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Analysis of Mixture Models Using Expected Posterior Priors,with Application to Classification of Gamma Ray BurstsJOSE M.PEREZ and JAMES O.BERGERUniversidad Simon Bolivar,Venezuela andDuke University,USAAbstract:Consider observations distributed according to a mixture of compo-nent densities with different parameters.In the Bayesian framework,it is notpossible to perform a statistical analysis of the mixture using an improperprior for the component parameters,since the posterior distribution does not ex-ist.To overcome this difficulty,we propose use of the expected posterior priorapproach of Perez and Berger(1999).Besides providing suitable default priorsfor general mixture models,a key advantage of the use of expected posteriorpriors is that they can be used in conjunction with Markov Chain Monte Carlomethods,even when the number of components is unknown.An application isconsidered involving Gamma Ray Bursts,modeled as arising from a bivariatenormal mixture model with measurement errors on the observations.Keywords:MIXTURE MODELS;DEFAULT PRIORS;CLASSIFICATION;MCMC.1.INTRODUCTIONMixture models have been used in situations including pseudo-parametric density estima-tion,clustering,change point problems and image analysis(McLachlan and Basford,1985; Escobar and West,1995;Roeder and Wasserman,1997).In Bayesian analysis,mixture models are typically analyzed using Markov Chain Monte Carlo simulation(MCMC).See, for example,Diebolt and Robert(1994),Escobar and West(1995)and Richardson and Green(1998).Unfortunately,with mixture models it is typically not possible to perform default anal-ysis with standard noninformative priors.In this work,we demonstrate that the expected posterior prior approach(Perez and Berger,1999)can be used to provide a default Bayesian analysis for this problem.In section2,we define the expected posterior prior for mixture models.In section3,a reversible jump MCMC scheme(Richardson and Green,1998)is presented for computation in mixture models with these priors.Application involving a bivariate normal mixture is considered in section4in the determination of the number of classes of Gamma Ray Bursts(Meegan et al.,1995).1.1NotationConsider thefinite mixture model,with independently distributed observationsfor.Here,represents the unknown number of components.Conditional on, the weights of the mixture are given by,where is the probability of an observation coming from component and.As indicated by the notation, we assume that each component of the mixture is a density of a given parametric form,but with different values of the parameters(typically multivariate).Let. (Strictly,we should write the components of and as and,but we will suppress the index to prevent notational overload.)For convenience,it is often useful to define latent variables,,represent-ing the unknown allocation of the observations into each component.In this way, indicates that observation comes(uniquely)from component.1.2BATSE gamma ray burstsThe proposed method will be applied in section4to analysis of the third BATSE catalog of gamma ray bursts.The third BATSE catalog contains1122measurements performed by the Compton Gamma Ray Observatory between1991and1994(Meegan et al.,1995).We utilize only the745observations that have no missing data and which have not beenflagged as being possibly erroneous in some way.The catalog also includes the known standard deviations of the measurement errors for the duration and hardness ratio of each burst,and these will be taken into account in the analysis in section4.Of interest here is classification of the gamma ray bursts in terms of their duration and hardness ratio.Hurley(1991)observed a bimodal distribution for the duration,in the log scale.A correlation between the hardness ratio and the duration had also been observed.It is thus natural to attempt to classify the gamma ray bursts based on the bivariate data.We assume that T90HR,where the are bivariate normal with mean zero and the standard deviations given in the BATSE catalog,and the arise from a mixture of bivariate normal distributions with completely unknown parameters.The number of components of the mixture,,is also considered unknown.This combination of a completely unknown mixture of bivariate normal distributions,with observations carrying measurement errors,has not,to our knowledge,been previously considered.1.3Mixture model and prior structureThe priors for the parameters of the mixture are typically defined in a hierarchical setup (Richardson and Green,1998).The joint density of all variables can be written in a hierar-chical form aswhere and.The following structure for the various probability distributions is assumed:is the probability that there are components;this is chosen to be uniform on in our default analysis.is a Dirichlet distribution with known parameter;in our default analysis,,resulting in a uniform distribution for.The allocation variables,,are i.i.d.withThe likelihood;in our examples,will be anormal density(of the appropriate dimension),and will denote the unknownmean and covariance parameters corresponding to component of the mixture.Choice of the prior for the component parameters,,is the major focus ofthis paper.1.4BackgroundThe component parameter priors,as specified above,must be proper or the posterior dis-tribution will be improper(Diebolt and Robert,1994;Shui,1996;Roeder and Wasserman, 1997).Unfortunately,the large number of parameters that typically arise in mixture models makes it very difficult to assess subjective proper priors.(Also,in our major example,the gamma ray bursts are completely unknown phenomena and so no subjective information concerning them is available.)It is thus desirable to have default or automatic priors for analysis of mixture models.Note that standard default priors in Bayesian analysis(e.g. Jeffreys-rule or reference priors)are typically improper,and so cannot be used directly here.In order to avoid this problem,Diebolt and Robert(1994)changed the probability distribution of in(2)so that each component contains sufficient observations so that the marginal isfinite.This method was given an alternative description in Wasserman(1998) and shown to have a number of desirable frequentist properties for large samples.On the whole,we feel that this method is very attractive but it does correspond to a presumption of more information than is actually present,which might cause concerns in smaller sample sizes.The use of arbitrary“vague proper priors”does not solve the problem,as has been pointed out by many(Diebolt and Robert,1994;Shui,1996).Thus,for the normal univari-ate case,Richardson and Green(1998)suggest that the degree of vagueness be chosen in a data-dependent way.Although a bit adhoc,the results produced by this approach seem to be sensible.It is unclear,however,how to make this a general approach,especially when the mixture components can be higher dimensional distributions.Several other approaches have been proposed in the literature(Robert,1996;Mengersen and Robert,1996;Roeder and Wasserman,1997;Shui,1996).Some of these approaches utilize a prior structure in which the components are linked to a common parameter with a flat prior,so that all observations contribute to its estimation.Shui(1996)proposes use of resampling from the observations in order to produce versions of the Intrinsic Bayes Factor (Berger and Pericchi,1996)and the Fractional Bayes Factor(O’Hagan,1995)which canthen be used for model selection.While attractive,this approach is extremely computa-tionally intensive.A direct‘intrinsic prior’approach,whose relationship to our approach is discussed later,is given in Moreno and Liseo(2000).2.EXPECTED POSTERIOR PRIORSExpected posterior priors(or EP priors)were introduced in Perez and Berger(1999)asa means to develop priors suitable for comparison among models,, having unknown parameters,when the standard noninformative priors,,areunsuitable because of impropriety.The EP prior approach considers(possibly imaginary) ,,such that is a proper density for all.It is recom-mended that the size of the training samples be as small as possible while guaranteeing thepropriety of these posteriors.The space of possible training samples,,is assumed to have a density,common to all models under consideration,and the EP prior for is defined asEven if is improper,the EP priors will be“well calibrated”for model compar-ison,because they are all based on the same(convolved with proper distributions). For further discussion of this,see Perez and Berger(1999).Since choice of in mixture models can be viewed as model selection,EP priors can be directly applied to mixture models with improper priors.Thus we consider the mixture model in(1),and it is natural to assume that the initial prior for,given,is a nonin-formative improper prior of the form.Let be the maximum number of components and consider imaginary training samples,where is viewed as being an imaginary sample arising from component of the mixture and is of minimal size such that exists.will denote the space of such imaginary minimal training samples.Perez and Berger(1999)consider a variety of possible choices of the marginal. The two main suggestions from that paper are the empirical marginal and the base model marginal.The empirical marginal is defined by randomly drawing training samples,with replacement,from the actual observations.(One could also do this without replacement, but this is somewhat more difficult computationally.)The empirical marginal is a proper discrete distribution,so that is clearly proper.(The integral in(3)is,of course, then to be interpreted as a sum.)The base model marginal is defined as the marginal distribution arising from the sim-plest model under consideration.The simplest model here corresponds to(i.e., the mixture consisting of a single component),and the resulting predictive for the training samples isThe EP prior under the base model marginal turns out to be equivalent to the‘intrinsic prior’for mixture models that was derived in Moreno and Liseo(2000);indeed,Perez and Berger(1999)show that this equivalence holds quite generally in model selection.The advantage of the EP prior representation is computational,since the can be introduced as latent variables for MCMC computation.We do not review all of the attractive properties of EP priors discussed in Perez and Berger(1999),but do highlight two of the properties that are particularly relevant to mixture models.First,the proposed EP priors are marginally coherent,in the sense that.This can easily be shown to hold for either the empirical or base model choices of.A second interesting property is that,for a single observation,the marginal densities corresponding to the EP priors,for different,are equal.This follows directly from the facts that we assume that the components of the mixture are of the same parametric form, and that the marginal EP priors for each are the same.The interest in this property is that it suggests that the EP priors are indeed predictively matched across,as discussed in, say,Berger et al.(1998).3.MCMC ESTIMATION OF THE MIXTURE MODELFor computation,we employ a variant of the reversible jump MCMC technique of Green (1995).The MCMC sampler will generate a chain,where the state variable consists of the unknown parameters.These samples can then be used to estimate posterior probabilities and expectations.Since is unknown,the statevariable assumes values among spaces of different dimensions,which motivates use of the reversible jump method.Let denote the current state of the Markov chain and consider a countable family of move types,indexed by.Within the reversible jump MCMC framework,a new state is proposed by choosing a move type and a destination,with joint density, where is arbitrary for the moment.The chain will move to with acceptance probabilitywhere is the probability of choosing move type when in state,is the density of and is the Jacobian of the transformation.A key advantage of using EP priors in analysis of mixture models is that one can introduce the training samples as latent variables,having density ,and thus simply generate from the larger Markov chain having state space.With some abuse of notation,the joint density for the observations ,the parameters and the training samples isTo improve the performance of the algorithm,it is beneficial to introduce additional latent variables,namely allocation variables for the training samples.Let, where indicates that training sample is(uniquely)assigned to component.De-fine and let denote the uniform distribution over all possible assignments of into components.For chosen as in(4),the reversible jump MCMC algorithm is described as follows:1.Generate weights from,Dirichlet,,,where is the number of observations in component.2.Generate component parameters from,hav-ing density3.Generate allocations from,withprobability4.Generate at time,conditional on,from the density5.Generate at time.6.Generate new assignments at time,for,with probabilities7.Generate.Steps1and3can be implemented using the Gibbs kernels described earlier.In step5,a new set of training samples,,is generated at simulation time,using a Metropolis-Hastings move,with proposal distributions, .The remaining training samples,,are generated in step4from their“posterior”.Step6uniquely assigns the training samples at time to the components.Step7involves changing by adding or deleting one component,and updating, ,and the assignment of the training samples,,accordingly.At this point the re-versible jump mechanism is needed.As in Richardson and Green(1998),we consider splitting or combining with probabilities and,respectively.Clearly, .A split move will produce components and by splitting a single componentin two.The weight and component parameters for these two components are obtained by independently generating a random variable,of the same dimension as,and deterministically assigning,where is an invertible, everywhere differentiable function.The allocation variables of those such thatare randomly set to either or,with probabilities determined by the Gibbs kernel given .A combine move entails merging two randomly chosen components and into a new component.The allocation variables are simply updated by setting for all such that is either or.The rest of the parameters,,are obtained deterministically from the inverse of at.The jump from to components in step7,is done conditionally on all the training samples.In this way,splitting a component into and only involves choosing training samples and from the available pool of training samples.Similarly,the combine move for the training samples only entails choosing from the pool of unassigned training samples.This ef-fectively avoids introducing unknown constants in the acceptance/rejection probabilities for the move.In section4,we consider a mixture of bivariate normal distributions.The th com-ponent is a bivariate normal distribution with parameters,with andbeing the mean and covariance matrix,respectively.The implementation of the reversible jump MCMC in this situation is patterned after Richardson and Green(1998).Indeed,the function is similar to the jump function used in their univariate case,but the bivariate mean is split in a random direction from the center.Similarly,auxiliary variables are introduced to split the component variances and correlation in an invertible way.The actual jump function is specified in full detail in Perez(1998).The mixture model components are permutation invariant,and hence their parameters are not identifiable.Although not completely satisfactory,for inference concerning individ-ual components,one usually imposes constraints on the parameter space(see,for example, Robert,1996;Mengersen and Robert,1996and Richardson and Green,1998.)When the constraints are not appropriate,the posterior estimates of the parameters resulting from the MCMC simulation can produce multimodal marginal posteriors.This multimodality is es-pecially noticeable when the posterior variances are not kept small,as can be the case with EP priors,since the posterior variances of empty components are not controlled.However, if the number of observations,,is large,the probability of observing empty components is reduced and use of a set of constraints is reasonable.4.BATSE GAMMA RAY BURST DATA SETWe now turn to analysis of the gamma ray burst data set discussed in the introduction,uti-lizing a bivariate normal mixture model(with all parameters unknown)for T90and HR.Let T90HR denote the th observation anddenote the standard deviations of the associated obser-vational errors.Then and are assumed to have independent normal distributions, with means and and variances and,respectively,where and repre-sent,respectively,the true values of T90and HR corresponding to observation. The true values,,are assumed to arise from a mixture of bivariate normaldistributions,having means and covariance matrices,.Thus,letting denote the diagonal matrix with diagonal,andwhere denotes the bivariate normal density with mean and covariance matrix .The initial prior distributions for the component parameters are chosen as in sections 1.3and2,with and.Analysis under the base model EP priors and under the empirical version of the EP priors was considered.The reversible jump MCMC algorithm was used in both cases. An additional step was added to the MCMC algorithm to deal with the generation of conditional on all other parameters and on.Indeed,conditional on,and,the posterior distribution for is given byThus,during each MCMC iteration,the must now themselves be generated according to the distribution in(6).The length of the run for the MCMC simulation was100,000sweeps,with only thelast60,000being used for inference.Identifiability was obtained by ordering thefirst co-ordinate of the means.The posterior probabilities for strongly indicate that only two components are present().Table1gives the corresponding estimatesof the location and covariance matrices for the two components,under the“base model”and the empirical versions of the EP prior.Both analyses essentially agree.Notice that no evidence of correlation between and was found.Table1.Estimates of log(T90)and log(HR).Base model EP priors Empirical EP priors=0.24=-0.85=1.61=1.04=0.50 =-0.03=0.76=3.31=0.95=1.10=0.49 =0.01Component1 Component2Figure1shows the allocation distribution for the gamma ray bursts,together with predictive confidence sets of levels90%,95%and99%for each of the two compo-nents.Empirical EP priorsFigure1.Burst classification probabilities when;the color bar indicates the valueof.ACKNOWLEDGMENTSThis work formed part of the second authors Ph.D.thesis at the Department of Statistics, Purdue University.This work was supported by the National Science Foundation(USA), Grants DMS-9303556and DMS-9802261.REFERENCESBerger,J.and Pericchi,L.(1996).The Intrinsic Bayes Factor for model selection and pre-diction.,91,109–122.Berger,J.,Pericchi,L.,and Varshavsky,J.(1998).Bayes factors and marginal distributions in invariant situations.,60, 307–321.Diebolt,J.and Robert,C.P.(1994).Estimation offinite mixture distributions through Bayesian sampling.,56,363–375.Escobar,M.and West,M.(1995).Bayesian density estimation and inference using mix-tures.,90,577–588.Green,P.(1995).Reversible jump Markov Chain Monte Carlo computation and Bayesian model determination.,82,711–732.Hastings,W.K.(1970).Monte Carlo sampling methods using Markov Chains and their applications.,57,97–109.Hurley,K.(1991).Gamma-ray bursts.In Paciesas,W.S.and Fishman,G.J.,editors,,265,New York.AIP.McLachlan,G.and Basford,K.E.(1985).Likelihood estimation with normal mixture mod-els.,34,282–289.Meegan,C.A.et al.(1995).The third BATSE Gamma-Ray Burst catalog.106.Mengersen,K.and Robert,C.(1996).Testing for mixtures:a Bayesian entropy approach.In Bernardo,J.et al.,editors,,London.Oxford University Press. Moreno,E.and Liseo,B.(2000).A default Bayesian test for the number of components ofa mixture.To appear in.O’Hagan,A.(1995).Fractional Bayes factors for model comparison.,57,99–138.Perez,J.M.(1998)..PhD thesis,Purdue University.Perez,J.M.and Berger,J.O.(1999).Expected posterior prior distributions for model selection.Technical Report99-09,CESMa.Universidad Simon Bolˇvar. Richardson,S.and Green,P.J.(1998).On Bayesian analysis of mixtures with an unknown number of components.,60. Robert,C.(1996).Practical Markov Chain Monte Carlo.In,pp.441-464,Chapman and Hall,London.Roeder,K.and Wasserman,L.(1997).Practical Bayesian density estimation using mixtures of normals.,92,894–902. Shui,C.(1996)..PhD thesis,Purdue University.Wasserman,L.(1998).Asymptotic inference for mixture models using data dependent pri-ors.Technical Report677,Carnegie-Mellon University.。

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脂肪组织材料本构模型参数力学响应灵敏度分析

脂肪组织材料本构模型参数力学响应灵敏度分析

河南科技Henan Science and Technology化工与材料工程总第875期第4期2024年2月收稿日期：2023-10-16作者简介：段海彤（1994—），女，硕士，工程师，研究方向：车辆被动安全及损伤生物力学。

脂肪组织材料本构模型参数力学响应灵敏度分析段海彤　郑光洁　韩菲菲　胡帛涛　梁亚妮（中汽研（天津）汽车工程研究院有限公司，天津 300300）摘　要：【目的】确定脂肪组织材料各材料本构的参数灵敏度，以便合理简化参数反求变量，提高材料对标效率。

【方法】采用对4种常用的脂肪组织材料本构模型进行无约束压缩试验，研究各材料参数对力学响应的影响情况。

【结果】线性黏弹性材料本构中短效剪切模量与衰减常数对接触力影响显著，Mooney-Rivlin 超弹性本构中材料常数对接触力值影响较大，Ogden 超弹性本构模型中Og⁃den 系数对接触力值影响最显著，软组织材料本构中材料常数对接触力影响显著。

【结论】为脂肪组织材料力学性能研究、参数反求及材料对标工作的合理简化提供了参考，极大地提高了计算效率。

关键词：脂肪组织；本构模型；材料参数；试验设计与分析中图分类号：U467.14；R318.01 文献标志码：A 文章编号：1003-5168（2024）04-0093-05DOI ：10.19968/ki.hnkj.1003-5168.2024.04.017Analysisof Constitutive Model Parameters of Adipose Tissue Materials on Mechanical Response SensitivityDUAN Haitong ZHENG Guangjie HAN Feifei HU Botao LIANG Yani（CATARC （Tianjin ） Automotive Engineering Research Institute Co., Ltd., Tianjin 300300, China ）Abstract: [Purposes ] The parameter sensitivity of each material constitutive of adipose material was de⁃termined in order to simplify the inverse parameter variables reasonably and improve the efficiency ofmaterial calibration. [Methods ] Unconstrained compression tests were carried out on four commonly used material constitutive models of adipose tissue, and the parameters of each material constitutive model were and analyzed to study the influence of each material parameter on the mechanical response of each material constitutive model. [Findings ] The results show that the short-time shear modulus and decay constant have significant effects on the maximum contact force in the linear viscoelastic material. In the Mooney-Rivlin hyperelastic constitutive, the material constant has a greater influence on the maxi⁃mum contact force, the Ogden coefficient has the most significant effect on the maximum contact force in the Ogden hyperelastic constitutive model.The influence of material constants on the maximum contact force is significant for soft issue constitutive model. [Conclusions ] It provides a reference for the rationalsimplification of adipose tissue material mechanical properties research and parameter inversion work,and greatly improves the calculation efficiency.Keywords: adipose tissue; constitutive mode; material parameters; design of experiment0 引言国内外对人体生物组织材料力学性能的研究已经开展了几十年［1］。

带空泡运动物体的附加质量研究

带空泡运动物体的附加质量研究

学位论文作者签名： 黄旋
日期：2009 年 3 月 2 日
上海交通大学 学位论文版权使用授权书
本学位论文作者完全了解学校有关保留、使用学位论文的规定，同意学 校保留并向国家有关部门或机构送交论文的复印件和电子版，允许论文被查 阅和借阅。本人授权上海交通大学可以将本学位论文的全部或部分内容编入 有关数据库进行检索，可以采用影印、缩印或扫描等复制手段保存和汇编本 学位论文。
In this work, a method of determining the added mass by “measuring” hydrodynamic force of a cavity swaying or rolling vehicle in numerical “tunnel” is proposed. Reasonable mathematical model and numerical simulation method are introduced in accordance with the characteristics of periodic swaying or rolling movement of a cavity vehicle, realize the solution of vehicle’s behavior of fluids dynamic; then according to the equation of motion, a method of separating the force of the added mass and calculating the added mass are brought up; investigate added mass of an underwater cavity running vehicle.
基于Mixture模型的乙醇汽油割嘴气液两相流模拟

基于Mixture模型的乙醇汽油割嘴气液两相流模拟

第48 卷第 6 期V〇1.48N〇.6 2018 年6 月Electric Welding Machine Jun. 2018本文参考文献引用格式:孔维宾，蒋应田，梁平.基于Mixture模型的乙醇汽油割嘴气液两相流模拟[J].电焊机，2018,48(06): 118-122.靄于_鼸乙__顬喔气鱺麗纏流靈拟孔维宾，蒋应田，梁平(辽宁石油化工大学机械工程学院，辽宁抚顺113001)摘要:通过Mixture模型对专用割嘴的内部进行速度场和压力场数值模拟，分析割嘴内部流场的气液两相 流流动状态，以及速度和压力在各部位的变化情况，研究不同压力下的氧气流对乙醇汽油燃料的速度变化 以及在一定压力下的氧气流对不同速度的乙醇汽油雾化的影响情况。

分析流体外部两侧的静压强的欠膨 胀，认为欠膨胀射流有利于空气与液体燃料的混合。

模拟结果表明，增加气液混合物的速度差可以提高割 嘴出口处乙醇汽油的雾化程度，有利于充分燃烧。

关键词：火焰切割；气液两相流；Mixture模型；数值模拟中图分类号:TG481 文献标志码:A文章编号：1001-2303(2018)06-0118-05DOI：10.7512/j.issn.1001-2303.2018.06.24B a s e d o n fluent o f e t h a n o l fuel n o z z l e i n t e r n a l f l o w field o f t h e c o l d state s i m u l a t i o nKONG Weibin,JIANG Yingtian,LIANG ping(School of Mechanical Engineering,Liaoning Shihua University,Fushun 113001,China)Abstract：In th is paper,the velocity field an d p ressu re field n u m erical sim u latio n of th e special cu ttin g n o zzle are carried o u t b y M ix tu re m odel,the flo w sta te of gas-liquid tw o p h ase flo w in cu ttin g n o zzle is analyzed,and th e ch an ge of velocity an d p ressu re a t d ifferen t p o sitio n s a re an alyzed.T o stu d y th e effect of oxygen flo w at d ifferen t p ressu res o n th e sp eed ch an ge of eth an o l fuel an d th e effect of o xygen flo w u n d er certain p ressu re o n th e a to m iza tio n of eth a n o l gasolin e at d ifferen t speeds.B y an alyzin g th e u n d erexp an sio n of static p ressu re o n b o th sid es of th e fluid,it is con cluded th a t th e un der-expan sion jet is con du cive to th e m ixin g of air an d liqu id fuel. T h e sim u latio n resu lts sh o w th a t in creasin g th e sp eed d ifferen ce of gas-liquid m ix tu re can im p ro v e th e a to m iza tio n d egree of eth an ol- gasolin e a t th e o u tlet of th e n ozzle,w h ich is ben eficial to th e full co m b u stio n.Key words：flam e cutting;gas-liquid tw o p h ase flow;M ix tu re m odel;n u m erical sim u la tio n0 刖目火焰切割一直利用以乙炔为代表的气体燃料 作为能源载体。

混凝土搅拌车搅拌实验系统仿真设计英文原文

混凝土搅拌车搅拌实验系统仿真设计英文原文

Properties of Fresh ConcreteEdited by H.-J. Wierig Fresh concrete is a mixture of water, cement, aggregate and admixture (if any). After mixing, operations such as transporting, placing, compacting and finishing of fresh concrete can all considerably affect the properties of hardened concrete. It is important that the constituent materials remain uniformly distributed within the concrete mass during the various stages of its handling and that full compaction is achieved. When either of these conditions is not satisfied the properties of the resulting hardened concrete, for example, strength and durability, are adversely affected.The characteristics of fresh concrete which affect full compaction are its consistency, mobility and compactability. In concrete practice these are often collectively known as workability. The ability of concrete to maintain its uniformity is governed by its stability, which depends on its consistency and its cohesiveness. Since the methods employed for conveying, placing and consolidatingd a concrete mix, as well as the nature of the section to be cast, may vary from job to job it follows that the corresponding workability and stability requirements will also vary. The assessment of the suitability of a fresh concrete for a particular job will always to some extent remain a matter of personal judgment.In spite of its importance, the behaviour of plastic concrete often tends to be overlooked. It is recommended that students should learn to appreciate the significance of the various characteristics of concrete in its plastic state and know how these may alter during operations involved in casting a concrete structure.13.1 WorkabilityWorkability of concrete has never been precisely defined. For practical purposes it generally implies the ease with which a concrete mix can be handled from the mixer to its finally compacted shape. The three main characteristics of the property are consistency, mobility and compactability. Consistency is a measure of wetness or fluidity. Mobility defines the ease with which a mix can flow into and completely fill the formwork or mould. Compactability is the ease with which a given mix can be fully compacted, all the trapped air being removed. In this context the required workability of a mix depends not only on the characteristics and relative proportions of the constituent materials but also on (1) the methods employed for conveyance and compaction, (2) the size, shape and surface roughness of formwork or moulds and (3) the quantity and spacing of reinforcement.Another commonly accepted definition of workability is related to the amount of useful internal work necessary to produce full compaction. It should be appreciated that the necessary work again depends on the nature of the section being cast. Measurement of internal work presents many difficulties and several methods have been developed for this purpose but none gives an absolute measure of workability.The tests commonly used for measuring workability do not measure the individual characteristics (consistency, mobility and compactability) of workability. However, they do provide useful and practical guidance on the workability of a mix. Workability affects the quality of concrete and has a direct bearing on cost so that, for example, anunworkable concrete mix requires more time and labour for full compaction. It is most important that a realistic assessment is made of the workability required for given site conditions before any decision is taken regarding suitable concrete mix proportions.13.2 Measurement of WorkabilityThree tests widely used for measuring workability are the slump, compacting factor and V-B consistometer tests (figure 13.1). These are standard tests in the United Kingdom and are described in detail in BS 1881: Part 2. Their use is also recommended in CP 110: Part 1. It is important to note that there is no single relationship between the slump, compacting factor and V-B results for different concretes. In the following sections the salient features of these tests together with their merits and limitations are discussed.Slump TestThis test was developed by Chapman in the United States in 1913. A 300 mm high concrete cone, prepared under standard conditions (BS 1881: Part 2) is allowed to subside and the slump or reduction in height of the cone is taken to be a measure of workability. The apparatus is inexpensive, portable and robustd and is the simplest of all the methods employed for measuring workability. It is not surprising that, in spite of its several limitations, the slump test has retained its popularity.Figure 13.1 Apparatus for workability measurement: (a) slump cone, (b) compacting factor and (c)V-B consistometerThe test primarily measures the consistency of plastic concrete and although it is difficult to see any significant relationship between slump and workability as defined previously, it is suitable for detecting changes in workability. For example, an increase in the water content or deficiency in the proportion of fine aggregate results in anincrease in slump. Although the test is suitable for quality-control purposes it should be remembered that it is generally considered to be unsuitable for mix design since concretes requiring varying amounts of work for compaction can have similar numerical values of slump. The sensitivity and reliability of the test for detecting variation in mixes of different workabilities is largely dependent on its sensitivity to consistency. The test is not suitable for very dry or wet mixes. For very dry mixes, with zero or near-zero slump, moderate variations in workability do not result in measurable changes in slump. For wet mixes, complete collapse of the concrete produces unreliable values of slump.Figure 13.2 Three main types of slumpThe three types of slump usually observed are true slump, shear slump and collapse slump, as illustrated in figure 13.2. A true slump is observed with cohesive and rich mixes for which the slump is generally sensitive to variations in workability. A collapse slump is usually associated with very wet mixes and is generally indicative of poor quality concrete and most frequently results from segregation of its constituent materials. Shear slump occurs more often in leaner mixes than in rich ones and indicates a lack of cohesion which is generally associated with harsh mixes (low mortar content). whenever a shear slump is obtained the test should be repeated and, ifpersistent, this fact should be recorded together with test results, because widely different values of slump can be obtained depending on whether the slump is of true or shear form.The standard slump apparatus is only suitable for concretes in which the maximum aggregate size does not exceed 37.5 mm. It should be noted that the value of slump changes with time after mixing owing to normal hydration processes and evaporation of some of the free water, and it is desirable therefore that tests are performed within a fixed period of time.Compacting Factor TestThis test, developed in the United Kingdom by Glanville et al. (1947), measures the degree of compaction for a standard amount of work and thus offers a direct and reasonably reliable assessment of the workability of concrete as previously defined. The apparatus is a relatively simple mechanical contrivance (figure 13.1) and is fully described in BS 1881: Part 2. The test requires measurement of the weights of the partially and fully compacted concrete and the ratio of the partially compacted weight to the fully compacted weight, which is always less than 1, is known as the compacting factor. For the normal range of concretes the compacting factor lies between 0.80 and 0.92. The test is particularly useful for drier mixes for which the slump test is not satisfactory. The sensitivity of the compacting factor is reduced outside the normal range of workability and is generally unsatisfactory for compacting factors greater than 0.92.It should also be appreciated that, strictly speaking, some of the basic assumptions of the test are not correct. The work done to overcome surface friction of the measuring cylinder probably varies with the characteristics of the mix. It has been shown by Cusens (1956) that for concretes with very low workability the actual work required to obtain full compaction depends on the richness of a mix while the compacting factor remains sensibly unaffected. Thus it follows that the generally held belief that concretes with the same compacting factor require the same amount of work for full compaction cannot always be justified. One further point to note is that the procedure for placing concrete in the measuring cylinder bears no resemblance to methods commonly employed on the site. As in the slump test, the measurement of compacting factor must be made within a certain specified period. The standard apparatus is suitable for concrete with a maximum aggregate size of up to 37.5 mm.V-B Consistometer TestThis test was developed in Sweden by B a hrner (1940) (see figure 13.1). Although generally regarded as a test primarily used in research its potential is now more widely acknowledged in industry and the test is gradually being accepted. In this test (BS 1881: Part 2) the time taken to transform, by means of vibration, a standard cone of concrete to a compacted flat cylindrical mass is recorded. This is known as the V-B time, in seconds, and is stated to the nearest 0.5 s. Unlike the two previous tests, the treatment of concrete in this test is comparable to the method of compacting concrete in practice. Moreover, the test is sensitive to change in consistency, mobility and compactability,and therefore a reasonable correlation between the test results and site assessment of workability can be expected.The test is suitable for a wide range of mixes and, unlike the slump and compacting factor tests, it is sensitive to variations in workability of very dry and also air-entrained concretes. It is also more sensitive to variation in aggregate characteristics such as shape and surface texture. The reproducibility of results is good. As for other tests its accuracy tends to decrease with increasing maximum size of aggregate; above 19.0 mm the test results become somewhat unreliable. For concretes requiring very little vibration for compaction the V-B time is only about 3 s. Such results are likely to be less reliable than for larger V-B times because of the difficulty in estimating the time of the end point (concrete in contact withd the whole of the underside of the plastic disc). At the other end of the workability range, such as with very dry mixes, the recorded V-B times are likely to be in excess of their true workability since prolonged vibration is required to remove the entrapped air bubbles under the transparent disc. To overcome this difficulty an automatic device which records the vertical settlement of the disc with respect to time can be attached to the apparatus. This recording device can also assist in eliminating human error in judging the end point. The apparatus for the V-B test is more expensive than that for the slump and compacting factor tests, requiring an electric power supply and greater experience in handling; all these factors make it more suitable for the precast concrete industry and ready-mixed concrete plants than for general site use.13.3 Factors Affecting WorkabilityVarious factors known to influence the workability of a freshly mixed concrete are shown in figure 13.3. From the following discussion it will be apparent that a change in workability associated with the constituent materials is mainly affected by water content and specific surface of cement and aggregate.Cement and WaterFigure 13.3 Factors affecting workability of fresh conreteTypical relationships between the cement-water ratio (by volume) and the volume fraction of cement for different workabilities are shown in figure 15.5. The change in workability for a given change in cement-water ratio is greater when the water content is changed than when only the cement content is changed. In general the effect of the cement content is greater for richer mixes. Hughes (1971) has shown that similar linear relationships exist irrespective of the properties of the constituent materials.For a given mix, the workability of the concrete decreases as the fineness of the cement increases as a result of the increased specific surface, this effect being more marked in rich mixtures. It should also be noted that the finer cements improve the cohesiveness of a mix. With the exception of gypsum, the composition of cement has no apparent effect on workability. Unstable gypsum is responsible for false set, which can impair workability unless prolonged mixing or remixing of the fresh concrete is carriedout. Variations in quality of water suitable for making concrete have no significant effect on workability.AdmixturesThe principal admixtures affecting improvement in the workability of concrete are water-reducing and air-entraining agents. The extent of the increase in workability is dependent on the type and amount of admixture used and the general characteristics of the fresh concrete.Workability admixtures are used to increase workability while the mix proportions are kept constant or to reduce the water content while maintaining constant workability. The former results in a slight reduction in concrete strength.Air-entraining agents are by far the most commonly used workability admixtures because they also improve both the cohesiveness of the plastic concrete and the frost resistance of the resulting hardened concrete. Two points of practical importance concerning air-entrained concrete are that for a given amount of entrained air, the increase in workability tends to be smaller for concretes containing rounded aggregates or low cement-water ratios (by volume) and, in general, the rate of increase in workability tends to decrease with increasing air content. However, as a guide it may be assumed that every 1 per cent increase in air content will increase the compacting factor by 0.01 and reduce the V-B time by 10 per cent.AggregateFor given cement, water and aggregate contents, the workability of concrete is mainly influenced by the total surface area of the aggregate. The surface area is governed by the maximum size, grading and shape of the aggregate. Workability decreases as the specific surface increases, since this requires a greater proportion of cement paste to wet the aggregate particles, thus leaving a smaller amount of paste for lubrication. It follows that, all other conditions being equal, the workability will be increased when the maximum size of aggregate increases, the aggregate particles become rounded or the overall grading becomes coarser. However, the magnitude of this change in workability depends on the mix proportions, the effect of the aggregate being negligible for very rich mixes (aggregate-cement ratios approaching 2). The practical significance of this is that for a given workability and cement-water ratio the amount of aggregate which can be used in a mix varies depending on the shape, maximum size and grading of the aggregate, as shown in figure 13.4 and tables 13.1 and 13.2. The influence of air-entrainment (4.5 per cent) on workability is shown also in figure 13.4.TABLE 13.1Effect of maximum size of aggregate of similar grading zone on aggregate-cement ratio of concrete having water-cement ratio of 0.55 by weight, based on McIntosh (1964)Maximum aggregatesize(mm)Aggregate-cement ratio (by weight)Low workability Medium workability High workability IrregulargravelCrushed rockIrregulargravelCrushed rockIrregulargravelCrushed rock9.5 5.3 4.8 4.7 4.2 4.4 3.719.0 37.56.27.65.56.45.46.54.75.54.95.94.45.2TABLE 13.2Effect of aggregate grading (maximum size 19.0 mm) on aggregate-cement ratio ofconcrete having medium workability and water-cement ratio of 0.55 by weight, based onMcIntosh (1964)Type of aggregateAggregate-cement ratioCoarse grading Fine gradingRounded gravel Irregular gravel Crushed rock 7.35.54.76.35.14.3Figure 13.4 Effect of aggregate shape on aggregate-cement ratio of concretes for different workabilities, based on Cornelius (1970)Several methods have been developed for evaluating the shape of aggregate, asubject discussed in chapter 12. Angularity factors together with grading modulus and equivalent mean diameter provide a means of considering the respective effects of shape, size and grading of aggregate (see chapter 15). Since the strength of a fully compacted concrete, for given materials and cement-water ratio, is not dependent on the ratio of coarse to fine aggregate, maximum economy can be obtained by using the coarse aggregate content producing the maximum workability for a given cement content (Hughes, 1960) (see figure 13.5). The use of optimum coarse aggregate content in concrete mix design is described in chapter 15. It should be noted that it is the volume fraction of an aggregate, rather than its weight, which is important.Figure 13.5 A typical relationship between workability and coarse aggregate content of concrete, based on Hughes (1960)The effect of surface texture on workability is shown in figure 13.6. It can be seen that aggregates with a smooth texture result in higher workabilities than aggregates with a rough texture. Absorption characteristics of aggregate also affect workability where dry or partially dry aggregates are used. In such a case workability drops, the extent of the reduction being dependent on the aggregate content and its absorption capacity.Ambient ConditionsEnvironmental factors that may cause a reduction in workability are temperature, humidity and wind velocityd. For a given concrete, changes in workability are governed by the rate of hydration of the cement and the rate of evaporation of water. Therefore both the time interval from the commencement of mixing to compaction and the conditions of exposure influence the reduction in workability. An increase in the temperature speeds up the rate at which water is used for hydration as well as its loss through evaporation. Likewise wind velocity and humidity influence the workability as they affect the rate of evaporation. It is worth remembering that in practice these factors depend on weather conditions and cannot be controlled.Figure 13.6 Effect of aggregate surface texture on aggregate-cement ratio of concretes for different workabilities, based on Cornelius (1970)TimeThe time that elapses between mixing of concrete and its final compaction depends onthe general conditions of work such as the distance between the mixer and the point of placing, site procedures and general management. The associated reduction in workability is a direct result of loss of free water with time through evaporation, aggregate absorption and initial hydration of the cement. The rate of loss of workability is affected by certain characteristics of the constituent materials, for example, hydration and heat development characteristics of the cement, initial moisture content and porosity of the aggregate, as well as the ambient conditions.For a given concrete and set of ambient conditions, the rate of loss of workability with time depends on the conditions of handling. Where concrete remains undisturbed after mixing until it is placed, the loss of workability during the first hour can be substantial, the rate of loss of workability decreasing with time as illustrated by curve A in figure 13.7. On the other hand, if it is continuously agitated, as in the case of ready-mixed concrete, the loss of workability is reduced, particularly during the first hour or so (see curve B in figure 13.7). However, prolonged agitation during transportation may increase the fineness of the solid particles through abrasion and produce a further reduction in workability. For concretes continuously agitated and undisturbed during transportation, the time intervals permitted (BS 1926) between the commencement of mixing and delivery on site are 2 hours and 1 hour respectively.For practical purposes, loss of workability assumes importance when concrete becomes so unworkable that it cannot be effectively compacted, with the result that its strength and other properties become adversely affected. Corrective measures frequently taken to ensure that concrete at the time of placing has the desired workability are eitheran initial increase in the water content or an increase in the water content with further mixing shortly before the concrete is discharged. When this results in a water content greater than that originally intended, some reduction in strength and durability of the hardened concrete is to be expected unless the cement content is increased accordingly. This important fact is frequently overlooked on site. It should be recalled that the loss of workability varies with the mix, the ambient conditions, the handling conditions and the delivery time. No restriction on delivery time is given in CP 110: Part 1 but the concrete must be capable of being placed and effectively compacted without the addition of further water. For detailed information on the use of ready-mixed concrete the reader is advised to consult the work of Dewar (1973).Figure 13.7 Loss of workability of concrete with time: (A) no agitation and (B)continuously agitated after mixing13.4 StabilityApart from being sufficiently workable, fresh concrete should have a composition such that its constituent materials remain uniformly distributed in the concrete during both the period between mixing and compaction and the period following compaction beforethe concrete stiffens. Because of differences in the particle size and specific gravities of the constituent materials there exists a natural tendency for them to separate. Concrete capable of maintaining the required uniformity is said to be stable and most cohesive mixes belong to this category. For an unstable mix the extent to which the constituent materials will separate depends on the methods of transportation, placing and compaction. The two most common features of an unstable concrete are segregation and bleeding.SegregationWhen there is a significant tendency for the large and fine particles in a mix to become separated, segregation is said to have occurred. In general, the less cohesive the mix the greater the tendency for segregation to occur. Segregation is governed by the total specific surface of the solid particles including cement and the quantity of mortar in the mix. Harsh, extremely wet and dry mixes as well as those deficient in sand, particularly the finer particles, are prone to segregation. As far as possible, conditions conducive to segregation such as jolting of concrete during transportation, dropping from excessive heights during placing and over-vibration during compaction should be avoided.Blemishes, sand streaks, porous layers and honeycombing are a direct result of segregation. These features are not only unsightly but also adversely affect strength, durability and other properties of the hardened concrete. It is important to realize that the effects of segregation may not be indicated by the routine strength tests on control specimens since the conditions of placing and compaction of the specimens differ fromthose in the actual structure. There are no specific rules for suspecting possible segregation but after some experience of mixing and handling concrete it is not difficult to recognize mixes where this is likely to occur. For example, if a handful of concrete is squeezed in the hand and then released so that it lies in the palm, a cohesive concrete will be seen to retain its shape. A concrete which does not retain its shape under these conditions may well be prone to segregation and this is particularly so far wet mixes.BleedingDuring compaction and until the cement paste has hardened there is a natural tendency for the solid particles, depending on size and specific gravity, to exhibit a downward movement. Where the consistency of a mix is such that it is unable to hold all its water some of it is gradually displaced and rises to the surface, and some may also leak through the joints of the formwork. Separation of water from a mix in this manner is known as bleeding. While some of the water reaches the top surface some may become trapped under the larger particles and under the reinforcing bars. The resulting variations in the effective water content within a concrete mass produce corresponding changes in its properties. For example, the strength of the concrete immediately underneath the reinforcing bars and coarse aggregate particles may be much less than the average strength and the resistance to percolation of water in these areas is reduced. In general, the concrete strength tends to increase with depth below the top surface. The water which reaches the top surface presents the most serious practical problems. If it is not removed, the concrete at and near the top surface will be much weaker andless durable than the remainder of the concrete. This can be particularly troublesome in slabs which have a large surface area. On the other hand, removal of the surface water will unduly delay the finishing operation on the site.The risk of bleeding increases when concrete is compacted by vibration although this may be minimized by using a correctly designed mix and ensuring that the concrete is not over-vibrated. Rich mixes tend to bleed less than lean mixes. The type of cement employed is also important, the tendency for bleeding to occur decreasing as the fineness of the cement or its alkaline and tricalcium aluminate (C3A) content increases. Air-entrainment provides another very effective means of controlling bleeding in, for example, wet lean mixes where both segregation and bleeding are frequently troublesome.。