Cement Concrete Mixture Performance Characterization

The cementitious composite nature of concrete makes very diffi cult directly ascertaining each mixture-factors’ contribution to a given concrete mixture performance characteristics but also doubly diffi cult to accurately balance mutually exclusive requirements for performance (workability, strength, durability) and sustainability (the economic and effi cient use of materials) for mixture proportioning based on recipes of previously produced concretes. This study sought to quantify individual mixture-factors’ contribution to a given concrete mixture’s performance characteristics. Proposed multi-parametric exponential mixture-response models were fi tted to available test-performance data sets of HPC mixtures proportioned based on the best combined grade aggregate (minimum void) to generate mixture-strength and mixture-porosity development (age-mixture response relationships) profi les of HPC mixtures and deemed robust enough to yield reliable determination of mixture-response rate-parameters So, Sp, Si and Po, Pp, Pi as functions of mixture-factors that permitted reliable quantifi cation of contributions to HPC mixture performance of individual mixture-factors and optimization of mixture properties under study over the study domain. Mixture-response sensitivity analysis models (or mixture response trace plots) to allow construction of mixture-factor envelopes and ultimately optimized mixture-response models to facilitate selection of optimal mixture-factors and optimal tailoring of HPC mixture requirements to HPC mixture performance were developed and used to obtain optimized adapted HPC mixtures from available high performance concrete (HPC) mixture design recipes investigated in the study over the study domain. Adapted HPC mixture design recipes yielded alternative mixture compositions with improved performance and effi ciency characteristics with statistical performance metrics MAPE, NMBE and RMSE values of 7.6%,–3.7% and 6.5 MPa, respectively.

f i (a) -maxima (or minima) of response trace-plot function, f ref (a) -maxima (or minima) of response trace-plot function for a reference mixture, w i -the weight for individual normalized response function of a mixture, w UB i -upper bound weight-value for individual normalized response function i, w LB i -lower bound weight-value for individual normalized response function i, π UB i -upper bound mixture-factor value i, π LB i -lower bound mixture-factor value i, σ -envelope standard deviation, α cement -packing density of cement, α sand -packing density of sand, α RHA -packing density of RHA, α silica-fume -packing density of silica-fume.

High Performance Concretes
The American Concrete Institute [1] defi nes high performance concrete (HPC) as concrete meeting special combinations of performance, durability and uniformity requirements that cannot always be achieved customarily using conventional constituents and production practices. It is basically constituted of the same materials as conventional (normal) concrete but also incorporates supplementary cementitious materials (SCM) and high-performance admixtures to obtain cold-cast HPC mixtures with the desired performance indicators. This family of cement concretes comprises of (on the basis of mixture-strength), high-strength (≥50-90 MPa), very high-strength (≥90-130 MPa) and ultra-high-strength (≥130-200 MPa) concrete (modifi ed from Büyüköztürk et al [2]).
Normal strength concrete structures have a mass to strength ratio of 40-120 kg/MNm while that for HPC structures averages 15 kg/MNm [3] but HPC mixtures have the disadvantage of high and expensive contents of cement and additives and hence the dominant HPC products on the market are almost exclusively pre-packaged, proprietary and expensive with patented aggregate and concrete formulation (information on their compositions not readily available) and almost impossible to modify or customize or adapt for alternative mixture designs for a specifi c individual design, construction or architectural description.
Currently, many HPC mixtures are proportioned based on prior history of production (historical information or experience) or the cost-prohibitive trial mixtures (trial-and-error) or by prescribing the limits (maximum or minimum) on the mixture-factors that circumscribe the desired performance indicators but rarely based on the actual needs of the mixture and the locally available materials and do not often involve properly balancing mutually exclusive requirements for performance (workability, strength, durability) and sustainability (the economic and effi cient use of materials) [4,5]. Most prescription-based design studies reported in the literature have not yielded expected end results because they tended to deliberately promote overdesigned mixtures by using cement content as a safety factor [6] while the trial-and-error modifi cation of existing HPC recipes although very popular for HPC mixture design in the HPC production sector is a hit-or-miss affair, expensive, wasteful and un-optimizable because of the many varied material-inputs involved and the different sources for the material-inputs [7,8]. Effective promotion of green concrete incorporating mineral additives is required in order to minimize the environment threat due to waste disposal from mineral additives and reduce cement consumption.

Cement Concrete Mixture Matrix Structure
Nearly a century after Abrams [9] proposed the water-to-binder ratio law, much has been contributed by cement concrete mixture researchers to broaden the understanding of how the fresh and hardened-state properties of concrete are controlled by the relative proportions of concrete constituent components-cement, coarse and fi ne aggregates, water, and various additives-while elevating concrete to the status of the most dominant construction material for 21 st century infrastructural needs. Important advances in admixture technology over the past decades and a recognition that coarse aggregates represent the weakest link in concrete and that they can be taken out to have only sand as the main aggregate ( Fig. 1) have led to the development of a new generation of cement concrete mixtures with low waterbinder ratio, low matrix porosity and high particle packing density that lead to far improved rheological, mechanical and durability (extended service lifetime) properties than obtains with conventional cement concretes (CCC) of similar unit weight [10]. Such cement concrete mixtures (collectively termed as high-performance concretes or engineered 'high-tech' concretes designed to meet project-specifi c needs) incorporate fi ne-grained additives (fga) also known as supplementary cementitious materials (SCMs) and high-range water reducers (see Fig. 1). The incorporation of fi ne-grained additives produces a wide and continuous grain size distribution that helps in optimizing packing density of the matrix, creates a more uniform stress distribution when the matrix is loaded and hence a strong matrix; the smaller grains serve as a lubricant that reduces the inter-particle friction and hence improved workability of the mixture; the fi ne-grains also lower the porosity of the system by fi lling the voids of the mixture with fi ne particles while expelling water from the voids to allow the water to be more homogeneously distributed in the system and hence, again, improve the workability of the mixture and produce a strong mixture matrix; the super-plasticizer helps disperse cement and fi ller particles, improve the lubrication and reduce the inter-particle friction and improve the workability of the mixture [10].
Nevertheless, predicting common time-dependent mechanical behaviours of cement concretes remains inherently complex because of the heterogeneous nature of cement concrete, forcing concrete practitioners (engineers, concrete producers and researchers) to rely heavily on prior history of production (historical information or experience) and/or the costprohibitive trial mixtures (trial-and-error).

Cement Concrete Mixture Performance Characterization
Non-optimized (under-designed or over-designed) HPC mixtures are almost always uneconomical and ineffi cient [11,12]. Effi cient HPC mixtures with cement content balanced to achieve the desired performance while minimizing the risk of problems arising from high cement content (such as the obvious increased cost, the negative environmental effects [13], shrinkage and cracking problems) and reduced resource requirements in infrastructural applications can only be achieved with optimization of HPC mixtures on a quantitative basis [14]. The traditional method for optimizing HPC mixtures to achieve the desired performance, involving systematically varying individual mixture-factors in small increments and studying the resultant effect, is time-consuming, requiring multi-trial batches and hence expensive and ineffi cient. In the traditional method, the basis for selecting SCM dosage is often arbitrary and often focuses on a specifi c set of requirements such as strength or durability but as the demand for large volumes of concrete and faster speeds of construction grows and budgets grow tighter, greater attention is beginning to be paid in designing mixtures that are more effi cient in their usage of materials without compromising engineering performance through optimisation of mixtures that makes it easy to take full advantage of the wide range of material and construction combinations and options not obtainable under non-optimized, high-energy and resource-consuming HPC mixtures. Optimized design of HPC mixtures involves consideration of more varied material constituents (and potentially more interactions among the material constituents), all optimized to determine the most economical and practical mixture-material constituent quantities to produce concrete of desired fresh-state properties (workability, pumpability, fi nishability, and consistency) and required hardened state properties (strength and durability-related properties such as water-tightness, wear and sulphate resistance, etc) consistent with particular conditions of use. Current mixture design of HPC production focuses on optimizing mixture properties (workability, strength and durability) of concrete in fresh and hardened states by optimizing the particle packing density of the granular ingredients of HPC as the accepted key mixture design consideration although the most common industry practice is to simply modify, by trial and error, existing HPC mixture recipes [12,15,16]. Sabir [17] has defi ned HPCs as cement concrete mixtures in which each granular ingredient performs effectively to contribute towards the HPC mixture's fresh and hardened state properties. The focus of this work is the quantifi cation of the contribution of each HPC mixture-factor towards an HPC mixture's fresh and hardened state properties and the optimization of these contributions to allow optimal tailoring of HPC mixture requirements to HPC mixture performance to achieve effi ciency and economy in HPC mixture design using cold-cast HPC mixture test-performance data sets available in cement concrete mixture research literature but also to allow modifi cation or customization of available high performance concrete (HPC) formulations to meet specifi c infrastructural applications.

Parameterized Mixture-Strength and Mixture-Porosity Response Models
One-hundred thirty-four (134) sets of test-data and test-results for HPC mixtures from available cement concrete mixture research literature were employed to construct parameterized mixture-strength and mixture-porosity response models following the determination of model rate-parameters S o , S p , S i and P o , P p , P i . Cement concrete mixture-composition optimization models were then built using the developed parameterized mixture-strength and mixture-porosity response models and constructed mixture-factor envelopes.

Proposed Parameterized Mixture-Strength and Mixture-Porosity Response Models
Porosity and compressive strength of cement concrete mixtures are interconnected and directly infl uence the failure behaviour of cement concretes. The time-dependent mechanical behaviour of concrete mixtures are characterized by an inner interval with early accelerated responses and a later outer interval with responses stabilized to a constant rate of increase which suggests a multiple-time scale problem and amenable to matching-approximations analysis [18]. The following uniform composite approximations are offered for the general response models for estimating concrete mixture-strength response S t and concrete mixtureporosity response P t after t days of curing: where: S o , S p , S i and P o , P p , P i are mixture-strength and mixture-porosity response rate-parameters, respectively, that are functions of mixture-factors of the cement concrete mixturematrix structure properties (see Fig. 1).
The proposed mixture-response models only require a determinations of the rate-parameters S o , S p , S i and P o , P p , P i from available test data or their evaluation through response prediction models based mixture-factors screened from the cement concrete mixture-matrix structure properties.

Proposed Mixture-Response Rate-Parameter Models
The aggregative effect of the enormous number of cement concrete mixture-constituents and properties contributing to the mixture-strength and mixture-porosity (and other desired concrete attributes) can be captured through the three rate-parameters as follows: where m is the number of independent parameterized mixture variables considered, π j is the j th independent parameterized mixture variable; α oj , α pj , α ij and β oj , β pj , β ij are the exponents to be determined for strength and porosity responses, respectively, through regression analysis. Independent parameterized mixture variables screened from the cement concrete mixturematrix structure (see Fig. 1) include: π 1 RHA surface area factor, π 2 water content factor, kg/cu. m π 3 cement content factor, kg/cu. m π 4 silica-fume factor, π 5 fga factor, π 6 super-plasticizer content factor, kg/cu. m π 7 coarse aggregate-to-binder ratio π 8 water-to-binder ratio factor, π 9 entrapped air factor, π 10 mixture-slump factor, mm π 11 sand percentage in the mixture-aggregate, % π 12 sand packing factor, π 13 RHA packing factor, π 14 silica-fume packing factor,

Mixture-Response Model Calibrations
HPC mixture test-data from tests by Azizinamini [19], Domone and Soutsos [20] and Nguyen [21] were used to calibrate the proposed models. MATLAB ® 's non-linear leastsquares (nlinfi t) regression analysis and MICROSOFT EXCEL ® 's linear least squares (linest) regression analysis programmes were employed to fi t concrete test-data to the proposed parameterized mixture-strength and mixture-porosity response models and to perform rate-parametric analyses to obtain model coeffi cients, respectively, as follows: Mixture-Strength Response Rate-Parameters Predicted results using the developed parameterized mixture-strength response and mixture-porosity response models and those predicted by mixture-strength response models proposed by Sarkar et al. [22] and Rajasekaran [23] are shown in Table 1, Table 2, Table 3 and Table 4.   [23] w/b: water-to-binder ratio c. aggr: coarse-aggregate fa: fl y-ash sf: silica-fume ggbs: ground-granulated blast--furnance slag sp: super-plasticizer sand content in aggregate: 37.5% †%: as a percentage of binder *28-day mix-compressive strength

Proposed Parameterized Mixture Optimization Algorithm
Using the developed parameterized mixture-strength and mixture-porosity response models and an optimization scheme based on a linearly weighted least-squares algorithm, mixture optimization models were constructed for mixtures investigated in the study as follows: Defi ning a linearly weighted summated mixture response (WSMR) function with individual normalized mixture response functions ′ ′ where are maxima (or minima) of mixture response trace plot (RTP) or mixture-response sensitivity analysis functions [25] shown in Fig. 2, Fig. 3, Fig. 4 and Fig. 5 to facilitate determination of the N mixture-factor weights, , provided and by minimizing the squared sum of deviations between the weighted-values and the target value, T Differentiating the squared sum of deviations with respect to corresponding weights, the following system of equations is obtained: that facilitates the determination of values of the mixture-factor weights, w i , and their envelope standard deviation The constructed mixture-factor envelopes for concrete mixtures investigated in the study are presented in Fig. 6.
The constructed mixture-factor envelopes readily yield optimized mixture-compositions and corresponding optimized mixture-responses S target and P target by simply interpolating between upper and lower bound values of mixture-factor weights and mixture-factor values ( for a known mixture-response S ref and P ref of a known concrete mixture as follows: For optimized mixture-compressive strength responses For optimized mixture-total porosity responses Predicted optimized mixture-compositions and corresponding optimized mixture responses are summarized in Table 5.

Results and Discussion
Results predicted by the proposed parameterized mixture-strength response and mixture-porosity response models are compared with mixture-test results and results of available models in Table 1, Table 2, Table 3 and Table 4. As can be observed, the proposed parameterized mixture-response models yield reasonably good results for most of the mixtures studied over the study domain (with statistical performance metrics for an unbiased estimate of the prediction ability of the models of mean absolute percentage error MAPE, normalized mean bias error NMBE and root mean square error RMSE values of 6.7%, -3.8% and 9.4MPa, respectively).
The computed mixture-factor weights and constructed mixture-factor envelopes were used to predict optimized mixture-compositions, with mixture-factors limited to the ranges in the study domain, and these were in turn used to predict optimized mixture-response(s) of interest. Optimized mixture-strength response results, presented in Table 5, indicate maximum performance of 8.7%, 12.6% and 12.6% over parent mixture-strengths of VHS, HS and UHS HPC mixtures recipes, respectively. HS concrete mixtures have the least mixture-factor envelope-area (see Fig. 6) and hence respond the least to mixture-composition optimization efforts. Optimized mixture-strength responses predicted by the mixture-factor envelope (MFE) model compare favourably with those obtained by the parameterized mixture-strength (PMS) response model (with statistical performance metrics for an unbiased estimate of the prediction ability of the models of mean absolute percentage error MAPE, normalized mean bias error NMBE and root mean square error RMSE values of 7.6%, -3.7% and 6.5MPa, respectively) and although the results for the parameterized mixture-porosity (PMP) response model are not provided, it has a lean mixture-factor envelope-area (see Fig. 6) and hence responds the least to mixture-porosity based mixture-composition optimization efforts.
The optimal values of mixture-factors for fl y-ash and silica-fume yield more strengtheffi cient T3 mixture while optimizing UHS (high-energy and resource-consuming) HPC mixtures through graded aggregates yielded strength-effi cient and cement-effi cient mixtures.
HPC mixtures can be similarly optimized for workability (through the mix-slump factor) and durability (through the mix entrained-air-content factor). Attempts at modelling optimization of HPC mixture-compositions in the larger mixture-response modelling research community-even with the non-traditional advanced machine learning optimization approaches like sequential learning neural network (SLNN) or neuro-fuzzy computing techniques such as the adaptive neuro-fuzzy inference system (ANFIS)-have thus far only yielded, at best, qualitative characterizations of concrete performance or mixture optimization [26,27].
It is acknowledged that these models are derived from laboratory-test strengths and therefore their application to fi eld cement concrete mixtures (in-situ concretes) suspect or uncertain. Experience has, however, shown that a high percentage (up to 90%) of laboratory test strengths are attainable in-situ concrete under good fi eld practices [28]. HPCs have already moved (transitioned) from laboratory research to industrial implementations (practical applications) and already occupy a sizeable share of the market although most of these applications have been limited to proprietary blends and non-in situ construction (commercial ready mix products and pre-cast applications) and even convenience blends but are in general more expensive (by an order of upwards of twenty) than non-proprietary conventional cement concretes mainly owing to proprietary specifi cations of mixture proportions of nonproprietary HPC mixtures usually being based on trial and error methods than any settled material/behavioural laws or some quantitative characterization of its performance [29,30].

Conclusions
Contributions to mixture-performance by identifi ed mixture-factors were quantitatively determined through models developed in the study which allowed optimal tailoring of mixture requirements to mixture performance of HPCs by explicitly relating performance (user) specifi cations to mixture (producer) requirements and make possible optimized trade-offs between them where the three main performance specifi cations-strength, workability, and durability variously specifi ed (the strength, through the desired compressive strength; workability, through the desired slump; and durability, through some given exposure condition)can be explicitly achieved through variously specifi ed mixture requirements (compressive strength, via some water/cement ratio; workability, via some indication of water content per unit volume of concrete; and durability, via some indication of some minimum cement content and maximum water/cement ratio). The study fi ndings suggest the performance of a known mixture (its strength, workability, and durability) can be improved by a determinable amount and an optimized mixture-composition reliably determined through mixturefactor envelopes largely by increasing the binder content of the mixture and/or the graded aggregate content of the mixture. Developed models were reliably used to optimally modify or customize available HPC formulations to yield alternative mixture compositions with improved performance and effi ciency characteristics (eliminating the cost-prohibitive need for undertaking multiple trials) with statistical performance metrics MAPE, NMBE and RMSE values of 7.6%, -3.7% and 6.5MPa, respectively.
Although the study considered only two mixture-performance parameters, compressive strength and porosity, the study approach can be applied for other mixture-performance parameters (tensile strength, shrinkage and other mixture-properties attributable to high performance concretes).