各种异形钢管混凝土的单轴应力应变关系Uniaxial stress–strain relationship of concrete confined by various shaped steel tubes.pdf

engineering structures 23 (2001) 13311347 /locate/engstruct uniaxial stressstrain relationship of concrete confi ned by various shaped steel tubes k.a.s. susantha, hanbin ge, tsutomu usami * department of civil engineering, nagoya university, chikusa-ku, nagoya 464-8603, japan received 31 may 2000; received in revised form 19 december 2000; accepted 14 february 2001 abstract a method is presented to predict the complete stressstrain curve of concrete subjected to triaxial compressive stresses caused by axial load plus lateral pressure due to the confi nement action in circular, box and octagonal shaped concrete-fi lled steel tubes. available empirical formulas are adopted to determine the lateral pressure exerted on concrete in circular concrete-fi lled 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 concretesteel interaction model. subsequently, an extensive parametric study is conducted to propose an empirical equation for the maximum average lateral pressure, which depends on the material and geometric properties of the columns. lateral pressure so calculated is correlated to confi ned concrete strength through a well known empirical formula. for determination of the post-peak stressstrain 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 confi ned concrete in columns having each type of sectional shapes. the predicted concrete strength and post-peak behavior are found to exhibit good agreement with the test results within the accepted limits. the proposed model is intended to be used in fi ber analysis involving beamcolumn elements in order to establish an ultimate state prediction criterion for concrete-fi lled steel columns designed as earthquake resisting structures. 2001 elsevier science ltd. all rights reserved. keywords: concrete-fi lled tubes; confi nement; concrete strength; ductility; stressstrain relation; fiber analysis 1. introduction concrete-fi lledsteeltubes(cft)arebecoming increasingly popular in recent decades due to their excel- lent 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 111. although the behavior of cft columns has been extensively examined, the concrete core con- fi nement is not yet well understood. many of the pre- vious research works have been mainly focused on investigating the performance of cft columns with vari- ous limitations. the main variables subjected to such limitations were the concrete strength, plate width-to- * corresponding author. tel.: +81-52-789-4617; fax: +81-52-789- 5461. e-mail address: usamicivil.nagoya-u.ac.jp (t. usami). 0141-0296/01/$ - see front matter 2001 elsevier science ltd. all rights reserved. pii: s0141-0296(01)00020-7 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 diffi cult and costly. this can be overcome by following a suitable numerical theoretical approach which is capable of hand- ling many experimentally unmanageable situations. at present, fi nite 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 fi ber analysis. in this procedure the cross section is discretized into many 1332k.a.s. susantha et al. / engineering structures 23 (2001) 13311347 small regions where a uniaxial constitutive relationship of either concrete or steel is assigned. this type of analy- sis can be employed to predict the loaddisplacement relationships of cft columns designed as earthquake resisting structures. the accuracy involved with the fi ber analysis is found to be quite satisfactory with respect to the practical design purposes. at present, an accurate stressstrain relationship for steel, which is readily applicable in the fi ber analysis, is currently available 12. however, in the case of con- crete, only a few models that are suited for such analysis can be found 3,8,9. among them, in tomii and saki- nos model 3, which is applicable to square shaped col- umns, the strength improvement due to confi nement 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 enhance- ment as well as the post-peak behavior. watanabe et al. 9 conducted model tests to determine a stressstrain relationship for confi ned concrete and subsequently pro- posed a method to analyze the ultimate behavior of con- crete-fi lled 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 col- umns. el-tawil and deierlein 11 reviewed and evalu- ated the concrete encased composite design provisions of the american concrete institute code (aci 318) 13, the aisc-lrfd specifi cations 14 and the aisc seis- mic provisions 15, based on fi ber section analyses con- sidering the inelastic behavior of steel and concrete. in this study, an analytical approach based on the existing experimental results is attempted to determine a complete uniaxial stressstrain law for confi ned con- crete in relatively thick-walled cft columns. the pri- mary objective of the proposed stressstrain model is its application in fi ber 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 concretesteel interaction model is employed to estimate the lateral pressure on concrete. then, the maximum lateral pressure is correlated to the strength of confi ned concrete through an empirical for- mula. a method based on the results of fi ber analysis using assumed concrete models is adopted to calibrate the post-peak behavior of the proposed model. finally, the complete axial loadaverage axial strain curves obtained through the fi ber analysis using the newly pro- posed 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 nishi- yama 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 col- umns. meanwhile, previous researches 17,18 indicate that the stressstrain relationship of concrete under com- pressive load histories produces an envelope curve ident- ical to the stressstrain curve obtained under monotonic loading. therefore, in further studies, the proposed con- fi ned uniaxial stressstrain law can be extended to a cyc- lic stressstrain relationship of confi ned concrete by including a suitable unloading/reloading stressstrain rule. 2. theoretical background 2.1. characteristic points on confi ned concrete stress strain curve referring to fig. 1, the following characteristic points have been identifi ed to defi ne a complete stressstrain curve when concrete is confi ned by surrounding steel tubes. the notation in the fi gure is as follows: f?cis the strength of unconfi ned concrete; f?ccis the strength of confi ned concrete; ecis the strain at the peak of uncon- fi ned concrete; ecc is the strain at the peak of confi ned concrete; eu is the ultimate strain of unconfi ned concrete; fu is the ultimate strength of unconfi ned concrete; ecuis the ultimate strain of confi ned concrete; and af?ccis the residual strength of confi ned concrete at very high strain levels. the expression for the complete stressstrain curve is defi ned as suggested by popovics 19, which was later modifi ed by mander et al. 20 and given by fc?f?cc xr r1+xr (1) fig. 1. general stressstrain curves for confi ned and unconfi ned con- crete. 1333k.a.s. susantha et al. / engineering structures 23 (2001) 13311347 x? e ecc (2) r? ec (ecf?cc/ecc) (3) ecc?ec?1?5?f?cc f?c ?1?(4) where fcand e denote the longitudinal compressive stress and strain, respectively; ecstands for the tangent modu- lus of elasticity of concrete. it should be noted that eq. (1) has been defi ned 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 stressstrain relation as will be discussed in section 4. 2.2. confi nement action in circular cft columns in 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 mono- tonically increasing axial load can be explained in terms of concretesteel interaction. the confi nement 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 load- ing, the circumferential steel hoop stresses are in com- pression and the concrete is under lateral tension pro- vided that no separation between concrete and steel occurs (i.e., 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 con- crete, and therefore a radial pressure develops at the con- cretesteel interface. at this stage, confi nement of the concrete core is achieved and the steel is in hoop tension. load 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 independently. in the triaxial stress state the uniaxial compressive concrete strength can be given by f?cc?f?c?mfrp(5) where frpis the maximum radial pressure on concrete and m is an empirical coeffi cient. in the past a lot of extensive experimental studies have been carried out to determine a value for coeffi cient m and it is found that for normal strength concrete, m is in the range of 46 21. in this study m is assumed to be 4.0. 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: fr? 2t d2tfsr (6) here, fsr, t and d denote the circumference stress in steel, the thickness and the outer diameter of the tube, respect- ively. 3. evaluation of confi nement in various shaped cft columns 3.1. circular section determination of the confi nement 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, b, is introduced for this purpose and subsequently the lateral pressure at the peak load is given by frp?b 2t d2tfy (7) factor b is defi ned as b?ne?ns(8) where neand nsare the poisson ratios of a steel tube with and without fi lled-in concrete, respectively. here, nsis taken as equal to 0.50 at the maximum strength point, and neis given by the following expressions: ne?0.2312?0.3582n?e?0.1524?f?c fy? (9) ?4.843n?e?f?c fy?9.169? f?c fy? 2 n?e?0.881?106(d/t)3?2.58?104(d/t)2?1.953(10) ?102(d/t)?0.4011 here, t, d and f?c are the same as previously defi ned and fystands for the yield stress of steel. the above equation is applicable for (f?c/fy) ranging from 0.04 to 0.20 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 frpgiven by eq. (7) depends on both the material properties and the geometry of the column. subsequently, frpcalculated from eq. (7) is substituted into eq. (5) to determine the confi ned concrete strength, f?cc. 3.2. box section 3.2.1. concretesteel interaction model in order to obtain the lateral pressure at the peak, a concretesteel interaction model is used, as shown in 1334k.a.s. susantha et al. / engineering structures 23 (2001) 13311347 fig. 2(a). the confi nement along the length of the col- umn is assumed uniform and hence a unit length of the column is considered for the analysis. here, concrete is discretized into a number of segments bounded by the lines joining the center point of the model and the mid- points of the adjacent steel beam elements, as shown in figs. 2(a, c). each of these segments is represented by an axial compressive bar element with an equal stiffness of the corresponding triangular concrete segments. lat- eral steel is also represented by a number of fi nite elements. a two-node cubic beam element (b23) and a two-node linear truss element (t2d2) included in the abaqus 22 program are employed to represent the steel beam elements and the concrete bar elements, respectively. uniform lateral strain is assumed for all of the concrete bar elements. consequently, for a pre- assumed lateral strain, corresponding displacements are computed and are applied incrementally at each node of the concrete bar elements at the center of the model. this leads to expansion of the outer steel cage since the steel elements are laterally pushed out by the concrete bar elements. at the end of each load increment, the reaction forces and displacements at the outer nodes are recorded. then, the average lateral stress, f r, and strain, e r, are computed according to the following equations: f r? n 1 fri ? n 1 sei (11) fig. 2.concretesteel interaction models for various sections. e r? n 1 (ui/lei) n (12) where friis the reaction force at node i; seiis the width of the ith concrete bar element as shown in fig. 3; ui is the displacement at node i; leiis the length of the ith concrete bar element; and n is the number of outer nodes. the maximum average lateral stress, f rp, obtained from eq. (11), is then substituted for frpin eq. (5) to predict the confi ned concrete strength. in this study, the uniform lateral pressure distribution along a steel element is assumed as shown in fig. 3. however, the lateral pressure along a side of a section is not uniform. obviously, higher values should be observed at around the corners of the section. this phenomenon has been well observed in many previous studies focused on reinforced concrete columns 16,20,23. since concrete elements are assumed to be in hydrostatic stress state, the lateral pressure at a distance l from the center point is constant. further, neglecting the poisson effect, an expression for the lateral stiffness of the concrete bar element can be derived as follows: k?aecr/lc(13) where a is the cross-sectional area of the idealized element, which is equal to sebecause a column length of unity is considered. ecris the tangent modulus of elas- ticity of concrete in the lateral direction. since the exact evaluation of ecrfor concrete in non-circular sections is diffi cult, an approximate method to determine the com- plete material behavior in the lateral direction is pro- posed as explained in section 3.2.2. to account for the separation between concrete and steel at locations where concrete is subjected to tensile stresses, it is assumed that concrete cannot bear any tensile stresses. it has to be mentioned here that a similar type of model has been employed by nishiyama et al. 16 in the case of reinforcedconcretecolumns.thebasicdifference between nishiyamas method and the one used in this fig. 3.idealization of a concrete element. 1335k.a.s. susantha et al. / engineering structures 23 (2001) 13311347 study is that the latter assigns a newly defi ned complete material behavior for concrete elements, and frpis directly determined from the analysis. 3.2.2. material models to be used in concretesteel interaction model to determine the concrete behavior in the lateral direction to be used in the concretesteel interaction model, a method based on the lateral stressstrain relationship of circular columns 8 is adopted. it is implied from eq. (6) that the maximum value of frcorre- sponds to the yield stress of steel. however, in practical situations, this is not the case. it has been observed that the actual maximum lateral pressure is less than the ana- lytical value in eq. (6). in fact, the actual maximum lat- eral pressure is dependent on geometry and properties of constituent materials of a particular column. so, the predicted lateral pressure through the concretesteel interaction model should refl ect all of these dependences. this can be guaranteed by a newly defi ned material behavior for concrete bar elements. it is also obvious that if the concretesteel interaction model is used for a circular tube, the maximum lateral pressure value coincides with the strength of concrete truss elements. so, it is reasonable to assume that th