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Uniaxial stress–strain relationship of concrete confined by various shaped steel . Susantha, Hanbin Ge, Tsutomu Usami *Department of Civil Engineering, Nagoya University, Chikusa-ku, Nagoya 464-8603, JapanReceived 31 May 2000; received in revised form 19 December 2000; accepted 14 February 2001AbstractA method is presented to predict the complete stress–strain curve of concrete subjected to triaxial compressive stresses caused by axial load plus lateral pressure due to the confinement action in circular, box and octagonal shaped concrete-filled steel tubes. Available empirical formulas are adopted to determine the lateral pressure exerted on concrete in circular concrete-filled steel columns. To evaluate the lateral pressure exerted on the concrete in box and octagonal shaped columns, FEM analysis is adopted with the help of a concrete–steel interaction model. Subsequently, an extensive parametric study is conducted to propose an empiricalequation for the maximum average lateral pressure, which depends on the material and geometric properties of the columns. Lateral pressure so calculated is correlated to confined concrete strength through a well known empirical formula. For determination of the post-peak stress–strain relation, available experimental results are used. Based on the test results, approximated expressions to predict the slope of the descending branch and the strain at sustained concrete strength are derived for the confined concrete in columns having each type of sectional shapes. The predicted concrete strength and post-peak behavior are found to exhibit goodagreement with the test results within the accepted limits. The proposed model is intended to be used in fiber analysis involving beam–column elements in order to establish an ultimate state prediction criterion for concrete-filled steel columns designed as earthquake resisting structures. •2001 Elsevier Science Ltd. All rights : Concrete-filled tubes; Confinement; Concrete strength; Ductility; Stress–strain relation; Fiber analysis1. IntroductionConcrete-filled steel tubes (CFT) are becoming increasingly popular in recent decades due to their excellent earthquake resisting characteristics such as high ductility and improved strength. As a result, numerous experimental investigations have been carried out in recent years to examine the overall performance of CFT columns [1–11]. Although the behavior of CFT columns has been extensively examined, the concrete core confinement is not yet well understood. Many of the previous research works have been mainly focused on investigating the performance of CFT columns with various limitations. The main variables subjected to such limitations were the concrete strength, plate width-to- thickness (or radius-to-thickness) ratios and shapes of the sections. Steel strength, column slenderness ratio and rate of loading were also additionally considered. It is understandable that examination of the effects of all the above factors on performances of CFTs in a wider range, exclusively on experimental manner, is difficult and costly. This can be overcome by following a suitable numerical theoretical approach which is capable of handling many experimentally unmanageable situations. At present, finite element analysis (FEM) is considered as the most powerful and accurate tool to simulate the actual behavior of structures. The accurate constitutive relationships for materials are essential for reliable results when such analysis procedures are involved. For example, CFT behavior may well be investigated through a suitable FEM analysis procedure, provided that appropriate steel and concrete material models are available. One of the simplest yet powerful techniques for the examination of CFTs is fiber analysis. In this procedure the cross section is discretized into many small regions where a uniaxial constitutive relationship of either concrete or steel is assigned. This type of analysis can be employed to predict the load–displacement relationships of CFT columns designed as earthquake resisting structures. The accuracy involved with the fiber analysis is found to be quite satisfactory with respect to the practical design present, an accurate stress–strain relationship for steel, which is readily applicable in the fiber analysis, is currently available [12]. However, in the case of concrete, only a few models that are suited for such analysis can be found [3,8,9]. Among them, in Tomii and Sakino’s model [3], which is applicable to square shaped columns, the strength improvement due to confinement has been neglected. Tang et al. [8] developed a model for circular tubes by taking into account the effect of geometry and material properties on strength enhancement as well as the post-peak behavior. Watanabe et al. [9] conducted model tests to determine a stress–strain relationship for confined concrete and subsequently proposed a method to analyze the ultimate behavior of concrete-filled box columns considering local buckling of component plates and initial imperfections. Among the other recent investigations, the work done by Schneider [10] investigated the effect of steel tube shape and wall thickness on the ultimate strength of the composite columns. El-Tawil and Deierlein [11] reviewed and evaluated the concrete encased composite design provisions of the American Concrete Institute Code (ACI 318) [13], the AISC-LRFD Specifications [14] and the AISC Seismic Provisions [15], based on fiber section analyses considering the inelastic behavior of steel and this study, an analytical approach based on the existing experimental results is attempted to determine a complete uniaxial stress–strain law for confined concrete in relatively thick-walled CFT columns. The primary objective of the proposed stress–strain model is its application in fiber analysis to investigate the inelastic behavior of CFT columns in compression or combined compression and bending. Such analyses are useful in establishing rational strength and ductility prediction procedures of seismic resisting structures. Three types of sectional shapes such as circular, box and octagonal are considered. A concrete–steel interaction model is employed to estimate the lateral pressure on concrete. Then, the maximum lateral pressure is correlated to the strength of confined concrete through an empirical formula. A method based on the results of fiber analysis using assumed concrete models is adopted to calibrate the post-peak behavior of the proposed model. Finally, the complete axial load–average axial strain curves obtained through the fiber analysis using the newly proposed material model are compared with the test results. It should be noted that a similar type of interaction model as used in this study has been adopted by Nishiyama et al. [16], which has been combined with a so called peak load condition line in order to determine the maximum lateral pressure on reinforced concrete , previous researches [17,18] indicate that the stress–strain relationship of concrete under compressive load histories produces an envelope curve identical to the stress–strain curve obtained under monotonic loading. Therefore, in further studies, the proposed confined uniaxial stress–strain law can be extended to a cyclic stress–strain relationship of confined concrete by including a suitable unloading/reloading stress–strain . Theoretical . Characteristic points on confined concrete stress–strain curveReferring to Fig. 1（General stress–strain curves for confined and unconfined concrete.）, the following characteristic points have been identified to define a complete stress–strain curve when concrete is confined by surrounding steel tubes. The notation in the figure is as follows: f ’c is the strength of unconfined concrete; f ’cc is the strength of confined concrete; εc is the strain at the peak of unconfined concrete; εcc is the strain at the peak of confined concrete; εu is the ultimate strain of unconfined concrete; fu is the ultimate strength of unconfined concrete; εcu is the ultimate strain of confined concrete; and αf ’cc is the residual strength of confined concrete at very high strain levels. The expression for the complete stress–strain curve is defined as suggested by Popovics [19], which was later modified by Mander et al. [20] and given by where fc and ε denote the longitudinal compressive stress and strain, respectively; Ec stands for the tangent modulus of elasticity of concrete. It should be noted that Eq. (1) has been defined even for the post-peak region, in this study, it is utilized only up to the peak point. The post-peak behavior is treated separately by assuming a linearly varied stress–strain relation as will be discussed in Section 4. 【1-4 Fig. 1】. Confinement action in circular CFT columnsIn short CFT columns with relatively thick-walled sections designed for seismic purposes, failure is mainly caused due to concrete crushing. The mode of failure is governed by the individual behavior of each component. The behavior of concrete in CFT columns under monotonically increasing axial load can be explained in terms of concrete–steel interaction. The confinement effect does not exist at the early stage of loading owing to the fact that the Poisson ratio of concrete is lower than that of steel at the initial loading stage. At this level of loading, the circumferential steel hoop stresses are in compression and the concrete is under lateral tension provided that no separation between concrete and steel occurs (., the bond between two materials does not break). However, as the axial load increases, the lateral expansion of concrete gradually becomes greater than the steel due to the change of the Poisson ratio of concrete, and therefore a radial pressure develops at the concrete– steel interface. At this stage, confinement of the concrete core is achieved and the steel is in hoop transferring from the steel tube to the concrete occurs at this stage. It is observed that the load at this stage is higher than the sum of loads that can be achieved by steel and concrete acting the triaxial stress state the uniaxial compressive concrete strength can be given by 【5】 where frp is the maximum radial pressure on concrete and m is an empirical coefficient. In the past a lot of extensive experimental studies have been carried out to determine a value for coefficient m and it is found that for normal strength concrete, m is in the range of 4–6 [21]. In this study m is assumed to be . The radial pressure, fr, can be expressed by the relationship given in Eq. (6), which is easily derived by considering the equilibrium of horizontal forces on a circular section: 【6】Here, fsr, t and D denote the circumference stress in steel, the thickness and the outer diameter of the tube, . Evaluation of confinement in various shaped CFT . Circular sectionDetermination of the confinement level in circular tubes is found in the method proposed by Tang et al. [8]. In this method, the change of the Poisson ratio of concrete and steel with column loading is investigated. An empirical factor, β, is introduced for this purpose and subsequently the lateral pressure at the peak load is given by 【7】 Factor β is defined as 【8】 where νe and νs are the Poisson ratios of a steel tube with and without filled-in concrete, respectively. Here, νs is taken as equal to at the maximum strength point, and νe is given by the following expressions: 【9 10】 Here, t, D and f ’c are the same as previously defined and fy stands for the yield stress of steel. The above equation is applicable for (f ’c/fy) ranging from to where most of the practically feasible columns are found within. A detailed description of the method can be found in Tang et al. [8]. It is clear that frp given by Eq. (7) depends on both the material properties and the geometry of the column. Subsequently, frp calculated from Eq. (7) is substituted into Eq. (5) to determine the confined concrete strength, f ’cc.摘要部分的翻译：各种断面形状钢管混凝土的单轴应力应变关系. Susantha ， Hanbin Ge, Tsutomu Usami*土木工程学院，名古屋大学, Chikusa-ku ，名古屋 464-8603, 日本收讫于2000年5月31日 ; 正式校定于2000年12月19日; 被认可于2001年2月14日¬¬摘要一种预测受三轴压应力混凝土的完全应力-应变曲线的方法被提出，这种三轴压应力是由环形、箱形和八角形的钢管混凝土中的限制作用导致的轴向荷载加测向压力所产生的。有效的经验公式被用来确定施加于环形钢管混凝土柱内混凝土的侧向压力。FEM（有限元）分析法和混凝土-钢箍交互作用模型已被用来估计施加于箱形和八角形柱的混凝土侧向压力。接着，进行了广泛的参数研究，旨在提出一个经验公式，确定不同的筒材料和结构特性下的最大平均侧向压力。如此计算出的侧向压力通过一个著名经验公式确定出侧向受限混凝土强度。对于高峰之后的应力-应变关系的确定，使用了有效的试验结果。基于这些测试结果，和近似表达式来推算下降段的斜度和各种断面形状的筒内侧向受限混凝土在确认的混凝土强度下的应变。推算出的混凝土强度和后峰值性能在允许的界限内与测试结果吻合得非常好。所提出的模型可用于包括梁柱构件在内的纤维分析，以确定抗震结构设计中混凝土填充钢柱筒的极限状态的推算标准。 •版权所有2001 Elsevier科学技术有限公司。关键词: 钢管混凝土；限制；混凝土强度；延性；应力应变关系；纤维分析这是当年毕业时我的翻译，因为原文有图表等原文也超过10000字，没法在这里发，如需要原文（pdf版及word版）及全部翻译（5000字，中文），请留下邮箱。