒体にある。The present invention also resides in a computer-readable recording medium on which a program for causing a computer to execute the analysis of the reinforced concrete member is recorded.
【０００９】この発明によれば、本来三次元である部材
を二次元の平面要素でモデル化しながら、部材の厚さ方
向の鉄筋の効果を適切に考慮することができるため、図
１に示されるように、コンクリートのポアソン効果によ
る厚さ方向の変形を、厚さ方向の鉄筋１，１が拘束する
メカニズムを定式化することができる。同図において、
円白抜き矢印は圧縮力、縦方向の矢印はひずみ、斜め方
向の矢印は厚さ方向の膨らみである。この厚さ方向の膨
らみは、鉄筋との付着力を通じ鉄筋に拘束されることと
なり、また、この拘束力は前記圧縮力に対する抵抗とな
って、コンクリートの剛性を高めることにつながる。According to the present invention, the effect of the reinforcing bar in the thickness direction of the member can be appropriately considered while modeling the member which is originally three-dimensional with a two-dimensional plane element, and is shown in FIG. As described above, it is possible to formulate a mechanism in which the deformation in the thickness direction due to the Poisson effect of concrete is restricted by the reinforcing bars 1 and 1 in the thickness direction. In the figure,
The hollow arrow indicates the compressive force, the vertical arrow indicates the strain, and the oblique arrow indicates the bulge in the thickness direction. The swelling in the thickness direction is restrained by the reinforcing bar through the adhesive force with the reinforcing bar, and the restraining force becomes a resistance to the compressive force, which leads to an increase in rigidity of the concrete.
【００１０】そして、この方法で誘導された要素は、平
面応力要素と平面ひずみ要素の中間の性質を有するもの
であって、厚さ方向の応力とひずみの両方を許容するも
のである。なお、要素形状は、図３に示す平面要素と同
一である。The element derived by this method has an intermediate property between the plane stress element and the plane strain element, and permits both stress and strain in the thickness direction. The element shape is the same as the plane element shown in FIG.
【００１１】このような解析方法は、コンピュータ読み
取り可能な記録媒体に記録したコンピュータプログラム
を実行することにより実現できる。Such an analysis method can be realized by executing a computer program recorded on a computer-readable recording medium.
【００１２】[0012]
【発明の実施の形態】以下、この発明の好ましい実施の
形態を、添付図面を参照しつつ詳細に説明する。Preferred embodiments of the present invention will be described below in detail with reference to the accompanying drawings.
【００１３】通常、有限要素法による鉄筋コンクリート
部材の解析は、例えば図６のフローチャートのような手
順で行われ、かかるフローチャートに関連して本発明に
おいては、このうち特に、要素剛性マトリクス、およ
び、要素ひずみと要素応力の計算について特徴をもつも
のである。以下、これらの特徴を中心に本実施形態を説
明する。Normally, the analysis of a reinforced concrete member by the finite element method is performed in accordance with, for example, a procedure shown in a flowchart of FIG. 6. In the present invention, in particular, the element rigidity matrix and the element It is characterized by the calculation of strain and element stress. Hereinafter, the present embodiment will be described focusing on these features.
【００１４】＝＝＝要素剛性マトリクスの作成＝＝＝ ３軸応力下のひずみから応力関係式は次式で表される。
添字の１，２は面内方向、３は厚さ方向を示す。=== Preparation of Element Stiffness Matrix === The stress-related equation is expressed by the following equation based on the strain under triaxial stress.
Subscripts 1 and 2 indicate in-plane directions, and 3 indicates a thickness direction.
【００１５】[0015]
【数３】 (Equation 3)
【００１６】ここで、εi ，σi ，Ｅi はそれぞれｉ方
向のひずみ、応力、剛性であり、νijはｉ−ｊ内面のポ
アソン比でｊ方向の応力によりｉ方向に生じるひずみの
比である。Here, εi, σi, and Ei are the strain, stress, and rigidity in the i direction, respectively, and νij is the ratio of the strain generated in the i direction by the stress in the j direction at the Poisson's ratio of the ij inner surface.
【００１７】厚さ方向の鉄筋比をρ3m、剛性をＥs3m 、
鉄筋の応力をσs3m とする。添字ｍは厚さ方向の鉄筋が
複数の場合（ｍ＝１，２，・・・など）を示す。The ratio of reinforcing bars in the thickness direction is ρ3m, the rigidity is Es3m,
The stress of the rebar is σs3m. The subscript m indicates a case where there are a plurality of reinforcing bars in the thickness direction (m = 1, 2,..., Etc.).
【００１８】厚さ方向の応力の釣り合いより、次式が成
立する。The following equation is established from the balance of the stress in the thickness direction.
【数４】 (Equation 4)
【００１９】鉄筋の応力は次式で計算される。The stress of the rebar is calculated by the following equation.
【数５】 (Equation 5)
【００２０】式（５）を式（４）に代入すると次式にな
る。When the equation (5) is substituted into the equation (4), the following equation is obtained.
【数６】 (Equation 6)
【００２１】式（６）より、厚さ方向のひずみは次式で
計算できる。From equation (6), the strain in the thickness direction can be calculated by the following equation.
【数７】 (Equation 7)
【００２２】式（７）を式（３）に代入し、ε3 を消去
すると次式となる。By substituting equation (7) into equation (3) and eliminating ε3, the following equation is obtained.
【数８】 (Equation 8)
【００２３】式（８）よりσ3 は次式で表される。From equation (8), σ 3 is represented by the following equation.
【数９】 (Equation 9)
【００２４】ここで、厚さ方向の等価剛性Ｅ3 * を次式
で定義する。Here, the equivalent rigidity E3 * in the thickness direction is defined by the following equation.
【数１０】 (Equation 10)
【００２５】Ｅ3 * を用いると、式（９）は次式とな
る。Using E3 * , equation (9) becomes:
【数１１】 [Equation 11]
【００２６】式（１１）を式（１）および（２）に代入
し、σ3 を消去すると次式となる。Substituting equation (11) into equations (1) and (2) and eliminating σ3 yields the following equation.
【数１２】 (Equation 12)
【００２７】ここで、ｎ＝Ｅ3 * ／Ｅ3 とおくと、式
（１２）、式（１３）は次式となる。Here, if n = E3 * / E3, equations (12) and (13) are as follows.
【数１３】 (Equation 13)
【００２８】ここで、等価ポアソン比μijを次式で定義
する。Here, the equivalent Poisson's ratio μij is defined by the following equation.
【数１４】 [Equation 14]
【００２９】Maxwell Betti の相反定理より、次式が成
立する。From Maxwell Betti's reciprocity theorem, the following equation holds.
【数１５】 (Equation 15)
【００３０】式（１７）を用いると、式（１４）、式
（１５）は次式となる。Using equation (17), equations (14) and (15) are as follows.
【数１６】 (Equation 16)
【００３１】式（１８）の第２項の係数と式（１９）の
第１項の係数は相反定理より等しくなければならない。
即ち、次式が成立する。The coefficient of the second term of the equation (18) and the coefficient of the first term of the equation (19) must be equal according to the reciprocity theorem.
That is, the following equation is established.
【数１７】 [Equation 17]
【００３２】よって、次式が得られる。Therefore, the following equation is obtained.
【数１８】 (Equation 18)
【００３３】ここで、式（１７）より、μ13とμ32は次
式で表される。Here, from the equation (17), μ13 and μ32 are represented by the following equations.
【数１９】 [Equation 19]
【００３４】式（２２）、式（２３）を用いると、式
（２１）の左辺は次式となる。Using equations (22) and (23), the left side of equation (21) is as follows.
【数２０】 (Equation 20)
【００３５】同様に、式（２１）の右辺は次式となる。Similarly, the right side of equation (21) is as follows.
【数２１】 式（２４）と式（２５）より、式（２１）の左辺と右辺
は一致することが分かる。(Equation 21) From Equations (24) and (25), it can be seen that the left side and the right side of Equation (21) match.
【００３６】以上の関係を用いて、式（１８）と式（１
９）をマトリクス表示すると次式となる。Using the above relations, equation (18) and equation (1)
When 9) is displayed in a matrix, the following expression is obtained.
【数２２】 (Equation 22)
【００３７】式（２６）を逆変換すると、次式が得られ
る。By inversely transforming equation (26), the following equation is obtained.
【数２３】 (Equation 23)
【００３８】よって、要素の応力〜ひずみマトリクス
［Ｄ］は次のように表される。Accordingly, the stress-strain matrix [D] of the element is expressed as follows.
【数２４】 (Equation 24)
【００３９】なお、上式でｎ＝１とすると、通常の異方
性平面ひずみの応力〜ひずみマトリクスと一致する。ま
た、μ12＝μ23＝μ31＝ν、Ｅ1 ＝Ｅ2 ＝Ｅとすると、
等方性平面ひずみの応力〜ひずみマトリクスと一致す
る。本要素の応力〜ひずみマトリクスの大きさは通常の
平面応力要素、あるいは平面ひずみ要素と同様に２×２
であり、要素剛性マトリクスの作成に際しては、応力〜
ひずみマトリクスに式（３２）を用いるだけで良い。When n = 1 in the above equation, the value is equal to the stress-strain matrix of ordinary anisotropic plane strain. Further, if μ12 = μ23 = μ31 = ν and E1 = E2 = E,
It is consistent with the stress-strain matrix of isotropic plane strain. The magnitude of the stress-strain matrix of this element is 2 × 2 as in a normal plane stress element or plane strain element.
When creating the element rigidity matrix, stress ~
It is only necessary to use equation (32) for the distortion matrix.
【００４０】＝＝＝要素ひずみと要素応力の計算＝＝＝ 通常の平面応力要素と同様に得られた変位から要素の面
内のひずみと応力を算定する。面内の応力σ1 、σ2 が
分かれば、厚さ方向の応力σ3 は式（１１）により算定
できる。=== Calculation of Element Strain and Element Stress === The in-plane strain and stress of the element are calculated from the displacement obtained in the same manner as a normal plane stress element. If the stresses σ1 and σ2 in the plane are known, the stress σ3 in the thickness direction can be calculated by equation (11).
【００４１】[0041]
【数２５】 (Equation 25)
【００４２】ここで、式（１７）よりν31とν32は次式
で表される。Here, from equation (17), ν31 and ν32 are represented by the following equations.
【数２６】 (Equation 26)
【００４３】これらを式（１１）に代入するとσ3 は次
式となる。By substituting these into the equation (11), σ3 becomes the following equation.
【数２７】 [Equation 27]
【００４４】厚さ方向のひずみε3 は式（７）により算
定できる。The strain ε3 in the thickness direction can be calculated by equation (7).
【数２８】 [Equation 28]
【００４５】上述した本実施形態の解析方法は、コンピ
ュータ読み取り可能な記録媒体に記録したコンピュータ
プログラムをコンピュータに実行させることにより実現
できる。The above-described analysis method of this embodiment can be realized by causing a computer to execute a computer program recorded on a computer-readable recording medium.
【００４６】[0046]
【発明の効果】以上詳細に説明したように、この発明に
よれば、二次元の平面要素でモデル化すればよいので手
間がかからず、厚さ方向の鉄筋比と降伏点を指定するこ
とにより部位毎に鉄筋量に応じた強度と変形を再現する
ことができながら、解析の結果には、厚さ方向の鉄筋に
よる厚さ方向のコンクリートを抑える効果が組み込まれ
るため、結果として、三次元解析に近い正確な計算結果
を得ることができる。As described in detail above, according to the present invention, since it is sufficient to model a two-dimensional plane element, no labor is required, and the rebar ratio and the yield point in the thickness direction can be specified. Can reproduce the strength and deformation according to the amount of rebar for each part, but the results of the analysis incorporate the effect of suppressing the concrete in the thickness direction by the rebar in the thickness direction. An accurate calculation result close to the analysis can be obtained.
【００４７】そして、鉄筋コンクリート柱などでは、帯
筋の効果によりコンクリートが三次元応力状態になるた
め、二次元解析では強度や変形を過小評価する惧れがあ
ったが、このような惧れがない。In a reinforced concrete column or the like, since the concrete is in a three-dimensional stress state due to the effect of the stirrups, there is a fear that the strength and deformation are underestimated in the two-dimensional analysis. .
【００４８】また、従来の一般化平面ひずみ要素のよう
に、モデル全域で厚さ方向の変形が同一となってしまう
ことはなく、要素毎に厚さ方向の鉄筋量や応力状態に応
じた変形が再現できる。Further, unlike the conventional generalized plane strain element, the deformation in the thickness direction does not become the same in the entire model, and the deformation in accordance with the amount of reinforcing bar in the thickness direction and the stress state for each element. Can be reproduced.
【図１】鉄筋によるコンクリートの変形拘束状態を示す
斜視図である。FIG. 1 is a perspective view showing a state in which concrete is restrained from deformation by a reinforcing bar.
【図２】（ａ），（ｂ）は鉄筋コンクリート部材を示す
縦断面図、横断面図である。FIGS. 2A and 2B are a longitudinal sectional view and a transverse sectional view showing a reinforced concrete member.
【図３】平面応力要素によるモデルを示す図である。FIG. 3 is a diagram showing a model based on a plane stress element.
【図４】六面体要素によるモデルを示す図である。FIG. 4 is a diagram showing a model based on hexahedral elements.
【図５】平面応力要素が厚さ方向の鉄筋の拘束効果を考
慮しない状態を示す図である。FIG. 5 is a diagram showing a state in which a plane stress element does not consider a restraining effect of a reinforcing bar in a thickness direction.
【図６】有限要素法による鉄筋コンクリート部材の解析
手順を示すフローチャートである。FIG. 6 is a flowchart showing an analysis procedure of a reinforced concrete member by a finite element method.
１ 鉄筋 1 rebar
Claims (2)
ル化して解析する二次元解析にあたり、平面要素に直交
する方向の鉄筋とコンクリートとの応力の釣り合い式か
ら厚さ方向の等価剛性 【数１】 を定義し、これを用いて要素剛性マトリクスを作成し、
荷重および変位を計算し、要素ひずみおよび要素応力の
計算にあたっても、厚さ方向の剛性として前記等価剛性
を用いて計算することを特徴とする鉄筋コンクリート部
材の解析方法。In a two-dimensional analysis in which a reinforced concrete member is modeled and analyzed as a plane element, an equivalent rigidity in a thickness direction is obtained from a balance equation of stress between a reinforcing bar and concrete in a direction orthogonal to the plane element. Is defined, and an element rigidity matrix is created using the
A method for analyzing a reinforced concrete member, comprising calculating a load and a displacement, and calculating the element strain and the element stress by using the equivalent rigidity as the rigidity in the thickness direction.
の解析をコンピュータに実行させるプログラムを記録し
たコンピュータ読み取り可能な記録媒体。2. A computer-readable recording medium in which a program for causing a computer to execute the analysis of the reinforced concrete member according to claim 1 is recorded.
Priority Applications (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| JP16514598A JP3405200B2 (en) | 1998-06-12 | 1998-06-12 | Analysis method for reinforced concrete member and recording medium therefor |
Applications Claiming Priority (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| JP16514598A JP3405200B2 (en) | 1998-06-12 | 1998-06-12 | Analysis method for reinforced concrete member and recording medium therefor |
Publications (2)
| Publication Number | Publication Date |
|---|---|
| JPH11352027A true JPH11352027A (en) | 1999-12-24 |
| JP3405200B2 JP3405200B2 (en) | 2003-05-12 |
Family
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Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| JP16514598A Expired - Lifetime JP3405200B2 (en) | 1998-06-12 | 1998-06-12 | Analysis method for reinforced concrete member and recording medium therefor |
Country Status (1)
| Country | Link |
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| JP (1) | JP3405200B2 (en) |
Cited By (3)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| JP2009250829A (en) * | 2008-04-08 | 2009-10-29 | Radioactive Waste Management Funding & Research Center | Method for simple three-dimensional analysis of welding deformation and residual stress |
| JP2019207486A (en) * | 2018-05-28 | 2019-12-05 | 日本車輌製造株式会社 | Design support device, design support method, design support program and method for manufacturing concrete structure |
| KR20210037524A (en) * | 2019-09-27 | 2021-04-06 | 경희대학교 산학협력단 | method for predicting movement of steel-rebar concrete composite columns considering concrete confinement provided by rebars and steel sections |
Citations (3)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| JPH04175637A (en) * | 1990-11-08 | 1992-06-23 | Shiro Takada | Method of calculating ground spring constant applied to buried corrugated pipe, and method of analyzing exhibition thereof with respect to seismic motion |
| JPH06305105A (en) * | 1993-04-22 | 1994-11-01 | Asics Corp | Method for forming quasi-three dimensional model for composite material laminate and method for analyzing composite material laminate using the model |
| JPH09128436A (en) * | 1995-10-30 | 1997-05-16 | Sony Corp | Element data forming method, object analytic method and recording medium |
-
1998
- 1998-06-12 JP JP16514598A patent/JP3405200B2/en not_active Expired - Lifetime
Patent Citations (3)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| JPH04175637A (en) * | 1990-11-08 | 1992-06-23 | Shiro Takada | Method of calculating ground spring constant applied to buried corrugated pipe, and method of analyzing exhibition thereof with respect to seismic motion |
| JPH06305105A (en) * | 1993-04-22 | 1994-11-01 | Asics Corp | Method for forming quasi-three dimensional model for composite material laminate and method for analyzing composite material laminate using the model |
| JPH09128436A (en) * | 1995-10-30 | 1997-05-16 | Sony Corp | Element data forming method, object analytic method and recording medium |
Cited By (3)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| JP2009250829A (en) * | 2008-04-08 | 2009-10-29 | Radioactive Waste Management Funding & Research Center | Method for simple three-dimensional analysis of welding deformation and residual stress |
| JP2019207486A (en) * | 2018-05-28 | 2019-12-05 | 日本車輌製造株式会社 | Design support device, design support method, design support program and method for manufacturing concrete structure |
| KR20210037524A (en) * | 2019-09-27 | 2021-04-06 | 경희대학교 산학협력단 | method for predicting movement of steel-rebar concrete composite columns considering concrete confinement provided by rebars and steel sections |
Also Published As
| Publication number | Publication date |
|---|---|
| JP3405200B2 (en) | 2003-05-12 |
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