EXPERIMENTAL AND NUMERICAL MODAL ANALYSIS OF A CONCRETE

LMS PolyMAXA Revolution in Modal Parameter EstimationCurrent Limitations of Experimental Modal Analysis Experimental Modal Analysis (EMA) is currently one of the key technologiesin structural dynamics analysis. Based on the academic fundaments of system identification, it has evolved to become a “standard” approach in mechanical product development. Essential inthis evolution is that modal analysis research has, from the start, taken the point of view of industrial applicability, focusing on solving the specific problems related to testing and modeling large industrial structures. The merit of each new method or new approach has always been checked against the added value it brought in terms of helping the application engineers to derive better models. The result is that EMA is now considered as a “commodity” tool, continuously expanding its application base.While the range of applications where EMA is used is continuously expanding, also the complexity of the tested structures is increasing. While inthe past isolated structures with low damping were analyzed, modal analysis is nowadays also explored on complex structures with high damping, such as on trimmed car bodies. Next to this, EMA has evolved to a standard tool for Finite Element Model updating and is exploited in combination with numerical technologies for hybrid engineering. These recent evolutions have highlighted the current limitations of the EMA process:The task of selecting the correct modelorder and discriminating betweenspurious and structural system poles isquite complex, in particular in the caseof high-order and/or highly dampedstructures. This inevitably results inhighly operator-dependant results, andrequires numerous iterations in theanalysis procedure.Whereas the quality of the currentmodal parameter estimationtechnology is satisfying for undampedor slightly damped structures, thereis an increasing need for better modalparameters estimations for highlydamped structures.Instead of a variety of parameterestimation techniques, each optimizedfor a specific test situation, there is aneed for a single reliable and robustmethod that can be used in a widevariety of applications.In an area where many critics claimno substantial advances were to beexpected, the new LMS LMS PolyMAXmethod brings a revolutionary modalparameter estimation technique thatis easy to use, quick to perform,substantially reduces operator-dependantjudgment, and that delivers high qualitymodal parameter estimations, even oncomplex data.In this application note, the LMS PolyMAXmethod was used on two “historically”difficult data sets: a trimmed car body(high damping) and flutter data (highdata noise). This application note alsoincludes a summary on the analyticalfoundation of the LMS PolyMAX method.•••Figure 1: Stabilization diagrams obtained byapplying different parameter estimation methodsto the Porsche data: (Top) FDPI; (Middle) LSCE;(Bottom) the new LMS PolyMAX method.Using LMS PolyMAX on a Trimmed Car BodyA typical example of a challenging modal analysis application is the identification of a trimmed car. The trim material turns a nicely resonating car body into a highly damped system with large modal overlap. In the present example, data from a Porsche 911 Targa Carrera 4 was used. The accelerations of the fully equipped car were measured at 154 degrees of freedom (DOF), while 4 shakers were simultaneously exciting the structure. This gives a total of 616 FRFs (Frequency Response Functions) used in the modal analysis procedure. More details about the test can be found in [1].The data were analyzed in a frequencyrange from 3.5 to 30 Hz using threetechniques:The frequency domain direct parameteridentification (FDPI) technique, whichhas traditionally been used to analyzedata from highly damped structures[2,3,4,5].The least squares complex exponential(LSCE) method, which is the “industry-standard” time domain estimationmethod [2,3,6].The new LMS PolyMAX method.Figure 1 shows the stabilization diagramsfor the 3 methods. FDPI yields aclearer diagram than LSCE, confirmingthe common assumption that FDPI is•••the preferred method in case of highdamping. However, both are clearlyoutperformed by the LMS PolyMAXmethod: especially at lower frequenciesit is much clearer than the FDPI diagramand also a larger number of stable polesis found.For the subset of LMS PolyMAX polesthat have an FDPI counterpart, Table 1shows that the resulting estimations forfrequency and damping are very close.Also the mode shapes are very similar,which is evidenced by the MAC valuesrepresented in Figure 2.The excellent identification resultsobtained with the LMS PolyMAXmethod are confirmed by comparing theFDPI method LMS PolyMAX method [Hz] [%] [Hz] [%]Figure 3: Comparison of the measured FRFs (grey/green) with FRFs synthesized from the identified modal model (black/red). The FRFs between 2 of the 4inputs and 1 typical output are shown.Figure 4 Stabilization diagrams obtained by applying different parameter estimation methods to data from a partially trimmed car: (Left) FDPI; (Right) the new LMS PolyMAX method.Whereas in the previous example, anexperienced user could have used FDPI to identify a subset of the structural modes, in this case the FDPI stabilization diagram is hardly to be interpreted at all. Again, LMS PolyMAX shows a much better, easy to interpret, stabilization diagram.measured FRFs with the FRFs that can be synthesized from the modal parameters. Figure 3 shows both measured and synthesized FRFs at one response location (FRFs with respect to 2 out of 4 input points). Overall, an impressive 93% correlation between measured and synthesized FRFs was reached.Figure 4 shows the FDPI and LMS PolyMAX stabilization diagrams ofanother trimmed body dataset, where 2 inputs and 500 outputs were measured.Figure 6: Stabilization diagrams obtained by applying different parameter estimation methods to the flight flutter test data: (Left) LSCE; (Right) the new LMS PolyMAX method.LMS PolyMAX: Historical backgroundThe LMS PolyMAX method is a further evolution of the least-squares complex frequency-domain (LSCF) estimation method. That method was first introduced to find initial values for the iterative maximum likelihood method [7]. The method estimates a so-called common-denominator transfer function model [8]. Quickly it was found that these “initial values” yielded already very accurate modal parameters witha very small computational effort [7, 9,10]. The most important advantage of the LSCF estimator over the available and widely applied parameter estimation techniques [2] is the fact that very clear stabilization diagrams are obtained. A thorough analysis of different variantsof the common-denominator LSCF method can be found in [10]. A complete background on frequency-domain system identification can be found in [11].It was found that the identified common-denominator model closely fitted the measured frequency response function (FRF) data. However, when converting this model to a modal model by reducing the residues to a rank-one matrix using the singular value decomposition (SVD), the quality of the fit decreased [9]. Another feature of the common-denominator implementation is thatthe stabilization diagram can only be constructed using pole information (eigenfrequencies and damping ratios). Neither participation factors nor mode shapes are available at first instance [12]. The theoretically associated drawback is that closely spaced poles will erroneously show up as a single pole.These two reasons provided the motivation for a polyreference versionof the LSCF method, using a so-called right matrix-fraction model. In this approach, also the participation factors are available when constructing the stabilization diagram. The main benefits of the polyreference method are the facts that the SVD step to decompose the residues can be avoided and that closely spaced poles can be separated. The method was introduced in [12, 13]. Herewe briefly review the theory.Figure 7: Comparison of the measured FRFs (green/grey) with FRFs synthesized from the identified modal model (black/red). (Top) Sensor at the wing tip; (Bottom) Sensor at the back of the plane.LMS PolyMAX: Theoretical Foundation Data modelJust like the FDPI (Frequency-domain direct parameter identification) method1 [4,5], the LMS PolyMAX method uses measured FRFs as primary data. Time-domain methods, such as the polyreference LSCE method2 [6], typically require impulse responses (obtained as the inverse Fourier transforms of the FRFs) as primary data. In the LMSPolyMAX method, following so-called right matrix-fractionmodel is assumed to represent the measured FRFs:(1)where is the matrix containing the FRFsbetween all m inputs and all l outputs; are thenumerator matrix polynomial coefficients; are the denominator matrix polynomial coefficients and is the model order. Please note that a so-called -domain model (i.e. a frequency-domain model that is derived from adiscrete-time model) is used in (1), with:(2)whereis the sampling time.Equation (1) can be written down for all values of thefrequency axis of the FRF data. Basically, the unknown modelcoefficientsare then found as the Least-Squares solution of these equations (after linearization). More details about this procedure can be found in [12,13].Poles and modal participation factorsOnce the denominator coefficients are determined, the poles and modal participation factors are retrieved as theeigenvalues and eigenvectors of their companion matrix:(3)The modal participation factors are the last m rows of; the matrix contains the (discrete-time) poles on its diagonal. They are related to theeigenfrequencies[rad/s] and damping ratios [-] asfollows ( •* denotes complex conjugate):(4)This procedure is similar to what happens in the time-domain LSCE method and allows constructing a stabilization diagram for increasing model orders and using stability criteria for eigenfrequencies, damping ratios and modal participation factors.Mode shapesAlthough theoretically, the mode shapes could be derivedfrom the model coefficients, we proceed in a different way.The mode shapes can be found by considering the so-calledpole-residue model:(5)where n is the number of modes; denotes complexconjugate transpose of a matrix; are the modeshapes;are the modal participation factors andare the poles (4). are respectively the lower and upper residuals modeling the influence of the out-of-band modes in the considered frequency band. The interpretationof the stabilization diagram yields a set of poles andcorresponding participation factors Since the modeshapes and the lower and upper residuals are the only unknowns, they are readily obtained by solving (5) in a linear least-squares sense. This second step is commonly called least-squares frequency-domain (LSFD) method [2,3]. The same mode-shape estimation method is normally also used in conjunction with the time-domain LSCE method.1This method is implemented as “Frequency – domain MDOF” in Cada-X [3].2This method is implemented as “Time – domain MDOF” both in b and Cada-X [3].,.Comparing LMS PolyMAX with other Modal Parameter Estimators LMS PolyMAX versus LSCEAs may be clear from previous section, the LMS PolyMAX method proceeds along similar lines as the polyreference LSCE time-domain method:Establishment of a set of linear equations for themaximum required model order, from which the matrixpolynomial coefficients can be computed in a least-squares sense.Construction of a stabilization diagram by solving the eigenvalue problem (3) for increasing model orders. The information regarding eigenfrequencies, damping ratios and modal participation factors is contained in this diagram.Based on the user-interpretation of the stabilizationdiagram, computation of the mode shapes and the lower and upper residuals by solving (5) in a least-squares sense.The difference between LSCE and LMS PolyMAX lies in the first step. LSCE uses impulse responses to find the polynomial coefficients, whereas LMS PolyMAX requires frequency response functions.However, this seemingly small difference has bigconsequences for the modal parameter estimation process. It turns out that the new LMS PolyMAX estimator yields extremely clear stabilization diagrams making it very simple to select the “physical” poles. In the LSCE method, the non-physical (and sometimes even the physical) poles tend to “wander” in the stabilization diagram, especially at large model orders. The LMS PolyMAX method has the interesting property that the non-physical poles are estimated with a negative damping ratio so that they can be excluded before plotting them. Such a clear diagram does not mean that some of the poles are missing; on the contrary, more poles can be found with the LMS PolyMAX method, as evidenced by the examples in this application note. Other validation studies also revealed that the LMS PolyMAX method has no problems in correctly estimating modes having a low damping ratio. It is sometimes stated that time-domain methods are preferred in case of low damping, andfrequency-domain methods in case of high damping. The LMS PolyMAX method excels in both cases.LMS PolyMAX versus other frequency-domain methodsTypical for many frequency-domain parameter estimation methods is that they involve the inversion of a matrix containing powers of the frequency-axis of the data.•••Therefore one quickly runs into numerical conditioningproblems and severe constraints apply to both the frequency range as the model order range of the analysis. In the past, it has been proposed to use an orthogonal polynomial basis for the frequency-domain model to solve the numerical problems. However this increases significantly the computational time and memory requirements.The LMS PolyMAX method does not suffer from numerical problems as it is formulated in the -domain (i.e. afrequency-domain model that is derived from a discrete-time model), whereas the existing frequency-domain methods use a Laplace-domain formulation (i.e. a frequency-domain model that is derived from a continuous-time model). In LMS PolyMAX, the frequency axis that extends betweenand is shifted and mapped into a half unit circle in thecomplex plane (2):, (6)Similar to other frequency-domain methods, the LMSPolyMAX method involves the inversion of a matrix containing powers of the frequency-axis of the data. The main advantage of LMS PolyMAX is that taking powers of the -variable does not increase the range of the values, as it boils down to a rotation in the complex plane: . As a result, the LMS PolyMAX method can deal with a large frequency range and very large model orders, speeding up the modal parameter estimation process considerably, as in many cases the complete frequency-band of interest can be processed at once.There was some common belief that the numericalconditioning of frequency-domain methods is worse than time domain methods and that broadband analyses are preferably performed in time-domain [2]. When using the LMS PolyMAX approach, these statements are definitely no longer true.Computational efficiencyThe advantages discussed here have no price in terms of computational time: LMS PolyMAX is as fast as LSCE. LSCE became the industry-standard because of its high speed even for a very large number of measured outputs. A lot of research efforts were spent to achieve this computational efficiency. On current PC platforms, calculation and display of the stabilization diagram for a typical full car body model (like the trimmed car body example discussed here) is in the order of some seconds.References [1] VAN DER AUWERAER H., C. LIEFOOGHE, K. WYCKAERT AND J.DEBILLE. Comparative study of excitation and parameter estimation techniques on a fully equipped car. In Proceedings of IMAC 11, the International Modal Analysis Conference, 627–633, Kissimmee (FL), USA, 1–4 February 1993[2] HEYLEN W., S. LAMMENS AND P. SAS. Modal Analysis Theory and Testing. Department of Mechanical Engineering, Katholieke Universiteit Leuven, Leuven, Belgium, 1995.[3] LMS INTERNATIONAL. The LMS Theory and Background Book, Leuven, Belgium, 2000.[4] LEMBREGTS F., J. LEURIDAN, L. ZHANG AND H. KANDA. Multiple input modal analysis of frequency response functions based direct parameter identification. In Proceedings of IMAC 4, the International Modal AnalysisConference, 589–598, Los Angeles (CA), USA, 1986.[5] LEMBREGTS F., R. SNOEYS AND J. LEURIDAN. Application and evaluation of multiple input modal parameter estimation. International Journal of Analytical and Experimental Modal Analysis, 2(1), 19–31, 1987.[6] BROWN D.L., R.J. ALLEMANG, R. ZIMMERMAN AND M. MERGEAY. Parameter estimation techniques for modal analysis. Society of Automotive Engineers, Paper No. 790221, 1979.[7] GUILLAUME P., P. VERBOVEN AND S. VANLANDUIT. Frequency-domain maximum likelihood identification of modal parameters with confidenceintervals. In Proceedings of ISMA 23, the International Conference on Noise and Vibration Engineering, Leuven, Belgium, 16–18 September 1998.[8] GUILLAUME P., R. PINTELON AND J. SCHOUKENS. Parametric identification of multivariable systems in the frequency domain - a survey. In Proceedings of ISMA 21, the International Conference on Noise and Vibration Engineering, 1069–1082, Leuven, Belgium, 18–20 September 1996.LMS PolyMAX: Executing Modal Testing Jobs at Unrivaled Speedand AccurayWith the new LMS PolyMAX method,a breakthrough in Experimental Modal Analysis has been achieved. Whereas the method equals or even outperforms the current standard LSCE technique on common test structures, it brings a solution for problems -like trimmed body and flutter data- where current EMA technology has shown its limits. By substantially simplifying the analysis process, LMS PolyMAX will be enjoyed by many new users in the field. For advanced applications, its powerfulclear stabilization, and the quality of the modal parameter estimation are real breakthroughs, widely expanding theapplication range and drastically reducing the number of iterations needed.LMS PolyMAX is not yet-another-parameter-estimation-technique, but a global solution for Experimental Modal Analysis.The new LMS PolyMAX is part of the LMS b Structures solution for Modal testing and analysis. LMS b Structures is an integrated suite of applications covering the range of structural dynamic engineering completely. Dedicated applications serve impact hammer measurements, single shaker testing and advanced multiple-input, multiple-output (MIMO) analysis. These starting points measure the motion/force transfer or Frequency Response Functions (FRF) required for modal analysis. A dedicated modal analysis module automatically accesses these measurements to compute the modal parameters: mode shape vector, resonant frequency, damping factor and modal mass. All applications are tightly integrated so that data streams smoothly from acquisition, through analysis, to display and reporting.[9] VAN DER AUWERAER H., P. GUILLAUME, P. VERBOVEN AND S. VANLANDUIT. Application of a fast-stabilizing frequency domain parameter estimation method. ASME Journal of Dynamic Systems, Measurement, and Control, 123(4), 651–658, 2001.[10] VERBOVEN, P. Frequency domain system identification for modal analysis. PhD Thesis, Vrije Universiteit Brussel, Belgium, 2002.[11] PINTELON R. AND J. SCHOUKENS. System Identification: a FrequencyDomain Approach. IEEE Press, New York, 2001.[12] GUILLAUME P., P. VERBOVEN, S. VANLANDUIT, H. VAN DER AUWERAER AND B. PEETERS. A poly-reference implementation of the least-squares complex frequency-domain estimator. In Proceedings of IMAC 21, the International Modal Analysis Conference, Kissimmee (FL), USA, February 2003.[13] PEETERS B., P. GUILLAUME, H. VAN DER AUWERAER, B. CAUBERGHE, P. VERBOVEN AND J. LEURIDAN.Automotive and aerospace applications of the LMS PolyMAX modal parameter estimation method. In Proceedings of IMAC 22, Dearborn (MI), USA, January 2004.LMS International, Empowering Engineering InnovationLMS enables customers to engineer functional performance targets intotheir products, creating and maintaining distinctive brand values. LMS’ unmatched understanding of the product development process is captured through a unique combination of products and services supporting physical and virtual product development: Test systems, CAE software products and engineering services. Critical attributes such as noise and vibration, ride, handling, motion, acoustics and fatigue are turned into a competitive advantage.LMS bLMS b is the world’s first integrated software environment for the functional performance engineering of critical design attributes, such as noise and vibration, ride, handling, comfort, safety and durability.LMS bLMS b provides the most complete portfolio of applications for test-based noise and vibration engineering. It comprises modules for structural, rotating machinery and acoustic testing and analysis, environmental testing, reporting and data sharing.LMS Engineering ServicesThrough its Engineering Services Division, LMS provides vehicle development support from overload contracting and troubleshooting, technology transfer, up to co-development projects.。

015 、017 时钢筋的锈蚀轮廓线 (图 2) 。图 2 清晰地展示了钢筋表面发生不
均匀锈蚀时 ,锈蚀范围逐步扩大到整个钢筋表面的过程 ,且 a 越大 ,钢筋 表面的锈层厚度越厚 。
图 1 钢筋不均匀锈蚀的 动态轮廓线模型
图 2 钢筋表面锈蚀轮廓线
2 混凝土中钢筋不均匀锈胀的数值模拟
锈蚀产物的体积膨胀使钢筋周围的混凝土产生相应的变形 。假设锈蚀产物仅沿钢筋断面的径向发
层厚度系数 ,得到了不同锈蚀程度的钢筋作用于孔洞周围混凝土的锈胀位移 ,用有限元软件 ANSYS810 模拟了锈胀
过程 ,得到了锈胀裂缝扩展到混凝土表面时的临界锈层厚度系数 , 在此基础上推导了胀裂时锈蚀产物量的预测公
式 。通过试验数据对本文提出的方法进行了验证 ,结果表明锈胀开裂的数值模拟和胀裂时锈蚀产物量的预测结果
的钢筋表面与周围混凝土的黏结良好 ,不发生相对位移 ,节点处作固支处理 ,由此来模拟钢筋表面的锈
蚀产物对周边混凝土的作用 。离散化的数值模型采用 ANSYS810 中具有开裂功能的 SOLID65 单元 。为
便于分析 ,不考虑钢筋变形 、锈蚀产物的体积压缩 、荷载作用及其它环境作用的影响 。
由小到大变化锈层厚度系数 a 的取值 ,得到若干组逐渐增大的锈胀位移 ,分别如上所述施加在数
由表 3 、图 3 的算例分析结果可以得出以下结论 : (1) 由
图 3 可见 ,当锈蚀产物量达到临界值时 ,钢筋顶部裂缝扩展 到其正上方的混凝土表面 ,形成了沿钢筋的通长裂缝 ,这与
图 3 试件 OA285916 混凝土保护层 胀裂时的裂纹扩展
文献 [ 4 ] 试验中观察到的现象相吻合 ,说明本文提出的基于
— 71 —
值模型上 ,观察在每组位移作用下混凝土保护层内裂缝的开展程度和规律 。当保护层内的裂缝由钢筋 周边扩展到混凝土表面 ,说明此时混凝土表面已出现了可见裂缝 ,此时的 a 值即为导致保护层胀裂的 临界锈层厚度系数 acrit 。

混凝土SHPB 试验的数值模拟秦佳俊1，2，夏 松2，陈贵炫3， 杨富莲2（1．广州大学 土木工程学院，广东 广州 510006；2．皖西学院 建筑与土木工程学院，安徽 六安 237012；3．广东工业大学 土木与交通工程学院 广东 广州 510006）摘 要：利用有限元软件ANSYS/LS-DYNA，运用HJC 本构模型，对混凝土试件的SHPB 试验进行数值模拟，重构了混凝土在低、中、中高以及高应变率下的应力应变曲线，并且对试件的破坏形态进行了模拟。

研究表明，HJC 模型可以很好地模拟混凝土材料的力学性能，重构的应力应变曲线在曲线形态和峰值应力上均与SHPB 试验拟合良好，数值模拟得到的混凝土的破坏形态也与SHPB 试验一致，确定该强度混凝土材料HJC 本构模型材料参数的方法具有通用性。

关键词：混凝土；应力应变曲线；SHPB 试验；HJC 本构模型中图分类号：TU502.6 文献标识码：A 文章编号：2095-8382(2016)06-048-05Numerical Simulation of SHPB Experiment for ConcreteQIN Jiajun 1,2，XIA Song 2，CHEN Guixuan 3， YANG Fulian 2(1. Civil Engineering School,Guangzhou University, Guangzhou ,510006,China;2.Architecture and Civil Engineering School,West Anhui University,Lu’an ,237012, China;3.Civil and Transportion Engineering School, Guangdong University of Technology,Guangzhou,510006, China;)Abstract ：By using finite element software ANSYS/ls-dyna,the SHPB test for concrete has been numerically simulated by using the HJC model,the stress strain curve of concrete at different strain rates are reconstructed,and the failure patterns of the specimens were simulated.The research shows that the HJC model can well simulate the mechanical properties of concrete,the stress strain curves are in agreement with the SHPB test on the curve shape and the peak stress,and the failure pattern of concrete obtained from the numerical simulation is also consistent with the SHPB test. The method for determining material parameters of HJC constitutive model of concrete material is universal.Keywords ：concrete;stress-strain curve;SHPB test;HJC model混凝土因为其较强的耐久性和较高的强度，广泛用于工业建筑、民用建筑、道路、桥梁以及防爆结构和军事设施中，这些建筑在承受静力荷载的同时，还要承受较大的动力荷载的作用，比如，工业建筑受到吊车荷载的作用、高层建筑受到地震荷载和风荷载的作用、路面受到汽车荷载的冲击作用等。

Experimental investigation of size effect on mechanical properties of carbon fiber reinforced polymer (CFRP)confined concrete circularspecimensJikai Zhou ⇑,Fengtong Bi,Zhiqiang Wang,Jian ZhangCollege of Civil and Transportation Engineering,Hohai University,Nanjing 210098,Chinah i g h l i g h t sStrength of unconfined specimen has apparent size effect.The mechanical properties of CFRP confined concrete cylinders have size effect. A modified stress-strain model with consideration of size effect was established.a r t i c l e i n f o Article history:Received 27April 2016Received in revised form 21September 2016Accepted 6October 2016Keywords:FRP confinementConfined concrete mechanical properties Confinement ratio Stress-strain modela b s t r a c tThe mechanical properties of concrete have significant size effect.Size effect of carbon fiber reinforced polymer (CFRP)confined concrete specimens should be investigated further.In this study,the uniaxial compressive tests of CFRP confined concrete cylinders in different sizes (70mm,100mm,150mm,190mm,310mm)under different confinement (CFRP of one layer,two layers and three layers)were con-ducted.Four parameters were obtained,including interception strength,uniaxial compressive strength,strain of turning point and peak strain.From the test results,it can be found that the uniaxial compressive strength of unconfined concrete cylinders has apparent size effect which accords with the Weibull size effect law.Under different ratios of CFRP confinement,the CFRP confined concrete cylinders have appar-ent size effect.The stress-strain model proposed by Lam and Teng fails to interpret how size effect acts on the mechanical properties of specimens.Thus,based on the stress-strain model proposed by Lam and Teng,a modified stress-strain model with the consideration of size effect was established.Ó2016Elsevier Ltd.All rights reserved.1.IntroductionFiber reinforced polymer (FRP)has excellent corrosion resis-tance and prominent tensile strength.This composite material becomes more and more popular in the field of structure strength-ening and maintenance.Some related applications in the practical engineering show that FRP confined concrete could reinforce the strength and ductil-ity [1]of plain concrete specimens.Stress-strain curve of FRP con-fined concrete [2–4]shows two phases:the initial part and the confining part (Fig.12).The initial part of the stress-strain curve indicates that the confinement of FRP at this stage has not been engaged.The confining part means the FRP confinement has been activated gradually by the expansion of confined concrete.An apparent turning point could divide the two different parts.Stress-strain curve of the FRP confined concrete at the confining part may present two main tendencies,the hardening one and the softening one.The distinction between two tendencies is the confinement degree of FRP.Wang and Wu [5]used the confine-ment ratio (FRP confining pressure to concrete compression strength)to differentiate the strengthened stage and the softened stage of stress-strain curve of the FRP confined concrete.Experimental results have shown that the mechanical properties of concrete vary with the size of specimens [6–16].The property may be called size effect.Size effect of strength is the intrinsic property of concrete.The main reason for size effect of concrete strength is the non-homogeneous internal structure character of concrete.Size effect may not only be caused by con-crete non-homogeneity but also the type of FRP,even wall effect [17].All the above factors make research conclusions discrepant.In addition,most of experiments are conducted on relatively/10.1016/j.conbuildmat.2016.10.0390950-0618/Ó2016Elsevier Ltd.All rights reserved.⇑Corresponding author.E-mail addresses:zhoujikaihhu@ (J.Zhou),273205236@ (F.Bi),wzq1603@ (Z.Wang),286106650@ (J.Zhang).small-scale specimens,while the research on relatively large-scale specimens is limited.Size effect of strength does exist.However,there is still a debate about size effect on the strength of concrete[18].It is obviously that the research on size effect of concrete strength[18–20],espe-cially the strength of FRP confined concrete specimens is just a beginning.The size effect of concrete strength can be explained by Weibull distribution[21].Few studies introduced the applica-tion of basic theory of size effect on strength.Many stress-strain relationships of FRP confined concrete were established by a lot of researchers[22–32].In the investigation of Lam and Teng[19,22],all together seven strength models were compared and analyzed.Model of Lam and Teng has different value in k1from other models.Moreover,Lam and Teng claimed that k1may be related with diameter or ratio of the height to the diameter,this is the basic assumption about the rule of size m and Teng collected the models and selected data in detail.The data of adhesive failure and FRP tube filled with concrete were excluded.Data selected by them are effective and consistent in condition.The stress-strain model pro-posed by Lam and Teng[22]is widely accepted by many research-ers[23–31,33–39].The model has advantage in applicability, which is adopted in this paper to research the size effect factor m and Teng proposed design-oriented stress-strain model [22]for FRP confined concrete and analysis-oriented stress-strain model[19]for FRP confined m and Teng modified their design-oriented stress-strain model for FRP confined concrete a few years later[32].The modified model can precisely describe the characteristics of the descending branch of the FRP confined concrete stress-strain curve.In order to investigate the size effect on mechanical properties of CFRP confined concrete specimens,the uniaxial compression exper-iment using concrete cylinders infive different sizes under three dif-ferent CFRP confinements were conducted.A control group of unconfined concrete specimens in each size were also set.Based on the experimental data and the design-oriented stress-strain model proposed by Lam and Teng[22],a stress-strain model of CFRP confined concrete considering size effect has been established.2.ExperimentsSpecimens under different confinement ratios with the same height-diameter ratio h/d=2(diameter70mm,100mm, 150mm,190mm,310mm)were used in the experiment.The CFRP confinement for specimens of each size CFRP were one layer CFRP,two layers CFRP,three layers CFRP,respectively.Also,uncon-fined specimens for each size were included.Concrete design level is C30(cylinder compressive strength). Water cement ratio is0.49and the water is205kg/m3.Cement is 32.5ordinary Portland cement with the weight of418kg/m3. Weight offine aggregate is613kg/m3.Coarse aggregate graduation varied from5mm to20mm with the weight of1164kg/m3. Moulds of specimens were made using polyvinyl chloride(PVC) tubes which have same sizes as the circular specimens.Concrete specimens cured for three weeks.A total of eight strain gages were pasted on the middle position of the specimen lateral surface.Eight strain gages were divided into four groups. Each group includes a vertical and a horizontal strain gages to obtain the vertical displacement and the horizontal displacement, respectively.Four groups of strain gages were distributed around the specimens.All the adjacent strain gages are equally-spaced. All the strain gages were sealed and insulated using silica gel. Wrapping CFRP around the specimens came after the solidification of the silica gel,as shown in Fig.1.The thickness(t)of CFRP is 0.167mm.The tensile strength of CFRP is3500MPa and elastic modulus of CFRP is242GPa.The testing system was calibrated before testing.Maximum load capacity of the loading equipment is5000kN.The data were collected automatically.Hydraulic valve was used to control the loading procedure.At the start,the speed of the load would be increased by2kN/s when the real-time curve was smooth.The speed of the load increased by1kN/s after the slope of the real-time curve declined.In the middle of height of specimens,an obvi-ous expansion showed up.The specimens were at ultimate condi-tion.Stress and strain(vertical and horizontal)would be highlighted and the speed of the load increased by0.5kN/s until the load could not increase any more.The development of load and strain performed in a mild way before the peak load.At the peak load,the destruction of unconfined concrete speci-mens began with vertical cracks.Concrete in the mid-height dilated then the diagonal crack appeared till the failure.Failure mode of CFRP confined concrete specimens had nothing to do with the layer of CFRP.CFRP jackets dilated in the middle of the CFRP surface with an obvious deformation.CFRP ruptured when the failure occurred.CFRP cohered well with concrete along the vertical line.A crisp and loud noise showed by the time of fail-ure.The destroyed specimens were divided into two conical parts, the upper ones and the below ones.Both parts damaged entirely. As shown in Figs.2and3.Nomenclatured150diameter of reference specimens(d150=150mm)d diameter of actual specimensE1initial elastic modulus of concreteE2tangent modulus of second stage in stress-strain curvef0 cc uniaxial compressive strengthf0 c;i interception strength(interception point between the opposite extension line of the second linear part in stress-strain curve and the stress axis)f0c0;d uniaxial compressive strength of unconfined concretef0c0;150uniaxial compressive strength of unconfined specimens with certain diameterf0c0;150uniaxial compressive strength of unconfined specimens with150mm diameterf0 cc;d uniaxial compressive strength of confined concrete with certain diameterf frp tensile strength of FRP f cu,k value of cube compressive strength of concrete k1coefficient of confinement effectivenessm degrees of homogeneity,m=38.51t thickness of FRPV volume of specimensa Ec ratio of elastic modulus,a E c¼E2E1b rtrelative confining pressure,b rt¼r tf0c0;de1turning point straine c0ultimate strain(corresponding to peak stress) r0stress of threshold value,r0=36.66MPar t stress of lateral confinementn parameter of Eq.(11)e cc strain of confined concreter cc stress of confined concrete644J.Zhou et al./Construction and Building Materials127(2016)643–6523.Experiment results and dataData from the experiment were classified into four tables(from Table1–4).Explanation of parameters in the Tables was just behind Table4.Notice:f cu,k–uniaxial compressive strength;E1–ultimate strain(corresponding to peak stress);E2–interception strength (interception point between the opposite extension line of the sec-ond linear part in stress-strain curve and the stress axis);e1–turn-ing point strain;According to the data from Tables1–4,the uniaxial compressive strength of all the conditions was summarized in Fig.4.The results of experiment are shown as following:(1)Uniaxial compressive strength of concrete specimens uncon-fined by FRP has significant size effect.The uniaxial com-pressive strength decreases with the increasing of size, while the range of variation is limited.(2)Uniaxial compressive strength of CFRP confined concretecylinders has a significant size effect.The range of variation is huge,but it becomes mild as the increasing of size.(3)The confinement degree of CFRP has a remarkable influenceon the uniaxial compressive strength of concrete cylinders, which increases the strength a lot.4.Analysis of size effect4.1.Weibull size effectWeibull used the weakest link theory to analyze and describe the size effect of strength[21].Concrete material has size effect and discreteness in strength[40].The size effect in strength can be explained by statistical theory of size effect.Statistical theory explain that the probability offlaw in concrete increase as the increasing of physical dimension.That is to say,the larger the size, the lower the strength and the more obvious the size effect to be. The relationship can be explained by the weakest link theory. Weibull established statistical theory to analyze the size effect [40].Weibull claimed that size effect may caused by random distri-bution of material strength.Due to the randomness of strength,the probability of low strength element appearance increases as the increasing of size.4.2.Analysis of size effect for unconfined concrete specimensAnalysis of compressive strength of unconfined concrete speci-mens(Application of the Weibull distribution law).Fitted equation(as Fig.5shows):ln f0c0¼ln r0Àln V mð1Þwhere f0c0;dis the uniaxial compressive strength of unconfined concrete,V is the volume of specimens,r0and m are the Weibull modulus:r0=36.66MPa,m=38.51.Substitute the Weibull modulus into Eq.(1),show as follows:f0c0;d¼expð3:59À0:078ln dÞð2Þwhere f0c0;dis the uniaxial compressive strength of unconfined specimens with a certain diameter.Value calculated by Eq.(2)is compared with experimental value from Tables1–4.According to Table5above,specimens’relative error with a diameter of150mm is much higher,while the rest of specimensa Pasting the strain gagesb Wrapping CFRPFig.1.Pasting the strain gages and wrapping CFRP.a Prior to testb After compressive failureFig.2.Unconfined concrete specimens.J.Zhou et al./Construction and Building Materials127(2016)643–652645maintain in a reasonable level.Based on the specimens with the diameter of 150mm,strength equation considering size effect is collected below.f 0c 0;d ¼f 0c 0;150exp ð0:39À0:078lnd Þð3Þ4.3.Analysis of size effect of CFRP confined concrete specimensCFRP can provide the concrete cylinders with adequate hoop direction confinement to enhance the uniaxial mechanical proper-ties of specimens.The principle is that the tri-axial stress state makes substantial improvement on uniaxial compressive of con-crete cylinders.Under the uniaxial load,the concrete dilates and then the Poisson effect occurs.CFRP presses the lateral surface,which is so-called the stress of lateral confinement.The rule of confinement is shown in Fig.6.An equation can be deduced by means of the static equilibrium.r t ¼2f frp t dð4ÞTable 1Unconfined concrete specimens.Size No.f 0c 0;d /MPaf 0c 0;d /MPae c 0/le e c 0/led/mm h/mm 70140S10626.025.814901508S10426.51532S11624.91504100200S22024.023.615941500S20223.11530S21823.71378150300S31528.328.119811670S30829.21536S32026.91493190380S41028.823.719661532S41423.71226S42023.51404310620S51027.622.215801544S50919.41296S50822.21756Table 2Concrete specimens confined with one layer CFRP.Size No.f 0cc /MPaf 0cc /MPaf 0c ;i /MPaf 0c ;i /MPae 1/le e 1/le e c 0/le e c 0/led/mm h/mm 70140S10782.961.941.240.33287429010,80213,179S11260.038.7597617,652S11856.337.3299213,504S12069.544.1490410,756100200S20150.853.630.631.92079262610,7059270S20368.634.5411912,965S20954.133.022408039S21251.429.320665369S21457.9///150300S30239.447.1/29.31642225431345963S30342.629.022834851S30647.724.914844313S30950.437.025839561S31947.535.332767958190380S40248.047.625.629.02143251561836616S40849.632.626246385S40948.224.822035867S41646.927.926445753S41845.134.029598891310620S51141.139.618.922.71828177024972576S51241.018.718912422S51340.621.819602572S51437.126.619313937S51538.127.412391450a Prior to testb After compressive failureFig.3.CFRP confined concrete specimens.646J.Zhou et al./Construction and Building Materials 127(2016)643–652Table3Concrete specimens confined with two layers CFRP.Size No.f0cc /MPa f0cc/MPa f0c;i/MPa f0c;i/MPa e1/le e1/le e c0/le e c0/led/mm h/mm70140S10391.2102.042.437.23439279915,33112,667 S110113.8///S111100.932.0215810,003100200S20773.871.9734.836.32232380419,45716,368 S20866.350.3635611,201S21975.823.8282518,446150300S30166.363.524.726.93216311511,83912,180 S31659.9///S31864.329.0301312,520190380S40555.157.824.922.32453238864615462 S40758.015.616303151S41260.326.530806774310620S51648.146.737.334.32193288363666832 S51947.730.936988780S52044.134.827585349Table4Concrete specimens confined with three layers CFRP.Size No.f0cc /MPa f0cc/MPa f0c;i/MPa f0c;i/MPa e1/le e1/le e c0/le e c0/led/mm h/mm70140S105122.5126.7/55.8/5581/34,567 S108120.243.8572858,360S114137.567.7543310,773100200S205105.9108.243.734.830722244963414,642 S215108.235.2208123,410S211130.925.5158010,882150300S30477.672.1/46.4/3690/12,086 S30772.754.6490616,507S31266.138.124747664190380S40164.459.531.236.039624408962011,425 S40462.635.4504017,944S41351.541.342216711310620S51754.354.531.428.03672294947316764 S51857.231.0337710,568S52152.121.717974994J.Zhou et al./Construction and Building Materials127(2016)643–652647where a t is the stress of lateral confinement,f frp is the tensile strength of FRP,t is the thickness of FRP,d is the diameter of concrete cylinder.Existing model of FRP confined concrete usually takes the following form:f 0cc ;d f 0c 0;d¼1þk 1r tf 0c 0;dð5Þwhere f 0cc ;d is the uniaxial compressive strength of confined concrete with certain diameter,f 0c 0;d is the uniaxial compressive strength of unconfined concrete with certain diameter,k 1is the coefficient of confinement effectiveness.Stress of lateral confinement was calculated by Eq.(5).A stress of lateral confinement (r t )–uniaxial compressivestrength of concrete (f 0cc ;d or f 0c 0;d )curve clearly described the rela-tionship between the two parameters.From Fig.7,it can be found that in a range of stress of lateral con-finement (r t ),uniaxial compressive strength (in different sizes)had an obvious liner relation with the stress of lateral confinement.The coefficient of confinement effectiveness k 1could be calculated by putting the data from Table 6to Eq.(5).Diameter d and coefficient of confinement effectiveness k 1would suit to the fitted Eq.(6):k 1¼2:14þ0:09dd 1503ð6Þwhere d 150is the diameter of reference specimens (d 150=150mm),d is the diameter of actual specimens.It is worth to notice that Eqs.(2),(5)and (6)can be used to deduce the stress-strain model of CFRP confined concrete cylin-ders.The model takes the size effect into account.5.Stress-strain model of CFRP confined concrete cylinders considering size effect5.1.Analyzing stress-strain model of CFRP confined concrete cylinders in different sizes5.1.1.Stress-strain curvesAll the data in Fig.8are from Tables 1–4.Stress-strain curves were depicted automatically by data acquisition instrument.The obtained representative curves were shown as following.The curve of confined specimens has two stages.In the first stage,the stress increases rapidly while the strain does not.But the growth trend of stress and strain in the second stage is contrary to it in the first stage.5.1.2.Interception strength of CFRP confined concrete,relative confining pressure and elastic modulusInterception strength f 0c ;i is the interception point between the opposite extension line of the second linear part (elastic modulus E 2)in the stress-strain curve and the stress axis.The interception strength to uniaxial compressive strength of unconfined concrete ratio is called the increase coefficient of interception strengthf 0c ;if 0c 0;d .While the ratio of stress of lateral confinement to uniaxial com-pressive strength of unconfined concrete is defined as relative con-fining pressure b r t ¼r tf 0c 0;d.E 1is the initial elastic modulus of concrete.All the parameters are shown in Table 7.5.1.3.Analysis of experiment data Fitted curve expression:Ultimate strain :e c 0¼2000þ12;600b r t ;R 2¼0:94ð7ÞStrain of turning point :e 1¼2000þ1400b r t ;R 2¼0:94ð8ÞAccording to the above figures,the strain increases with the increasing of relative confining pressure.Fitted curve expression (comparison between fitted expression and data from Figs.9–11):Increase coefficient interception strength:Increase coefficient interception strength :f 0c ;if 0c 0;d ¼1þ0:49b r t ;R 2¼0:98ð9ÞRatio of elastic modulus :E 2E 1¼0:14ð1Àe À5:2b r t Þ;R 2¼0:98ð10ÞTable 6Stress of lateral confinement (r t ).Layer of CFRPStress of lateral confinement r t /MPa 70mm100mm 150mm 190mm 310mm 116.7011.697.79 6.15 3.77233.4023.3815.5912.307.54350.1035.0723.3718.4511.31Table 5Comparison of unconfined concrete uniaxial compressive strength between calculated value and experimental value.d/mm70100150190310Experimental value/MPa 25.823.628.123.722.2Calculated value/MPa 26.0225.3124.5224.0823.17Relative error/%À0.85À6.7614.60À1.58À4.19648J.Zhou et al./Construction and Building Materials 127(2016)643–652Coefficient of interception strength increases with the increasing of relative confining pressure.Meanwhile,the ratio of elastic modu-lus increases with the increasing of relative confining pressure.5.2.Establishing stress-strain model of CFRP confined concrete cylinders(considering size effect)5.2.1.Stress-strain model of CFRP confined concrete cylinders considering size effectThe experiment results showed that the stress-strain curve of CFRP confined concrete has a feature of m and Teng proposed the design-oriented stress-strain model[22]for FRP-confined concrete.In this study,the design-oriented stress-strain model[22]for FRP-confined concrete from Lam and Teng was taken as a reference to propose a stress-strain model of CFRP con-fined concrete cylinders considering size effect.The curve is shown in Fig.12.The model proposed in this study hasfive assumptions.(1)The line-type of the curve in thefirst stage is a parabola.(2)The line-type of the curve in the second stage is a straightline.(3)Interception strength,turning point strain and elastic modu-lus of the second stage is related to relative confining pressure.(4)The curve is connected smoothly between thefirst stage andthe second.(5)Peak strength and ultimate strain were reached at the end ofthe second stage in the curve.Based on the assumptions above,the new stress-strain model is shown as following,r cc¼f0c;i1À1Àe cc ec0nh ic06e cc6e1r cc¼f0c0þE2e cc e16e cc6e c0ð11Þn¼2À160ðf cu;kÀ50ÞE2¼0r t¼00:15E1r t>0&whereJ.Zhou et al./Construction and Building Materials127(2016)643–652649f 0c ;i is the interception strength,f 0c ;i ¼f 0c 0;d ð1þ0:49b r t Þ,d is the diameter of specimens,f 0c 0;d is the uniaxial compressive strength ofunconfined concrete (with the diameter d),f 0c 0;d ¼f 0c 0;150exp ð0:39À0:078lnd Þ,f cu ,k is the value of cube compres-sive strength of concrete (see notation).E 1is initial elastic mod-ulus of concrete and E 2is tangent modulus of second stage in stress-strain curve,e 1is the strain of turning point,e c0is theultimate strain,b r t is the relative confining pressure,b r t ¼r t f 0c 0;d,r t is the stress of lateral confinement.The ultimate strain e c0can be either estimated by Eq.(7)or deduced accurately by Eq.(12).e c0¼f 0cc ;d Àf 0c ;i a Ec E 1ð12Þwhere f 0cc ;dis the uniaxial compressive strength of CFRP confined concrete (with the diameter d)and a Ec is the ratio of elastic modu-lus,a E c ¼E 2E 1.Substitute Eqs.(3),(5),(6)–(12),thene c0¼1:65þ0:09d 150ÀÁ3a Ec E 1r tð13ÞThe parameters’meanings are the same as the ones above.Table 7Interception strength,Elastic modulus and Relative confining pressure (Average in each size and confinement).Size t/mmf 0c ;i /MPaf 0c ;if 0c 0;dE 1/GPa E 2/GPaE 2E 1r tc 0;dd/mm h/mm 7014025.8123.7––00.16740.3 1.5621.7 2.40.110.650.33437.2 1.4424.0 5.10.21 1.290.50155.8 2.1625.9 4.00.15 1.9410020023.6125.8––00.16731.9 1.3528.4 3.00.110.500.33436.3 1.5426.4 3.60.140.990.50134.8 1.4831.2 3.90.13 1.4915030028.1124.4––00.16729.3 1.0430.4 3.50.120.280.33426.90.9633.1 3.20.100.550.50146.4 1.6527.0 2.80.100.8319038023.7127.3––00.16729.0 1.2328.0 3.50.130.260.33422.30.9526.7 3.70.140.520.50136.0 1.5323.2 2.20.090.7831062022.2123.9––00.16722.7 1.0229.4 2.20.070.170.33434.3 1.5525.4 1.70.070.340.50128.01.2625.94.40.170.51650J.Zhou et al./Construction and Building Materials 127(2016)643–652The calculated values of the ultimate strain with different rela-tive confining pressure and sizes were depicted in Fig.13.Accord-ing to Fig.13,it is clear that the calculated values have a good agreement with the experimental results.parison and discussion between the proposed model and Lam and Teng modelSince size effect has been considered in the modified model,there is necessity to verify the reliability of the modified model.The model proposed by Lam and Teng [22](as reference shows)did not take the size as a input value,which means the size effect is ignored.In the modified model,the size effect is taken as the parameter in the stress strain model,which leads to significant change.And the model is also different from other models.In addition,the interception strength f 0c ;i (interception point between the opposite extension line of the second linear part in stress-strain curve and the stress axis)has an essential differencefrom f 0c 0;d (compressive strength of unconfined concrete speci-mens)in the model proposed by Lam and pared with f 0c 0;d ,the parameter f 0c ;i accords with the actual situation better.So the model proposed in the study may instruct the structuralengineers to design and calculate the CFRP confined concrete struc-ture more accurately.6.ConclusionsThe purpose of the paper is to investigate the mechanical prop-erties of CFRP confined concrete cylinders with different sizes under different confinements.On the basis of the stress-strain model proposed by Lam and Teng,a modified stress-strain model (considering the size effect)of CFRP confined concrete cylinder was proposed.The conclusions can be obtained as following,(1)Lam and Teng design-oriented stress-strain model has goodaccuracy and applicability in design.So the modification also has reliability for confined specimens with ascending branch at the plastic stage.Stress-strain model of CFRP confined con-crete cylinders under different confinement (considering size effect)was pared with the original model,the modified model makes the size parameter as the input value.(2)The confinement degree of CFRP has a significant influence onthe uniaxial compressive strength of concrete cylinders.The confinement degree of CFRP,can increase the strength greatly.Uniaxial compressive strength of CFRP confined con-crete cylinders under different confinement pressure showed evident size effect.While the ductility showed slight size effect.(3)The uniaxial compressive strength model of CFRP confinedconcrete cylinders under various confinements was established.(4)There still have been some divergences about the size effect sofar.Main reason for this point is that the material property of concrete is non-homogeneous.Meanwhile,size effect of FRP confined large-scale specimens should be investigated further.7.FundingThe work was supported by the National Natural Science Foun-dation of China (Grant No.51479048).References[1]R.Abbasnia,F.Hosseinpour,M.Rostamian,H.Ziaadiny,Cyclic and monotonicbehavior of FRP confined concrete rectangular prisms with different aspect ratios,Constr.Build.Mater.40(2013)118–125.[2]R.K.Akogbe,M.Liang,Z.M.Wu,Size effect of axial compressive strength ofCFRP confined concrete cylinders,Int.J.Concr.Struct.Mater.5(2011)49–55.[3]S.B.Batdorf,H.L.Heinisch,Weakest link theory reformulated for arbitraryfracture criterion,J.Am.Ceram.Soc.61(1978)355–358.[4]Z.P.Bazant,E.P.Chen,Scaling of structural failure,Appl.Mech.Rev.50(10)(1997)593–627.[5]Y.F.Wang,H.L.Wu,Size effect of concrete short columns confined with aramidFRP jackets,pos.Constr.15(4)(2011)535–544.[6]S.Jie,Experimental study on the size effect of concrete uniaxial compressivebehavior,Ind.Constr.42(6)(2012)122–126.[7]B.Csuka,L.P.Kollár,Analysis of FRP confined columns under eccentric 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