Abstract

An approach was presented in this manuscript for displacement-based analysis of reinforced concrete elements such as columns, beams and shear walls. The method is based on axial-shear-flexure interaction concept, considering equilibrium and compatibility conditions. The approach is entitled ASFI method, which is stand for axial shear flexure interaction. ASFI element can be considered as an element between two subsequent sections of a reinforced concrete element. The element is a combination of two models, axial-shear model and axial-flexure model. Axial-shear model is an in-plane shear element, applying average stresses and strains considering smeared rotating crack method. Axial-flexure model is based on conventional section analyses at the two end sections of the ASFI element. One-stress-block, two-stress-block and fiber model are three approaches, presented in this study, for axial-flexure model, applicable in ASFI method. Compatibility of axial-shear and axial-flexure models in ASFI method is satisfied when axial strains due to axial mechanism in the two axial-shear and axial-flexure models are identical. In other words, at any steps of an analysis after unloading lateral load to zero, considering the degraded concrete strength, axial strains due to axial load in the two axial-shear and axial-flexure models should become identical. Hence, compatibility of axial deformation for the two models can be simply satisfied by considering identical material properties and material constitutive laws. Total axial strain of ASFI element is obtained based on summation of axial strains due to axial, flexural, and shear mechanisms and total drift ratio is determined based on summation of lateral shear and flexural deformations.

Equilibrium of the two models is satisfied in stress field for normal and shear stresses. As for axial stress, simply equilibrium is satisfied considering the same axial stress in the two models. Moment in axial-flexure mechanism is converted into shear stress to be in equilibrium with shear stress of axial-shear model. It is important to consider that shear stress for axial-flexure mechanism is obtained based on the effective depth of the section, d, which is related to an average level arm for the moment, and width of the section, B. However, shear stress for axial-shear mechanism is determined based on total depth of the section, D, and width of the section, B. Studies on analysis by ASFI method indicated that, d, is a proper value for the equivalent level arm of the moment in axial-flexure model for columns and beams sections. However, for shear walls with boundary columns, it is considered equal to the distance between axes of the two boundary columns.

Axial deformation due to flexure is simply determined by considering linear strain distribution between two subsequent sections. Then centroidal strain in the section with lower moment is deducted from the centroidal strain in the section with higher moment and the result is multiplied by 0.5. In another way, axial deformation due to flexure can be determined by obtaining relative centroidal displacement between the two sections divided by the distance between the two sections by means of integration. Axial strain due to flexure can be simply added to the axial deformation in axial-shear model applying flexibility relationships. Stresses in perpendicular directions to the ASFI element plane, or clamping stresses, are considered zero, satisfying the equilibrium between concrete and transverse reinforcement stresses; in columns it is equilibrium between confinement pressure and hoops stresses.

One of the important roles of axial-shear model in ASFI method is to degrade concrete strength in concrete fiber of the axial-flexure model. In case of a reinforced concrete column with dominant flexure behavior, first at the conventional yielding point, axial-shear element is in the pre-crack state. As flexure deformation increases, axial deformation due to flexure mechanism is increased and contributed into axial deformation of the axial-shear element. As the result, shear deformation is increased and compression softening factor is decreased, which follows by degradation of concrete strength in axial-flexure fibers. This phenomenon is continuing until whether a shear-tension or shear-compression failure is dominated. From the onset of the post-peak response, if shear-tension is the dominant failure, axial-shear model rules the ultimate drift. However, if shear-compression failure is dominated, then each of the two models can govern the post-peak responses. Since, axial-flexure model has many sub-elements, fibers, it can give more proper post-peak results comparing to the shear-axial model, which has only one integration point. Applying secant stiffness models in ASFI method, lateral load, axial deformation and drift ratios are estimated for five reinforced concrete columns, a reinforced concrete beam, one bay reinforced concrete frame, and a reinforced concrete shear wall with two boundary columns. Consequently, the analytical results were compared with the experimental data for all the specimens. As the result, consistent correlations were achieved between the analytical results and test outcomes. Experimental results of a reinforced concrete column with bond failure pointed out that bond failure affected the drift-load response of the column and as the result ultimate lateral load capacity was obtained at the lower drift ratio comparing to that of a column with a perfect bond. Therefore modification should be applied on ASFI method to model bond failure mechanism. Based on analytical results by ASFI method, it was found also that slipping and buckling of compression bars （after inclination of post-peak compression strength of concrete in the extreme fibers） have considerable effects on estimation of ultimate drift at the ultimate lateral load capacity. In this study, for elements with high transverse reinforcement ratio, compression stresses of the main bars were degraded after strength of concrete fiber （next to the bars） reached twenty percent of the concrete compression strength. Then it was linearly declined corresponding to degrading compression strength of the concrete fiber. As for columns with low transverse bar ratio, reduction of compression bars was started after concrete fiber strength reached thirty percent of the concrete compression strength.

Basically, in many cases of analyses by ASFI method, axial-shear and axial-flexure models works mutually until post-peak compression strength in the extreme concrete fiber declines to about thirty percent of the maximum concrete strength. Afterward, post-peak response was governed by the axial-flexure model.

Displacement-based response was estimated by ASFI method for a reinforced concrete frame （with two columns and top & bottom stubs）, prior to the test of the frame. Prediction was done also based on AIJ and ACI design equations. Results from both AIJ and ACI indicated that a shear behavior would dominate the performance of the specimen. However, a flexure behavior followed by shear failure was estimated by ASFI method. In the test, experimental results showed a clear flexural behavior followed by shear failure （at the drift ratio almost the same as that of the predicted one by ASFI method）, however with 10% lower lateral load capacity. After analysis of the test data, it was found that loading and test setup of the test had also 10% declining effects on the ultimate lateral load capacity （due to displacing inflection point from the center of the columns）. This was found from both strain gage and curvature transducer test data. Considering the calculated effective lengths of the columns, satisfactory correlations were obtained between experimental and analytical results for lateral load of the frame as well. The result of this study also indicates highly consideration of preventing shear failures by AIJ and ACI design codes. In the experimental and analytical study of a reinforced concrete frame infilled with a masonry wall, it was found that infill wall reduced the flexural flexibility of the boundary columns, due to interaction of columns and wall in the compression zones. As the result, columns performed as a shorter column with higher shear capacity comparing to that of the column with the original length. However, columns responded a smaller ultimate drift ratio at the ultimate capacity due to forwarding shear failure. A model was described for infill wall and combined with models of ASFI method for the reinforced concrete frame, and lateral load-drift ratio relationship of the specimen was estimated, analytically. Consequently, the analytical results showed acceptable correlation with the experimental data. Based on analytical and experimental results of a shear wall, a modification was applied on the tension constitutive law of concrete for the axial-shear model in ASFI method. Then, displacement-based response of the shear wall was estimated by ASFI approach and compared with experimental outcomes. As the result, a reasonable correlation was obtained between the analytical and experimental results for the shear wall specimen. The modification was done in order to consider the effect of different shear stresses in shear wall and boundary columns due to different thickness. At a loading state in which shear stress in the wall is higher than shear stress in the boundary columns and concrete tensile stress in the columns is lower and in the wall is higher than concrete tensile strength, shear cracks occur first on the shear wall and then its concrete tensile stress starts to decrease. As the result, average tensile strain in the equivalent model is lower than concrete tensile strain in the wall and higher than that in the columns. Hence, in the modified constitutive law, lower peak tension strength at larger strain is applied as average tensile strength and average peak strain, however in the same original constitutive law. Without above modification, ASFI method gives nearly satisfactory results for drift ratio-axial deformation relationship and ultimate lateral load capacity, however higher stiffness is estimated for load-displacement response. In case of reinforced concrete beams, a specimen tested in the University of Tokyo was selected for model verification of ASFI method for beams. In order to apply ASFI method to estimate load-deflection response of a beam, the only consideration in the analysis is to apply an enough negligible axial stress to avoid creating infinitive value in the flexibility matrix. A satisfactory correlation was obtained for displacement-based estimated results by ASFI method with the experimental outcomes of the beam specimen.

FEM analyses were implemented by two programs; VecTor2 program, developed at Civil Engineering Department of University of Toronto and UC-win/WCOMD program, developed at Civil Engineering Department of the University of Tokyo. In order to study on modified compression field theory and to implement FEM analysis by VecTor2 program, the author had a great opportunity to join, as a visiting student, VecTor Analysis Group at the University of Toronto, and hospitalized by Professor Vecchio, who developed Modified Compression Field Theory and VecTor2 program. The author received also valuable advised from Professor Maekawa at the University of Tokyo, who developed UC-win/WCOMD program, in order to do the FEM analysis. FEM analyses were done by the two programs for four reinforced concrete columns with dominant failures of shear, shear-flexure and flexure failure. The analytical results of both programs showed satisfactory correlations with the experimental data in terms of load-drift relationships until ultimate lateral load capacities and drifts. The results of FEM and ASFI analyses were derived as envelope load-drift curves for the four column specimens and compared with the experimental results. Comparable and even more correlated responses to the test data were obtained by ASFI method comparing to the FEM results. ASFI method showed more correlation to post-peak test data comparing to the FEM methods, which might be due to the average strain-stress concept of ASFI method for the entire element. In FEM analysis local collapse mechanisms may occur for the elements in the high stress zone, and leading the analysis to fail before attaining the dominant failure mode.

本論文は，鉄筋コンクリート造の柱，はりおよび耐震壁部材を対象とする変位に基づく解析手法に関する研究をまとめたものである．本解析手法は釣り合い条件および適合条件を考慮した曲げ-せん断-軸力の相互作用概念に基づいており，それをASFI法と名付けている．概要は以下のとおりである．

ASFI法は鉄筋コンクリート部材を軸−せん断モデルと軸−曲げモデルの2つの連続する断面要素モデルで構成する．軸−せん断モデルは離散および回転ひび割れモデルの平均応力−平均ひずみ関係に基づいており，軸−曲げモデルは部材両端面での従来の断面曲げ解析法に基づいている．

ASFI法では，軸−曲げモデルとして応力ブロック法またはファイバーモデルを用いるが，2つのモデル（ 軸−せん断モデルと軸−曲げモデル）における軸ひずみが一致するように適合条件を満足させることに特徴がある．すなわち，水平力の除加後，すべての解析ステップにおいて，コンクリート圧縮強度の軟化効果を考慮した同一材料特性および構成則を用いることによって2つのモデルの軸力による軸ひずみを一致させる．両モデルの釣り合い条件は垂直およびせん断成分の応力場で軸応力を考慮することによって満足させる．

曲げによる軸変形は部材中間で断面内の線形分布を仮定して縁ひずみから算定する．横方向応力はゼロと仮定して，コンクリートと横補強筋の応力の釣り合い条件を満足する．特に柱においては横方向の拘束力とフープ応力の釣り合い状態を表す．軸−せん断要素の重要な役割のひとつは，軸−曲げモデルのコンクリートファイバーの圧縮強度を低下させることである．曲げ変形の増加によって軸変形が増加するとともに軸−せん断要素の軸変形の増加にも影響を与える．したがって，せん断変形が増加し，また圧縮強度軟化係数が減少することによって軸−曲げファイバ−のコンクリート強度が低下する結果となる．これによりポストピークでせん断−引張もしくはせん断−圧縮破壊が卓越するまでの解析が可能になる．

Secant stiffness法に基づいたASFIモデルを用いて，5本のRC柱，1本のはり，1スパンのフレームそして側柱付き耐震壁に対する水平荷重，軸変形および水平変形の評価を行い，解析と実験の比較が行われている．

付着破壊した柱の実験結果は付着破壊が柱の変形−荷重関係に影響を与えており，完全付着仮定の解析結果と比べてより低い変形で終局強度が得られている．これはASFIモデルに付着破壊メカニズムを考慮する必要があることを示している．また，ASFI法による解析結果は圧縮縁のコンクリートファイバーの最大圧縮強度後，圧縮鉄筋の抜け出しや座屈現象が終局強度時の水平変形評価に大きい影響があることを示している．ASFI法による解析結果では，圧縮縁コンクリートファイバ−のポストピーク強度が最大圧縮強度の30％に低下するまで軸−せん断および軸−曲げモデルが影響を与え合っており，その後の挙動は軸−曲げモデルによって支配される結果が得られている．

2本の柱と上下端が剛な梁で構成されている1スパンフレームの実験が行われているが，実験前にAIJおよびACIの設計式による評価とともにASFI法による事前解析が行われている．AIJおよびACI設計式ではせん断挙動が卓越するのに対して，ASFI法からは曲げ降伏後のせん断破壊が予測されている．実験結果は明らかに曲げ降伏後のせん断破壊を示しており，試験体の終局強度に関してはASFI法による解析結果より10％低い結果となったものの水平変形の評価は実験結果とほぼ一致している．実験結果の分析から，実験装置の加力フレームが原因で柱の反曲点高さが中央からずれることにより，終局強度が10％低下する結果となった．この影響を考慮して柱の有効長さを用いた解析結果はフレームの水平荷重に対する実験結果との良好な対応関係を示した．

コンクリ−トブロック（CB）壁を有する鉄筋コンクリート造架構に対する実験および解析では，圧縮域における柱と壁の相互作用により粗積造壁が側柱の曲げ変形能を低減させることを明らかにして．CBモデルおよびASFIモデルとの組み合わせにより，実験結果との良好な対応の解析結果が得られている．また，ASFI法による耐震壁の解析では，軸−せん断モデルの引張構成則を修正することにより，実験結果と良好な相関関係が得られている．さらに， 過去に実施されたはりの実験を対象に検討を行い，はりに対する解析結果でも実験結果と良好な対応関係を得ている．

FEMによる解析結果とも比較してASFI法の有効性を確認している．FEM解析プログラムを用いてせん断型，せん断−曲げ型，曲げ型RC柱に対する解析を行い，終局強度および終局変形までの荷重−変形関係は概ね実験結果とも良好な相関関係が得られている．解析による荷重−変形関係の包絡線と実験結果との比較からASFI法による結果がFEM解析に比べて，むしろ良好な対応関係である、としている．特に，ASFI法はポストピーク挙動を精度よく再現することが可能であるが，FEM解析では応力が集中した要素に局部破壊が生じ，その影響で破壊モードの形成前に解析が発散する傾向がある．

以上のように，本論文は，軸力−曲げ−せん断の相互作用を考慮して耐力低下を含む鉄筋コンクリ−ト部材の崩壊挙動を合理的に評価しうる新しいモデルを提案して，既往の実験的研究により広範にその適用精度を確認したものであるが，提案されたモデルは今後の解析手法や設計基準および診断基準の普遍的な背景にもなりうる独創的かつ簡易なモデルであり，耐震工学の発展に大きく貢献している．

よって本論文は博士（工学）の学位請求論文として合格と認められる．